🧠:[<think>Okay, let's try to figure out this problem step by step. The question is about finding the smallest positive integer ( n ) such that less than 1% of the pairs ( (a, b) ) with ( 0 leq a, b leq n ) are primitive. A primitive pair is defined as a pair where there exists a positive integer ( ell ) such that ( (a + bi)^ell ) is real. First, I need to understand what makes a pair ( (a, b) ) primitive. The condition is that some power of ( a + bi ) is real. So, geometrically, this means that the complex number ( a + bi ) can be rotated by some integer multiple of angles (through multiplication) to land on the real axis. In other words, the angle of ( a + bi ) must be a rational multiple of ( pi ). Because if the angle is ( theta = frac{p}{q} pi ) for some integers ( p ) and ( q ), then multiplying it by ( q ) would give a multiple of ( pi ), resulting in a real number (either positive or negative). Therefore, the key insight here is that the argument ( theta ) of the complex number ( a + bi ) must be such that ( theta = frac{2pi k}{m} ) for some integers ( k ) and ( m ), where ( m ) is the minimal positive integer for which ( (a + bi)^m ) is real. This is because the argument of a complex number raised to the power ( ell ) is ( ell times theta ). For this to be an integer multiple of ( pi ), ( theta ) must be a rational multiple of ( pi ). Specifically, ( theta = frac{pi k}{ell} ) for some integers ( k ) and ( ell ).But how does this relate to the integers ( a ) and ( b )? Well, if ( a + bi ) has such an argument, then the ratio ( frac{b}{a} ) must be equal to ( tan(theta) ), which would be ( tanleft(frac{pi k}{ell}right) ). However, since ( a ) and ( b ) are integers, the tangent of the angle must be a rational number. Therefore, the problem reduces to finding pairs ( (a, b) ) where ( frac{b}{a} ) is a rational multiple of ( pi ), but since ( a ) and ( b ) are integers, this implies that the angle ( theta ) must have a rational tangent. Wait, but not exactly. Because even if ( tan(theta) ) is rational, the angle ( theta ) might not be a rational multiple of ( pi ). For example, ( tan^{-1}(1) = frac{pi}{4} ), which is a rational multiple of ( pi ), but ( tan^{-1}(1/2) ) is not a rational multiple of ( pi ). So there's a distinction here. Therefore, perhaps another approach is needed. Let's recall that for a complex number ( z = a + bi ), ( z^ell ) is real if and only if ( z ) lies on a line through the origin making an angle with the real axis that is a rational multiple of ( pi ). However, if ( z ) is a Gaussian integer (i.e., ( a ) and ( b ) are integers), then ( z ) must lie on such a line. But how can we characterize such Gaussian integers?Another way to look at this is that ( z^ell ) is real implies that ( z ) is a root of unity times a real number. However, in the ring of Gaussian integers ( mathbb{Z}[i] ), the only roots of unity are ( pm 1 ) and ( pm i ). Therefore, if ( z ) is a root of unity multiplied by a real number, then ( z ) must be either purely real, purely imaginary, or a multiple by ( pm 1 ) or ( pm i ). But this seems too restrictive, because for example, ( (1 + i)^2 = 2i ), which is purely imaginary, so ( (1 + i)^4 = -4 ), which is real. Hence, ( (1 + i) ) is a primitive pair because ( ell = 4 ) works. Therefore, there must be more Gaussian integers that are primitive besides the axes.Hence, the key idea is that a Gaussian integer ( a + bi ) is primitive if and only if the argument of ( a + bi ) is a rational multiple of ( pi ). Since when raised to some power ( ell ), the argument becomes an integer multiple of ( pi ), making the result real. Therefore, the problem reduces to counting the number of Gaussian integers ( a + bi ) within the square ( [0, n] times [0, n] ) such that the argument ( theta = tan^{-1}(b/a) ) is a rational multiple of ( pi ). Then, we need to find the smallest ( n ) where such pairs constitute less than 1% of all pairs.First, let's note that the total number of pairs ( (a, b) ) with ( 0 leq a, b leq n ) is ( (n + 1)^2 ). We need to count the number of primitive pairs and find when this count divided by ( (n + 1)^2 ) is less than 0.01.So, first, let's try to characterize all primitive pairs. For a pair ( (a, b) ), the complex number ( a + bi ) must have an argument that is a rational multiple of ( pi ). The possible angles ( theta ) where ( tan(theta) ) is rational are known as "rational angles," but not all such angles have ( theta ) as a rational multiple of ( pi ). However, in our case, we need precisely those angles where ( theta = frac{p}{q} pi ) for integers ( p, q ). From number theory, it's known that the only rational multiples of ( pi ) with rational tangent are those where ( tan(theta) = 0, 1, ) or undefined (i.e., ( theta = 0, pi/4, pi/2, ) etc.). Wait, that might not be accurate. For example, ( tan(pi/3) = sqrt{3} ), which is irrational. Similarly, ( tan(pi/6) = 1/sqrt{3} ), also irrational. So actually, the only angles with rational tangent are ( 0 ), ( pi/4 ), and ( pi/2 ), but even ( pi/4 ) gives ( tan(pi/4) = 1 ). Wait, but ( tan(0) = 0 ), ( tan(pi/4) = 1 ), ( tan(pi/2) ) is undefined. So, if we have a complex number with argument ( pi/4 ), like ( 1 + i ), then ( (1 + i)^4 = -4 ), which is real. Similarly, ( (1 + 2i) ) has some argument, but does that argument correspond to a rational multiple of ( pi )? Let's check.Wait, for ( z = 1 + 2i ), its argument is ( tan^{-1}(2) ). Is ( tan^{-1}(2) ) a rational multiple of ( pi )? I don't think so. If it were, then there would be some integer ( ell ) such that ( (1 + 2i)^ell ) is real. Let's try computing some powers:( (1 + 2i)^2 = (1 - 4) + 4i = -3 + 4i )( (1 + 2i)^3 = (-3 + 4i)(1 + 2i) = -3(1) - 6i + 4i + 8i^2 = -3 - 2i - 8 = -11 - 2i )( (1 + 2i)^4 = (-11 - 2i)(1 + 2i) = -11(1) - 22i - 2i -4i^2 = -11 -24i +4 = -7 -24i )Continuing:( (1 + 2i)^5 = (-7 -24i)(1 + 2i) = -7 -14i -24i -48i^2 = -7 -38i +48 = 41 -38i )It seems like the powers are not becoming real. So maybe only specific slopes ( b/a ) correspond to angles that are rational multiples of ( pi ). In fact, the only time when ( tan(theta) ) is rational and ( theta ) is a rational multiple of ( pi ) is when ( theta ) is a multiple of ( pi/4 ), because those are the only angles with rational tangents that correspond to constructible angles. Wait, but ( tan(pi/6) = 1/sqrt{3} ), which is irrational. So perhaps even angles with irrational tangent can be rational multiples of ( pi ). For example, ( pi/3 ) has tangent ( sqrt{3} ), which is irrational, but ( pi/3 ) is a rational multiple of ( pi ). Therefore, the key is that even if the tangent is irrational, as long as the angle is a rational multiple of ( pi ), then some power of the complex number will be real. But how can we determine whether a given Gaussian integer ( a + bi ) has an argument that is a rational multiple of ( pi )? This seems related to roots of unity. If ( z = a + bi ) is such that ( z^ell ) is real, then ( z ) must be a complex root of a real number. In other words, ( z ) lies on a line through the origin with angle ( theta = frac{2pi k}{ell} ) for some integers ( k ) and ( ell ). Therefore, ( theta ) must be a rational multiple of ( 2pi ), i.e., ( theta = frac{2pi k}{ell} ), which simplifies to ( theta = frac{pi k'}{ell'} ) where ( k' ) and ( ell' ) are coprime. Therefore, ( z ) must lie on a line with angle ( theta = frac{pi m}{n} ), for some integers ( m, n ). But how do we characterize such Gaussian integers?Alternatively, consider that if ( (a + bi)^ell ) is real, then ( (a + bi)^ell = overline{(a + bi)^ell} ), since the conjugate of a real number is itself. Therefore, ( (a + bi)^ell = (a - bi)^ell ), which implies that ( left( frac{a + bi}{a - bi} right)^ell = 1 ). Therefore, ( frac{a + bi}{a - bi} ) must be a root of unity. Let ( frac{a + bi}{a - bi} = frac{(a + bi)^2}{a^2 + b^2} = frac{a^2 - b^2 + 2abi}{a^2 + b^2} ). Let this be equal to ( e^{itheta} ), so ( theta = 2 tan^{-1}(b/a) ). For this to be a root of unity, ( theta ) must be a rational multiple of ( 2pi ). Therefore, ( 2 tan^{-1}(b/a) = frac{2pi k}{m} ), which implies ( tan^{-1}(b/a) = frac{pi k}{m} ), so ( tan(frac{pi k}{m}) = frac{b}{a} ). Therefore, the ratio ( frac{b}{a} ) must be equal to ( tan(frac{pi k}{m}) ) for some integers ( k, m ). So, for a pair ( (a, b) ) to be primitive, ( frac{b}{a} ) must be the tangent of a rational multiple of ( pi ). But the question is, for integer pairs ( (a, b) ), when is ( frac{b}{a} ) equal to ( tan(frac{pi k}{m}) )? The possible values of ( tan(frac{pi k}{m}) ) that are rational are limited. For example, when ( m = 4 ), ( tan(pi/4) = 1 ); for ( m = 8 ), ( tan(pi/8) = sqrt{2} - 1 approx 0.414 ), which is irrational; ( tan(pi/6) = 1/sqrt{3} approx 0.577 ), also irrational. Wait, but in the example earlier, ( (1 + i) ) works because ( tan(pi/4) = 1 ). Similarly, ( (1 + 0i) ) or ( (0 + 1i) ) trivially work because their powers are real immediately. So perhaps the only primitive pairs are those lying on the axes (i.e., ( a = 0 ) or ( b = 0 )) and those lying on the lines ( a = b ), ( a = -b ), etc., but given the domain ( 0 leq a, b leq n ), negative slopes like ( a = -b ) would only be relevant if ( a ) or ( b ) could be negative, which they can't here. Wait, but even ( a = b ), like ( (1,1) ), which we saw gives ( (1 + i)^4 = -4 ), which is real. Similarly, ( (2, 2) ): ( (2 + 2i)^4 = (2(1 + i))^4 = 16(1 + i)^4 = 16(-4) = -64 ), which is real. So in general, any pair where ( a = b ) would work. Similarly, what about pairs where ( b/a = sqrt{3} )? But since ( a ) and ( b ) are integers, ( sqrt{3} ) is irrational, so such pairs can't exist. Therefore, only ratios ( b/a ) that are rational and correspond to a tangent of a rational multiple of ( pi ) would work. But from what we know, the only rational values of ( tan(theta) ) where ( theta ) is a rational multiple of ( pi ) are 0, 1, and undefined (i.e., infinity). Because other angles like ( pi/3 ) have irrational tangents. Therefore, perhaps the only primitive pairs are those on the axes (i.e., ( a = 0 ) or ( b = 0 )) and those on the lines ( y = x ) and ( y = -x ). But since ( a ) and ( b ) are non-negative, ( y = -x ) would only pass through the origin here. Wait, but in the first quadrant (since ( a, b geq 0 )), the lines with angles that are rational multiples of ( pi ) with rational slope would only be the axes (0 or ( pi/2 )) and the line ( y = x ) (angle ( pi/4 )). Because those correspond to slopes 0, infinity, and 1, which are rational and correspond to angles that are rational multiples of ( pi ). So, is that all? Are the only primitive pairs those with ( a = 0 ), ( b = 0 ), or ( a = b )? Let's test this. For example, take ( (1, 2) ). If ( (1 + 2i)^ell ) is real for some ( ell ), then the argument of ( 1 + 2i ), which is ( tan^{-1}(2) ), must be a rational multiple of ( pi ). But I don't think ( tan^{-1}(2) ) is a rational multiple of ( pi ). Similarly, take ( (3, 4) ). The angle is ( tan^{-1}(4/3) ), which is not a rational multiple of ( pi ), so likely ( (3, 4) ) is not primitive. Therefore, perhaps the only primitive pairs are those on the axes and the line ( a = b ). Let's check another example. Take ( (1, sqrt{3}) ), but since ( sqrt{3} ) is irrational, this is not an integer pair. So, in integer pairs, the only way to get a rational tangent that is a rational multiple of ( pi ) is when the slope is 0, infinity, or 1. Hence, perhaps the primitive pairs are:1. All pairs where ( a = 0 ) (the vertical axis). For these, ( (0 + bi)^ell ) is real if ( ell ) is even (giving ( (-b^2)^{ell/2} )) or if ( ell ) is a multiple of 1 for ( b = 0 ).Wait, actually, if ( a = 0 ), then ( z = bi ). So ( z^ell = (bi)^ell ). This is real if ( ell ) is even, because ( i^2 = -1 ), so ( (bi)^2 = -b^2 ), which is real. For ( ell = 2 ), it's real. Similarly, if ( b = 0 ), then ( z = a ), which is already real, so any power ( ell ) will keep it real. Similarly, pairs where ( a = b ). For example, ( z = a + ai = a(1 + i) ). Then, ( z^4 = a^4 (1 + i)^4 = a^4 (-4) ), which is real. So, any pair with ( a = b ) will have ( z^4 ) real. Therefore, these are primitive. Are there other lines? For example, consider a slope of ( sqrt{3} ), but again, since ( sqrt{3} ) is irrational, there are no integer pairs (except (0,0)) on that line. Similarly, a slope of 1/1, which is 1, gives the line ( a = b ). Therefore, perhaps the primitive pairs are exactly those on the axes (including the origin) and those on the line ( a = b ). Wait, but let's check another angle. Suppose ( theta = pi/6 ), which has a tangent of ( 1/sqrt{3} approx 0.577 ). If there were integer pairs ( (a, b) ) with ( b/a = 1/sqrt{3} ), that would mean ( a = bsqrt{3} ), but since ( sqrt{3} ) is irrational, ( a ) and ( b ) cannot both be integers unless ( b = 0 ), which gives ( a = 0 ). So, no such pairs except the origin. Similarly, ( theta = pi/3 ), tangent ( sqrt{3} approx 1.732 ), which would require ( b/a = sqrt{3} ), again impossible for integers ( a, b ) except ( a = b = 0 ). Therefore, the only non-axis primitive pairs are those on the line ( a = b ). Wait, but what about other slopes? For example, take ( (a, b) = (1, 2) ). The angle is ( tan^{-1}(2) ), which is not a rational multiple of ( pi ), so raising ( (1 + 2i) ) to any power won't make it real. Similarly, for ( (3, 4) ), the angle is ( tan^{-1}(4/3) ), which is also not a rational multiple of ( pi ). So, in that case, the only primitive pairs are those on the axes and the line ( a = b ). But wait, let's check another example. Take ( z = sqrt{3} + i ). The angle here is ( tan^{-1}(1/sqrt{3}) = pi/6 ), which is a rational multiple of ( pi ). However, ( sqrt{3} ) is not an integer, so this is not a Gaussian integer. Therefore, in Gaussian integers, such pairs do not exist except the origin. Therefore, consolidating, the only primitive pairs are:1. All pairs on the x-axis: ( (a, 0) ) for ( 0 leq a leq n ).2. All pairs on the y-axis: ( (0, b) ) for ( 0 leq b leq n ).3. All pairs on the line ( a = b ): ( (k, k) ) for ( 0 leq k leq n ).But wait, is that all? Let me check another example. Take ( z = 1 + sqrt{3}i ). Then, ( z^6 ) is real. Indeed:( z^2 = (1 + sqrt{3}i)^2 = 1 + 2sqrt{3}i - 3 = -2 + 2sqrt{3}i )( z^3 = (-2 + 2sqrt{3}i)(1 + sqrt{3}i) = -2(1) -2sqrt{3}i + 2sqrt{3}i + 6i^2 = -2 - 6 = -8 ), which is real. So ( z^3 = -8 ), hence ( z = 1 + sqrt{3}i ) is primitive. However, again, this is not a Gaussian integer because ( sqrt{3} ) is not an integer. Therefore, such pairs do not exist in our problem's constraints.Therefore, returning to Gaussian integers, perhaps the only primitive pairs are those on the axes and the line ( a = b ). Let's confirm this with another example. Take ( (2, 2) ). Then, ( (2 + 2i)^4 = 16(1 + i)^4 = 16(-4) = -64 ), which is real. Similarly, ( (3, 3)^4 = (3(1 + i))^4 = 81(-4) = -324 ), real. So those work. What about ( (1, 3) )? Let's compute ( (1 + 3i)^2 = 1 + 6i -9 = -8 + 6i ), then ( (-8 + 6i)^2 = 64 - 96i -36 = 28 -96i ), not real. Continuing, ( (28 -96i)(1 + 3i) = 28 + 84i -96i -288i^2 = 28 -12i + 288 = 316 -12i ), still not real. So, it seems that ( (1, 3) ) is not primitive.Therefore, tentatively, the primitive pairs are those on the axes and the line ( a = b ). Let's count them. First, the x-axis: ( (a, 0) ) for ( a = 0, 1, ..., n ). That's ( n + 1 ) pairs.Similarly, the y-axis: ( (0, b) ) for ( b = 0, 1, ..., n ). That's another ( n + 1 ) pairs. However, note that the origin ( (0, 0) ) is counted in both axes, so total axis pairs are ( 2(n + 1) - 1 = 2n + 1 ).Then, the line ( a = b ): pairs ( (k, k) ) for ( k = 0, 1, ..., n ). That's ( n + 1 ) pairs. However, the origin is already included here, so the number of new primitive pairs from the line ( a = b ) is ( n ), excluding the origin. Wait, no: when we count ( a = b ), it's ( k = 0 ) to ( n ), so ( n + 1 ) pairs, but ( (0, 0) ) is already counted in the axes. Therefore, the total number of primitive pairs would be the axis pairs plus the diagonal pairs minus the overlaps. Specifically:Number of axis pairs (including origin): ( 2(n + 1) - 1 = 2n + 1 ).Number of diagonal pairs (excluding origin): ( n ), since ( (1,1), (2,2), ..., (n, n) ).Therefore, total primitive pairs would be ( 2n + 1 + n = 3n + 1 ). But wait, if we include the diagonal pairs from ( k = 0 ) to ( n ), which is ( n + 1 ) pairs, but ( (0,0) ) is already in the axis count. Therefore, total primitive pairs would be ( (2n + 1) + (n + 1) - 1 = 3n + 1 ). Wait, let's do this step by step:1. Axis pairs: x-axis has ( n + 1 ) points, y-axis has ( n + 1 ), but they overlap at (0,0). So total axis pairs: ( (n + 1) + (n + 1) - 1 = 2n + 1 ).2. Diagonal pairs: ( a = b ), from (0,0) to (n,n). That's ( n + 1 ) points. However, (0,0) is already counted in the axes. So if we add the diagonal pairs, the new unique points added are ( n ), since (1,1) to (n,n) are not on the axes.Therefore, total primitive pairs: axis pairs (2n + 1) + diagonal pairs (n) = 3n + 1.Wait, but let's test this with a small n. Let n = 1.n = 1:- Axis pairs: (0,0), (1,0), (0,1). Total 3, which is 2*1 + 1 = 3.- Diagonal pairs: (0,0), (1,1). But (0,0) is already counted. So unique diagonal pair added: (1,1). So total primitive pairs: 3 + 1 = 4.Total possible pairs: (1 + 1)^2 = 4. Wait, so when n = 1, 4 pairs, all of which are primitive? But according to the count, axis pairs are 3, diagonal adds 1, so 4 total. Indeed, all pairs (0,0), (1,0), (0,1), (1,1). Let's check if they are all primitive.- (0,0): z = 0, which is real. So, any power is 0, which is real. So primitive.- (1,0): z = 1. Real, so primitive.- (0,1): z = i. Then, i^2 = -1, which is real. So primitive.- (1,1): z = 1 + i. As before, (1 + i)^4 = -4, real. So primitive.Therefore, indeed, for n = 1, 100% of pairs are primitive. For n = 2:- Axis pairs: (0,0), (1,0), (2,0), (0,1), (0,2). Total 5, which is 2*2 + 1 = 5.- Diagonal pairs: (0,0), (1,1), (2,2). Excluding (0,0), we add (1,1), (2,2). So 2 new pairs.Total primitive pairs: 5 + 2 = 7.Total pairs: (2 + 1)^2 = 9.Percentage: 7 / 9 ≈ 77.78%.So, 77.78% primitive. Similarly, for n = 3:- Axis pairs: 2*3 +1 = 7.- Diagonal pairs: 3 (excluding origin). So total primitive pairs: 7 + 3 = 10.Total pairs: 16.Percentage: 10 / 16 = 62.5%.Continuing:n=4:- Axis: 2*4 +1=9- Diagonal: 4Total primitive: 13Total pairs:2513/25=52%n=5:Primitive: 2*5+1 +5=16Total:3616/36≈44.44%n=10:Primitive:2*10 +1 +10=31Total:12131/121≈25.62%Wait, but this seems to be a linear growth in the number of primitive pairs (3n +1) versus quadratic growth in total pairs (n+1)^2. Therefore, the percentage is roughly (3n +1)/(n+1)^2, which for large n approximates to 3n / n^2 = 3/n. So, 3/n ≈ 0.01 when n ≈ 300. However, we need the exact n where (3n +1)/(n+1)^2 < 0.01.But wait, according to the current logic, the number of primitive pairs is 3n +1. However, wait a minute. Earlier, we considered that the diagonal pairs from (1,1) to (n,n). However, in the case when a=0 or b=0, those are on the axes. But are there other lines with rational slopes that could contribute? For example, suppose a line like a=2b. If (a,b) is on such a line, would that give a primitive pair?Wait, for example, take (2,1). Then z = 2 + i. Let's compute some powers:z^1 = 2 + i (not real)z^2 = (2 + i)^2 = 4 +4i -1 = 3 +4i (not real)z^3 = (3 +4i)(2 +i) = 6 +3i +8i +4i^2 = 6 +11i -4 = 2 +11i (not real)z^4 = (2 +11i)(2 +i) = 4 +2i +22i +11i^2 = 4 +24i -11 = -7 +24i (not real)z^5 = (-7 +24i)(2 +i) = -14 -7i +48i +24i^2 = -14 +41i -24 = -38 +41i (not real)Not real. So (2,1) is not primitive. Therefore, even though the slope is rational, unless it's 0, infinity, or 1, the angle is not a rational multiple of pi. Therefore, perhaps only the axes and the main diagonal contribute.But let me check another example. Take z = 1 + 0i. It's already real. z = 0 + 1i. Then, z^2 = -1, which is real. z = 1 + 1i, z^4 = -4. So, these work. What about z = 2 + 2i. z^4 = (2 + 2i)^4 = 16*(1 + i)^4 = 16*(-4) = -64. Real. Similarly, z = 3 + 3i, z^4 = 81*(-4) = -324. Real. So, all pairs on the line a = b work.But what about other slopes? For example, take z = 1 + 3i. Is there any power that makes it real? Let's check:(1 + 3i)^1 = 1 + 3i(1 + 3i)^2 = 1 + 6i + 9i^2 = -8 + 6i(1 + 3i)^3 = (-8 + 6i)(1 + 3i) = -8 -24i +6i +18i^2 = -8 -18i -18 = -26 -18i(1 + 3i)^4 = (-26 -18i)(1 + 3i) = -26 -78i -18i -54i^2 = -26 -96i +54 = 28 -96i(1 + 3i)^5 = (28 -96i)(1 + 3i) = 28 +84i -96i -288i^2 = 28 -12i +288 = 316 -12iStill not real. So, unless we can find a higher power where it becomes real, which seems unlikely, this pair is not primitive. Therefore, it seems that only the axes and the main diagonal contribute.However, there's a problem here. Let me check another case. For example, consider z = sqrt(3) + i. As mentioned earlier, z^6 is real. But sqrt(3) is not an integer, so this is not a Gaussian integer. Similarly, with other angles. So, in the Gaussian integers, only the axes and the main diagonal have angles that are rational multiples of pi. Therefore, the count of primitive pairs is indeed 3n +1.Wait, but when n = 1, this formula gives 3(1) +1 = 4, which matches the 4 pairs. For n = 2, 3(2)+1=7, but when n=2, total primitive pairs are 7: 5 axis pairs and 2 diagonal pairs. Wait, 5 + 2 =7. Then, for n=2, total pairs are 9, so 7/9≈77.78%. So that works. Similarly, for n=3, 3*3 +1=10 primitive pairs, and total pairs 16, 10/16=62.5%. Therefore, general formula: number of primitive pairs is 3n +1. However, let's confirm for n=0. n=0: only (0,0). 3(0)+1=1, which is correct. So this formula holds.Wait, but wait: in the case of the line a = b, when n = 0, we have (0,0), which is included in the axes. For n = 1, we have (1,1) added. So yes, the formula seems to hold.Therefore, the fraction of primitive pairs is (3n +1)/(n +1)^2. We need to find the smallest n where this fraction is less than 0.01.Therefore, solve (3n +1)/(n +1)^2 < 1/100.Multiply both sides by (n +1)^2 (assuming n positive, which it is):3n +1 < (n +1)^2 /100Multiply both sides by 100:100(3n +1) < (n +1)^2Expand RHS:300n + 100 < n^2 + 2n +1Bring all terms to left:n^2 + 2n +1 -300n -100 > 0Simplify:n^2 -298n -99 >0This is a quadratic inequality. Let's solve n^2 -298n -99 >0.First, find the roots of n^2 -298n -99 =0.Using quadratic formula:n = [298 ± sqrt(298^2 +4*99)] /2Compute discriminant:298^2 = (300 -2)^2 = 300^2 -4*300 +4 =90000 -1200 +4=888044*99=396So discriminant =88804 +396=89200sqrt(89200)=sqrt(100*892)=10*sqrt(892). Let's compute sqrt(892):892=4*223. So sqrt(892)=2*sqrt(223)=approximately 2*14.93=29.86Therefore, sqrt(89200)=10*29.86≈298.6Therefore, roots are approximately:[298 ±298.6]/2Positive root:(298 +298.6)/2≈596.6/2≈298.3Negative root:(298 -298.6)/2≈(-0.6)/2≈-0.3Therefore, the quadratic is positive when n > ~298.3 or n < -0.3. Since n is positive integer, the inequality n^2 -298n -99 >0 holds when n >298.3. Therefore, the smallest integer n where this holds is n=299.But wait, let's verify this. Let's test n=298:Check if 3*298 +1 =895, and (298 +1)^2=299^2=89401So 895/89401≈0.01001, which is just over 1%. So at n=298, the fraction is ~1.001%, which is still above 1%.At n=299:3*299 +1=898(299 +1)^2=300^2=90000898/90000≈0.009977, which is ~0.9977%, less than 1%. Therefore, the answer would be n=299.But wait, according to the quadratic solution, the critical point is around 298.3, so n=299 is the first integer where the fraction drops below 1%.However, let's confirm with exact calculation.For n=298:Left side: 3*298 +1=895Right side: (298 +1)^2=299^2=89401Check 895 <89401/100=894.01? 895 <894.01? No, 895>894.01. Therefore, inequality not satisfied.n=298: 895>894.01 → Not less than 1%.n=299:Left side:3*299 +1=898Right side: (299 +1)^2=90000Check 898 <90000/100=900. 898 <900? Yes. Therefore, 898 <900 → 898/90000≈0.009977<0.01.Therefore, the smallest n is 299.But wait, according to our previous logic, the number of primitive pairs is 3n +1. However, this assumes that the only primitive pairs are the axes and the main diagonal. However, is this actually the case? Are there other primitive pairs that we are missing?Wait, let's think again. For example, take the pair (1, 2). Is there any power of (1 + 2i) that is real? Let's compute (1 + 2i)^2 = -3 + 4i, (1 + 2i)^3 = (-3 + 4i)(1 + 2i) = -3 -6i +4i +8i^2 = -3 -2i -8 = -11 -2i, (1 + 2i)^4 = (-11 -2i)(1 + 2i) = -11 -22i -2i -4i^2 = -11 -24i +4 = -7 -24i, (1 + 2i)^5 = (-7 -24i)(1 + 2i) = -7 -14i -24i -48i^2 = -7 -38i +48 = 41 -38i, etc. It seems like it's not becoming real. So, (1,2) is not primitive.Another test: (3,4). Let's compute (3 + 4i)^2 = 9 + 24i +16i^2 = 9 +24i -16 = -7 +24i, (3 +4i)^3 = (-7 +24i)(3 +4i) = -21 -28i +72i +96i^2 = -21 +44i -96 = -117 +44i, (3 +4i)^4 = (-117 +44i)(3 +4i) = -351 -468i +132i +176i^2 = -351 -336i -176 = -527 -336i, still not real. So, (3,4) is not primitive.What about (5,12)? Let's compute (5 +12i)^2 =25 +120i +144i^2=25 +120i -144= -119 +120i, which is not real. (5 +12i)^3= (-119 +120i)(5 +12i)= -595 -1428i +600i +1440i^2= -595 -828i -1440= -2035 -828i, still not real. So no.Therefore, it seems that only the axes and the main diagonal pairs are primitive. Hence, our initial count of 3n +1 is accurate.Therefore, the answer should be n=299. However, let's check the problem statement again: it says "less than 1% of the pairs (a, b) with 0 ≤ a, b ≤ n are primitive." So we need to ensure that (3n +1)/(n +1)^2 <0.01. Solving this gives n=299.But let me check with n=298 and n=299:For n=298:(3*298 +1)/(298 +1)^2 = (895)/89401 ≈0.010013, which is ~1.0013%, just over 1%.For n=299:(3*299 +1)/(299 +1)^2= (898)/90000≈0.009977, which is ~0.9977%, less than 1%.Therefore, the smallest positive integer n required is 299.But wait, is there a possibility that there are other primitive pairs that we haven't considered? For instance, pairs where a or b is zero, but in higher multiplicity? Wait, no. The axes and the main diagonal are the only ones. However, maybe pairs where a + bi is a root of unity multiplied by a real number. But in Gaussian integers, the roots of unity are only 1, -1, i, -i. So if a + bi is a real number multiplied by a root of unity, then it must lie on the axes or on the lines a = ±b. However, since a and b are non-negative, only a = b and the axes are relevant. Therefore, indeed, only those pairs are primitive.Therefore, the answer is boxed{299}.Final Answerboxed{299}</think>To determine the smallest positive integer ( n ) such that less than 1% of the pairs ((a, b)) with (0 leq a, b leq n) are primitive, we need to understand the structure of primitive pairs. A pair ((a, b)) is primitive if there exists a positive integer (ell) such that ((a + bi)^ell) is real. This happens when the argument of (a + bi) is a rational multiple of (pi). Key insights:1. The only Gaussian integers (complex numbers with integer coordinates) that can be raised to a power to become real are those lying on the coordinate axes (either (a = 0) or (b = 0)) or the line (a = b).2. These pairs are counted as follows: - Pairs on the x-axis: (n + 1) - Pairs on the y-axis: (n + 1) - Pairs on the line (a = b): (n + 1) - Overlaps: The origin ((0, 0)) is counted three times, so subtract 2.The total number of primitive pairs is (3n + 1). The total number of pairs ((a, b)) is ((n + 1)^2). We need to find the smallest (n) such that:[frac{3n + 1}{(n + 1)^2} < 0.01]Solving this inequality:1. Multiply both sides by ((n + 1)^2): (3n + 1 < 0.01(n + 1)^2)2. Multiply both sides by 100: (100(3n + 1) < (n + 1)^2)3. Expand and simplify: (n^2 - 298n - 99 > 0)Solving the quadratic equation (n^2 - 298n - 99 = 0) using the quadratic formula, we find the critical points. The positive root is approximately 298.3. Testing (n = 298) and (n = 299):- For (n = 298), the fraction is approximately 1.0013%, which is just over 1%.- For (n = 299), the fraction is approximately 0.9977%, which is less than 1%.Thus, the smallest positive integer (n) such that less than 1% of the pairs are primitive is (boxed{299}).,<think>Okay, let's see. The problem is about finding the smallest positive integer n such that less than 1% of the pairs (a, b) with 0 ≤ a, b ≤ n are primitive. A primitive pair (a, b) is defined as one where there exists a positive integer ℓ such that (a + bi)^ℓ is real. Hmm, interesting.First, I need to understand what makes a pair (a, b) primitive. So, (a + bi) is a complex number, and raising it to some power ℓ makes it real. That means that (a + bi)^ℓ has to be a real number, so its imaginary part must be zero. When do complex numbers raised to a power become real? Well, if the complex number is a real number or a purely imaginary number, then raising it to some power will keep it real or imaginary. But since we allow any positive integer ℓ, maybe even exponents could turn purely imaginary numbers into real ones. For example, (bi)^2 = -b², which is real. So, if a = 0 or b = 0, then (a + bi) is either purely real or purely imaginary, so those pairs (a, b) are definitely primitive because you can take ℓ = 1 if it's already real, or ℓ = 2 if it's purely imaginary.But there might be other complex numbers that aren't purely real or imaginary but still have some power that's real. For example, complex numbers on the unit circle with angles that are rational multiples of π. Wait, but those would have arguments that are multiples of π/2, right? Because if their angle is θ, then raising them to the power ℓ would give an angle ℓθ. For this to be a multiple of π (so that the result is real), we need ℓθ = kπ for some integer k. So θ must be a rational multiple of π. But in our case, the complex numbers a + bi are Gaussian integers, right? So their angles might not necessarily be rational multiples of π, but maybe for some specific ratios of a and b, they could be.Alternatively, maybe we can think in terms of roots of unity. If a complex number is a root of unity multiplied by some real number, then raising it to the appropriate power would make it real. But Gaussian integers are complex numbers with integer coordinates, so unless they lie on the coordinate axes or on the lines y = ±x, maybe? Wait, let's see. For example, 1 + i has an angle of π/4, so raising it to the 4th power gives (1 + i)^4 = (sqrt(2))^4 * e^(iπ) = 4 * (-1) = -4, which is real. So (1 + i)^4 is real. Therefore, the pair (1,1) is primitive because ℓ = 4 works. Similarly, maybe other pairs where a and b are such that the angle is a multiple of π/2 divided by some integer.But how do we characterize all such primitive pairs? Let's try to formalize this.Suppose z = a + bi is a Gaussian integer (a, b non-negative integers). We need z^ℓ to be real for some ℓ ≥ 1. That means z must lie on a line through the origin making an angle θ with the real axis such that ℓθ is a multiple of π/2. Because when you raise a complex number to a power, you multiply its angle by that power. So, for z^ℓ to be real, its angle must be 0 or π modulo π. So ℓθ ≡ 0 mod π. Therefore, θ must be a rational multiple of π where the denominator divides 2ℓ. Wait, maybe it's better to say that θ must be a multiple of π/(2ℓ). Hmm.Alternatively, θ = (kπ)/(2ℓ) for some integer k. Then ℓθ = kπ/2. But we need ℓθ to be an integer multiple of π/2. Wait, maybe θ must be a multiple of π/(2m) for some integer m. Then taking ℓ = m would make the angle mθ = π/2 * (k), which is either 0, π/2, π, 3π/2, etc. But to get a real number, we need the angle to be 0 or π mod 2π. So, ℓθ must be congruent to 0 or π mod 2π. Therefore, θ must be congruent to 0 or π/ℓ mod 2π/ℓ. Hmm, this is getting a bit tangled.Perhaps a better approach is to note that if z = a + bi is not zero, then z^ℓ is real if and only if z is either real, purely imaginary, or lies on a line through the origin with angle θ such that θ = π/(2k) for some integer k. Wait, but even that might not capture all cases. Let's think in terms of ratios. If a + bi is a complex number, then its angle θ is arctan(b/a) (if a ≠ 0). For z^ℓ to be real, we need tan(ℓθ) = 0 or undefined. Because the imaginary part over the real part must be zero, which happens when the angle is 0 or π. So, tan(ℓθ) = 0, meaning ℓθ is a multiple of π. Therefore, θ must be a multiple of π/ℓ. So, arctan(b/a) must be a multiple of π/ℓ. Therefore, b/a = tan(kπ/ℓ) for some integer k.But since a and b are integers, this ratio must be rational. So tan(kπ/ℓ) must be rational. But tan(kπ/ℓ) is rational only for certain angles. For example, tan(π/4) = 1, tan(π/3) = sqrt(3), which is irrational, tan(π/6) = 1/sqrt(3), also irrational. The only angles where tan(θ) is rational with θ being a rational multiple of π are angles where tan(θ) is 0, 1, or undefined (i.e., θ = 0, π/4, π/2, etc.). Wait, but tan(π/4) = 1, which is rational, but tan(π/3) is irrational. So perhaps the only angles where tan(kπ/ℓ) is rational are those where kπ/ℓ is 0, π/4, π/2, 3π/4, etc. So that would correspond to lines with slopes 0, 1, undefined, -1. So in that case, the pairs (a, b) where either a = 0, b = 0, a = b, or a = -b. But since a and b are non-negative, a = -b would only be possible if b = 0. So, the only possible slopes would be 0, infinity, or 1. So, does that mean the only primitive pairs are those on the axes (a=0 or b=0) or on the line a = b?Wait, but we saw earlier that (1 + i)^4 is real. So (1,1) is a primitive pair. Similarly, (2,2) would also be primitive because (2 + 2i)^4 = (2(1 + i))^4 = 16*(1 + i)^4 = 16*(-4) = -64, which is real. Similarly, (1,2) might not be primitive. Let's check. Take z = 1 + 2i. Let's compute some powers. z^2 = (1 + 2i)^2 = 1 - 4 + 4i = -3 + 4i. Not real. z^3 = (-3 + 4i)(1 + 2i) = -3 -6i +4i +8i² = -3 -2i -8 = -11 -2i. Not real. z^4 = (-11 -2i)^2 = 121 + 44i +4i² = 121 +44i -4 = 117 +44i. Not real. z^5 would be even worse. Maybe there's no ℓ such that z^ℓ is real. So perhaps (1,2) is not primitive.Therefore, maybe the only primitive pairs are those on the axes (a=0 or b=0) or on the line a=b. But wait, let's check another pair not on the axes or a=b. For example, (3,1). Let's compute (3 + i)^ℓ. Let's try ℓ=4. (3 + i)^2 = 9 +6i +i² = 9 +6i -1 = 8 +6i. Then (8 +6i)^2 = 64 + 96i +36i² = 64 +96i -36 = 28 +96i. Not real. How about ℓ=3? (3 + i)^3 = (3 + i)(8 +6i) = 24 +18i +8i +6i² = 24 +26i -6 = 18 +26i. Not real. Hmm. Maybe there is no ℓ for which this becomes real. So perhaps only the axes and the line a = b have primitive pairs.Wait, but let's check (2,1). (2 + i)^2 = 4 +4i +i² = 4 +4i -1 = 3 +4i. (3 +4i)^2 = 9 +24i +16i² = 9 +24i -16 = -7 +24i. Not real. (2 + i)^3 = (2 + i)(3 +4i) = 6 +8i +3i +4i² = 6 +11i -4 = 2 +11i. Still not real. (2 + i)^4 = (2 +11i)^2 = 4 +44i +121i² = 4 +44i -121 = -117 +44i. Not real. So again, maybe not primitive.Alternatively, let's try (1,3). (1 +3i)^2 = 1 +6i +9i² = 1 +6i -9 = -8 +6i. Not real. (1 +3i)^3 = (-8 +6i)(1 +3i) = -8 -24i +6i +18i² = -8 -18i -18 = -26 -18i. Not real. (1 +3i)^4 = (-26 -18i)^2 = 676 + 936i + 324i² = 676 +936i -324 = 352 +936i. Still not real. So maybe only the axes and a=b lines work.Wait, but let's check another angle. Suppose a=1, b=√3. Then angle is π/3, tan(θ)=√3. Then (a + bi)^6 would be real, because angle becomes 6*(π/3) = 2π, which is real. But since a and b must be integers, we can't have b=√3. So maybe in the case of integer coordinates, the only way for tan(theta) to be rational is if tan(theta) is 0, 1, or undefined. Because tan(theta) = b/a must be rational, and theta must be such that ℓ*theta is a multiple of pi/2. So if theta is pi/4, then tan(theta)=1, which is rational, and ℓ=4 gives theta*4=pi, which is real. Similarly, if theta is 0 or pi/2, which are the axes.Therefore, perhaps the only primitive pairs are those on the axes (a=0 or b=0) or on the line a = b. Because those have angles 0, pi/2, or pi/4. Then, raising to power 4 for the a = b case gives a real number. For the axes, raising to power 1 (if on the real axis) or power 2 (if on the imaginary axis) gives real numbers.Therefore, primitive pairs are (a, 0), (0, b), and (a, a) for non-negative integers a, b. Wait, but we need to confirm this. For example, (a, a): (a + ai)^ℓ. Let's compute ℓ=4: (a + ai)^4 = a^4*(1 + i)^4 = a^4*(-4) which is real. So yes, ℓ=4 works. For (a, 0), z = a is already real, so ℓ=1. For (0, b), z = bi, so (bi)^2 = -b² which is real, so ℓ=2.But are there other primitive pairs not on these lines? For example, let's take (3, 4). (3 + 4i) has magnitude 5 and angle arctan(4/3). Let's see if any power of this is real. If we take (3 + 4i)^2 = -7 + 24i, not real. (3 +4i)^4 = (-7 +24i)^2 = 49 - 336i + 576i² = 49 - 336i -576 = -527 -336i. Not real. Maybe higher exponents? Let's try ℓ=5. Hmm, this seems tedious. Alternatively, note that the angle is arctan(4/3), which is not a rational multiple of pi. So, unless arctan(4/3) is a rational multiple of pi, which I don't think it is, then there's no ℓ such that ℓ*arctan(4/3) is a multiple of pi/2. Therefore, (3,4) is not primitive.Thus, perhaps the only primitive pairs are those on the axes or the line a = b. If that's the case, then the number of primitive pairs up to n would be the number of pairs on the axes plus the number on the line a = b.But wait, let's check another example. Suppose (a, b) is (1, 2). Then (1 + 2i). Let's see if any power makes it real. (1 + 2i)^2 = -3 + 4i. (1 + 2i)^3 = (-3 + 4i)(1 + 2i) = -3 -6i +4i +8i² = -3 -2i -8 = -11 -2i. (1 + 2i)^4 = (-11 -2i)^2 = 121 + 44i +4i² = 121 +44i -4 = 117 +44i. Not real. Seems like it's not becoming real. So perhaps only axes and a = b.But let's check a case where a and b are not equal but maybe have some common factor. For example, (2, 4). Then z = 2 + 4i = 2(1 + 2i). Then z^2 = 4(1 + 2i)^2 = 4*(-3 +4i) = -12 +16i. Not real. z^3 = 8*(1 +2i)^3 = 8*(-11 -2i) = -88 -16i. Not real. So still not primitive.Alternatively, let's consider a pair where a = kb for some k. For example, (1, 3). Then z = 1 + 3i. Raising to powers: z^2 = -8 + 6i, z^3 = -26 - 18i, z^4 = 352 + 936i (wait, that doesn't seem right). Wait, (1 + 3i)^4 is [(1 + 3i)^2]^2 = (-8 + 6i)^2 = 64 - 96i + 36i² = 64 -96i -36 = 28 -96i. Not real. So, again, not primitive.Therefore, it seems like only the axes (a=0 or b=0) and the diagonal (a=b) are primitive. Wait, but let me check (a, b) where a = -b. But since a and b are non-negative, the only case would be b = 0, which is already covered by the real axis. So, in our case, since a and b are non-negative integers, the only lines we can have are the axes and the line a = b.Therefore, the number of primitive pairs (a, b) with 0 ≤ a, b ≤ n is:- Number of pairs on the x-axis: (n + 1) pairs (since a can be 0 to n, b=0)- Number of pairs on the y-axis: (n + 1) pairs (since b can be 0 to n, a=0)- Number of pairs on the line a = b: (n + 1) pairs (since a = b from 0 to n)However, we have to subtract the overlapping pairs. The pair (0, 0) is counted in all three sets. So total primitive pairs would be:(n + 1) + (n + 1) + (n + 1) - 2 (for the overlaps: (0,0) is counted three times, so we need to subtract 2 to count it once)Wait, let's be precise. The three sets are:1. X-axis: all (a, 0) for 0 ≤ a ≤ n2. Y-axis: all (0, b) for 0 ≤ b ≤ n3. Diagonal: all (k, k) for 0 ≤ k ≤ nThe overlaps are:- (0,0) is in all three sets- No other overlaps because if a ≠ 0 and b ≠ 0, then (a, 0) is only on x-axis, (0, b) only on y-axis, and (k, k) only on diagonal unless k=0.Therefore, using inclusion-exclusion:Total primitive pairs = |X-axis| + |Y-axis| + |Diagonal| - |X ∩ Y| - |X ∩ Diagonal| - |Y ∩ Diagonal| + |X ∩ Y ∩ Diagonal|But X ∩ Y is only (0,0). X ∩ Diagonal is also (0,0). Y ∩ Diagonal is also (0,0). And X ∩ Y ∩ Diagonal is (0,0).Therefore:Total primitive pairs = (n + 1) + (n + 1) + (n + 1) - 1 -1 -1 +1 = 3(n +1) -3 +1 = 3n +3 -2 = 3n +1.Wait, let me check:|X-axis| = n+1, |Y-axis|=n+1, |Diagonal|=n+1.The intersections:X ∩ Y: only (0,0), so 1.X ∩ Diagonal: only (0,0), so 1.Y ∩ Diagonal: only (0,0), so 1.X ∩ Y ∩ Diagonal: only (0,0), so 1.Therefore, inclusion-exclusion gives:Total = |X| + |Y| + |D| - |X∩Y| - |X∩D| - |Y∩D| + |X∩Y∩D|= (n+1)+(n+1)+(n+1) -1 -1 -1 +1= 3(n+1) -3 +1= 3n +3 -3 +1= 3n +1So total primitive pairs are 3n +1.Wait, for example, when n=1:Primitive pairs are (0,0), (1,0), (0,1), (1,1). So 4 pairs. According to formula, 3*1 +1 = 4. Correct.n=2: primitive pairs are (0,0), (1,0), (2,0), (0,1), (0,2), (1,1), (2,2). So 7 pairs. Formula: 3*2 +1=7. Correct. So formula holds.Therefore, the number of primitive pairs is 3n +1.Wait, but hold on. Let's check n=0: 3*0 +1=1. Which is correct, only (0,0).So seems like the formula is correct.Therefore, the number of primitive pairs is 3n +1.But the total number of pairs (a, b) with 0 ≤ a, b ≤ n is (n +1)^2.Therefore, the proportion of primitive pairs is (3n +1)/(n +1)^2.We need to find the smallest positive integer n such that this proportion is less than 1%, i.e., (3n +1)/(n +1)^2 < 1/100.So solving for n in (3n +1)/(n +1)^2 < 1/100.Let me set up the inequality:(3n +1) < (n +1)^2 / 100Multiply both sides by 100:100(3n +1) < (n +1)^2Expand the right side:300n +100 < n² + 2n +1Bring all terms to one side:n² + 2n +1 -300n -100 > 0Simplify:n² -298n -99 > 0This is a quadratic inequality: n² -298n -99 > 0To find where this is positive, find the roots of the equation n² -298n -99 = 0.Using quadratic formula:n = [298 ± sqrt(298² + 4*99)] / 2Compute discriminant:298² = (300 -2)² = 300² -2*300*2 +4 = 90000 -1200 +4 = 888044*99 = 396So discriminant = 88804 + 396 = 89200sqrt(89200). Let's see, 298^2 = 88804, so sqrt(89200) ≈ 298.66 (since 298.66^2 ≈ (298 +0.66)^2 ≈ 298² + 2*298*0.66 +0.66² ≈ 88804 +393.36 +0.4356≈ 89197.8≈ 89200). So approximately 298.66.Therefore, roots:[298 ± 298.66]/2Positive root: (298 +298.66)/2 ≈ 596.66/2 ≈ 298.33Negative root: (298 -298.66)/2 ≈ (-0.66)/2 ≈ -0.33So the quadratic is positive when n < -0.33 or n > 298.33. Since n is a positive integer, we need n > 298.33, so n ≥ 299.But wait, we need to check if n=299 satisfies the original inequality.Let me check n=299:Left side: 3*299 +1 = 897 +1 = 898Right side: (299 +1)^2 = 300^2 = 90,000So 898 /90,000 ≈ 0.009977, which is approximately 0.9977%, which is less than 1%. So n=299 works.Check n=298:Left side: 3*298 +1 = 894 +1=895Right side: (298 +1)^2=299²=89401895 /89401 ≈ 0.010011, which is approximately 1.0011%, which is just over 1%. So n=298 does not satisfy the inequality.Therefore, the smallest n is 299.Wait, but before finalizing this answer, let me double-check my reasoning.First, we assumed that the only primitive pairs are those on the axes and the diagonal. Is that actually the case?Earlier, we considered (1 + i)^4 being real, which is correct. Similarly, (2 + 2i)^4 is real, etc. But are there other primitive pairs not on the axes or diagonal?Suppose (a, b) is such that a/b = tan(kπ/(2ℓ)) for some integers k and ℓ. If such a ratio is rational, then maybe (a, b) is primitive. For example, take tan(π/6) = 1/√3. But since a and b must be integers, this would require a =1, b=√3, which is not possible. Similarly, tan(π/3)=√3, which would require a=√3, b=1, again not integers. The only angles where tan(theta) is rational and theta is a rational multiple of π are 0, π/4, π/2, etc. Therefore, only slopes 0, 1, infinity. So yes, only the axes and the diagonal.Therefore, our initial conclusion that primitive pairs are those on the axes and diagonal is correct. Hence, the number of primitive pairs is indeed 3n +1.Therefore, the proportion is (3n +1)/(n +1)^2 < 1/100, leading to n=299 as the minimal solution.Therefore, the answer is boxed{299}Final Answerboxed{299}</think>To determine the smallest positive integer ( n ) such that less than 1% of the pairs ((a, b)) with (0 le a, b le n) are primitive, we first need to characterize primitive pairs. A pair ((a, b)) is primitive if there exists a positive integer (ell) such that ((a + bi)^ell) is real. Through analysis, we found that the only primitive pairs are those on the axes ((a = 0) or (b = 0)) or on the line (a = b). This is because these are the only cases where the complex number (a + bi) can be raised to a power to become real. The number of primitive pairs is calculated as follows:- The number of pairs on the x-axis: (n + 1)- The number of pairs on the y-axis: (n + 1)- The number of pairs on the line (a = b): (n + 1)Using inclusion-exclusion principle to account for overlaps (specifically the pair ((0, 0)) counted three times), the total number of primitive pairs is:[3(n + 1) - 3 + 1 = 3n + 1]The total number of pairs ((a, b)) with (0 le a, b le n) is ((n + 1)^2). We need this proportion to be less than 1%:[frac{3n + 1}{(n + 1)^2} < frac{1}{100}]Solving the inequality:[100(3n + 1) < (n + 1)^2][300n + 100 < n^2 + 2n + 1][n^2 - 298n - 99 > 0]Finding the roots of the quadratic equation (n^2 - 298n - 99 = 0) using the quadratic formula, we determine the positive root is approximately 298.33. Thus, the smallest integer (n) satisfying the inequality is 299. Verification shows that for (n = 299), the proportion is approximately 0.9977%, which is less than 1%, while for (n = 298), the proportion is approximately 1.0011%, which is more than 1%. Therefore, the smallest positive integer (n) is (boxed{299}).]

🔑:1. Understanding the Problem: We need to find the smallest positive integer ( n ) such that less than ( 1% ) of the pairs ((a, b)) with ( 0 le a, b le n ) are primitive. A pair ((a, b)) is called primitive if there exists a positive integer (ell) such that ((a + bi)^ell) is real.2. Characterizing Primitive Pairs: To determine when ((a + bi)^ell) is real, we need to consider the argument of the complex number (a + bi). The argument (theta) of (a + bi) is given by (theta = tan^{-1}left(frac{b}{a}right)). For ((a + bi)^ell) to be real, (ell theta) must be an integer multiple of (pi). This implies that (theta) must be a rational multiple of (pi).3. Primitive Pair Conditions: The argument (theta) is a rational multiple of (pi) if and only if (frac{b}{a}) is rational. This means that (a) and (b) must be such that (frac{b}{a}) is rational, or equivalently, (a) and (b) must be proportional to each other. Therefore, the primitive pairs ((a, b)) are those where (a = 0), (b = 0), or (a = b).4. Counting Primitive Pairs: - Pairs of the form ((0, k)) where (0 le k le n): There are (n + 1) such pairs. - Pairs of the form ((k, 0)) where (0 le k le n): There are (n + 1) such pairs. - Pairs of the form ((k, k)) where (0 le k le n): There are (n + 1) such pairs. - Note that the pair ((0, 0)) is counted three times, so we need to subtract 2 from the total count. Therefore, the total number of primitive pairs is: [ (n + 1) + (n + 1) + (n + 1) - 2 = 3n + 1 ]5. Total Number of Pairs: The total number of pairs ((a, b)) with (0 le a, b le n) is: [ (n + 1)^2 ]6. Finding (n): We need to find the smallest (n) such that the proportion of primitive pairs is less than (1%): [ frac{3n + 1}{(n + 1)^2} < 0.01 ] Solving this inequality: [ 3n + 1 < 0.01 (n + 1)^2 ] [ 3n + 1 < 0.01 (n^2 + 2n + 1) ] [ 3n + 1 < 0.01n^2 + 0.02n + 0.01 ] [ 3n + 1 - 0.02n - 0.01 < 0.01n^2 ] [ 2.98n + 0.99 < 0.01n^2 ] [ 0.01n^2 - 2.98n - 0.99 > 0 ] Solving the quadratic inequality: [ n = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] where (a = 0.01), (b = -2.98), and (c = -0.99): [ n = frac{2.98 pm sqrt{(2.98)^2 - 4 cdot 0.01 cdot (-0.99)}}{2 cdot 0.01} ] [ n = frac{2.98 pm sqrt{8.8804 + 0.0396}}{0.02} ] [ n = frac{2.98 pm sqrt{8.92}}{0.02} ] [ n = frac{2.98 pm 2.985}{0.02} ] [ n = frac{5.965}{0.02} quad text{or} quad n = frac{-0.005}{0.02} ] [ n approx 298.25 quad text{or} quad n approx -0.25 ] Since (n) must be a positive integer, we take (n = 299).The final answer is (boxed{299})

🧠:[<think>Okay, let's see. The question is about a regression equation, right? The equation given is y = 3 - 5x. And the question is asking what happens to y when x increases by one unit. The options are about whether y increases or decreases by 5 or 3 units. Hmm.First, I remember that in a linear regression equation of the form y = a + bx, the coefficient b represents the slope. That means for every one unit increase in x, y changes by b units. If b is positive, then y increases, and if b is negative, y decreases. So in this case, the equation is y = 3 - 5x. Let me rewrite that in the standard form. It would be y = -5x + 3. So the intercept is 3, and the slope is -5.So the slope here is -5. That means for each one unit increase in x, the predicted y changes by -5 units. Since it's negative, that's a decrease. Therefore, when x increases by one unit, y decreases by 5 units on average. Let me check the options again.Option A says "y decreases by an average of 5 units." That seems right because the coefficient is -5. Option B talks about increasing by 3 units, which doesn't make sense because the 3 is the intercept, not the slope. Option C is decrease by 3, which would be if the slope was -3, but here it's -5. Option D is increase by 5, which is the opposite sign. So the correct answer should be A.Wait, just to make sure I didn't mix up anything. The intercept is where the line crosses the y-axis when x is 0, right? So when x is 0, y is 3. But the question is about the change when x increases. So yeah, the slope is definitely the key here. Each unit increase in x leads to a change of the slope coefficient in y. Since the coefficient is -5, that's a decrease of 5. So yes, A is correct. The other options are either using the intercept or have the wrong direction or magnitude. So I think I can confirm that A is the answer.Final Answerboxed{A}</think>The given regression equation is ( y = 3 - 5x ). In a linear regression equation of the form ( y = a + bx ), the coefficient ( b ) (the slope) represents the change in ( y ) for each one unit increase in ( x ). Here, the equation can be rewritten as ( y = -5x + 3 ), where the slope ( b ) is -5. This means that for each one unit increase in ( x ), the predicted value of ( y ) decreases by 5 units. Thus, the correct answer is (boxed{A}).,<think>Okay, so I have this regression equation here: y = 3 - 5x. The question is asking what happens to y when x increases by one unit. The options are about whether y increases or decreases by 3 or 5 units. Hmm, let's think.First, I remember that in a linear regression equation of the form y = a + bx, the coefficient b represents the slope. That means for every one unit increase in x, y changes by b units. If b is positive, y increases, and if b is negative, y decreases. Right?So in the given equation, y = 3 - 5x, the intercept is 3 and the coefficient for x is -5. That coefficient is the slope. So when x increases by 1, y should change by -5 units. Which would mean a decrease of 5 units. Let me verify that.Let me plug in some numbers. Suppose x is 0. Then y = 3 - 5(0) = 3. If x increases by 1 to 1, then y = 3 - 5(1) = 3 - 5 = -2. So y went from 3 to -2, which is a decrease of 5. If x increases by another unit to 2, y = 3 - 5(2) = 3 -10 = -7. So from -2 to -7 is again a decrease of 5. So each time x increases by 1, y decreases by 5.Looking at the options, option A says y decreases by an average of 5 units. Option B says increases by 3, which doesn't make sense because the coefficient is negative. Option C is decrease by 3, but we saw it's 5. Option D is increase by 5, which is the opposite. So the correct answer should be A.Wait, let me make sure I didn't mix up anything. The equation is definitely y = 3 -5x. So the slope is -5. So for each unit increase in x, y is predicted to decrease by 5. Yeah, that's right. So A is correct. The other options don't fit. B and D have the direction wrong (increase vs decrease) and the wrong magnitude, C has the magnitude wrong. So definitely A.</think>The regression equation is given as ( y = 3 - 5x ). In linear regression, the coefficient of ( x ) (which is (-5) here) indicates the change in ( y ) for a one-unit increase in ( x ). A negative coefficient means ( y ) decreases as ( x ) increases. Specifically, when ( x ) increases by 1 unit, ( y ) decreases by 5 units on average. Testing with example values confirms this relationship (e.g., ( x = 0 rightarrow y = 3 ), ( x = 1 rightarrow y = -2 ), showing a decrease of 5). Answer: boxed{A}]

🔑:AnalysisSince there is a regression equation y = 3 - 5x, when the variable x increases by one unit, y decreases by an average of 5 units. Therefore, the correct choice is boxed{text{A}}.

🧠:[<think>Okay, let's see. I need to find the equation that describes the acceleration of the ball when it's attached to the spring and oscillating. The options are given, and I need to pick the correct one. Hmm. Let me start by recalling some physics concepts related to springs and oscillations.First, when a mass is attached to a spring, the motion is governed by Hooke's Law and Newton's Second Law. Hooke's Law states that the force exerted by the spring is proportional to the displacement from its equilibrium position, right? So F_spring = -kx, where x is the displacement. The negative sign indicates that the force is restoring, meaning it acts in the opposite direction of the displacement.But in this problem, the spring is vertical, not horizontal. I remember that for vertical springs, the equilibrium position isn't where the spring is unstretched; instead, the weight of the mass stretches it a bit. So when the ball is attached to the spring, the spring will stretch until the force from the spring equals the weight of the ball. That equilibrium point is where the net force is zero.However, in this case, the ball is oscillating, so we need to consider the forces at any position y above the original top end of the spring. Let me visualize the setup. The original position of the top end is the point where the spring was before the ball was attached. Once the ball is attached, it will stretch the spring, but during oscillation, the ball moves up and down around the equilibrium position.Wait, the problem says "when it is at a height y above the original position of the top end of the spring." So y is measured from the original, unstretched position. Let's be careful here. If the ball is oscillating, its motion is around the equilibrium position, which is lower than the original top end due to the weight of the ball. So when the ball is at height y above the original position, that might actually be above or below the equilibrium point, depending on the oscillation.But the key is to find the acceleration at that position y. Let's think about the forces acting on the ball when it's at position y. There's the gravitational force pulling down, which is mg, and the spring force. The spring force depends on how much the spring is compressed or stretched from its original position. Since y is measured from the original position, if the ball is at height y above that original position, then the spring's displacement from its original position is y' = -y (since if the ball is above the original position, the spring is compressed by y, so the displacement is in the negative direction). Wait, maybe I need to be precise with coordinates here.Let me set up a coordinate system. Let's take the original position of the top end of the spring as the origin (y=0). The downward direction is negative, as per the problem statement. So when the ball is attached to the spring and starts oscillating, its position y can be positive (above the original position) or negative (below the original position). However, if the ball is above the original position, that means the spring is compressed; if it's below, the spring is stretched.But Hooke's Law says the spring force is F = -kx, where x is the displacement from the equilibrium position. Wait, no, actually, Hooke's Law is from the natural length of the spring. In the vertical case, the equilibrium position is where the spring is stretched by some amount due to the weight of the mass. So in this problem, the original position (before the mass is attached) is the natural length of the spring.When the mass is attached, the spring stretches until mg = kx_0, where x_0 is the stretch at equilibrium. So the equilibrium position is x_0 = mg/k below the original position. Therefore, if we take the original position as y=0, the equilibrium position is at y = -x_0 = -mg/k.But the problem states that y is the height above the original position. So when the ball is at position y, relative to the original position, which is our coordinate origin. The displacement from the equilibrium position would be y - (-mg/k) = y + mg/k. Wait, maybe I need to clarify.Wait, the coordinate system: Let's define y as the position relative to the original top end of the spring (y=0). The positive direction is upwards, but the problem says "down is negative," so maybe I should define downward as negative. Let me check the problem statement again: "Let down be negative." So, if the ball is above the original position, that's positive y, and below is negative y. But when the ball is attached to the spring, the spring will stretch downward, so the equilibrium position is at some negative y.But when considering the displacement for Hooke's Law, we need to measure the displacement from the original, natural length of the spring. Since the spring is mounted with the bottom end fixed and the top end originally at y=0. When the ball is attached, it can move the top end to y positions. So the displacement of the spring from its natural length is actually |y|, but since downward is negative, if the ball is at y (which could be positive or negative), the displacement from the original position is negative if it's stretched downward. Wait, this is confusing.Wait, let's get precise. The spring's natural length is with the top end at y=0. If the ball is attached and pulls the spring down, then the top end is at some negative y (since down is negative). The displacement from the natural length is then Δy = y_final - y_initial. If the top end moves down to y = -d, then the displacement from natural length is -d (since it's stretched downward by d). So the spring force is F = -k * (-d) = kd upward. Which makes sense because if the spring is stretched downward, it pulls upward.Alternatively, if the ball is at a position y above the original position (so y is positive), then the spring is compressed. The displacement from natural length is y (since it's compressed by y), so the spring force is F = -k * y (since displacement is upward, the force is downward). But since the ball is attached to the top end, if the spring is compressed, it would push downward on the ball.Wait, maybe I need to model this.Let me take the coordinate system with y=0 at the original top end of the spring, and downward is negative. So:- If the ball is above y=0, that's positive y, meaning the spring is compressed.- If the ball is below y=0, that's negative y, meaning the spring is stretched.But Hooke's Law: the force exerted by the spring is F_spring = -k * displacement from natural length.Displacement from natural length is the amount the spring is compressed or stretched. So if the spring is compressed by y (ball is at +y), then displacement is +y, so F_spring = -k*y (since the spring wants to return to natural length, it exerts a force downward if compressed). Wait, no. If you compress a spring, it exerts a force in the opposite direction. If the spring is compressed (ball is above y=0), the spring pushes downward. If the spring is stretched (ball is below y=0), the spring pulls upward.But in the coordinate system, downward is negative. So if the ball is at position y (which could be positive or negative), then:- If y is positive: the spring is compressed by y. The spring exerts a force downward (negative direction) with magnitude ky. So F_spring = -ky.- If y is negative: the spring is stretched by |y|. The spring exerts a force upward (positive direction) with magnitude k|y|. So F_spring = -k y (because y is negative, so -k y becomes positive).So in both cases, F_spring = -k y. That seems right. Because if you define the displacement from the natural length as y (with the sign according to the coordinate system), then Hooke's Law gives F = -k y. For example, if the spring is stretched downward (y negative), then displacement is y (negative), so F = -k y (positive force upwards). If compressed (y positive), displacement is y, so F = -k y (negative force downwards). Perfect.So the spring force is always F_spring = -k y. Then, the other force acting on the ball is gravity, which is pulling downward. The gravitational force is F_gravity = -mg (since down is negative). Wait, but in the problem statement, g is the magnitude of free-fall acceleration, so acceleration due to gravity is -g (since down is negative). Therefore, the gravitational force is F_gravity = m * (-g) = -mg.So total force on the ball is F_total = F_spring + F_gravity = -k y - mg.By Newton's Second Law, F_total = m a. Therefore:m a = -k y - mgDivide both sides by m:a = -(k/m) y - gWhich is option (e): a = -(k/m) y - g.Wait, but let me check if this makes sense. Let's think about the equilibrium position. At equilibrium, the net force is zero. So:0 = -k y_eq - mg => y_eq = - (mg)/kSo the equilibrium position is at y = -mg/k, which is below the original position (since down is negative). That makes sense. When the ball is at the equilibrium position, the acceleration is zero. If the ball is displaced from equilibrium, it oscillates around that point. However, the equation we derived is a = -(k/m)y - g. At equilibrium, substituting y = -mg/k:a = -(k/m)(-mg/k) - g = (g) - g = 0. Correct.If the ball is at y = 0 (original position), then a = -(k/m)(0) - g = -g. That makes sense because at the original position, the spring is neither stretched nor compressed, so the only force is gravity, so acceleration is -g (downward). If the ball is above y=0, say y positive (spring compressed), then the spring is pushing down, so acceleration is more negative (since a = -(k/m)y - g). If the ball is below y=0 (spring stretched), then the spring is pulling up, so the acceleration is -(k/m)y - g. Since y is negative there, -(k/m)y becomes positive, so a = positive term - g. Depending on how much it's stretched, the acceleration could be upward or downward.This seems to make sense. Let me check the answer options again.Option (e) is a = -(k/m)y - g. That's exactly what we derived. So the answer should be (e).But wait, let me check the other options to make sure none of them could be correct.Option (c): a = (k/m)y - g. If that were the case, at y=0, a = -g, which is correct, but when y is negative (stretched), a = (k/m)(negative) - g, which would be more negative, but actually when stretched, the spring should be pulling up, so acceleration should be less negative. Wait, no. If y is negative (stretched), then in option (c), (k/m)y is negative, so a = negative - g, which is more negative. But actually, if the spring is stretched (y negative), the spring force is upward, opposing gravity, so the acceleration should be upward (positive). But according to (c), it's even more downward. So (c) can't be right.Option (d): a = -(k/m)y + g. At y=0, a = g, which is upward, but at y=0, only gravity is acting, so acceleration should be -g. So (d) is wrong here.Option (e) gives a = - (k/m)y - g. At y = 0, that's -g. When y is negative (stretched), a = - (k/m)(negative) - g = positive term - g. So if the positive term is greater than g, acceleration is upward; if less, downward. That makes sense. For example, if the spring is stretched such that k/m |y| > g, then acceleration is upward. Which is correct because the spring force is stronger than gravity. If it's less, then net acceleration is downward.Another check: When the ball is at the highest point (maximum compression), y is positive. Then acceleration is -(k/m)y - g. Both terms are negative (since y is positive), so acceleration is downward, which makes sense because the spring is compressed and pushing down, and gravity is also down. So the net acceleration is downward. Which would cause the ball to start moving down, which is correct.Alternatively, when the ball is at the lowest point (maximum stretch), y is negative. Then acceleration is -(k/m)(negative) - g = positive (k/m)|y| - g. If |y| is large enough, this is positive, meaning upward acceleration, which is correct as the spring pulls up more than gravity pulls down, causing the ball to decelerate and start moving back up.Therefore, option (e) seems correct.But let me just think again if the sign in Hooke's Law was correctly applied. The problem states that down is negative. So when the spring is stretched downward (displacement from natural length is downward), the displacement y (from original position) is negative. Then Hooke's Law says F_spring = -k y. Since y is negative (stretched downward), F_spring = -k*(negative) = positive, which is upward. Correct. If y is positive (compressed upward), F_spring = -k*(positive) = negative, which is downward. Correct. So Hooke's Law gives F_spring = -k y in this coordinate system. Then adding the gravitational force F_gravity = -mg. Therefore, total force F = -k y - mg. Acceleration a = F/m = -(k/m)y - g. So yes, option (e) is correct.I don't see any mistakes in this reasoning. So the answer should be (e).Final Answerboxed{E}</think>To determine the equation describing the acceleration ( a ) of the ball when it is at a height ( y ) above the original position of the top end of the spring, we start by analyzing the forces acting on the ball. 1. Hooke's Law: The spring force ( F_{text{spring}} ) is given by ( -ky ), where ( y ) is the displacement from the original position of the spring (with down being negative).2. Gravitational Force: The gravitational force ( F_{text{gravity}} ) is ( -mg ) (downward direction is negative).Using Newton's Second Law, the total force ( F_{text{total}} ) acting on the ball is the sum of the spring force and the gravitational force:[F_{text{total}} = F_{text{spring}} + F_{text{gravity}} = -ky - mg]By Newton's Second Law, ( F = ma ), we equate the total force to ( ma ):[ma = -ky - mg]Solving for acceleration ( a ):[a = -frac{k}{m}y - g]This equation accounts for both the spring force and gravitational force acting on the ball. The acceleration ( a ) is thus given by ( -frac{k}{m}y - g ), which corresponds to option (e).[boxed{E}],<think>Okay, let's try to figure out this physics problem step by step. So, the question is about a massless spring with spring constant k, vertically mounted. A ball of mass m falls onto the top end of the spring, sticks to it, and then oscillates. We need to find the equation that describes the acceleration a of the ball when it's at a height y above the original position of the top end of the spring. The options are given from (a) to (e), and we have to pick the right one.First, let me visualize the setup. The spring is standing vertically, fixed at the bottom. The ball falls down, hits the spring, and then starts oscillating. Since down is considered negative, we have to be careful with the directions of forces and acceleration.The key here is to analyze the forces acting on the ball when it's attached to the spring and at position y above the original top end of the spring. The original position of the top end is the equilibrium position of the spring before the ball is attached. But once the ball is attached, the new equilibrium position would be lower because the weight of the ball stretches the spring. However, the problem mentions the height y is measured from the original position, not the new equilibrium. So that's important.Wait, when the ball is attached, the spring will compress or extend. Let's see, since the ball is falling onto the spring, the spring will compress. But actually, no. Wait, when the ball is placed on the spring, due to gravity, it will stretch the spring downward. But since the spring is fixed at the bottom, the top end can move. Wait, no, if you attach a mass to the top end of a vertically mounted spring, the mass's weight will cause the spring to stretch downward. But in this case, the ball is falling onto the spring, so initially, the spring is at its natural length, right? The top end is free. Then the ball hits it, sticks, and starts oscillating.So the original position of the top end is the unstretched position. When the ball attaches, the spring will start to compress or extend? Hmm, when you push down on a spring fixed at the bottom, the spring compresses. So if the ball is moving downward and hits the spring, the spring will compress. So the ball's motion is downward, and when it hits the spring, it starts compressing it. Then the ball and spring system will oscillate. So the coordinate system here is such that down is negative, so the displacement y is measured from the original top end. If the ball is above the original position, that means the spring is compressed, so y is positive upwards, but the displacement from the original position would be in the negative direction? Wait, maybe I need to be careful here.The problem states: "when it is at a height y above the original position of the top end of the spring". So, if the ball is at height y above the original position, and down is negative, then y is positive upwards. But if the spring is compressed, the ball would actually be below the original position, right? Wait, no. Let's think. Suppose the original position of the top end of the spring is at point O. After the ball falls and attaches, when it's oscillating, if it's at a height y above O, that would mean the spring is stretched upwards? But that doesn't make sense because the spring can't be stretched upwards if it's fixed at the bottom. So perhaps when the ball is attached, the spring can either be compressed or stretched from its original position. But since the ball is on top of the spring, when it moves down, the spring compresses, and when it moves up, the spring extends. Wait, but a spring fixed at the bottom can only be compressed or stretched from its natural length. If the ball is attached to the top, moving up would stretch the spring beyond its natural length? Wait, no. If the spring is mounted vertically with the bottom fixed, then the natural length is when there's no mass attached. When you put a mass on top, it will stretch the spring downward due to gravity. But here, the ball is falling onto the spring, so it's an impact, and then they start oscillating.Wait, perhaps the oscillation happens around the new equilibrium position. The new equilibrium position is where the spring is stretched by an amount Δy = mg/k, due to the weight of the ball. So when the ball is attached, it will oscillate around that new equilibrium point. However, the question is asking about the acceleration when the ball is at height y above the original position (the top of the unstretched spring). So that position could be either above or below the new equilibrium point.But regardless, the acceleration of the ball is determined by the net force acting on it. The forces are gravity acting downward (which is -mg in sign if down is negative) and the spring force. The spring force depends on the displacement from the original position. Wait, or is it from the equilibrium position?Hmm, Hooke's Law says that the spring force is F = -kx, where x is the displacement from the equilibrium position. But in this case, if we take the original position as the reference, then when the spring is compressed or stretched, the displacement is from the original position. But actually, when the ball is attached, the equilibrium position is shifted by Δy = mg/k. So the displacement from the original position is x = y (since y is measured from the original position), but the displacement from the new equilibrium would be x' = y - Δy.But maybe the question is considering the original position as the reference point, not the equilibrium position. Let me think. The problem says "height y above the original position of the top end of the spring". So y=0 is the original position. When the ball is oscillating, it moves above and below this original position. But wait, when the ball is attached, due to its weight, the spring will stretch until the force from the spring equals the weight. So the equilibrium position is at y = -Δy (since down is negative). So if the ball is at some position y relative to the original position, then the displacement from the equilibrium position is y - (-Δy) = y + Δy. But maybe we don't need to involve equilibrium position here. Let's try to model the forces directly.At any position y (displacement from original position), the spring is compressed or stretched by an amount |y|. Wait, no. If the original position is y=0, then if the ball is at y above the original position, then the spring is compressed by y (since the ball is above the original position, meaning the spring is shorter). But since down is negative, if the ball is displaced downward from the original position, that would be negative y. Wait, this is confusing.Wait, let's clarify the coordinate system. Let me set up a coordinate system where the original position of the top of the spring is at y=0. The positive y-direction is upward, and downward is negative. So when the ball is attached to the spring, if it moves downward, it's going into negative y values, and if it moves upward, it's positive y. However, the problem states "height y above the original position", so y is measured as a height, meaning upward is positive. So if the ball is at height y above the original position, it's at position +y in this coordinate system. But physically, when the ball is attached to the spring, if it's oscillating, it can move both above and below the original position. But how can it move above the original position? If the spring is fixed at the bottom, stretching it would require pulling the top end up, but the spring is fixed at the bottom. Wait, no, a spring fixed at the bottom can have its top end move up and down. If the spring is massless, then when you attach a mass to the top, it can oscillate. So if you pull the mass upward, the spring would stretch, but since it's fixed at the bottom, stretching it would require the top to move up. So in that case, the coordinate system allows for positive y (upward) and negative y (downward) displacements from the original position.But in reality, when the mass is attached, the spring will stretch until the spring force equals the weight. So the equilibrium position is at y = -mg/k, because the displacement is downward (negative direction). So at equilibrium, the spring is stretched by Δy = mg/k downward, so the position is y = -Δy.Now, when the ball is oscillating, its position y is measured from the original position (y=0). So when the ball is at position y, the displacement from the equilibrium position is y - (-Δy) = y + Δy. The acceleration is determined by the net force. The net force is the spring force plus gravity? Wait, but gravity is already accounted for in the spring force at equilibrium. Wait, maybe not. Let's think again.The two forces acting on the ball are gravity (mg downward) and the spring force (ky upward or downward depending on displacement). Wait, Hooke's law: the spring force is F_spring = -k*y, where y is the displacement from the original position. But direction matters. If the ball is displaced upward from the original position (y positive), the spring is compressed, so the spring force is downward (negative). If the ball is displaced downward (y negative), the spring is stretched, so the spring force is upward (positive). So the spring force is F_spring = -k*y. Then the net force on the ball is F_net = F_spring + F_gravity. Since down is negative, F_gravity = -mg. Therefore, F_net = -k*y - mg.Wait, but then Newton's second law says F_net = m*a, so:m*a = -k*y - mgTherefore, a = -(k/m)*y - gWhich is option (e): a = - (k/m) y - gBut wait, let me verify again. If the displacement y is upward (positive), then the spring is compressed, so the spring exerts a downward force (negative). So F_spring = -k*y. The gravitational force is also downward, which is -mg. So total force is -k*y - mg. Therefore, acceleration a = (-k*y - mg)/m = - (k/m) y - g. So that's option (e). However, let's check if this makes sense.At the new equilibrium position, which is y = - mg/k (since that's where the net force is zero: -k*(-mg/k) - mg = mg - mg = 0). So at equilibrium, the acceleration is zero, which makes sense. If the ball is displaced upward from the original position (y positive), then the spring is compressed, so the spring force is downward, and gravity is also downward, so the acceleration is downward, which would be negative. So a = - (k/m)y - g. If y is positive, then acceleration is more negative. If y is negative (below original position), then the spring is stretched, spring force is upward, gravity is downward. So for y negative, - (k/m)y becomes positive (since y is negative), and then subtract g. So acceleration would be positive (upward) if the spring force is greater than g, or still downward otherwise. Wait, but at equilibrium, when y = -mg/k, the acceleration is zero. If you displace the ball upward from equilibrium, then the net force is downward, and vice versa.But according to our expression, a = - (k/m) y - g. Let's plug in the equilibrium position y = - mg/k:a = - (k/m)(-mg/k) - g = (g) - g = 0. Correct.If the ball is at the original position y=0, then a = -0 - g = -g. Which makes sense, because at y=0, the spring is at its natural length, so the only force acting is gravity, so the acceleration is -g downward. Wait, but when the ball is attached to the spring, wouldn't it immediately start stretching the spring? But maybe at the instant it's at y=0, the spring isn't stretched or compressed, so the only force is gravity, hence acceleration is -g. That seems right.Alternatively, if the ball is at some position y above the original position, meaning the spring is compressed, then the spring is pushing down, and gravity is also pulling down, so acceleration is downward (negative) with magnitude (k/m)y + g.If the ball is below the original position (y negative), then the spring is stretched, so the spring is pulling up with force -k*y (since y is negative, this becomes positive), and gravity is pulling down. So the net acceleration would be (k/m)(-y) - g. Wait, but y is negative, so -y is positive. So a = (k/m)|y| - g. Depending on how much it's stretched, this could be positive or negative.But in any case, the equation given is a = - (k/m)y - g, which is option (e). Let me check the options again:(a) a = mv²/y + g(b) a = mv²/k - g(c) a = (k/m)y - g(d) a = - (k/m)y + g(e) a = - (k/m)y - gSo according to our analysis, it's option (e). But let me think again. Is there any chance that the displacement should be measured from the equilibrium position instead of the original position? If that's the case, then the acceleration would be a = - (k/m) x, where x is the displacement from equilibrium. But the question specifically says "at a height y above the original position", so y is from the original, not the equilibrium. Therefore, our previous conclusion holds.Alternatively, maybe there's a different approach. Let's consider energy, but since the question is about acceleration, which is a force question, energy might not be necessary. Acceleration is directly related to forces via Newton's second law.Another way: The net force on the ball is the sum of the spring force and gravitational force. The spring force depends on the displacement from the original position. Since down is negative, if the ball is displaced upward (y positive), spring is compressed, spring force is downward (negative). If displaced downward (y negative), spring is stretched, spring force is upward (positive). So spring force is -k*y. Gravitational force is -m*g. Therefore, total force F = -k*y - m*g. Then acceleration a = F/m = - (k/m)*y - g. Which is exactly option (e). So this must be the correct answer.I don't see any flaws in this reasoning. The key points are correctly identifying the direction of the spring force based on displacement and remembering to include gravity as a separate force. Since the problem says the spring is massless, we don't have to consider its mass, which simplifies things. Also, air resistance is neglected, so no other forces are involved. Therefore, the correct answer should be option (e).Final Answerboxed{E}</think>To determine the acceleration (a) of the ball when it is at a height (y) above the original position of the top end of the spring, we need to consider the forces acting on the ball. The forces are gravity and the spring force.1. Gravitational Force: The gravitational force acting downward is ( -mg ) (since down is negative).2. Spring Force: The spring force is given by Hooke's Law, ( F_{text{spring}} = -ky ), where ( y ) is the displacement from the original position of the spring. If the ball is above the original position (positive ( y )), the spring is compressed, and the force is downward (negative). If the ball is below the original position (negative ( y )), the spring is stretched, and the force is upward (positive).The net force ( F_{text{net}} ) on the ball is the sum of the spring force and the gravitational force:[ F_{text{net}} = -ky - mg ]Using Newton's second law, ( F = ma ), we can find the acceleration:[ ma = -ky - mg ][ a = -frac{k}{m}y - g ]Thus, the equation describing the acceleration ( a ) of the ball is:[ a = -left( frac{k}{m} right)y - g ]The correct answer is (boxed{E}).,<think>Okay, so I need to figure out which equation describes the acceleration of the ball when it's attached to the spring and oscillating. Let me start by visualizing the problem. There's a vertical spring attached to the ground, massless, with spring constant k. A ball of mass m falls onto the top end of the spring, sticks to it, and then they oscillate together. The question is about the acceleration a when the ball is at a height y above the original position of the top end of the spring. Down is considered negative, and air resistance is neglected. First, let's recall some physics concepts. When dealing with springs and oscillations, Hooke's Law comes into play, which states that the force exerted by a spring is F = -kx, where x is the displacement from the equilibrium position. The negative sign indicates that the force is restorative, acting in the opposite direction to the displacement. But wait, in this case, the equilibrium position isn't the original position of the spring. Because once the ball is attached to the spring, the weight of the ball will stretch the spring to a new equilibrium point. So when the ball is oscillating, its motion will be around this new equilibrium position. However, the problem mentions that the height y is measured from the original position of the top end of the spring, not the new equilibrium. Hmm, that might be important.Let me consider the forces acting on the ball when it's at position y. Since down is negative, if the ball is above the original position, y is positive. But depending on whether the spring is compressed or extended, the displacement from the original position might be different. Wait, the spring was originally at some position, let's say y_original. When the ball is attached, the weight of the ball (mg) will stretch the spring by some amount. Let's call that stretch delta. So delta = mg/k, because at equilibrium, the spring force k*delta equals the weight mg. So the new equilibrium position is y_original - delta, since stretching the spring moves the end downward. But in the problem, they're measuring y from the original position. So if the ball is oscillating around the new equilibrium, its displacement from the original position would be y. So when the ball is at a height y above the original position, the displacement from the new equilibrium is y - (-delta)? Wait, maybe I need to clarify the coordinates.Let me set up a coordinate system. Let’s take the original position of the top of the spring as y = 0. Downwards is the negative direction. When the ball is attached, the spring stretches until the force from the spring equals the weight of the ball. The new equilibrium position is at y = -delta, where delta = mg/k. So when the ball is at some position y (which is a displacement from the original top of the spring), the displacement from the equilibrium position is y - (-delta) = y + delta. But the acceleration of the ball is determined by the net force. The net force on the ball is the spring force plus gravity? Wait, when the ball is attached to the spring, the forces acting on it are the spring force and gravity. But since the spring is massless, the only mass is the ball. So applying Newton's second law: F_net = ma. The spring force is -k times the displacement from the original position? Or from the equilibrium? Let me think. If we take y as the displacement from the original position, then when the spring is stretched beyond the original position, the displacement is negative (since down is negative). Wait, if the original position is y=0, and down is negative, then when the spring is compressed, the displacement is positive (since the top end is above y=0), and when it's stretched, the displacement is negative. Wait, no, that might not be the case. Hold on, perhaps I need to clarify. If the original position is y=0, and the positive direction is upwards, then stretching the spring (pulling it down) would correspond to negative y. But if down is considered negative, then moving downward from y=0 is negative. So if the ball is attached to the spring and moves down, its position becomes negative. However, the problem states that the height y is above the original position, so if the ball is above the original position, y is positive; if it's below, y is negative. But since the ball is oscillating, it can be both above and below the original position. Wait, but when the ball is attached to the spring, it can't really go above the original position unless the spring is compressed. But initially, the spring was at rest, so the original position is the equilibrium of the spring without the mass. When the mass is attached, the spring stretches, so the equilibrium position shifts downward. So during oscillation, the ball moves around the new equilibrium. Therefore, relative to the original position, the ball can be either above or below, depending on the phase of the oscillation.But the problem states that the height y is measured from the original position of the top end. So if the ball is at a height y above the original position, that would mean the spring is compressed by y, because the top end was originally at y=0, and if the ball is above that, it's compressing the spring. Conversely, if the ball is below the original position, y is negative, and the spring is stretched.However, in the problem, they mention "height y above the original position", so y is positive when above, negative when below. The question is asking for the acceleration a when the ball is at height y. Since down is negative, acceleration downward is negative, upward is positive.So, to find acceleration, we need to consider the net force on the ball at position y. The forces are:1. The spring force: which depends on the displacement from the original position. If the ball is at position y (above original position), then the spring is compressed by y. So the spring force will be upward (since compressed spring pushes back), which would be positive in our coordinate system. Wait, no. Wait, if the spring is compressed, meaning the top end is pushed down, so the displacement is negative? Wait, maybe I need to be precise.Wait, the coordinate system: Let's define y as the vertical position relative to the original top end of the spring (y=0). Positive y is upward, negative y is downward. So if the ball is at position y, which is a height y above the original position, then the displacement of the spring's top end from its original position is y. Wait, but if the ball is above the original position, that would mean the spring is compressed. So the displacement x (in Hooke's law) is the amount the spring is compressed or stretched. If the ball is at y above the original position, the spring is compressed by |y|, but since y is positive upward, the displacement x (from original position) is y. But Hooke's law states F_spring = -k x, where x is the displacement from the equilibrium. Wait, no. Wait, Hooke's law is F = -k x, where x is the displacement from the natural length. But in this case, the original position of the spring's top end is its natural length, because when nothing is attached, it's at rest. So when the ball is attached, the spring's natural length is at y=0. Therefore, if the ball is at position y, the displacement from the natural length is y (if y is positive, the spring is compressed by y; if y is negative, it's stretched by |y|). Therefore, the spring force is F_spring = -k * y. Because if y is positive (compression), the spring pushes downward? Wait, no. Wait, if the spring is compressed (ball is above original position), the spring exerts a downward force to push back, right? Wait, no. If you compress a spring, it pushes back in the opposite direction. So if the spring is compressed, meaning the top end is pushed upward relative to its natural length (wait, original position is natural length). If the ball is attached and is at y above original position, that means the spring is compressed by y. So the spring exerts a force downward (trying to return to original length). So in terms of our coordinate system, where down is negative, the force would be negative. So F_spring = -k * y. Wait, but Hooke's law is F = -k x, where x is the displacement from equilibrium. But here, the natural length is the original position. So if we take x as the displacement from natural length, then F_spring = -k x. So if the spring is compressed by y (x = y), then F_spring = -k y. But in our coordinate system, down is negative. So if the spring is compressed (x = y > 0), the spring exerts a force downward, which is negative. So that makes sense: F_spring = -k y.But there's also the gravitational force acting on the ball, which is F_gravity = m * g, directed downward. Wait, but in the coordinate system, down is negative. So F_gravity = -m g.Therefore, the net force on the ball is F_net = F_spring + F_gravity = -k y - m g.Wait, but according to Newton's second law, F_net = m a. Therefore:- k y - m g = m aSolving for a:a = (-k y - m g)/m = - (k/m) y - gBut looking at the options, option (e) is a = - (k/m) y - g. So that seems like the answer. But wait, let me check again.Alternatively, maybe I made a mistake in considering the displacement. Let's think again. Suppose the ball is attached to the spring, and when the spring is at its original position (y=0), the only force acting on the ball is gravity. Wait, but once the ball is attached, the spring will start to compress or stretch. Wait, actually, when the ball is attached, it will begin to stretch the spring because of its weight. But the problem states that the ball falls vertically down onto the top end of the spring, becomes attached, and then oscillates. So initially, the ball is falling, hits the spring, and then they start oscillating. So the system will oscillate around the equilibrium position, which is lower than the original position due to the weight of the ball.But in the problem, we are told to express the acceleration when the ball is at height y above the original position. So regardless of the equilibrium, we need to express a in terms of y measured from the original position.Wait, perhaps another approach: Consider the forces at position y. When the ball is at position y (above original position), the spring is compressed by y. So the spring exerts a force upward? Wait, no, if the spring is compressed, it exerts a force in the direction opposite to the compression. If the spring is compressed upwards (i.e., the top is pushed up), then the spring would push down. But in this case, the original position is y=0. If the ball is at y above y=0, then the spring is compressed by y, so the spring force is downward. Since down is negative, that would be -k y. Then, the gravitational force is also downward, which is -m g. So total force is -k y - m g, leading to a = (-k y - m g)/m = -(k/m) y - g. So that's option (e). Hmm. That seems correct. But wait, let's check the equilibrium position. At equilibrium, the net force is zero. So setting F_net = 0 = -k y - m g. So y = - m g / k. Which makes sense because the equilibrium position is below the original position (y negative), which is correct. So when the ball is at y = - mg/k, the acceleration is zero, which is the equilibrium. Then, when displaced from that equilibrium, the acceleration would be proportional to the displacement from equilibrium. But in the problem, the displacement is measured from the original position, not the equilibrium. So the displacement from equilibrium is y - (- mg/k) = y + mg/k. Then, the acceleration would be -k/m times the displacement from equilibrium. Let's see:If we take x = y + mg/k as the displacement from equilibrium, then a = - (k/m) x = - (k/m)(y + mg/k) = - (k/m)y - g. Which is the same as before. So indeed, the acceleration is a = - (k/m)y - g. Therefore, option (e). Alternatively, if you don't consider the equilibrium shift, you might mistakenly think the acceleration is just - (k/m)y. But you have to remember that even at the equilibrium position (which is shifted), the acceleration is zero. Therefore, when expressing acceleration in terms of displacement from the original position, you must account for both the spring force and the gravitational force. However, in this case, since we are given y from the original position, both forces contribute. Therefore, the answer is (e). Wait, but let me check the options again:(a) a = mv²/y + g(b) a = mv²/k - g(c) a = (k/m)y - g(d) a = -(k/m)y + g(e) a = -(k/m)y - gYes, option (e) is - (k/m)y - g, which matches our result. So the answer should be (e).But let me just think again. When the spring is compressed by y (ball at y above original position), the spring force is downward (negative) with magnitude ky. Gravity is also downward (negative) with magnitude mg. So total force is -ky - mg. Therefore, acceleration is (-ky - mg)/m = - (k/m)y - g. So yes, option (e). That seems correct. Therefore, I believe the answer is (e).Final Answerboxed{E}</think>To determine the equation describing the acceleration (a) of the ball when it is at a height (y) above the original position of the top end of the spring, we need to consider the forces acting on the ball. The key steps are as follows:1. Forces Acting on the Ball: - The spring force (F_{text{spring}}) given by Hooke's Law: (F_{text{spring}} = -ky), where (y) is the displacement from the original position of the spring (negative sign indicates the force is restorative). - The gravitational force (F_{text{gravity}} = -mg) (downward direction is negative).2. Net Force and Acceleration: - The net force (F_{text{net}}) on the ball is the sum of the spring force and the gravitational force: [ F_{text{net}} = -ky - mg ] - Using Newton's second law (F = ma), we equate the net force to (ma): [ -ky - mg = ma ] - Solving for acceleration (a): [ a = frac{-ky - mg}{m} = -left(frac{k}{m}right)y - g ]3. Conclusion: - The acceleration (a) of the ball when it is at a height (y) above the original position of the spring is given by: [ a = -left(frac{k}{m}right)y - g ]Thus, the correct answer is (boxed{E}).,<think>Okay, let me try to figure out this physics problem. So, the question is about a massless spring with spring constant k that's attached vertically. A ball of mass m falls onto the top end of the spring, sticks to it, and then oscillates. We need to find the equation for the acceleration a of the ball when it's at a height y above the original position of the top end of the spring. The options are given from (a) to (e), and we need to pick the right one.First, I need to recall some concepts about springs and oscillations. I remember that when a mass is attached to a spring, the system undergoes simple harmonic motion if there's no damping. The acceleration in such motion is proportional to the displacement from the equilibrium position and is directed opposite to the displacement. The formula for acceleration in SHM is a = - (k/m)x, where x is the displacement from equilibrium. But here, gravity is also acting on the ball, so I need to consider that as well.Let me visualize the scenario. The spring is initially at its natural length, attached to the ground. The ball falls onto the spring, compresses it, and then starts oscillating. When the ball is attached to the spring, the spring will be compressed until the force from the spring equals the weight of the ball. Wait, but when the ball is oscillating, it's moving around the equilibrium position. So, the equilibrium position isn't the original position of the spring's top end; it's lower because the spring is compressed by the weight of the ball.So, the equilibrium position is where the spring force balances gravity. Let me denote the equilibrium compression as y_eq. At that point, k*y_eq = m*g, so y_eq = (m*g)/k. That makes sense. So, when the ball is oscillating, its displacement from the equilibrium position will determine the acceleration. But in the problem, they're asking for the acceleration when the ball is at a height y above the original position of the spring's top end. Hmm, so y is measured from the original, uncompressed position. So, when the ball is at y, that's actually a displacement from the original position, but the equilibrium position is at y = -y_eq (since down is negative). So, if the ball is oscillating around y = -y_eq, then the displacement from equilibrium would be y - (-y_eq) = y + y_eq. But maybe I need to think in terms of coordinates.Let me set up a coordinate system. Let's take the original position of the top end of the spring as y = 0. Downward direction is negative. So, when the spring is compressed, the ball's position is negative. The equilibrium position is at y = -y_eq = - (m*g)/k. So, if the ball is at some position y (which could be positive or negative relative to the original top of the spring), then the displacement from the equilibrium position is y - (-y_eq) = y + y_eq. Therefore, the acceleration in SHM is given by a = - (k/m)*(displacement from equilibrium). But also, there's gravity acting on the ball. Wait, but when we consider the net force on the ball, it's the spring force plus the gravitational force. Maybe I need to write the equation of motion.The forces acting on the ball are the spring force and gravity. The spring force is -k*y, since when the spring is compressed (y negative), the spring force is upward (positive). Wait, maybe I need to be careful with the signs. Let me think. If the ball is at position y, which is displacement from the original top of the spring. If y is positive, that means the ball is above the original position (spring is extended), which isn't possible because the spring is mounted vertically with the bottom attached. So, the ball can only move downward from the original position, so y would be negative. But maybe the problem states that y is the height above the original position, so even if the spring is compressed, y would still be measured as positive above the original position? Wait, no. The problem says, "when it is at a height y above the original position". So if the ball is below the original position (compressed spring), then y would be negative because down is negative. So y is positive when above, negative when below.So, let's clarify. The coordinate system: y=0 is the original position of the top end of the spring. Down is negative. So when the ball is attached and compresses the spring, its position y is negative. The problem states "height y above the original position", so if it's above, y is positive; if it's below, y is negative. So, when the ball is oscillating, it moves between some maximum compression (y negative) and some extension (if possible). But since the spring can't be extended beyond its original length (unless the ball is somehow pulled up), but in this case, the ball is just falling onto the spring, so the maximum position would be y=0 (original position) when the spring is fully extended. Wait, but once the ball is attached, can the spring extend beyond its original length? Hmm, probably not, because the spring is massless and the ball is attached to the top. So, when the spring is extended, the ball would be above the original position, but since the spring is massless, it can't really extend beyond its original length unless there's a force pulling it up. But in this case, the only forces are gravity and the spring force. Once the ball is attached, if the spring is at its original length (y=0), the only force acting on the ball is gravity (mg downward). So, if the ball is at y=0, the spring isn't exerting any force, so the net force is mg downward, hence acceleration g downward. But that might be the case when the ball is passing through y=0 during oscillation.Wait, but when the spring is compressed, the spring force is upward (positive), and when it's extended (if possible), the spring force is downward (negative). But since the spring is mounted vertically, maybe it can't extend beyond its original length. So maybe the spring can only be compressed, not extended. But in reality, if the ball is attached to the spring, it can oscillate both above and below the equilibrium position. Hmm, this is getting a bit confusing. Maybe I need to approach it with equations.Let me consider the forces on the ball. At any position y, where y is the displacement from the original top of the spring (y=0). If the ball is at position y, then the spring is compressed by |y| if y is negative (since down is negative). The spring force is given by Hooke's Law: F_spring = -k*y. Wait, but if y is negative (below the original position), then the spring is compressed, so the spring force is upward, which would be positive. So F_spring = -k*y. If y is negative, F_spring is positive (upward). If y is positive (above original position, which may not be possible if the spring can't extend), then F_spring is negative (downward). But let's assume the spring can extend, even though in reality it's attached to the ground. Wait, if the spring is extended, that would mean the top end is above its original position. But since the bottom end is attached to the ground, the spring can be compressed or extended. However, in the problem statement, the ball is falling onto the top end, so maybe after the collision, the spring can be both compressed and extended as the ball oscillates. So, when extended, the spring is pulling the ball down, and when compressed, it's pushing the ball up.But the problem says "height y above the original position", so if the ball is above the original position (y positive), the spring is extended, so the spring force is downward (since the spring is stretched, it's pulling the ball down). If the ball is below the original position (y negative), the spring is compressed, so the spring force is upward.So, the net force on the ball is F_net = F_spring + F_gravity. Since down is negative, F_gravity = -m*g (acting downward). The spring force is F_spring = -k*y. Wait, if y is positive (above original position), then F_spring = -k*y (downward). If y is negative (below original position), F_spring = -k*y (upward). So, the net force is F_net = -k*y - m*g. Then, acceleration a = F_net / m = (-k*y - m*g)/m = - (k/m)*y - g. Therefore, the acceleration a is equal to - (k/m)y - g. Looking at the options, option (e) is a = - (k/m)y - g. So, is that the answer?Wait, but let me verify. Let's consider the equilibrium position. At equilibrium, the net force is zero. So, setting F_net = 0: -k*y_eq - m*g = 0 => y_eq = - (m*g)/k. Which is the equilibrium position. So, if we write the acceleration in terms of displacement from equilibrium, we would have a = - (k/m)(y - y_eq). But the problem states y is measured from the original position, not from equilibrium. So, perhaps when you substitute y = y_eq + x, where x is displacement from equilibrium, then a = - (k/m)x. But the problem is not asking in terms of displacement from equilibrium, but from the original position. Therefore, the acceleration as a function of y (from original position) is indeed a = - (k/m)y - g. So, option (e) is the correct answer.But wait, let me check with a specific case. Suppose the ball is at the equilibrium position, y = - (m*g)/k. Plugging into option (e): a = - (k/m)(-m*g/k) - g = g - g = 0. Which makes sense, since at equilibrium, the acceleration is zero. If the ball is at the original position, y = 0, then the acceleration is a = -0 - g = -g. Which also makes sense, because at the original position, the spring is not compressed or extended, so the only force is gravity, hence acceleration -g (downward). So that's correct.If the spring is compressed further below equilibrium, say y = -A (where A > y_eq), then the acceleration would be a = - (k/m)(-A) - g = (kA/m) - g. But since at equilibrium, k*y_eq = -m*g => k*(-m*g/k) = -m*g, so if A is greater in magnitude than y_eq, then (kA/m) would be greater than g, so acceleration upward. That makes sense because if you compress the spring more, the spring force is stronger upward, overcoming gravity, leading to upward acceleration.Alternatively, if the ball is above the original position (y positive), which would mean the spring is extended, then the acceleration is a = - (k/m)y - g. Since y is positive, the first term is negative, and combined with -g, the total acceleration is more negative (downward). So, if the spring is stretched upward, the spring is pulling down, adding to gravity, resulting in a larger downward acceleration. That seems correct.So, putting it all together, the acceleration is given by a = - (k/m)y - g. Therefore, the answer should be option (e).Final Answerboxed{E}</think>To determine the equation describing the acceleration ( a ) of the ball when it is at a height ( y ) above the original position of the top end of the spring, we need to consider the forces acting on the ball. The two forces are the spring force and the gravitational force. 1. Spring Force: The spring force is given by Hooke's Law, ( F_{text{spring}} = -ky ), where ( y ) is the displacement from the original position of the spring's top end. The negative sign indicates that the spring force is in the opposite direction of the displacement.2. Gravitational Force: The gravitational force acting downward is ( F_{text{gravity}} = -mg ), where ( g ) is the acceleration due to gravity.The net force on the ball is the sum of these two forces:[ F_{text{net}} = -ky - mg ]Using Newton's second law, ( F = ma ), we can find the acceleration ( a ):[ ma = -ky - mg ][ a = -frac{k}{m}y - g ]This equation shows that the acceleration ( a ) is the sum of the acceleration due to the spring force ( -frac{k}{m}y ) and the acceleration due to gravity ( -g ).Verification with specific cases:- At the equilibrium position ( y = -frac{mg}{k} ), the acceleration is zero.- At the original position ( y = 0 ), the acceleration is ( -g ).Thus, the correct answer is:[boxed{E}]]

🔑:1. Identify the forces acting on the ball: - The gravitational force acting downward: ( F_g = -mg ) - The spring force acting according to Hooke's Law: ( F_s = -ky ), where ( y ) is the displacement from the equilibrium position (original position of the top end of the spring).2. Apply Newton's Second Law: According to Newton's Second Law, the net force ( F_{text{net}} ) acting on the ball is equal to the mass ( m ) of the ball times its acceleration ( a ): [ F_{text{net}} = ma ]3. Sum the forces: The net force is the sum of the gravitational force and the spring force: [ F_{text{net}} = F_g + F_s = -mg - ky ]4. Set up the equation: Substitute the net force into Newton's Second Law: [ ma = -mg - ky ]5. Solve for acceleration ( a ): Divide both sides of the equation by ( m ): [ a = -g - frac{k}{m}y ]Thus, the equation that describes the acceleration ( a ) of the ball when it is at a height ( y ) above the original position of the top end of the spring is:[a = -frac{k}{m}y - g]The final answer is (boxed{e}).

🧠:[<think>Okay, so I need to find the maximum value of the function y = -1/(x + 1) on the interval [1, 2]. Let me think about how to approach this. First, I remember that to find the maximum or minimum values of a function on a closed interval, we can use the Extreme Value Theorem, which says that any continuous function on a closed interval [a, b] will have both a maximum and a minimum value. So, is this function continuous on [1, 2]? Well, the function is -1 divided by (x + 1). The denominator x + 1 can't be zero, but in the interval [1, 2], x + 1 ranges from 2 to 3, so there's no division by zero here. That means the function is continuous on [1, 2], right? So by the Extreme Value Theorem, there should be both a maximum and a minimum value on this interval.But the question is asking specifically for the maximum value. So I need to find where this function reaches its highest point in the interval [1, 2]. To do that, I think I should check the critical points inside the interval and also evaluate the function at the endpoints. Critical points are where the derivative is zero or undefined. Since the function is continuous on [1, 2], the derivative should exist there? Let me check.Let's compute the derivative of y with respect to x. The function is y = -1/(x + 1). Let me rewrite that as y = - (x + 1)^(-1). Using the power rule, the derivative dy/dx would be (-1) * (-1) * (x + 1)^(-2) * 1, right? Wait, that's a bit confusing. Let me do it step by step.If y = - (x + 1)^(-1), then dy/dx = -1 * (-1) * (x + 1)^(-2) * derivative of (x + 1) with respect to x. The derivative of (x + 1) is 1. So putting it all together, dy/dx = 1/(x + 1)^2. So the derivative is 1 over (x + 1) squared. Now, critical points occur where the derivative is zero or undefined. But since the denominator here is (x + 1)^2, which is always positive for x in [1, 2], the derivative is never zero because the numerator is 1, which is a constant. Therefore, the derivative is always positive on the interval [1, 2], meaning the function is increasing on this interval. Wait, if the derivative is positive, that means the function is increasing. So if the function is increasing on [1, 2], then its minimum value would be at the left endpoint x = 1, and its maximum value would be at the right endpoint x = 2. Let me confirm that. So, let's compute the function at x = 1: y = -1/(1 + 1) = -1/2. At x = 2: y = -1/(2 + 1) = -1/3. Since the function is increasing from x = 1 to x = 2, the value at x = 1 (-1/2) is less than the value at x = 2 (-1/3). So, the maximum value on the interval should be at x = 2, which is -1/3. Looking at the options, A is -1/3, B is -1/2, C is -1, D is does not exist. So the answer should be A.But wait, let me double-check to make sure I didn't make a mistake in the derivative. The original function is y = -1/(x + 1). Let's compute the derivative again. If y = -1/(x + 1), then dy/dx = derivative of -1*(x + 1)^(-1) which is -1*(-1)*(x + 1)^(-2)*1 = 1/(x + 1)^2. Yes, that's correct. So the derivative is positive, so the function is indeed increasing on [1, 2]. Therefore, the maximum is at x = 2. But just to be thorough, let's check if there are any critical points in the interval. Since the derivative is 1/(x + 1)^2, which is defined for all x ≠ -1. Since our interval is [1, 2], there's no x where the derivative is undefined in this interval. And the derivative is never zero because the numerator is 1. Therefore, there are no critical points in the interval. So the only candidates for maxima and minima are the endpoints. Therefore, since the function is increasing, the right endpoint x = 2 gives the maximum, which is -1/3. So the answer is A. Wait, but let me visualize the function to make sure. The function y = -1/(x + 1) is a hyperbola. Since there's a negative sign, it's a reflection of the hyperbola 1/(x + 1) over the x-axis. The original function 1/(x + 1) is decreasing on its domain (since x + 1 is in the denominator and as x increases, the denominator increases, so the value decreases). Reflecting over the x-axis would mean that the function becomes increasing? Wait, if the original function 1/(x + 1) is decreasing, then multiplying by -1 would make it increasing. Let me verify this with numbers. For example, at x = 1: original function is 1/2, so y = -1/2. At x = 2: original is 1/3, so y = -1/3. So as x increases from 1 to 2, y increases from -1/2 to -1/3. Yes, that's an increasing function. Therefore, the function is increasing on [1, 2], confirming that the maximum is at x = 2. Therefore, the answer should be A: -1/3. But just to be absolutely sure, let's check other points in the interval. Let's pick x = 1.5. Then y = -1/(1.5 + 1) = -1/2.5 = -0.4. Which is between -1/2 (-0.5) and -1/3 (-0.333...). So -0.4 is indeed greater than -0.5 and less than -0.333. Wait, no, -0.4 is actually greater than -0.5 because it's closer to zero. Wait, numbers on the number line: -0.5, -0.4, -0.333. So from left to right: -0.5 (smallest), then -0.4, then -0.333 (largest). So yes, as x increases, y increases from -0.5 to -0.4 to -0.333. So the function is indeed increasing. Therefore, the maximum is at x = 2. So, the answer is A: -1/3. All the steps check out. I think that's correct.</think>To find the maximum value of the function ( y = -frac{1}{x+1} ) on the interval ([1, 2]), we first note that the function is continuous on this interval. According to the Extreme Value Theorem, it must attain both a maximum and a minimum value.Next, we compute the derivative to determine critical points:[frac{dy}{dx} = frac{d}{dx} left( -frac{1}{x+1} right) = frac{1}{(x+1)^2}]The derivative ( frac{1}{(x+1)^2} ) is always positive for ( x in [1, 2] ), indicating the function is increasing on this interval. Since there are no critical points (the derivative is never zero or undefined within ([1, 2])), the extrema must occur at the endpoints.Evaluating the function at the endpoints:- At ( x = 1 ): ( y = -frac{1}{1+1} = -frac{1}{2} )- At ( x = 2 ): ( y = -frac{1}{2+1} = -frac{1}{3} )Since the function is increasing, the maximum value occurs at ( x = 2 ), which is ( -frac{1}{3} ).Answer: boxed{A},<think>Okay, so I need to find the maximum value of the function y = -1/(x + 1) on the interval [1, 2]. Let's see... Hmm, first, I remember that to find maximum or minimum values of a function on a closed interval, you can use the Extreme Value Theorem, which says that continuous functions on closed intervals have both a maximum and a minimum. So, I should check if this function is continuous on [1, 2].The function y = -1/(x + 1) is a rational function, right? And rational functions are continuous wherever the denominator isn't zero. In this case, the denominator is x + 1. So, when x + 1 = 0, that's x = -1. But our interval is [1, 2], which doesn't include -1. So, the function is definitely continuous on [1, 2]. That means by the Extreme Value Theorem, there should be both a maximum and a minimum on this interval.Alright, so the next step is to find critical points in the interval where the derivative is zero or undefined, and then evaluate the function at those critical points as well as at the endpoints of the interval. Then compare all those values to see which is the largest and which is the smallest.Let me compute the derivative of y with respect to x. The function is y = -1/(x + 1). Let's rewrite that as y = - (x + 1)^(-1). Using the power rule, the derivative dy/dx = -1 * (-1)(x + 1)^(-2) * 1, which simplifies to 1/(x + 1)^2. Wait, so dy/dx = 1/(x + 1)^2. Let me check that again. Yes, derivative of -(x + 1)^(-1) is (-1)(-1)(x + 1)^(-2) * derivative of (x + 1), which is 1. So, yeah, dy/dx = 1/(x + 1)^2. Now, critical points are where the derivative is zero or undefined. But dy/dx is 1 over (x + 1) squared. The denominator is (x + 1)^2, which is never zero on the interval [1, 2], so the derivative is never undefined there. And the numerator is 1, so the derivative can never be zero either. That means there are no critical points inside the interval [1, 2].Therefore, according to the Extreme Value Theorem, the maximum and minimum must occur at the endpoints of the interval. So, we just need to evaluate the function at x = 1 and x = 2.Let's compute y at x = 1: y = -1/(1 + 1) = -1/2.At x = 2: y = -1/(2 + 1) = -1/3.So, comparing these two values: -1/2 and -1/3. Since we're dealing with negative numbers, the larger value is the one closer to zero. So, -1/3 is greater than -1/2. Therefore, the maximum value of the function on [1, 2] is -1/3 at x = 2, and the minimum is -1/2 at x = 1.Looking at the options, A is -1/3 and B is -1/2. So, the maximum value is A: -1/3.Wait, but let me make sure I didn't make a mistake here. The function is -1/(x + 1), so as x increases, x + 1 increases, so 1/(x + 1) decreases, but with a negative sign in front, the function y would increase as x increases. So, on the interval [1, 2], as x increases from 1 to 2, y increases from -1/2 to -1/3. Therefore, the maximum is at x = 2, which is -1/3, and the minimum is at x = 1, which is -1/2. That seems right.So, the answer should be option A: -1/3.Wait, but let me check if there's any other consideration. Since there are no critical points, we only check the endpoints. So, yeah, I think that's correct.Alternatively, maybe someone could think that the function approaches zero as x approaches infinity, but here we're restricted to [1, 2], so the maximum can't be more than -1/3. So, yeah, A is correct.Final Answerboxed{A}</think>To find the maximum value of the function ( y = -frac{1}{x+1} ) on the interval ([1, 2]), we first note that the function is continuous on this interval because the denominator ( x + 1 ) does not equal zero for ( x ) in ([1, 2]). According to the Extreme Value Theorem, the function must attain both a maximum and a minimum on this closed interval.Next, we find the critical points by taking the derivative of the function:[frac{dy}{dx} = frac{d}{dx} left( -frac{1}{x+1} right) = frac{1}{(x+1)^2}]The derivative ( frac{1}{(x+1)^2} ) is never zero and is defined for all ( x ) in ([1, 2]). Therefore, there are no critical points within the interval.We then evaluate the function at the endpoints of the interval:- At ( x = 1 ): [ y = -frac{1}{1+1} = -frac{1}{2} ]- At ( x = 2 ): [ y = -frac{1}{2+1} = -frac{1}{3} ]Comparing the values ( -frac{1}{2} ) and ( -frac{1}{3} ), we see that ( -frac{1}{3} ) is the larger value. Therefore, the maximum value of the function on the interval ([1, 2]) is ( -frac{1}{3} ).[boxed{A}]]

🔑:The function y = -frac{1}{x+1} is increasing in the interval [1, 2], which means the maximum value is taken at f(2), and it is -frac{1}{3}. Therefore, the correct answer is boxed{A}. Since the function y = -frac{1}{x+1} is increasing in the interval [1, 2], the maximum value can be obtained at f(2). This question tests the method of finding the maximum value of a function, emphasizing the use of monotonicity to solve the problem and testing computational skills. It is a basic question.

🧠:[<think>Okay, let's see. I have this problem here: Given that (x/a) + (y/b) + (z/c) equals the square root of 2, and (a/x) + (b/y) + (c/z) equals 0. I need to find the value of (x²/a²) + (y²/b²) + (z²/c²). Hmm. Alright, let's break this down step by step.First, let's denote the first equation as Equation (1): (x/a) + (y/b) + (z/c) = √2. The second equation is Equation (2): (a/x) + (b/y) + (c/z) = 0. And the expression we need to find is E = (x²/a²) + (y²/b²) + (z²/c²).I need to relate these equations somehow. Maybe if I square Equation (1)? Let me try that.Squaring Equation (1): [(x/a) + (y/b) + (z/c)]² = (√2)². That simplifies to (x²/a²) + (y²/b²) + (z²/c²) + 2[(xy)/(ab) + (xz)/(ac) + (yz)/(bc)] = 2.So, expanding the square gives E plus twice the sum of the products of the variables over the products of the constants equals 2. So, E + 2[(xy)/(ab) + (xz)/(ac) + (yz)/(bc)] = 2. Therefore, if I can find the value of the term [(xy)/(ab) + (xz)/(ac) + (yz)/(bc)], then I can solve for E.But how can I find that term? Let's look at Equation (2): (a/x) + (b/y) + (c/z) = 0. Maybe if I multiply these two equations together? Wait, multiplying (x/a + y/b + z/c) by (a/x + b/y + c/z). That might give me something. Let me try that.Multiplying Equation (1) and Equation (2):[(x/a) + (y/b) + (z/c)] * [(a/x) + (b/y) + (c/z)] = √2 * 0 = 0.Let's compute the left-hand side. Multiplying each term:(x/a)*(a/x) + (x/a)*(b/y) + (x/a)*(c/z) + (y/b)*(a/x) + (y/b)*(b/y) + (y/b)*(c/z) + (z/c)*(a/x) + (z/c)*(b/y) + (z/c)*(c/z).Simplify each term:First term: (x/a)*(a/x) = 1.Second term: (x/a)*(b/y) = (b x)/(a y).Third term: (x/a)*(c/z) = (c x)/(a z).Fourth term: (y/b)*(a/x) = (a y)/(b x).Fifth term: (y/b)*(b/y) = 1.Sixth term: (y/b)*(c/z) = (c y)/(b z).Seventh term: (z/c)*(a/x) = (a z)/(c x).Eighth term: (z/c)*(b/y) = (b z)/(c y).Ninth term: (z/c)*(c/z) = 1.So, adding all these terms up:1 + (b x)/(a y) + (c x)/(a z) + (a y)/(b x) + 1 + (c y)/(b z) + (a z)/(c x) + (b z)/(c y) + 1.Combine the constants: 1 + 1 + 1 = 3.Then, the remaining terms are:[(b x)/(a y) + (a y)/(b x)] + [(c x)/(a z) + (a z)/(c x)] + [(c y)/(b z) + (b z)/(c y)].Each pair looks like (k + 1/k) multiplied by something. For example, (b x)/(a y) + (a y)/(b x) = (b x)/(a y) + (a y)/(b x) = ( (b x)^2 + (a y)^2 ) / (a y b x ). But I don't know if that helps. Alternatively, notice that each pair is of the form m + 1/m, which is always greater than or equal to 2 by AM ≥ GM, but since the sum is zero here, maybe these terms must be negative? But that seems impossible because each term is positive if variables are positive. Wait, but we have cross terms here. Hmm, but the original product is supposed to equal zero.Wait, but the product of Equation (1) and Equation (2) is zero. So, even though each term in the expansion is positive or some combination, the total sum equals zero. But how can the sum of 3 plus those positive terms equal zero? That seems impossible unless... Wait, unless some variables are negative? The problem doesn't specify that a, b, c, x, y, z are positive. So maybe they can be negative. However, even then, the product terms like (x/a)*(a/x) would be 1, which is positive, and the cross terms would be (b x)/(a y) + (a y)/(b x) which is 2 times sqrt( (b x)/(a y) * (a y)/(b x) ) = 2 by AM-GM, but only if both terms are positive. If they are negative, then the terms could be negative. Wait, but if (x/a), (y/b), (z/c) can be negative, but then (a/x), (b/y), (c/z) would also have the same sign. So when multiplied, the cross terms could be positive or negative?Wait, this is getting complicated. Maybe there's another approach. Let's think again.We have:Equation (1): S = x/a + y/b + z/c = √2.Equation (2): T = a/x + b/y + c/z = 0.We need to find E = (x/a)^2 + (y/b)^2 + (z/c)^2.We saw that squaring S gives E + 2(xy/(ab) + xz/(ac) + yz/(bc)) = 2.So if we can find the value of the cross term, then E can be found.But how? Let's denote u = x/a, v = y/b, w = z/c. Then the problem becomes:u + v + w = √2,and (1/u) + (1/v) + (1/w) = 0.We need to find u² + v² + w².So, this substitution might simplify things. Let's proceed with that.Let u = x/a, v = y/b, w = z/c. Then:Equation (1): u + v + w = √2.Equation (2): (1/u) + (1/v) + (1/w) = 0.We need to find E = u² + v² + w².So, perhaps we can relate these equations. Let's see.From Equation (2): (1/u) + (1/v) + (1/w) = 0. Let's combine these fractions:(v w + u w + u v)/(u v w) = 0.Therefore, the numerator must be zero (assuming denominator is not zero, which we can assume since a/x, etc., are defined, so u, v, w ≠ 0).So, v w + u w + u v = 0.So, uv + uw + vw = 0.Now, recall that (u + v + w)² = u² + v² + w² + 2(uv + uw + vw).We know that u + v + w = √2, so (√2)^2 = 2 = E + 2(0), since uv + uw + vw = 0.Therefore, 2 = E + 0 => E = 2.Wait, so that's the answer? 2? Let me check that again.Yes, substituting variables:Let u = x/a, v = y/b, w = z/c.Given u + v + w = √2,and 1/u + 1/v + 1/w = 0. Which leads to uv + uw + vw = 0.Then, (u + v + w)^2 = u² + v² + w² + 2(uv + uw + vw) => 2 = E + 2*0 => E = 2.Therefore, the expression (x²/a²) + (y²/b²) + (z²/c²) equals 2.Hmm, that seems straightforward once we make the substitution. I wonder if there was a trick I missed initially, but this substitution simplifies everything. So, the key insight was to let u, v, w be the ratios of variables to constants, which allows us to express both equations in terms of u, v, w and then use algebraic identities to relate them. Then, using the square of the sum and the given condition from the second equation, we can eliminate the cross terms and solve directly for E.So, the answer is 2. Therefore, I should box that.Final Answerboxed{2}</think>Given the equations:1. (frac{x}{a} + frac{y}{b} + frac{z}{c} = sqrt{2})2. (frac{a}{x} + frac{b}{y} + frac{c}{z} = 0)We need to find the value of (frac{x^2}{a^2} + frac{y^2}{b^2} + frac{z^2}{c^2}).First, we introduce substitutions (u = frac{x}{a}), (v = frac{y}{b}), and (w = frac{z}{c}). This transforms the given equations into:1. (u + v + w = sqrt{2})2. (frac{1}{u} + frac{1}{v} + frac{1}{w} = 0)From the second equation, combining the fractions gives:[frac{vw + uw + uv}{uvw} = 0]Since the denominator (uvw) is non-zero, the numerator must be zero:[uv + uw + vw = 0]Next, we square the first equation:[(u + v + w)^2 = (sqrt{2})^2]Expanding the left-hand side, we get:[u^2 + v^2 + w^2 + 2(uv + uw + vw) = 2]Substituting (uv + uw + vw = 0) into the equation, we have:[u^2 + v^2 + w^2 + 2 cdot 0 = 2]Thus, the value of (frac{x^2}{a^2} + frac{y^2}{b^2} + frac{z^2}{c^2}) is:[boxed{2}],<think>Okay, let's see. I have this problem here: given that (x/a) + (y/b) + (z/c) equals the square root of 2, and (a/x) + (b/y) + (c/z) equals zero. I need to find the value of (x²/a²) + (y²/b²) + (z²/c²). Hmm, interesting. I wonder how to approach this. First, let me write down the given equations to have them clear:1. (x/a) + (y/b) + (z/c) = √22. (a/x) + (b/y) + (c/z) = 0And I need to find (x²/a²) + (y²/b²) + (z²/c²). So, maybe I can square the first equation? Because squaring a sum often gives cross terms that might relate to the expression we need to find. Let me try that.Squaring the left side of the first equation:[(x/a) + (y/b) + (z/c)]² = (x²/a²) + (y²/b²) + (z²/c²) + 2[(xy)/(ab) + (xz)/(ac) + (yz)/(bc)]And that equals (√2)² = 2.So, we have:(x²/a²) + (y²/b²) + (z²/c²) + 2[(xy)/(ab) + (xz)/(ac) + (yz)/(bc)] = 2Let me denote S = (x²/a²) + (y²/b²) + (z²/c²), which is the quantity we need to find. Then the equation becomes:S + 2[(xy)/(ab) + (xz)/(ac) + (yz)/(bc)] = 2So, if I can find the value of the term (xy)/(ab) + (xz)/(ac) + (yz)/(bc), then I can solve for S. How can I relate that to the second given equation, which is (a/x) + (b/y) + (c/z) = 0?Let me denote T = (a/x) + (b/y) + (c/z) = 0. Maybe if I multiply T by something?Wait, let's look at the term (xy)/(ab). If I multiply (a/x) and (b/y), I get (a/x)*(b/y) = (ab)/(xy). Similarly, (a/x)*(c/z) = (ac)/(xz), and (b/y)*(c/z) = (bc)/(yz). Hmm, but these are reciprocals of the terms we have in the cross terms.Alternatively, if I consider the product of the two equations, but the second equation is zero. Let's see:[(x/a) + (y/b) + (z/c)] * [(a/x) + (b/y) + (c/z)] = ?Expanding this product:(x/a)*(a/x) + (x/a)*(b/y) + (x/a)*(c/z) + (y/b)*(a/x) + (y/b)*(b/y) + (y/b)*(c/z) + (z/c)*(a/x) + (z/c)*(b/y) + (z/c)*(c/z)Simplifying each term:1. (x/a)*(a/x) = 12. (x/a)*(b/y) = (b x)/(a y)3. (x/a)*(c/z) = (c x)/(a z)4. (y/b)*(a/x) = (a y)/(b x)5. (y/b)*(b/y) = 16. (y/b)*(c/z) = (c y)/(b z)7. (z/c)*(a/x) = (a z)/(c x)8. (z/c)*(b/y) = (b z)/(c y)9. (z/c)*(c/z) = 1So adding all these up:1 + 1 + 1 + [(b x)/(a y) + (a y)/(b x)] + [(c x)/(a z) + (a z)/(c x)] + [(c y)/(b z) + (b z)/(c y)]So the total is 3 + [ (b x)/(a y) + (a y)/(b x) ) + (c x)/(a z) + (a z)/(c x) ) + (c y)/(b z) + (b z)/(c y) ) ]But since the second equation is (a/x) + (b/y) + (c/z) = 0, when we multiplied the two equations, the left side is [√2] * 0 = 0. Therefore:3 + [ (b x)/(a y) + (a y)/(b x) + (c x)/(a z) + (a z)/(c x) + (c y)/(b z) + (b z)/(c y) ] = 0Wait, but this sum equals zero? But 3 plus those terms equals zero? That seems impossible because the terms like (b x)/(a y) + (a y)/(b x) are each of the form t + 1/t, which is always greater than or equal to 2 (by AM ≥ GM). Therefore, each pair like (b x)/(a y) + (a y)/(b x) is ≥ 2, so the sum would be 3 + (something ≥ 6) which is ≥9, but the product is supposed to be zero. That contradicts. Hmm, maybe I made a mistake here.Wait, let me check. The product of the two equations:First equation: (x/a) + (y/b) + (z/c) = √2Second equation: (a/x) + (b/y) + (c/z) = 0Therefore, the product of the left sides is [ (x/a) + (y/b) + (z/c) ] * [ (a/x) + (b/y) + (c/z) ] = √2 * 0 = 0But when expanding, as I did, I get 3 + sum of terms each of which is t + 1/t. But 3 + (sum of t + 1/t) = 0. But t + 1/t is always ≥ 2, so sum of three such terms would be ≥6, so total is 3 + 6 = 9. But 9 = 0? That's impossible. Therefore, there must be a mistake in my reasoning.Wait, but how? Maybe I made a miscalculation when expanding the product. Let me check again.First term: (x/a)*(a/x) = 1Second term: (x/a)*(b/y) = (b x)/(a y)Third term: (x/a)*(c/z) = (c x)/(a z)Fourth term: (y/b)*(a/x) = (a y)/(b x)Fifth term: (y/b)*(b/y) = 1Sixth term: (y/b)*(c/z) = (c y)/(b z)Seventh term: (z/c)*(a/x) = (a z)/(c x)Eighth term: (z/c)*(b/y) = (b z)/(c y)Ninth term: (z/c)*(c/z) = 1Adding them up: 1 + 1 + 1 + [ (b x)/(a y) + (c x)/(a z) + (a y)/(b x) + (c y)/(b z) + (a z)/(c x) + (b z)/(c y) ]Yes, so that's 3 + [ three pairs of terms ]Each pair is like (b x)/(a y) + (a y)/(b x). Similarly for the others.But in the expansion, are these grouped as such?Wait, (b x)/(a y) + (a y)/(b x) is one pair.Similarly, (c x)/(a z) + (a z)/(c x) is another pair.And (c y)/(b z) + (b z)/(c y) is the third pair.Each pair is of the form t + 1/t.Therefore, each pair is ≥2, so total sum is 3 + 3*2 = 9, but the total is supposed to be 0. Contradiction. Therefore, that suggests that my initial assumption that these terms are all real numbers might be wrong? Or perhaps the problem is set in such a way that these terms are complex? But the problem statement doesn't specify. Hmm.Wait, but the problem is given with variables x, y, z, a, b, c, but no constraints. So maybe the variables are complex? But if they are real, then this leads to a contradiction. Therefore, perhaps there's a different approach.Alternatively, maybe I misapplied some step here. Let me check the equations again.First equation: (x/a) + (y/b) + (z/c) = √2Second equation: (a/x) + (b/y) + (c/z) = 0Product of these two equations: 0But when expanded, gives 3 + sum of t + 1/t = 0, which is impossible if variables are real positive numbers. So, perhaps variables can take negative values? But even so, t + 1/t can be negative if t is negative. Wait, if t is negative, then t + 1/t is ≤ -2. Because if t is negative, let t = -k, k > 0, then t + 1/t = -k -1/k = -(k + 1/k) ≤ -2. So each pair can be ≤ -2 if t is negative.Therefore, if all the pairs (b x)/(a y), (c x)/(a z), (c y)/(b z) are negative, then each pair would contribute ≤ -2, so the total sum would be 3 + (something ≤ -6) = something ≤ -3. But the product is supposed to be 0, which is in contradiction again.Therefore, how is this possible? Maybe there's a miscalculation. Let me check again.Wait, actually, in the product, the cross terms are (x/a)*(b/y) + (x/a)*(c/z) + (y/b)*(a/x) + (y/b)*(c/z) + (z/c)*(a/x) + (z/c)*(b/y). Which is indeed:(b x)/(a y) + (c x)/(a z) + (a y)/(b x) + (c y)/(b z) + (a z)/(c x) + (b z)/(c y)These can be grouped into three pairs:[(b x)/(a y) + (a y)/(b x)] + [(c x)/(a z) + (a z)/(c x)] + [(c y)/(b z) + (b z)/(c y)]Each pair is of the form t + 1/t.So each pair is ≥2 if t >0 or ≤-2 if t <0.But if each pair is ≤-2, then the total would be 3 + (-6) = -3, which is still not 0. If some pairs are positive and others are negative, maybe the sum cancels out.But in order for the total sum to be zero, we need 3 + sum = 0 => sum = -3.But sum is the sum of three terms each of which is either ≥2 or ≤-2. If all three are ≤-2, sum ≤ -6, so 3 + sum ≤ -3. If two are ≤-2 and one is ≥2, sum ≤ -4 + 2 = -2, so 3 + sum ≤1. If one ≤-2 and two ≥2, sum ≥ -2 +4=2, 3 + sum ≥5. If all three ≥2, sum ≥6, 3 + sum ≥9. Therefore, there is no way for 3 + sum =0. So this seems impossible.Therefore, the conclusion is that the initial equations cannot hold in real numbers, unless variables are allowed to be complex. But if variables are complex, then t +1/t can take other values. However, the problem statement doesn't specify. But since the answer is likely to be a real number (since √2 is real), perhaps the variables are real and there's a different approach here.Alternatively, maybe squaring the first equation is the right approach, but how to connect the cross terms with the second equation.Wait, let's denote u = x/a, v = y/b, w = z/c. Then the first equation becomes u + v + w = √2. The second equation becomes (1/u) + (1/v) + (1/w) =0. And we need to find u² + v² + w².So, maybe this substitution simplifies things.Let me set u = x/a, v = y/b, w = z/c. Then we have:u + v + w = √21/u + 1/v + 1/w =0Find u² + v² + w².Hmm, that seems more manageable.So, given u + v + w = √2 and 1/u + 1/v + 1/w =0, find u² + v² + w².Let me think about this. Let's write 1/u +1/v +1/w = (vw + uw + uv)/(uvw) =0. Therefore, the numerator must be zero (assuming uvw ≠0). So, vw + uw + uv =0.Therefore, we have:u + v + w = √2uv + uw + vw =0And we need to find u² + v² + w².But u² + v² + w² = (u + v + w)² - 2(uv + uw + vw) = (√2)² - 2*0 = 2 -0=2.So the answer is 2. Wait, that's straightforward?Wait, but let me confirm. If we let u + v + w = S, uv + uw + vw = P, and uvw = Q. Then, the first equation is S = √2, the second equation gives P =0. Then, u² + v² + w² = S² - 2P = (√2)^2 - 2*0 = 2. So yes, that's correct.But how did I not see that earlier? Because I got confused by the product of the two equations leading to a contradiction. But when I substituted variables, it simplified to something straightforward. So perhaps the key was to substitute u, v, w. So, the answer is 2. Therefore, boxed{2}.Wait, let me check again to make sure.Given u + v + w = √2 and uv + uw + vw =0, then u² + v² + w² = (u + v + w)^2 - 2(uv + uw + vw) = (√2)^2 - 2*0 = 2 -0=2. Correct. Therefore, the answer is 2. But earlier, when I tried multiplying the two equations, I encountered a contradiction. How does that reconcile?Ah, because in that approach, multiplying the two equations gave a product that equals zero, but expanding it led to an equation that seems impossible. However, by substitution, we found a valid answer. Therefore, there must be an error in the first approach. Let me think why.In the substitution approach, we considered u =x/a, v = y/b, w =z/c, which is valid as long as a, b, c, x, y, z are non-zero. Then, we translated the given equations into u + v + w = √2 and 1/u +1/v +1/w =0. Then, from 1/u +1/v +1/w =0, we derived that uv + uw + vw =0. Then, using the identity for u² + v² + w², we found the answer. But when we tried to multiply the original equations, we ended up with a contradiction. Why is that? Because when we multiply (u + v + w)(1/u +1/v +1/w) =0, which expands to 3 + sum of (t +1/t) =0. But from substitution, we have:(u + v + w)(1/u +1/v +1/w) =0, but according to substitution, (u + v + w)=√2 and (1/u +1/v +1/w)=0, so their product is zero, which is consistent.But according to expansion, (u + v + w)(1/u +1/v +1/w) =3 + sum (t +1/t). However, from substitution, we have uv + uw + vw =0, so (uv + uw + vw) =0. But how does that relate to the sum of (t +1/t)?Wait, perhaps there's another way to express sum (t +1/t). Let me write:Let me denote t1 = u/(v) + v/(u), t2 = u/w + w/u, t3 = v/w + w/v. Then sum (t +1/t) is equivalent to t1 + t2 + t3. But how does that relate to uv + uw + vw?Alternatively, maybe there's a different relation here. Wait, but in the substitution approach, we have uv + uw + vw=0, which is given by 1/u +1/v +1/w=0 multiplied by uvw. So, uv + uw + vw=0. Then, u² + v² + w² = (u +v +w)^2 - 2(uv + uw + vw) =2 -0=2. So that works.But when expanding (u +v +w)(1/u +1/v +1/w), we get 3 + sum of (u/v + v/u + u/w + w/u + v/w + w/v). Which is 3 + [ (u/v + v/u) + (u/w + w/u) + (v/w + w/v) ].But since uv + uw + vw =0, is there a relation between these terms?Let me try expressing u/v + v/u. Let's denote u/v + v/u = (u² + v²)/(uv). Similarly, u/w + w/u = (u² + w²)/(uw), and v/w + w/v = (v² + w²)/(vw).Therefore, the sum becomes:[ (u² + v²)/(uv) ) + (u² + w²)/(uw) ) + (v² + w²)/(vw) ) ]Which can be written as:(u² + v²)/uv + (u² + w²)/uw + (v² + w²)/vw= (u²)/(uv) + (v²)/(uv) + (u²)/(uw) + (w²)/(uw) + (v²)/(vw) + (w²)/(vw)= u/v + v/u + u/w + w/u + v/w + w/vWhich is the same as before. But since uv + uw + vw =0, maybe we can substitute that into this expression. Let me see.If uv + uw + vw =0, then for example, uw = -uv - vw. Let me try substituting that into the terms:Take the term u/w. Since uw = -uv - vw, then u/w = (-uv - vw)/w². Wait, this might not be helpful.Alternatively, let's consider u² + v² + w² =2. Then, since uv + uw + vw =0, we have:(u + v + w)^2 = u² + v² + w² + 2(uv + uw + vw) =2 +0=2. So, u +v +w =√2, which is consistent.But how does this relate to the product (u + v +w)(1/u +1/v +1/w)=0.But given that uv + uw + vw=0, so (u +v +w)(1/u +1/v +1/w)=3 + sum(t +1/t)=0.But sum(t +1/t)= -3.But how can sum(t +1/t) = -3?Each term t +1/t is (u/v + v/u) + (u/w + w/u) + (v/w + w/v). If each pair sums to -1, for example, then total is -3. But is this possible?For example, if u/v + v/u = -1, similarly for others. Let me suppose that u/v + v/u = -1.Let me set k = u/v, then k +1/k = -1. Multiply both sides by k: k² +1 = -k => k² +k +1=0. The discriminant is 1 -4 = -3, so complex roots. Therefore, complex numbers.Therefore, if u, v, w are complex numbers, then this is possible. But since the problem didn't specify whether variables are real or complex, but the answer is a real number 2, perhaps the substitution approach bypasses the need to consider the product of equations, and directly gives the answer through algebraic manipulation. So, despite the seeming contradiction in the product expansion, the substitution gives a valid answer.Therefore, the answer is 2.Final Answerboxed{2}</think>Given the equations:1. (frac{x}{a} + frac{y}{b} + frac{z}{c} = sqrt{2})2. (frac{a}{x} + frac{b}{y} + frac{c}{z} = 0)We need to find the value of (frac{x^2}{a^2} + frac{y^2}{b^2} + frac{z^2}{c^2}).First, we introduce the substitutions (u = frac{x}{a}), (v = frac{y}{b}), and (w = frac{z}{c}). This transforms the given equations into:1. (u + v + w = sqrt{2})2. (frac{1}{u} + frac{1}{v} + frac{1}{w} = 0)The second equation can be rewritten as (frac{vw + uw + uv}{uvw} = 0), which implies (uv + uw + vw = 0) (assuming (u, v, w neq 0)).We need to find (u^2 + v^2 + w^2). Using the identity for the square of a sum:[(u + v + w)^2 = u^2 + v^2 + w^2 + 2(uv + uw + vw)]Substituting the known values from the equations:[(sqrt{2})^2 = u^2 + v^2 + w^2 + 2 cdot 0]Simplifying this, we get:[2 = u^2 + v^2 + w^2]Thus, the value of (frac{x^2}{a^2} + frac{y^2}{b^2} + frac{z^2}{c^2}) is (boxed{2}).]

🔑:11. Answer: 2.2=left(frac{x}{a}+frac{y}{b}+frac{z}{c}right)^{2}=frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}+frac{z^{2}}{c^{2}}+2left(frac{x y}{a b}+frac{y z}{b c}+frac{z x}{c a}right)=frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}+frac{z^{2}}{b^{2}}+2 frac{x y z}{a b c}left(frac{c}{z}+frac{a}{x}+frac{b}{y}right) .So frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}+frac{z^{2}}{c^{2}}=2.

🧠:[<think>Okay, let's try to solve this problem step by step. The problem says that in the arithmetic sequence {a_n}, a_3 and a_{4017} are two different extreme points of the function f(x) = x³ - 6x² + 4x - 1. Then we need to find log base 1/4 of a_{2010} and choose the correct answer from the options given.First, let me recall that an arithmetic sequence has a common difference between consecutive terms. So, if the first term is a_1 and the common difference is d, then the nth term a_n = a_1 + (n - 1)d. That's the formula for arithmetic sequences.Now, the function given is f(x) = x³ - 6x² + 4x - 1. The extreme points of a function occur where the first derivative is zero. So, I need to find the critical points by taking the derivative of f(x) and setting it equal to zero.Let's compute f'(x):f'(x) = 3x² - 12x + 4.So, the critical points are the solutions to 3x² - 12x + 4 = 0.Wait, but the problem says that a_3 and a_{4017} are two different extreme points. That means these terms in the arithmetic sequence correspond to the x-values where the function f(x) has local maxima or minima. Since they are two different extreme points, these x-values must be the two distinct solutions to the equation f'(x) = 0. Let me solve this quadratic equation to find those x-values.So, the equation is 3x² - 12x + 4 = 0. Let's divide both sides by 3 to simplify:x² - 4x + (4/3) = 0.Now, using the quadratic formula, x = [4 ± sqrt(16 - 16/3)] / 2.Wait, let me compute the discriminant first. The discriminant D is 16 - 4*(1)*(4/3) = 16 - 16/3 = (48/3 - 16/3) = 32/3.So, sqrt(D) = sqrt(32/3) = (4*sqrt(6))/3. Hmm, because sqrt(32) is 4*sqrt(2), but 32/3 is 32/3. Wait, let's check again.Wait, sqrt(32/3) can be written as sqrt(32)/sqrt(3) = (4*sqrt(2))/sqrt(3) = (4*sqrt(6))/3 after rationalizing the denominator. Yes, that's correct.So, x = [4 ± (4*sqrt(6)/3)] / 2.Let's simplify that:x = [4/2 ± (4*sqrt(6)/3)/2] = 2 ± (2*sqrt(6)/3).Therefore, the two critical points are x = 2 + (2√6)/3 and x = 2 - (2√6)/3.Wait, so these are the two extreme points. The problem states that a_3 and a_{4017} are these two different extreme points. Since the arithmetic sequence is ordered, the terms a_3 and a_{4017} must correspond to these two x-values. But since they are two different terms in the arithmetic sequence, and the arithmetic sequence is linear, these two x-values must be terms in the sequence. So, perhaps one is a maximum and the other is a minimum? Let me check the second derivative to confirm which is which.But maybe that's not necessary for the problem. Let's see. The key point is that in the arithmetic sequence, the 3rd term and the 4017th term are these two critical points. So, we can model the arithmetic sequence such that a_3 and a_{4017} are equal to these two x-values. But since an arithmetic sequence is linear, the difference between a_3 and a_{4017} is (4017 - 3) times the common difference d. That is, a_{4017} = a_3 + (4017 - 3)d = a_3 + 4014d. But since these two terms are different, d cannot be zero. So, we have two equations here.But we need to figure out which term is which. The problem says they are "two different extreme points", so maybe it's possible that one is a maximum and the other is a minimum, but maybe their order in the sequence depends on the common difference. Hmm. Let me think.Alternatively, since the arithmetic sequence is ordered, either a_3 comes before a_{4017} (which it does, since 3 < 4017) and depending on the common difference d, the sequence could be increasing or decreasing. But the two critical points are x = 2 + (2√6)/3 and x = 2 - (2√6)/3. Let me compute their approximate numerical values to see which one is larger.Calculating 2 + (2√6)/3: √6 ≈ 2.449, so 2*2.449 ≈ 4.898. Divided by 3: ≈1.632. So, 2 + 1.632 ≈3.632.Similarly, 2 - (2√6)/3 ≈2 -1.632≈0.368. So, the two critical points are approximately 3.632 and 0.368. Therefore, one is at around 3.6 and the other at around 0.37. So, one is greater than 2, the other is less than 2.Now, in the arithmetic sequence, if the common difference d is positive, then the sequence is increasing, so a_3 would be less than a_{4017}. If d is negative, then the sequence is decreasing, so a_3 would be greater than a_{4017}.But from the approximate values, the two critical points are 0.368 and 3.632. So, if the arithmetic sequence is increasing (d>0), then a_3 would be 0.368 and a_{4017} would be 3.632. Alternatively, if the sequence is decreasing (d<0), then a_3 would be 3.632 and a_{4017} would be 0.368. Either way, the arithmetic sequence must have these two terms as a_3 and a_{4017}.But the problem says "two different extreme points", so perhaps we don't need to worry about which is which, just that they correspond to those two x-values. Let me set up equations for a_3 and a_{4017}.Let me denote the arithmetic sequence as a_n = a_1 + (n -1)d.Therefore, a_3 = a_1 + 2d.a_{4017} = a_1 + 4016d.Given that a_3 and a_{4017} are the two critical points, which we found to be 2 + (2√6)/3 and 2 - (2√6)/3. Let me denote them as x1 and x2, where x1 = 2 + (2√6)/3 and x2 = 2 - (2√6)/3.So, we can set up the equations:a_1 + 2d = x1a_1 + 4016d = x2Or vice versa. So, depending on which term is which, but since the problem says they are different, the two equations will give us a system to solve for a_1 and d.Alternatively, if we don't know which term is which, we can just set up two possibilities. Let's first assume that a_3 = x1 and a_{4017} = x2. Then:a1 + 2d = x1a1 + 4016d = x2Subtract the first equation from the second:(4016d - 2d) = x2 - x14014d = x2 - x1But x2 - x1 = [2 - (2√6)/3] - [2 + (2√6)/3] = - (4√6)/3.Thus, 4014d = - (4√6)/3Therefore, d = [- (4√6)/3] / 4014But 4014 is a large number, but maybe we can simplify it. Let's note that 4014 = 4017 -3, but perhaps 4014 factors into something? Let me check.4014 divided by 2 is 2007, which is divisible by 3: 2007 ÷ 3 = 669, which is again divisible by 3: 669 ÷3=223. 223 is a prime number. So, 4014 = 2×3×3×223. Hmm, not sure if that helps. Anyway, perhaps we can keep it as 4014 for now.So, d = - (4√6)/(3×4014) = - (2√6)/(3×2007). Hmm, maybe.Alternatively, maybe if we take the other possibility where a_3 = x2 and a_{4017}=x1, then:a1 +2d = x2a1 +4016d =x1Subtracting first equation from the second:4014d = x1 -x2 = (4√6)/3Thus, d = (4√6)/(3×4014) = (2√6)/(3×2007)So, depending on the order, d is positive or negative. But since the problem says they are two different extreme points, maybe the ordering doesn't matter. However, we need to find a_{2010}, which is a term in the arithmetic sequence, so we need to know a1 and d. But since we have two equations, we can solve for a1 and d.But perhaps there is a smarter way. Let's notice that the two critical points are symmetric around x=2. Since x1 = 2 + (2√6)/3 and x2 = 2 - (2√6)/3. Therefore, their average is (x1 + x2)/2 = [2 + 2]/2 = 2. So, the two critical points are symmetric with respect to x=2. Therefore, the arithmetic sequence has a_3 and a_{4017} as two points symmetric around 2. Since arithmetic sequences are linear, the midpoint between a_3 and a_{4017} is 2. So, the average of a_3 and a_{4017} is 2. Therefore, (a_3 + a_{4017}) /2 = 2. So, a_3 + a_{4017} =4.Alternatively, since arithmetic sequences have the property that the average of two terms equidistant from the ends is equal. But in this case, the terms a_3 and a_{4017} are separated by 4017 -3 =4014 terms. Wait, but in an arithmetic sequence, the difference between term k and term m is (m -k)d. So, the difference between a_{4017} and a_3 is (4017 -3)d =4014d. So, the difference between the two terms is 4014d. Also, since their average is 2, the sum is 4, so a_3 + a_{4017}=4.But also, a_{4017} = a_3 +4014d. So substituting into a_3 + a_{4017}=4:a_3 + (a_3 +4014d) =42a_3 +4014d=4But a_3 is one of the critical points, say x1 or x2, but perhaps we can avoid substituting the exact values here. Wait, but maybe we can express a_{2010} in terms of a_3 and d. Let's see.Since a_{2010} = a_1 +2009d. But we need to express this in terms of a_3. Since a_3 =a1 +2d, so a1 =a3 -2d. Therefore, a_{2010}= a3 -2d +2009d= a3 +2007d.Similarly, a_{4017}=a3 +4014d. From earlier, we have a_{4017} =4 -a3. Therefore:a3 +4014d =4 -a3So, 2a3 +4014d=4. But from the previous equation when we considered the sum, we also have 2a3 +4014d=4. So, that's consistent. Therefore, this gives us no new information. So, we need another equation. Wait, but we also know that a_3 is either x1 or x2.Wait, but maybe we can find a_{2010} by noting that 2010 is the midpoint between 3 and 4017? Let's check: (3 +4017)/2=4020/2=2010. Yes! So, 2010 is the midpoint between 3 and 4017. Therefore, in an arithmetic sequence, the term at the midpoint is the average of the two terms. Therefore, a_{2010} = (a_3 +a_{4017})/2=4/2=2. Therefore, a_{2010}=2. Therefore, log base 1/4 of a_{2010} is log_{1/4}(2).Wait, is that right? So, since 2010 is the midpoint of 3 and 4017 in terms of term numbers, then yes, the arithmetic sequence's term at the midpoint would be the average of a_3 and a_{4017}, which is 2. So, a_{2010}=2. Then, log_{1/4}(2). Let's compute that.Let me recall that log_{1/4}(2) is the exponent to which we raise 1/4 to get 2. Let me write it as x where (1/4)^x =2. But (1/4)^x =4^(-x)=2^(-2x). So, 2^(-2x)=2^1. Therefore, -2x=1, so x= -1/2. Therefore, log_{1/4}(2)= -1/2. But let me check that again.Alternatively, using the change of base formula: log_{1/4}(2)= ln(2)/ln(1/4)= ln(2)/(ln(1)-ln(4))= ln(2)/(-ln4)= -ln(2)/(2ln2)= -1/2. Yes, that's correct. So, log base 1/4 of 2 is -1/2, which is option D.But wait, let me confirm if the midpoint idea is correct. In an arithmetic sequence, the term exactly halfway between term k and term m (where m >k) is term (k + (m -k)/2). But if m -k is even, then it's an integer. In this case, m=4017 and k=3, so m -k=4014. Then, halfway would be 3 +4014/2=3 +2007=2010. So, yes, term 2010 is the midpoint. Therefore, a_{2010}=(a_3 +a_{4017})/2=4/2=2. Therefore, a_{2010}=2. Therefore, the logarithm is log_{1/4}(2)= -1/2, which is option D. So, the answer should be D.Wait, but let me check if there's any mistake here. For example, does the midpoint property hold for arithmetic sequences? Yes, because in an arithmetic sequence, each term is equally spaced. So, if you have two terms a_k and a_m, then the term halfway between them in position is the average of the two terms. Therefore, a_{(k+m)/2} = (a_k +a_m)/2. But only if (k +m) is even, which in this case, 3 +4017=4020, which is even, so (3+4017)/2=2010. Therefore, yes, a_{2010} is the average of a_3 and a_{4017}, which is 2.Therefore, the value we need is log_{1/4}(2). As calculated earlier, that's -1/2. So, the correct answer is D.But let me verify once more to be thorough. Let me check the steps again.1. Found the derivative f'(x) =3x² -12x +4. Correct.2. Solved 3x² -12x +4=0, divided by 3: x² -4x +4/3=0. Quadratic formula: x=(4 ±sqrt(16 -16/3))/2= [4 ±sqrt(32/3)]/2= [4 ± (4√6)/3]/2=2 ± (2√6)/3. Correct.3. Realized that a_3 and a_{4017} are these two roots. Since arithmetic sequence is linear, the difference between a_{4017} and a_3 is 4014d. The average of the two roots is 2, so a_3 +a_{4017}=4, hence their average is 2. Therefore, since 2010 is the midpoint term between 3 and 4017, a_{2010}=2. Then, log_{1/4}(2)= -1/2. All steps seem correct.Therefore, the answer is D.Final Answerboxed{D}</think>Given the arithmetic sequence ({a_n}), where (a_3) and (a_{4017}) are two different extreme points of the function (f(x) = x^3 - 6x^2 + 4x - 1), we need to find (log_{frac{1}{4}} a_{2010}).1. Find the critical points of (f(x)): - The first derivative is (f'(x) = 3x^2 - 12x + 4). - Solving (3x^2 - 12x + 4 = 0) using the quadratic formula: [ x = frac{12 pm sqrt{144 - 48}}{6} = frac{12 pm sqrt{96}}{6} = frac{12 pm 4sqrt{6}}{6} = 2 pm frac{2sqrt{6}}{3} ] - The critical points are (x_1 = 2 + frac{2sqrt{6}}{3}) and (x_2 = 2 - frac{2sqrt{6}}{3}).2. Relate the critical points to the arithmetic sequence: - The terms (a_3) and (a_{4017}) correspond to the critical points (x_1) and (x_2). - The average of the critical points is (frac{x_1 + x_2}{2} = 2), so (a_3 + a_{4017} = 4).3. Determine (a_{2010}): - Since (2010) is the midpoint between (3) and (4017) in the arithmetic sequence, (a_{2010}) is the average of (a_3) and (a_{4017}): [ a_{2010} = frac{a_3 + a_{4017}}{2} = frac{4}{2} = 2 ]4. Calculate (log_{frac{1}{4}} 2): - Let (y = log_{frac{1}{4}} 2). This means (left(frac{1}{4}right)^y = 2). - Rewriting (frac{1}{4}) as (4^{-1}) and (4) as (2^2): [ (4^{-1})^y = 2 implies 4^{-y} = 2 implies (2^2)^{-y} = 2 implies 2^{-2y} = 2^1 ] - Equating the exponents: (-2y = 1 implies y = -frac{1}{2}).Thus, the value of (log_{frac{1}{4}} a_{2010}) is (boxed{D}).,<think>Okay, so I need to solve this problem where we have an arithmetic sequence {a_n}, and they tell us that a_3 and a_4017 are two different extreme points of the function f(x) = x³ - 6x² + 4x - 1. Then we need to find log base 1/4 of a_{2010} and choose the correct option from A to D.First, let me recall what an arithmetic sequence is. An arithmetic sequence is a sequence where each term after the first is obtained by adding a constant difference. So, in general, an arithmetic sequence can be written as a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.The problem mentions that a_3 and a_{4017} are two different extreme points of the function f(x). Extreme points of a function occur where the first derivative is zero. So, I need to find the critical points of f(x) by taking its derivative and setting it equal to zero.Let me compute f'(x):f(x) = x³ - 6x² + 4x - 1So, f'(x) = 3x² - 12x + 4.To find the critical points, set f'(x) = 0:3x² - 12x + 4 = 0.This is a quadratic equation. Let me solve for x using the quadratic formula.Quadratic formula: x = [12 ± sqrt(144 - 4*3*4)] / (2*3)Compute the discriminant:sqrt(144 - 48) = sqrt(96) = sqrt(16*6) = 4*sqrt(6)Therefore, the solutions are:x = [12 ± 4√6] / 6 = [12 ± 4√6]/6 = simplify by dividing numerator and denominator by 2:x = [6 ± 2√6]/3 = 2 ± (2√6)/3 = 2 ± (2/3)√6.Wait, that seems correct? Let me check:Wait, 3x² - 12x + 4 = 0. So:x = [12 ± sqrt(144 - 48)] / 6 = [12 ± sqrt(96)] / 6 = [12 ± 4√6]/6 = [6 ± 2√6]/3 = 2 ± (2√6)/3. Yes, that's right.So, the critical points are x = 2 + (2√6)/3 and x = 2 - (2√6)/3. These are the two extreme points. Since the problem states that a_3 and a_{4017} are two different extreme points, that means these two x-values correspond to a_3 and a_{4017}. Since they are two different points, one must be the maximum and the other the minimum. But maybe we don't need to know which is which.Given that {a_n} is an arithmetic sequence, let's express a_3 and a_{4017} in terms of the first term a_1 and common difference d.In an arithmetic sequence, a_n = a_1 + (n-1)d.Therefore:a_3 = a_1 + 2da_{4017} = a_1 + 4016dNow, these two terms are the two critical points we found earlier: 2 + (2√6)/3 and 2 - (2√6)/3. But since the arithmetic sequence is linear, and these two terms are different, the common difference d must be non-zero.But how do we relate a_3 and a_{4017} to those two critical points?Wait, the problem says "two different extreme points of the function f(x)". So, the values of a_3 and a_{4017} are the x-coordinates of the two critical points, right?Wait, hold on. Wait, the problem says "a_3 and a_{4017} are two different extreme points of the function f(x)". Hmm, the wording is a bit ambiguous. Does it mean that the terms a_3 and a_{4017} in the sequence are the x-values where the extremes occur, or that they are the y-values?But in calculus, extreme points are points (x, f(x)). But here, the problem says "extreme points of the function f(x)", so those are the x-values where the function has a local maximum or minimum. So, the x-coordinates. So, that would mean that a_3 and a_{4017} are the x-values where f has extremes. Therefore, the terms of the arithmetic sequence {a_n} correspond to the critical x-values.Therefore, the critical points are x = a_3 and x = a_{4017}. So, the two critical x-values (which we found to be 2 ± (2√6)/3) must be equal to a_3 and a_{4017} in some order.Therefore, the two critical points are 2 + (2√6)/3 and 2 - (2√6)/3, and these correspond to a_3 and a_{4017}. Since the arithmetic sequence is linear, we can set up equations.Given that a_3 and a_{4017} are two terms in the arithmetic sequence, separated by 4017 - 3 = 4014 terms. The difference between these two terms in the sequence is (4017 - 3)d = 4014d. So, the difference between the two critical points is equal to 4014d.But the two critical points are 2 + (2√6)/3 and 2 - (2√6)/3. The difference between them is [2 + (2√6)/3] - [2 - (2√6)/3] = (4√6)/3. Therefore, 4014d = (4√6)/3. Therefore, d = (4√6)/(3*4014). Let me compute that.But before proceeding, let me confirm: if a_3 and a_{4017} are two different critical points, which are 2 + (2√6)/3 and 2 - (2√6)/3, then the difference between them is (4√6)/3. Since the difference in the arithmetic sequence between term 3 and term 4017 is (4017 - 3)d = 4014d. Therefore, 4014d = (4√6)/3 => d = (4√6)/(3*4014). Simplify numerator and denominator.First, note that 4014 is 4014. Let's see if 4014 and 4 have a common factor. 4014 divided by 2 is 2007, so 4014 = 2*2007. So, d = (4√6)/(3*2*2007) = (4√6)/(6*2007) = (2√6)/(3*2007). But 2007 divided by 3 is 669, so 2007 = 3*669. Therefore, d = (2√6)/(3*3*669) = (2√6)/(9*669). Hmm, not sure if this helps.Alternatively, maybe we can think of the arithmetic sequence's terms as being these two critical points. So, a_3 is one critical point, and a_{4017} is the other. Since the arithmetic sequence progresses by a constant difference, the distance between a_3 and a_{4017} is (4017 - 3)d = 4014d. So, the difference between the two critical points is 4014d. Let me compute that difference.The two critical points are:x₁ = 2 + (2√6)/3x₂ = 2 - (2√6)/3Difference: x₁ - x₂ = (4√6)/3So, 4014d = (4√6)/3 => d = (4√6)/(3*4014) = (2√6)/(3*2007) = (2√6)/(6021). Hmm, okay.But maybe there's another way to approach this. Since the arithmetic sequence has terms a_3 and a_{4017} as the two critical points, then if we can express a_3 and a_{4017} in terms of the sequence's first term and common difference, and set them equal to the critical points, we can solve for a_1 and d.So, let's set:a_3 = a₁ + 2d = 2 + (2√6)/3a_{4017} = a₁ + 4016d = 2 - (2√6)/3Or vice versa. Depending on which critical point is assigned to which term.But since arithmetic sequences can go either increasing or decreasing, depending on the common difference. Since a_3 and a_{4017} are two different terms, 4017 is much larger than 3, so depending on whether d is positive or negative, the sequence could be increasing or decreasing. However, the two critical points are one greater than 2 and one less than 2. So, depending on the common difference, either a_3 is the larger one and a_{4017} is the smaller, or vice versa.But let's not assume. Let's suppose that a_3 is 2 + (2√6)/3 and a_{4017} is 2 - (2√6)/3. Then, let's write the two equations:1) a₁ + 2d = 2 + (2√6)/32) a₁ + 4016d = 2 - (2√6)/3Subtract equation 1 from equation 2:(4016d - 2d) = [2 - (2√6)/3] - [2 + (2√6)/3]4014d = - (4√6)/3Therefore, d = - (4√6)/ (3*4014) = - (2√6)/(3*2007) as before.Alternatively, if we had assigned a_3 to be the smaller critical point and a_{4017} the larger, then we would get d positive. But let's check.Suppose a_3 = 2 - (2√6)/3 and a_{4017} = 2 + (2√6)/3.Then equations:1) a₁ + 2d = 2 - (2√6)/32) a₁ + 4016d = 2 + (2√6)/3Subtract 1 from 2:4014d = (4√6)/3So, d = (4√6)/(3*4014) = same magnitude but positive. So, d could be positive or negative. But since 4017 is a large index, if d is positive, then a_{4017} would be larger, but if d is negative, a_{4017} would be smaller.But the problem states that a_3 and a_{4017} are two different extreme points. Since both critical points are different, so regardless of the sign of d, as long as the common difference is non-zero, which it is, because they are different.Therefore, we can proceed with either case. However, since the answer is to compute log_{1/4}(a_{2010}), which is the same regardless of the sign of d? Wait, no. Because a_{2010} depends on a_1 and d, which would differ based on the assignment. Wait, but let's see.Wait, but actually, the arithmetic sequence is uniquely determined once we fix a_1 and d. However, depending on which critical point we assign to a_3 and a_{4017}, we would get different a_1 and d. But let's see if both assignments lead to the same a_{2010}.Alternatively, maybe not. Let's check both possibilities.First, let's suppose that a_3 = 2 + (2√6)/3 and a_{4017} = 2 - (2√6)/3.Then, we can solve for a₁ and d.From equation 1: a₁ + 2d = 2 + (2√6)/3From equation 2: a₁ + 4016d = 2 - (2√6)/3Subtract equation 1 from equation 2:4014d = - (4√6)/3So, d = - (4√6)/(3*4014) = - (2√6)/(3*2007)Therefore, a₁ = 2 + (2√6)/3 - 2d = 2 + (2√6)/3 - 2*(-2√6)/(3*2007)Wait, let me compute that:a₁ = 2 + (2√6)/3 - 2dBut d = - (2√6)/(3*2007)Therefore, -2d = (4√6)/(3*2007)So, a₁ = 2 + (2√6)/3 + (4√6)/(3*2007)Hmm, complicated expression. Maybe there is a smarter way.Alternatively, maybe note that in an arithmetic sequence, the terms are equally spaced. So, the average of a_3 and a_{4017} should be equal to the average of the two critical points.The average of the two critical points is [ (2 + (2√6)/3 ) + (2 - (2√6)/3 ) ] / 2 = (4)/2 = 2.Similarly, the average of a_3 and a_{4017} is (a_3 + a_{4017}) / 2. But in an arithmetic sequence, the average of two terms equidistant from the center is equal to the middle term. However, 3 and 4017 are not equidistant from the center unless the total number of terms is 4017 + 3 - 1 = 4019, which we don't know. Wait, actually, the arithmetic sequence can be infinite. The problem doesn't specify the number of terms.But perhaps since a_{4017} is mentioned, the sequence must have at least 4017 terms. Anyway, the average of a_3 and a_{4017} is equal to the average of the two critical points, which is 2. Therefore:(a_3 + a_{4017}) / 2 = 2 => a_3 + a_{4017} = 4.But since a_3 and a_{4017} are terms in the arithmetic sequence, the average of these two terms is also equal to the term halfway between them. The number of terms between a_3 and a_{4017} is 4017 - 3 = 4014, so the halfway term would be 3 + 2007 = 2010. Wait, 4014 terms between 3 and 4017, so the middle term is 3 + 2007 = 2010. Therefore, a_{2010} is the average of a_3 and a_{4017}.But we already know that the average is 2, so a_{2010} = 2.Therefore, regardless of the values of a_1 and d, the term a_{2010} is the average of a_3 and a_{4017}, which is 2. Therefore, log base 1/4 of a_{2010} is log_{1/4}(2).Then, we need to compute log_{1/4}(2).Recall that log_b(a) = ln(a)/ln(b). So, log_{1/4}(2) = ln(2)/ln(1/4) = ln(2)/(-ln(4)) = - ln(2)/(2 ln 2) = -1/2. Wait, but let me check:Alternatively, since (1/4)^x = 2. Solve for x.(1/4)^x = 2 => 4^{-x} = 2 => (2^2)^{-x} = 2 => 2^{-2x} = 2^1 => -2x = 1 => x = -1/2.Therefore, log_{1/4}(2) = -1/2. Therefore, the answer is D: -1/2.But wait, that seems straightforward. Wait, but did I make a mistake here? Let me recap.Because if a_{2010} is the average of a_3 and a_{4017}, which are the two critical points, and the average of the two critical points is 2, then a_{2010} = 2. Then log base 1/4 of 2 is log_{1/4}(2) = -1/2. So the answer is D.But wait, let me confirm this step again. The key insight is that in an arithmetic sequence, the term midway between two terms is their average. Since a_{2010} is the term midway between a_3 and a_{4017} (since 2010 - 3 = 2007 and 4017 - 2010 = 2007), then yes, a_{2010} = (a_3 + a_{4017}) / 2. Since the average of the two critical points is 2, then a_{2010} = 2.Therefore, log_{1/4}(2) is indeed -1/2, which is option D.Alternatively, let me check if there's a possibility that the average might not hold. Wait, in an arithmetic sequence, the difference between consecutive terms is constant. So, the terms are equally spaced. Therefore, the average of two terms that are symmetric around a central term is equal to the central term. In this case, a_{2010} is the central term between a_3 and a_{4017}, so it must be the average. Therefore, this holds regardless of the values of a_1 and d. Therefore, a_{2010} = 2.Therefore, log base 1/4 of 2 is computed as follows:We know that (1/4)^x = 2. Since 1/4 is 4^{-1} or 2^{-2}, so (2^{-2})^x = 2 => 2^{-2x} = 2^1 => -2x = 1 => x = -1/2. Therefore, the answer is D: -1/2.So, this seems to check out. Therefore, the correct answer is D.Final Answerboxed{D}</think>Given the arithmetic sequence ({a_n}), where (a_3) and (a_{4017}) are two different extreme points of the function (f(x) = x^3 - 6x^2 + 4x - 1), we need to find (log_{frac{1}{4}} a_{2010}).First, we find the critical points of (f(x)) by solving (f'(x) = 0):[f'(x) = 3x^2 - 12x + 4 = 0]Using the quadratic formula:[x = frac{12 pm sqrt{144 - 48}}{6} = frac{12 pm 4sqrt{6}}{6} = 2 pm frac{2sqrt{6}}{3}]Thus, the critical points are (2 + frac{2sqrt{6}}{3}) and (2 - frac{2sqrt{6}}{3}).Since (a_3) and (a_{4017}) are these critical points, we note that in an arithmetic sequence, the term midway between (a_3) and (a_{4017}) is their average. The midpoint term is (a_{2010}) because (2010 - 3 = 2007) and (4017 - 2010 = 2007). The average of the critical points is:[frac{left(2 + frac{2sqrt{6}}{3}right) + left(2 - frac{2sqrt{6}}{3}right)}{2} = frac{4}{2} = 2]Thus, (a_{2010} = 2).To find (log_{frac{1}{4}} 2):[log_{frac{1}{4}} 2 = x implies left(frac{1}{4}right)^x = 2 implies 4^{-x} = 2 implies (2^2)^{-x} = 2 implies 2^{-2x} = 2^1 implies -2x = 1 implies x = -frac{1}{2}]Therefore, the answer is (boxed{D}).]

🔑:AnalysisThis problem examines the use of derivatives to study the extreme values of a function, the properties of an arithmetic sequence, and the rules for logarithmic operations. By using derivatives, we can determine the extreme points of the function, and then by applying the properties of an arithmetic sequence and logarithmic operations, we can find the result.SolutionFirst, we find the derivative of the function: f′(x)=3x^2-12x+4.Since a_3 and a_{4017} are extreme points of the function f(x)=x^3-6x^2+4x-1, they are the two real roots of the equation 3x^2-12x+4=0. Therefore, a_3 + a_{4017} = 4. Given that {a_n} is an arithmetic sequence, we have a_3 + a_{4017} = 2a_{2010}, which implies a_{2010} = 2.Thus, {log }_{ frac{1}{4}}{a}_{2010}={log }_{ frac{1}{4}}2= -dfrac{1}{2}.Therefore, the correct answer is boxed{text{D}}.