🧠:[<think>Okay, so I need to simplify the expression (x - 3)² + (x + 4)(x - 4) given that x² - 3x + 1 = 0. Hmm, let me start by expanding the expression first. First, let's expand (x - 3)². I remember that (a - b)² = a² - 2ab + b², so applying that here, it would be x² - 6x + 9. Then, the next part is (x + 4)(x - 4). That's a difference of squares, right? So that simplifies to x² - 16. So now, combining both parts, the entire expression becomes (x² - 6x + 9) + (x² - 16). Let me add those together. Combining like terms: x² + x² is 2x², -6x remains as is because there's no other x term, and then 9 - 16 is -7. So the simplified expression is 2x² - 6x - 7.Okay, so now I need to find the value of this simplified expression 2x² - 6x - 7, but we know from the given equation that x² - 3x + 1 = 0. Maybe I can use this equation to substitute for x² or x terms in the simplified expression. Let me see.From the given equation x² - 3x + 1 = 0, I can solve for x². If I rearrange that, x² = 3x - 1. That might be useful. Let me substitute x² in the simplified expression with 3x - 1. Let's try that.The simplified expression is 2x² - 6x - 7. Replacing x² with 3x - 1, we get:2*(3x - 1) - 6x - 7. Let's compute that. First, multiply 2 into (3x - 1): 6x - 2. Then subtract 6x and 7. So, 6x - 2 - 6x - 7. Combining like terms here: 6x - 6x cancels out, and -2 -7 is -9. So, does that mean the entire expression simplifies to -9? Let me check my steps again to make sure I didn't make a mistake.Original expression: (x - 3)^2 + (x + 4)(x - 4) becomes x² -6x +9 +x² -16. That adds up to 2x² -6x -7. Yes, that's correct. Then since x² -3x +1 =0, so x² =3x -1. Substitute into 2x² gives 2*(3x -1) =6x -2. So, 6x -2 -6x -7 = (6x -6x) + (-2 -7) = 0 -9 = -9. That seems right. Wait, but let me verify with another approach just to be sure. Sometimes substituting can lead to errors if not careful. Maybe plug in the roots of the equation x² -3x +1=0 into the original expression. Let's see. But that might be more complicated. Alternatively, maybe express everything in terms of the given equation.Alternatively, notice that if x² = 3x -1, then in the expression 2x² -6x -7, substituting x² gives 2*(3x -1) -6x -7 = 6x -2 -6x -7 = -9. So that's the same result. So that seems consistent. Alternatively, maybe we can compute the value directly by solving for x. The equation x² -3x +1=0 has solutions x = [3 ± sqrt(9 -4)] /2 = [3 ± sqrt(5)]/2. So x is (3 + sqrt(5))/2 or (3 - sqrt(5))/2. Then plug those into the simplified expression 2x² -6x -7 and see if it's indeed -9.Let's pick one of the roots, say x = [3 + sqrt(5)]/2. Let's compute 2x² -6x -7.First, compute x². x = [3 + sqrt(5)]/2, so x² = ([3 + sqrt(5)]/2)^2 = (9 +6sqrt(5) +5)/4 = (14 +6sqrt(5))/4 = (7 +3sqrt(5))/2.Then 2x² = 2*(7 +3sqrt(5))/2 =7 +3sqrt(5). Now compute -6x: -6*(3 + sqrt(5))/2 = -3*(3 + sqrt(5)) = -9 -3sqrt(5).So adding 2x² -6x -7: (7 +3sqrt(5)) + (-9 -3sqrt(5)) -7 = (7 -9 -7) + (3sqrt(5) -3sqrt(5)) = (-9) +0 = -9. Same result. Let's check with the other root x = [3 - sqrt(5)]/2.Compute x²: [3 - sqrt(5)]/2 squared: (9 -6sqrt(5) +5)/4 = (14 -6sqrt(5))/4 = (7 -3sqrt(5))/2.Then 2x² = 2*(7 -3sqrt(5))/2 =7 -3sqrt(5).-6x = -6*(3 - sqrt(5))/2 = -3*(3 - sqrt(5)) = -9 +3sqrt(5).So 2x² -6x -7 = (7 -3sqrt(5)) + (-9 +3sqrt(5)) -7 = (7 -9 -7) + (-3sqrt(5) +3sqrt(5)) = -9 +0 = -9. Same result. So that confirms that regardless of which root we take, the value is -9. Therefore, the answer is -9.But just to make sure, let me recap the steps once again.Original problem: Simplify (x -3)^2 + (x +4)(x -4) given that x² -3x +1 =0.Step 1: Expand the expression.(x -3)^2 = x² -6x +9.(x +4)(x -4) = x² -16.Add them: x² -6x +9 +x² -16 = 2x² -6x -7.Step 2: Use the given equation x² -3x +1=0 to substitute for x².From x² =3x -1.So replace x² in 2x² -6x -7 with 3x -1:2*(3x -1) -6x -7 = 6x -2 -6x -7 = -9.Therefore, the value is -9.Seems consistent. All steps check out. So I think the answer is indeed -9.Final Answerboxed{-9}</think>To simplify the expression ((x - 3)^2 + (x + 4)(x - 4)) given that (x^2 - 3x + 1 = 0), we start by expanding and combining like terms.First, expand ((x - 3)^2):[(x - 3)^2 = x^2 - 6x + 9]Next, expand ((x + 4)(x - 4)) using the difference of squares:[(x + 4)(x - 4) = x^2 - 16]Combine the expanded terms:[x^2 - 6x + 9 + x^2 - 16 = 2x^2 - 6x - 7]Given the equation (x^2 - 3x + 1 = 0), we solve for (x^2):[x^2 = 3x - 1]Substitute (x^2 = 3x - 1) into the simplified expression (2x^2 - 6x - 7):[2(3x - 1) - 6x - 7]Simplify the expression:[6x - 2 - 6x - 7 = -9]Thus, the value of the expression is:[boxed{-9}],<think>Okay, so I need to simplify the expression (x - 3)² + (x + 4)(x - 4) given that x² - 3x + 1 = 0. Hmm, let's start by expanding and simplifying the expression first. First, let me expand (x - 3)². That's (x - 3)(x - 3). Using the formula (a - b)² = a² - 2ab + b², that should be x² - 6x + 9. Right? Let me check: x times x is x², x times -3 is -3x, -3 times x is another -3x, and -3 times -3 is +9. So adding those up: x² - 6x + 9. Yeah, that's correct.Next, (x + 4)(x - 4). That's a difference of squares, so it should simplify to x² - 16. Let me verify: x times x is x², x times -4 is -4x, 4 times x is +4x, and 4 times -4 is -16. The middle terms (-4x and +4x) cancel out, leaving x² - 16. Yep, that's right.So now, combining both parts of the original expression: (x² - 6x + 9) + (x² - 16). Let's combine like terms. The x² terms: x² + x² = 2x². The x terms: -6x. The constants: 9 - 16 = -7. So altogether, the simplified expression is 2x² - 6x - 7. Now, the problem says to find the value of this expression given that x² - 3x + 1 = 0. Hmm, so I need to evaluate 2x² - 6x - 7, but I know that x² = 3x - 1 from the given equation. Maybe I can substitute that into the expression to reduce it.Let's see. The expression is 2x² - 6x - 7. If x² = 3x - 1, then 2x² would be 2*(3x - 1) = 6x - 2. So substituting that in: 6x - 2 - 6x - 7. The 6x and -6x terms cancel out, leaving -2 - 7 = -9. Wait, so does that mean the value is -9? Let me check my steps again to make sure I didn't make a mistake.Starting with the original expression:(x - 3)² + (x + 4)(x - 4) expands to (x² - 6x + 9) + (x² - 16) which combines to 2x² - 6x - 7. That seems correct. Then, given x² - 3x + 1 = 0, so x² = 3x - 1. Substitute into 2x²: 2*(3x - 1) = 6x - 2. Then, the entire expression becomes 6x - 2 - 6x -7. The 6x terms cancel, -2 -7 is -9. That seems right.Alternatively, maybe there's another way to check. Let's suppose we don't substitute x² = 3x -1, but instead use the original equation. Let me see. The expression is 2x² -6x -7. If I can express this in terms of the given equation x² -3x +1 =0, which is equal to zero. Let's see.Let me note that 2x² -6x -7 can be written as 2(x² - 3x) -7. From the equation x² -3x +1 =0, so x² -3x = -1. Therefore, substituting into the expression: 2*(-1) -7 = -2 -7 = -9. Oh, that's another way to look at it, and still get -9. So that's consistent. So that seems to confirm that the answer is indeed -9.Alternatively, maybe I can solve for x in the equation x² -3x +1 =0 and plug those values into the original expression. Let me see what the roots are. The quadratic equation x² -3x +1 =0 can be solved using the quadratic formula: x = [3 ± sqrt(9 - 4)] / 2 = [3 ± sqrt(5)] /2. So x = (3 + sqrt(5))/2 or x = (3 - sqrt(5))/2. Then, if I substitute these values into the original expression (x -3)² + (x +4)(x -4), let's compute each part. Let's take x = [3 + sqrt(5)]/2 first. Compute (x -3)^2:x -3 = [3 + sqrt(5)]/2 - 3 = [3 + sqrt(5) -6]/2 = [ -3 + sqrt(5) ] /2. Then squared: [ (-3 + sqrt(5))/2 ]² = (9 -6 sqrt(5) +5)/4 = (14 -6 sqrt(5))/4 = (7 -3 sqrt(5))/2.Next, compute (x +4)(x -4) = x² -16. Since x² =3x -1 from the original equation. So x² -16 = (3x -1) -16 =3x -17. Then, 3x =3*( [3 + sqrt(5)]/2 ) = (9 +3 sqrt(5))/2. Therefore, 3x -17 = (9 +3 sqrt(5))/2 -17 = (9 +3 sqrt(5) -34)/2 = (-25 +3 sqrt(5))/2.So adding the two parts: (7 -3 sqrt(5))/2 + (-25 +3 sqrt(5))/2 = (7 -25 + (-3 sqrt(5) +3 sqrt(5)) )/2 = (-18)/2 = -9. Similarly, if we take the other root x = [3 - sqrt(5)]/2, let's check:First part: (x -3)^2. x -3 = [3 - sqrt(5)]/2 -3 = [3 - sqrt(5) -6]/2 = [ -3 - sqrt(5) ] /2. Squared: [ (-3 - sqrt(5))/2 ]² = (9 +6 sqrt(5) +5)/4 = (14 +6 sqrt(5))/4 = (7 +3 sqrt(5))/2.Second part: (x +4)(x -4)=x² -16. Again, x²=3x -1. So x² -16=3x -17. 3x=3*( [3 - sqrt(5)]/2 )=(9 -3 sqrt(5))/2. So 3x -17=(9 -3 sqrt(5))/2 -17=(9 -3 sqrt(5) -34)/2=(-25 -3 sqrt(5))/2.Adding both parts: (7 +3 sqrt(5))/2 + (-25 -3 sqrt(5))/2 = (7 -25 +3 sqrt(5) -3 sqrt(5))/2=(-18)/2=-9. So both roots give the same result, -9. Therefore, regardless of the value of x satisfying the equation x² -3x +1=0, the expression simplifies to -9. Therefore, the answer is -9. Wait, so all three methods give -9. The substitution of x²=3x -1 into the original expression, factoring out x² -3x and substituting, and even plugging in the actual roots and computing. All lead to -9. So I can be confident that the answer is indeed boxed{-9}. Final Answerboxed{-9}</think>To simplify the expression ((x - 3)^2 + (x + 4)(x - 4)) given that (x^2 - 3x + 1 = 0), we start by expanding and combining like terms:1. Expand ((x - 3)^2): [ (x - 3)^2 = x^2 - 6x + 9 ]2. Expand ((x + 4)(x - 4)): [ (x + 4)(x - 4) = x^2 - 16 ]3. Combine the results: [ (x^2 - 6x + 9) + (x^2 - 16) = 2x^2 - 6x - 7 ]Given the equation (x^2 - 3x + 1 = 0), we know (x^2 = 3x - 1). Substitute (x^2) into the simplified expression:4. Substitute (x^2 = 3x - 1) into (2x^2 - 6x - 7): [ 2x^2 - 6x - 7 = 2(3x - 1) - 6x - 7 = 6x - 2 - 6x - 7 = -9 ]Thus, the value of the expression is (boxed{-9}).]

🔑:To simplify the given expression and then find its value given that x^{2}-3x+1=0, we proceed as follows:First, we simplify the expression left(x-3right)^{2}+left(x+4right)left(x-4right) step by step:1. Expand left(x-3right)^{2}: [ left(x-3right)^{2} = x^{2} - 6x + 9 ]2. Expand left(x+4right)left(x-4right): [ left(x+4right)left(x-4right) = x^{2} - 16 ]3. Add the expanded forms: [ x^{2} - 6x + 9 + x^{2} - 16 = 2x^{2} - 6x - 7 ]Given the equation x^{2}-3x+1=0, we can manipulate it to find a relation that matches part of our simplified expression:4. Rearrange the given equation: [ x^{2} - 3x = -1 ]Now, we substitute x^{2} - 3x = -1 into our simplified expression 2x^{2} - 6x - 7:5. Substitute and simplify: [ 2(x^{2} - 3x) - 7 = 2(-1) - 7 = -2 - 7 = -9 ]Therefore, the value of the given expression, after simplification and substitution, is boxed{-9}.

🧠:[<think>Okay, let's tackle this problem step by step. We have three variables a, b, and c defined by different equations, and we need to compare their values to determine the correct order. Let's break down each one individually.First, let's look at variable a. It's given as a = 24/7. Hmm, 24 divided by 7 is approximately... Well, 7 times 3 is 21, and 24 minus 21 is 3, so 24/7 is 3 and 3/7. Calculating that, 3/7 is approximately 0.4286, so a ≈ 3.4286. Got that.Next up is b, which is defined by the equation b times e to the power of b equals 7 times the natural logarithm of 7. That is, b e^b = 7 ln 7. Hmm, this looks a bit tricky. I know that equations of the form x e^x = y can be solved using the Lambert W function, which is the inverse function of f(x) = x e^x. But I don't remember the exact values or how to compute this without a calculator. Let me think if there's another way to approximate b.First, let's compute the right-hand side: 7 ln 7. Let's recall that ln 7 is approximately... Since ln 2 ≈ 0.693, ln 3 ≈ 1.098, ln 5 ≈ 1.609, and e^2 is about 7.389, so ln 7 is between 1.9 and 2. Let me check: e^1.9 is approximately e^(1.6) * e^(0.3) ≈ 4.953 * 1.349 ≈ 6.68, which is less than 7. e^2 is about 7.389. So ln 7 is between 1.9 and 2. Let's use a calculator approximation. Actually, I remember that ln 7 is approximately 1.9459. So 7 times that is approximately 7 * 1.9459 ≈ 13.6213.So the equation becomes b e^b ≈ 13.6213. Now, we need to solve for b. Let's try some values. Let's start with b=2: 2*e^2 ≈ 2*7.389 ≈ 14.778, which is higher than 13.6213. What about b=1.9: 1.9*e^1.9. Let's approximate e^1.9 first. Since e^1.9 is approximately 6.68 as before. So 1.9*6.68 ≈ 12.692. That's lower than 13.6213. So between 1.9 and 2.Let's try b=1.95: e^1.95. Hmm, e^1.9 is 6.68, e^2 is 7.389. The difference between 1.9 and 2 is 0.1, so each 0.1 increment in the exponent multiplies the result by e^0.1 ≈ 1.105. So from 1.9 to 2 is multiplying by 1.105. Let's do linear approximation. Wait, maybe it's better to approximate e^1.95 as e^(2 - 0.05) = e^2 / e^0.05 ≈ 7.389 / 1.0513 ≈ 7.027. Then, 1.95 * 7.027 ≈ 1.95 * 7 + 1.95 * 0.027 ≈ 13.65 + 0.05265 ≈ 13.70265. That's pretty close to 13.6213. So b ≈ 1.95 gives us about 13.702, which is slightly higher than 13.6213, so maybe b is slightly less than 1.95.Let's try b=1.94. Then, e^1.94. Again, using the same method, e^1.94 = e^(1.9 + 0.04) = e^1.9 * e^0.04 ≈ 6.68 * 1.0408 ≈ 6.68 * 1.04 ≈ 6.9472. Then, 1.94 * 6.9472 ≈ 1.94 * 6 + 1.94 * 0.9472 ≈ 11.64 + 1.84 ≈ 13.48. That's still higher than 13.6213? Wait, wait, no, 13.48 is less than 13.6213. Wait, no: 1.94 * 6.9472 ≈ 13.48. Hmm, but we need 13.6213. So between 1.94 and 1.95.Wait, at b=1.94, we have ~13.48; at b=1.95, we have ~13.70. Our target is 13.6213. Let's find the linear approximation between these two points. The difference between 1.94 and 1.95 is 0.01 in b, leading to an increase from 13.48 to 13.70, which is a difference of about 0.22. We need to cover 13.6213 - 13.48 = 0.1413. So the fraction is 0.1413 / 0.22 ≈ 0.642. Therefore, b ≈ 1.94 + 0.01 * 0.642 ≈ 1.94 + 0.00642 ≈ 1.9464. So approximately 1.9464. Let's check that.Compute e^1.9464. Since 1.9464 is 1.9 + 0.0464. e^0.0464 ≈ 1 + 0.0464 + (0.0464)^2/2 ≈ 1 + 0.0464 + 0.00107 ≈ 1.0475. Therefore, e^1.9464 ≈ e^1.9 * 1.0475 ≈ 6.68 * 1.0475 ≈ 6.68 + 6.68*0.0475 ≈ 6.68 + 0.3173 ≈ 6.9973. Then, 1.9464 * 6.9973 ≈ 1.9464*7 - 1.9464*0.0027 ≈ 13.6248 - 0.00525 ≈ 13.6195. That's very close to 13.6213. So b ≈ 1.9464. Therefore, we can approximate b ≈ 1.946. So b is approximately 1.95. Let's just note that b ≈ 1.95.Now, moving on to c. The equation given is 3^{c-1} = 7 / e. Let's solve for c. First, take the natural logarithm of both sides: ln(3^{c-1}) = ln(7/e). That simplifies to (c - 1) ln 3 = ln 7 - ln e. Since ln e = 1, so (c - 1) ln 3 = ln 7 - 1. Therefore, c - 1 = (ln 7 - 1) / ln 3. Hence, c = 1 + (ln 7 - 1)/ln 3.We can compute this numerically. Let's plug in the approximate values. ln 7 ≈ 1.9459, ln 3 ≈ 1.0986. Therefore, ln 7 - 1 ≈ 0.9459. Then, (ln 7 - 1)/ln 3 ≈ 0.9459 / 1.0986 ≈ 0.861. Therefore, c ≈ 1 + 0.861 ≈ 1.861. So c is approximately 1.861.Now, let's summarize:a ≈ 3.4286b ≈ 1.946c ≈ 1.861So comparing them: a is the largest, followed by b, then c. So the order is a > b > c. Therefore, the correct option is A: a > b > c. Wait, but let me double-check my calculations to make sure I didn't make a mistake.First, a = 24/7. 24 divided by 7: 7*3 = 21, 24 -21 = 3, so 3/7 ≈ 0.4286, so total a ≈ 3.4286. Correct.For b: b e^b = 7 ln7 ≈ 13.6213. We approximated b ≈ 1.946. Let's check again with b = 1.946, e^b ≈ e^1.946. Since e^1.9459 ≈ 7 (Wait, no, e^1.9459 is approximately 7? Wait, e^2 ≈ 7.389, so e^1.9459 is slightly less than e^2. Wait, actually, ln 7 ≈ 1.9459. Wait, hold on! Wait, ln 7 is approximately 1.9459. Wait, so e^{1.9459} = 7. Therefore, if ln 7 is 1.9459, then e^{ln7} =7. Therefore, if b e^b = 7 ln7 ≈ 13.6213. But if b is ln7 ≈1.9459, then e^b =7. Therefore, b e^b = ln7 *7 ≈1.9459*7≈13.6213. Therefore, that's exactly the equation. Therefore, b = ln7 ≈1.9459. Wait, hold on! Wait, the equation is b e^b =7 ln7. But if b = ln7, then e^b =7, so b e^b = ln7 *7, which is exactly 7 ln7. Therefore, b = ln7. Therefore, b is exactly ln7 ≈1.9459. Therefore, I didn't need to approximate it numerically earlier. I could have recognized that if x e^x = k, then x = W(k), but in this case, since k =7 ln7, and recognizing that if we set x = ln7, then x e^x = ln7 * e^{ln7} = ln7 *7 =7 ln7. Therefore, x=ln7 is the solution. Therefore, b=ln7≈1.9459. So my previous approximation was spot on. So no problem there.For c: 3^{c-1}=7/e. So c-1=log_3(7/e). Therefore, c=1 + log_3(7/e). Which is the same as c=1 + (ln(7/e))/ln3=1 + (ln7 -1)/ln3≈1 + (1.9459 -1)/1.0986≈1 +0.9459/1.0986≈1+0.861≈1.861. So that's correct.Therefore, a≈3.4286, b≈1.9459, c≈1.861. So ordering is a > b > c, which is option A.But wait, hold on. Let me check if the problem is maybe presented differently. The options are:A: a > b > cB: a > c > bC: c > b > aD: c > a > bSo according to our calculations, A is correct. But wait, let me check again if I miscalculated c. Let's recalculate c step by step.Given 3^{c-1} = 7/e. Taking natural logs:ln(3^{c-1}) = ln(7/e)Which is (c -1) ln3 = ln7 - lneSince lne=1, so (c -1) ln3 = ln7 -1Therefore, c -1 = (ln7 -1)/ln3Therefore, c =1 + (ln7 -1)/ln3Compute ln7 ≈1.9459, ln3≈1.0986Therefore, ln7 -1≈0.9459Then, 0.9459 divided by 1.0986≈0.861Therefore, c≈1 +0.861≈1.861. Correct. So c≈1.861. So a≈3.4286, b≈1.9459, c≈1.861. Therefore, a > b > c. So option A.Wait, but in the options, option B is a > c > b. But according to our calculations, c≈1.861, which is less than b≈1.9459. So c < b. Therefore, the correct order is a > b > c, which is option A.But let me verify once more because sometimes these problems can have tricks. For example, could there be multiple solutions for b in the equation b e^b =7 ln7?The function f(b) =b e^b is strictly increasing for b >0, since its derivative is e^b +b e^b =e^b (1 +b) >0 for b >0. Therefore, there is only one solution, which is b=ln7, as we found earlier. So b is uniquely ln7≈1.9459.Similarly, for c, solving 3^{c -1}=7/e gives a unique solution. Therefore, c is uniquely determined as≈1.861.Therefore, the order is a >b >c. So the answer should be A.Wait, but let me check again the problem statement to make sure I didn't misread anything."Given a=24/7, be^b=7 ln7, 3^{c-1}=7/e (where e is the base of natural logarithm), then which of the following is true?"Yes, that's correct. The equations are as I solved. So a≈3.4286, b≈1.9459, c≈1.861. So a >b >c. Therefore, the correct answer is A.But wait, just to make sure, maybe the options were misread. The options are:A: a > b > cB: a > c > bC: c > b > aD: c > a > bYes, so A is the correct one.But just to cross-verify, let's compute all three with more precise values.First, a=24/7. Let's compute 24 divided by 7. 7*3=21, 24-21=3, so 3/7=0.4285714285..., so a=3.4285714285...b=ln7. Let's compute ln7. Using more precise value: ln7≈1.9459101490553...c=1 + (ln7 -1)/ln3. Let's compute ln7≈1.9459101490553, ln3≈1.09861228866811.Compute ln7 -1≈0.9459101490553.Divide by ln3: 0.9459101490553 /1.09861228866811≈0.9459101490553 ÷1.09861228866811.Let's compute that division:1.09861228866811 *0.8=0.8788898309344880.9459101490553 -0.878889830934488=0.067020318120812Now, 1.09861228866811 *0.06=0.06591673732008660.067020318120812 -0.0659167373200866≈0.0011035808007254So total so far is 0.8 +0.06=0.86, and the remaining is≈0.0011035808007254 /1.09861228866811≈≈0.001004So total≈0.86 +0.001004≈0.861004. Therefore, c≈1 +0.861004≈1.861004.Therefore, precise values:a≈3.4285714...b≈1.9459101...c≈1.861004...Therefore, a >b >c. So the answer is A.Therefore, the correct option is A.Final Answerboxed{A}</think>Given (a = frac{24}{7}), (be^b = 7 ln 7), and (3^{c-1} = frac{7}{e}), we need to determine the order of (a), (b), and (c).1. Calculating (a): [ a = frac{24}{7} approx 3.4286 ]2. Solving for (b): The equation (be^b = 7 ln 7) can be recognized as a form solved by the Lambert W function. Here, (b = ln 7) because (e^{ln 7} = 7), so: [ b = ln 7 approx 1.9459 ]3. Solving for (c): The equation (3^{c-1} = frac{7}{e}) is solved by taking the natural logarithm of both sides: [ ln(3^{c-1}) = lnleft(frac{7}{e}right) ] Simplifying, we get: [ (c - 1) ln 3 = ln 7 - 1 ] Solving for (c): [ c = 1 + frac{ln 7 - 1}{ln 3} ] Substituting the values (ln 7 approx 1.9459) and (ln 3 approx 1.0986): [ c approx 1 + frac{1.9459 - 1}{1.0986} approx 1 + 0.861 approx 1.861 ]4. Comparing the values: - (a approx 3.4286) - (b approx 1.9459) - (c approx 1.861)Thus, the order is (a > b > c), which corresponds to option A.[boxed{A}],<think>Okay, so I need to figure out the relationships between a, b, and c given the equations provided, and then determine which of the options A to D is correct. Let me start by writing down all the given information clearly.First, a is given as 24/7. Let me calculate that value. 24 divided by 7 is approximately 3.4286. So, a ≈ 3.4286. That's straightforward.Next, the equation for b is be^b = 7 ln 7. Hmm, that looks a bit more complicated. Let me parse that. So, it's b multiplied by e raised to the power of b equals 7 times the natural logarithm of 7. I remember that equations of the form x e^x = k can be solved using the Lambert W function, which is the inverse function of f(x) = x e^x. But I don't remember the exact details of how to apply that here. Maybe I can approximate the value of b numerically?Similarly, for c, we have 3^{c-1} = 7/e. Let's solve that equation for c. Since it's an exponential equation, I can take the logarithm of both sides. Let me do that step by step.Starting with 3^{c - 1} = 7/e. Taking the natural logarithm of both sides gives (c - 1) ln 3 = ln(7/e). Since ln(7/e) is ln 7 - ln e, and ln e is 1, so that's ln 7 - 1. Therefore, (c - 1) ln 3 = ln 7 - 1. Solving for c gives c = [(ln 7 - 1)/ln 3] + 1. Let me compute that value.First, let's compute ln 7. Ln 7 is approximately 1.9459. Then ln 7 - 1 is 1.9459 - 1 = 0.9459. Ln 3 is approximately 1.0986. So, 0.9459 / 1.0986 ≈ 0.861. Then adding 1 gives c ≈ 0.861 + 1 = 1.861. So, c is approximately 1.861.Now, back to b. The equation is b e^b = 7 ln 7. Let's compute the right-hand side first. 7 times ln 7, which is 7 * 1.9459 ≈ 13.6213. So, the equation becomes b e^b ≈ 13.6213. Now, I need to solve for b in this equation. Since this is a transcendental equation, I might need to use numerical methods or approximation techniques.Let me recall that the Lambert W function satisfies W(z) e^{W(z)} = z. So, if we have x e^x = z, then x = W(z). Therefore, in this case, b = W(13.6213). However, I don't remember the exact values for the Lambert W function, and I might not have a calculator that can compute it here. So, perhaps I can approximate the value of b.Alternatively, I can make an educated guess and test values for b to see where b e^b gets close to 13.6213.Let me try b = 2. Then, e^2 ≈ 7.389, so 2 * 7.389 ≈ 14.778, which is higher than 13.6213.Next, try b = 1.8. e^1.8 ≈ e^(1.7918) ≈ 6.05 (Wait, no. Let me compute e^1.8 more accurately. e^1 is 2.718, e^0.8 is approximately 2.2255. So, e^1.8 ≈ e^1 * e^0.8 ≈ 2.718 * 2.2255 ≈ 6.05. Then, 1.8 * 6.05 ≈ 10.89, which is lower than 13.6213.So, at b=1.8, we get 10.89; at b=2, we get 14.778. So, the value of b is between 1.8 and 2. Let's try b=1.9. Compute e^1.9. e^1.9 ≈ e^1 * e^0.9. e^0.9 ≈ 2.4596. Therefore, e^1.9 ≈ 2.718 * 2.4596 ≈ 6.686. Then, 1.9 * 6.686 ≈ 12.7034. That's still lower than 13.6213.So, b=1.9 gives approximately 12.7, which is less than 13.6213. Let's try b=1.95. e^1.95. e^0.95 ≈ 2.585, so e^1.95 ≈ 2.718 * 2.585 ≈ 7.025. Then, 1.95 * 7.025 ≈ 13.699. That's a bit higher than 13.6213. So, between 1.9 and 1.95.So, b=1.9 gives ~12.7, b=1.95 gives ~13.7. Wait, actually, wait a second. Wait, 1.95 * e^1.95. Wait, perhaps my approximation for e^1.95 is off.Wait, actually, perhaps I should compute e^1.95 directly. Let's remember that e^1.95 = e^(2 - 0.05) = e^2 / e^0.05. e^2 ≈ 7.389, e^0.05 ≈ 1.0513. Therefore, e^1.95 ≈ 7.389 / 1.0513 ≈ 7.027. Then, 1.95 * 7.027 ≈ 13.699. So, that's correct.So, at b=1.95, the product is ~13.7, which is a bit higher than 13.6213. So, we need to find a b between 1.9 and 1.95 such that b e^b ≈13.6213.Let me try b=1.93. Let's compute e^1.93. Let's break it down. 1.93 is 1 + 0.93. e^1.93 = e * e^0.93. e^0.93. Let's approximate e^0.9 is 2.4596, e^0.03 is approximately 1.03045. So, e^0.93 ≈ e^0.9 * e^0.03 ≈ 2.4596 * 1.03045 ≈ 2.534. Therefore, e^1.93 ≈ 2.718 * 2.534 ≈ 6.887. Then, 1.93 * 6.887 ≈ 13.303, which is still less than 13.6213.Next, try b=1.94. e^1.94 = e * e^0.94. e^0.94. Let's compute e^0.94. Since e^0.9 is 2.4596, e^0.04 is 1.0408. So, e^0.94 ≈ 2.4596 * 1.0408 ≈ 2.56. Therefore, e^1.94 ≈ 2.718 * 2.56 ≈ 6.96. Then, 1.94 * 6.96 ≈ 13.52. Closer, but still a bit low.Now, b=1.945. Let's compute e^1.945. Similarly, 1.945 = 1 + 0.945. e^0.945. Let's approximate e^0.945. Since e^0.9 is 2.4596, and e^0.045 ≈ 1 + 0.045 + (0.045)^2/2 + ... ≈ 1.046. Therefore, e^0.945 ≈ 2.4596 * 1.046 ≈ 2.573. Then, e^1.945 ≈ 2.718 * 2.573 ≈ 6.99. Then, 1.945 * 6.99 ≈ 13.60. That's very close to 13.6213.So, maybe b≈1.945. Let's check with b=1.947. e^1.947 ≈ e * e^0.947. Let's compute e^0.947. Since e^0.9 is 2.4596, e^0.047 ≈ 1 + 0.047 + (0.047)^2/2 ≈ 1.047 + 0.0011 ≈ 1.0481. Therefore, e^0.947 ≈ 2.4596 * 1.0481 ≈ 2.578. Then, e^1.947 ≈ 2.718 * 2.578 ≈ 7.006. Multiply by 1.947: 1.947 * 7.006 ≈ 13.64. That's very close to 13.6213. So, 1.947 gives approximately 13.64, which is just a bit over. Therefore, the actual value of b is approximately 1.945 to 1.947. Let's take an approximate average and say b≈1.946.So, b≈1.946. Then, we have a≈3.4286, c≈1.861. Therefore, the order from largest to smallest is a > b > c. So, option A: a > b > c. But let me check the options again.Wait, options are:A: a > b > cB: a > c > bC: c > b > aD: c > a > bSo, according to my calculations, a≈3.43, b≈1.946, c≈1.861. Therefore, a is the largest, then b, then c. So, order is a > b > c, which is option A.But wait, let me double-check my calculations for c. Maybe I made a mistake there.Given 3^{c-1} = 7/e. Let's compute 7/e. e≈2.718, so 7 / 2.718 ≈2.575. Therefore, 3^{c-1}=2.575. Taking natural log: (c-1) ln 3 = ln(2.575). Compute ln(2.575). Let's compute ln(2)≈0.6931, ln(2.575)=?Well, ln(2.575). Let's use linear approximation or remember that ln(2.718)=1. So, 2.575 is a bit less than e. Let's compute ln(2.575). Since ln(2.5)≈0.9163, ln(2.6)≈0.9555, so 2.575 is between 2.5 and 2.6. Let's compute ln(2.575):Using the Taylor series around 2.5:Let me denote x=2.5, and Δx=0.075.ln(2.5 + 0.075) ≈ ln(2.5) + (0.075)/2.5 - (0.075)^2/(2*(2.5)^2) + ...But maybe this is too time-consuming. Alternatively, since 2.575 = 2.5 * 1.03. So, ln(2.575) = ln(2.5) + ln(1.03). ln(2.5)≈0.9163, ln(1.03)≈0.02956. So, ln(2.575)≈0.9163 + 0.0296≈0.9459. Which matches the previous calculation. So, (c - 1) ln 3 ≈0.9459. Therefore, c -1≈0.9459 / 1.0986≈0.861, so c≈1.861. So that's correct.Therefore, c≈1.861, which is less than b≈1.946, which is less than a≈3.4286. Therefore, a > b > c, which is option A.Wait, but in the problem statement, the options given are A: a > b > c, B: a > c > b, C: c > b > a, D: c > a > b. So, according to this, A is the correct answer.But wait, hold on. Let me check the value of b again. Because when I approximated b, I found that at b=1.946, the product is approximately 13.62, which matches the right-hand side. So, b≈1.946.But let me verify using a different approach. Maybe using the Lambert W function.The equation is b e^b = 13.6213. So, the solution is b = W(13.6213). The Lambert W function W(z) is defined such that z = W(z) e^{W(z)}. For z > e, the function W(z) is a single-valued function in the principal branch. The value of W(13.6213) can be approximated.I remember that for large z, W(z) ≈ ln z - ln ln z. Let's test this approximation.Compute ln(13.6213) ≈ 2.612. Then ln(ln(13.6213)) ≈ ln(2.612)≈0.96. Therefore, W(13.6213) ≈ 2.612 - 0.96 ≈1.652. But that's not matching our earlier approximation. Wait, maybe this approximation is for z tending to infinity. Maybe for z=13.6, which is not that large, this approximation is not so accurate.Alternatively, maybe I can use iterative method. Let me try Newton-Raphson method to solve b e^b = 13.6213.Let f(b) = b e^b - 13.6213.We need to find root of f(b)=0.f'(b) = e^b + b e^b = e^b (1 + b).Starting with an initial guess. Let's take b0=1.9, as before. f(1.9)=1.9 e^1.9 -13.6213. As calculated earlier, e^1.9≈6.686, so 1.9*6.686≈12.703. 12.703 -13.6213≈-0.9183. f(1.9)= -0.9183.f'(1.9)=e^1.9*(1 + 1.9)=6.686*2.9≈19.3894.Next iteration: b1 = b0 - f(b0)/f'(b0) = 1.9 - (-0.9183)/19.3894≈1.9 + 0.0473≈1.9473.Compute f(1.9473): 1.9473*e^{1.9473} -13.6213.Compute e^{1.9473}. Let's compute it as e^{1.9}*e^{0.0473}. e^{1.9}≈6.686, e^{0.0473}≈1 +0.0473 +0.0473^2/2≈1.0484. So, e^{1.9473}≈6.686 *1.0484≈7.01. Therefore, 1.9473*7.01≈13.66. Then, 13.66 -13.6213≈0.0387. So, f(b1)=0.0387.f'(b1)=e^{1.9473}*(1 +1.9473)≈7.01*(2.9473)≈20.66.Next iteration: b2 = b1 - f(b1)/f'(b1)=1.9473 -0.0387/20.66≈1.9473 -0.00187≈1.9454.Compute f(b2)=1.9454*e^{1.9454} -13.6213.Compute e^{1.9454}≈e^{1.9}*e^{0.0454}≈6.686*1.0465≈6.686*1.04≈6.953. Then, 1.9454*6.953≈13.53. Wait, 1.9454*6.953≈1.9454*7 -1.9454*0.047≈13.6178 -0.0914≈13.5264. So, 13.5264 -13.6213≈-0.0949. Wait, this seems conflicting with the previous step. Maybe my approximations for e^{1.9454} are off.Alternatively, perhaps I need to compute e^{1.9454} more accurately.Wait, 1.9454 can be written as 1.9 + 0.0454. So, e^{1.9454}=e^{1.9} * e^{0.0454}. e^{1.9}≈6.686, and e^{0.0454}=1 +0.0454 +0.0454^2/2 +0.0454^3/6≈1 +0.0454 +0.00103 +0.00003≈1.04646. Therefore, e^{1.9454}=6.686 *1.04646≈6.686 *1.04≈6.953 +6.686*0.00646≈6.953 +0.043≈6.996. Then, 1.9454 *6.996≈1.9454*7 -1.9454*0.004≈13.6178 -0.00778≈13.610. Therefore, 13.610 -13.6213≈-0.0113. So, f(b2)= -0.0113.Compute f'(b2)=e^{1.9454}*(1 +1.9454)=6.996*2.9454≈6.996*2 +6.996*0.9454≈13.992 +6.618≈20.61.Next iteration: b3 = b2 - f(b2)/f'(b2)=1.9454 - (-0.0113)/20.61≈1.9454 +0.000548≈1.9459.Compute f(b3)=1.9459*e^{1.9459} -13.6213.Compute e^{1.9459}=e^{1.9454 +0.0005}=e^{1.9454}*e^{0.0005}≈6.996*(1 +0.0005)≈6.996*1.0005≈6.9995. Then, 1.9459*6.9995≈1.9459*7 -1.9459*0.0005≈13.6213 -0.00097≈13.6203. Therefore, f(b3)=13.6203 -13.6213≈-0.001.f'(b3)=e^{1.9459}*(1 +1.9459)=6.9995*2.9459≈20.61.Next iteration: b4 = b3 - f(b3)/f'(b3)=1.9459 - (-0.001)/20.61≈1.9459 +0.000048≈1.94595.Compute f(b4)=1.94595*e^{1.94595} -13.6213. Similarly, e^{1.94595}≈6.9995*1.00005≈7.000. So, 1.94595*7≈13.62165. Therefore, f(b4)=13.62165 -13.6213≈0.00035.So, we can see that b≈1.94595 gives a value very close to 13.6213. Therefore, b≈1.946.Therefore, confirming that b≈1.946. So, a≈3.4286, b≈1.946, c≈1.861. Therefore, the order is a > b > c. Therefore, the correct answer should be option A.But wait, let me check once again. The problem says "which of the following is true". The options are A, B, C, D. So, according to the calculations, A is correct.But hold on a second. The problem states that be^b=7 ln7. Wait, let's confirm that 7 ln7 is indeed approximately 13.6213. Let's compute 7*ln7. Since ln7≈1.9459, so 7*1.9459≈13.6213. Correct.So, the equation is correctly set up, and the solution for b is approximately 1.946, as we found. Then, c≈1.861. Therefore, a≈3.43, which is the largest, then b≈1.946, then c≈1.861. So, order is a > b > c, which is option A.Therefore, the correct answer is A.Wait, but wait, the options given in the problem: the user listed the options as A: a > b > c, B: a > c > b, C: c > b > a, D: c > a > b.But according to the calculated values, a≈3.43, b≈1.95, c≈1.86. So, a > b > c. Hence, option A is correct.But the original problem might have a typo? Let me check again.Wait, no. The problem states:a = 24/7 ≈3.4286be^b =7 ln7 ≈13.6213, solved for b≈1.9463^{c -1} =7/e≈2.575, solved for c≈1.861So, all steps correct.Therefore, answer is A. But wait, I just checked an online calculator for the equation be^b=13.6213. Let me confirm with Wolfram Alpha.[Imagining accessing Wolfram Alpha]Typing "solve b*e^b = 13.6213" into Wolfram Alpha gives the result b ≈1.945. Which matches our calculation. So, correct.Therefore, the order is a > b > c. Hence, option A.But why is the option B: a > c > b? That would require c > b, but c≈1.86 and b≈1.95, so c < b. So, B is not correct. Therefore, A is correct.Wait, but let me check once again if I miscalculated c. Let me recompute c:Given 3^{c -1} =7/e. Let's compute 7/e ≈7/2.71828≈2.575. So, 3^{c -1}=2.575. Take natural logarithm: (c -1) ln3 = ln(2.575). So, ln(2.575)≈0.9459. Then, c -1≈0.9459 /1.0986≈0.861. Hence, c≈1.861. Correct.Yes, that's accurate. So, c≈1.861. So, indeed, c is less than b≈1.946, which is less than a≈3.4286. Therefore, order is a > b > c. So, answer is A. Therefore, the correct answer is A.But the options given in the problem are A, B, C, D as listed. So, the answer must be A.Wait, but hold on. Wait, in some countries, the options might be labeled differently. Wait, no, the user listed options as A: a > b > c, B: a > c > b, etc. So, as per the user's description, A is a > b > c.Therefore, the conclusion is option A is correct.But wait, I have a doubt. Let me check once again. Is the value of b indeed approximately 1.946? Because 1.946* e^{1.946}≈1.946*7.0≈13.622, which is close to 13.6213. So, yes.Alternatively, if we use the Lambert W function, W(13.6213). Let me check the approximate value of W(13.6213). Since W(z) is increasing, and we know that W(13.6213)≈1.946. For example, W(10)≈2.065, but wait, no. Wait, actually, Lambert W function has different branches, but in the principal branch, W(z) increases with z. Wait, actually, when z increases, W(z) also increases. For example, W(1)=0.567, W(e)=1, W(10)≈1.7455, W(20)≈2.205, W(50)≈2.895. Wait, so according to this, W(13.6213) should be around 2.0? Wait, but that conflicts with our numerical calculation. Hmm, this is confusing.Wait, no. Wait, according to the numerical calculations, when z=13.6213, we found that b≈1.946, but according to the example values I just thought of, W(10)≈1.7455, W(20)≈2.205. So, W(13.6213)≈1.946 seems reasonable, between 1.7455 and 2.205. Therefore, our calculation is correct.Alternatively, maybe the example values I thought of are incorrect. Let me check with Wolfram Alpha again. [Imagining checking]Compute Lambert W(10): Result is approximately 1.7455.Lambert W(13.6213): approximately 1.945. Correct. Therefore, my previous thought that W(10)=2.065 was wrong.Therefore, Lambert W(13.6213)=1.945, which matches our calculation. Therefore, the value of b≈1.945 is correct. Therefore, the order is a > b > c. Therefore, answer is A.But why is there an option B? Maybe a common mistake is to miscalculate b and c? For example, someone might miscalculate c as higher than b. But according to the correct calculation, c≈1.861 and b≈1.946, so b > c.Alternatively, maybe there's a mistake in solving for c? Let me check again.Given 3^{c-1}=7/e. Solve for c. Take natural logarithm:(c -1) ln3 = ln(7) - ln(e) = ln7 -1≈1.9459 -1=0.9459.Then, c -1=0.9459 / ln3≈0.9459 /1.0986≈0.861. Hence, c≈1.861. That's correct.Alternatively, if someone uses logarithm base 3 instead of natural logarithm, they might make a mistake. Let's see:3^{c -1}=7/e. If someone takes log base 3: c -1 = log_3(7/e). Then, c =1 + log_3(7/e). Which is the same as before, because log_3(7/e)=ln(7/e)/ln3=(ln7 -1)/ln3. So, same result. So, that's correct.Alternatively, if someone miscalculates log_3(7/e) as log_3(7) - log_3(e), but that's still correct because log_b(a/c)=log_b a - log_b c. So, correct.Therefore, the calculation for c is correct. Therefore, c≈1.861.Therefore, conclusion remains a > b > c, option A. So, answer is A.Final Answerboxed{A}</think>Given the problem, we need to determine the relationships between (a), (b), and (c) given the following equations:1. (a = frac{24}{7})2. (be^b = 7 ln 7)3. (3^{c-1} = frac{7}{e})Step-by-Step Calculations:1. Value of (a): [ a = frac{24}{7} approx 3.4286 ]2. Value of (b): The equation (be^b = 7 ln 7) can be solved using numerical methods. - Calculate (7 ln 7 approx 13.6213). - Using the Newton-Raphson method, we approximate (b approx 1.946).3. Value of (c): The equation (3^{c-1} = frac{7}{e}) can be solved using logarithms. - Calculate (frac{7}{e} approx 2.575). - Take the natural logarithm of both sides: ((c-1) ln 3 = lnleft(frac{7}{e}right)). - Simplify: (lnleft(frac{7}{e}right) = ln 7 - 1 approx 0.9459). - Solve for (c): (c - 1 = frac{0.9459}{ln 3} approx 0.861), so (c approx 1.861).Comparison:- (a approx 3.4286)- (b approx 1.946)- (c approx 1.861)Ordering:(a > b > c)Thus, the correct answer is (boxed{A}).]

🔑:To solve the given problem, we start by analyzing each of the given equations and expressions to determine the relationships between a, b, and c.Step 1: Solve for bGiven be^b=7ln 7, we take the natural logarithm on both sides to find the value of b:begin{align*}ln(be^b) &= ln(7ln 7) ln b + b &= ln 7 + ln(ln 7)end{align*}Since the function g(x) = x + ln x is monotonically increasing for x > 0, we deduce that b = ln 7.Step 2: Compare a and bGiven a = frac{24}{7}, we compare a and b using two methods:*Method 1:*Define f(x) = ln x - frac{1}{2}left(x - frac{1}{x}right). The derivative f'(x) = -frac{1}{2}left(frac{1}{x} - 1right)^2 leq 0, indicating f(x) is monotonically decreasing. Therefore, f(7) b.*Method 2:*Direct comparison shows a = frac{24}{7} > 2 = ln e^2 > ln 7 = b. Thus, a > b.Step 3: Compare b and cGiven 3^{c-1} = frac{7}{e}, we solve for c:begin{align*}c &= log_3frac{7}{e} + 1end{align*}Comparing b and c, we have:begin{align*}b - c &= ln 7 - (log_3frac{7}{e} + 1) &= lnfrac{7}{e} - log_3frac{7}{e} > 0end{align*}This implies b > c.Conclusion:Since we have established that a > b and b > c, the correct relationship among a, b, and c is a > b > c.Therefore, the correct answer is boxed{A}.

🧠:[<think>Okay, let's see. The question is asking for the symbol of the element with atomic number 14. The options are A: He, B: Li, C: Si, D: Ar. Hmm, I need to remember the periodic table here.First, atomic number refers to the number of protons in an atom's nucleus. Each element has a unique atomic number. So, the element with atomic number 14 must have 14 protons. I need to recall the elements in order up to at least 14.Starting from the beginning: Hydrogen (H) is 1, Helium (He) is 2. Then Lithium (Li) is 3, Beryllium (Be) 4, Boron (B) 5, Carbon (C) 6, Nitrogen (N) 7, Oxygen (O) 8, Fluorine (F) 9, Neon (Ne) 10. Then comes Sodium (Na) 11, Magnesium (Mg) 12, Aluminum (Al) 13, Silicon (Si) 14. Oh wait, so Silicon is number 14. Let me check again to be sure I didn't skip any.Hydrogen (1), Helium (2), Lithium (3), Beryllium (4), Boron (5), Carbon (6), Nitrogen (7), Oxygen (8), Fluorine (9), Neon (10), Sodium (11), Magnesium (12), Aluminum (13), Silicon (14). Yep, that's correct. So atomic number 14 is Silicon, which has the symbol Si. Let's check the options. Option C is Si. So the answer should be C.</think>The atomic number corresponds to the number of protons in an element. Listing elements sequentially:1. H (Hydrogen) 2. He (Helium) 3. Li (Lithium) 4. Be (Beryllium) 5. B (Boron) 6. C (Carbon) 7. N (Nitrogen) 8. O (Oxygen) 9. F (Fluorine) 10. Ne (Neon) 11. Na (Sodium) 12. Mg (Magnesium) 13. Al (Aluminum) 14. Si (Silicon) The element with atomic number 14 is Silicon, whose symbol is Si. Answer: C: Si boxed{C},<think>Okay, let's see. The question is asking for the symbol of the element with atomic number 14. The options are A: He, B: Li, C: Si, D: Ar. Hmm, atomic numbers... I remember that the atomic number corresponds to the number of protons in an atom's nucleus. So each element has a unique atomic number.I need to recall the periodic table. Let me start from the beginning. Atomic number 1 is Hydrogen (H), 2 is Helium (He), which is option A. But that's atomic number 2, so A is out. Then 3 is Lithium (Li), which is option B. But the question is about 14, so B is for 3. Not the answer here. Then 4 is Beryllium (Be), 5 Boron (B), 6 Carbon (C), 7 Nitrogen (N), 8 Oxygen (O), 9 Fluorine (F), 10 Neon (Ne), 11 Sodium (Na), 12 Magnesium (Mg), 13 Aluminum (Al), and 14... what's next? After 13 is 14, which should be Silicon. Silicon's symbol is Si, which is option C. Then 15 is Phosphorus (P), 16 Sulfur (S), 17 Chlorine (Cl), 18 Argon (Ar), which is option D. So D is 18. So atomic number 14 is Silicon, symbol Si, so the correct answer is C. Let me double-check. Starting from the first few elements: H, He, Li, Be, B, C, N, O, F, Ne, Na, Mg, Al, Si. Yeah, that's 14. So yes, Si is correct. The options given have C as Si, which matches. So the answer should be C.</think>The element with atomic number 14 corresponds to its position in the periodic table. Starting from atomic number 1 (Hydrogen, H), we count up: 1. H, 2. He (A), 3. Li (B), 4. Be, 5. B, 6. C, 7. N, 8. O, 9. F, 10. Ne, 11. Na, 12. Mg, 13. Al, 14. Si (C). Options D (Ar) is atomic number 18. Thus, the correct symbol for atomic number 14 is Si. boxed{C}]

🔑:Answer: The element with atomic number 14 is silicon, and its symbol is Si; Therefore, the correct choice is boxed{C}. Analysis: The element with atomic number 14 is silicon. Correctly writing the symbol for silicon provides the solution.

🧠:[<think>Okay, so I have this problem here where I need to find the complex conjugate of z, which is given by z = (2 + i) divided by (1 + i² + i⁵). The options are A through D, and I need to figure out which one is the correct answer. Let me start by recalling some basic concepts about complex numbers and their conjugates.First, the complex conjugate of a complex number a + bi is a - bi. So, if I can find z in the standard form a + bi, then the conjugate should be straightforward. But before that, I need to simplify the expression for z. The denominator here is 1 + i² + i⁵. Let me compute each term step by step.I remember that i is the imaginary unit, where i² = -1. So, substituting that in, the denominator becomes 1 + (-1) + i⁵. Let's compute that: 1 - 1 is 0, so the denominator simplifies to just i⁵. Hmm, okay, so z = (2 + i)/i⁵.Now, I need to compute i⁵. Since i has a cycle of 4: i¹ = i, i² = -1, i³ = -i, i⁴ = 1, i⁵ = i⁴ * i = 1 * i = i. So, i⁵ is just i. Therefore, the denominator simplifies to i. So now z is (2 + i)/i.Dividing by a complex number can be tricky, but I remember that to simplify expressions like this, I can multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of i is -i (since the conjugate of a + bi is a - bi, and here a=0, b=1, so conjugate is -i). Let me try that.So, multiply numerator and denominator by -i:z = [(2 + i) * (-i)] / [i * (-i)]Let me compute the denominator first. i * (-i) = -i². Since i² = -1, this becomes -(-1) = 1. So the denominator is 1, which is nice because that simplifies things.Now the numerator: (2 + i) * (-i). Let's distribute the multiplication:2*(-i) + i*(-i) = -2i - i².But i² is -1, so this becomes -2i - (-1) = -2i + 1. Therefore, the numerator is 1 - 2i.So putting it all together, z = (1 - 2i)/1 = 1 - 2i.Wait, but hold on. The question is asking for the complex conjugate of z, which is denoted as (overline{z}). Since z is 1 - 2i, then its conjugate would be 1 + 2i. So, looking at the options, that's option B.But let me double-check my steps to make sure I didn't make any mistakes. Let's go through each part again.Starting with z = (2 + i)/(1 + i² + i⁵).First, compute the denominator: 1 + i² + i⁵. As before, i² = -1, so that term is -1. Then, i⁵: since i⁴ is 1, i⁵ = i⁴ * i = 1 * i = i. So the denominator is 1 + (-1) + i = 0 + i = i. So denominator is i.Then z = (2 + i)/i. To simplify, multiply numerator and denominator by -i: [(2 + i)(-i)] / [i*(-i)].Denominator: i*(-i) = -i² = -(-1) = 1. Correct.Numerator: (2 + i)(-i) = 2*(-i) + i*(-i) = -2i - i². Then, since i² = -1, that becomes -2i - (-1) = -2i + 1 = 1 - 2i. So z = 1 - 2i. Therefore, the conjugate is 1 + 2i, which is option B. So that's correct.Alternatively, another way to approach this could be to express the original fraction in terms of real and imaginary parts by rationalizing the denominator.Let me try that method to confirm.Starting with z = (2 + i)/i. To rationalize, multiply numerator and denominator by i:z = [(2 + i)*i]/[i*i] = (2i + i²)/i².Compute numerator: 2i + i² = 2i + (-1) = -1 + 2i.Denominator: i² = -1. So z = (-1 + 2i)/(-1) = (1 - 2i)/1 = 1 - 2i. Same result.Therefore, conjugate is 1 + 2i. So option B.But wait, let me check if there was a mistake in the initial simplification of the denominator. The denominator was given as 1 + i² + i⁵. Maybe I should re-express that step.Given: 1 + i² + i⁵.i² is -1, so that's 1 - 1 + i⁵ = 0 + i⁵. Then i⁵ is i^(4 + 1) = (i^4)*(i^1) = 1 * i = i. So yes, denominator is i. That seems right.So everything checks out. Therefore, z is 1 - 2i, conjugate is 1 + 2i, option B.But let me look at the answer choices again. The options are:A: 1 - 2iB: 1 + 2iC: 2 - iD: 2 + iSo the conjugate of z is B: 1 + 2i. Therefore, the answer is B.Wait, but let me just verify once more if z is indeed 1 - 2i. Because if the answer options include 1 - 2i as option A, maybe there's a trick here where the question is asking for z itself and not the conjugate? Wait, no. The question specifically says "Then (overline{z}) = (...)", so we need the conjugate. Since z is 1 - 2i, the conjugate is 1 + 2i, which is B. So that's correct.Alternatively, maybe I made a mistake in the multiplication steps? Let me check.Multiplying (2 + i) by (-i):First term: 2*(-i) = -2i.Second term: i*(-i) = -i² = -(-1) = 1.Therefore, total numerator is -2i + 1 = 1 - 2i. Correct.Denominator: i*(-i) = -i² = 1. Correct.So z = 1 - 2i, conjugate is 1 + 2i. Therefore, answer is B. Therefore, the correct option is B.Alternatively, perhaps the original problem was written differently? Let me check again.Problem statement: Let z = (2 + i)/(1 + i² + i⁵). Then conjugate of z is which of the following.Yes, so steps:1. Compute denominator: 1 + i² + i⁵.2. Simplify denominator: 1 + (-1) + i = 0 + i = i.3. Therefore, z = (2 + i)/i.4. Multiply numerator and denominator by -i: [ (2 + i)(-i) ] / [ i*(-i) ] = ( -2i -i² ) / ( -i² ) = ( -2i - (-1) ) / ( -(-1) ) = ( -2i +1 ) / 1 = 1 - 2i.5. Therefore, z = 1 - 2i.6. Therefore, conjugate of z is 1 + 2i, which is option B.Therefore, the answer is B.Alternatively, maybe I can check using another method. For example, express z in terms of real and imaginary parts and then take the conjugate.Alternatively, use rectangular form:Given z = (2 + i)/i.We can write this as z = (2 + i)/i.Multiply numerator and denominator by i: z = (2i + i²)/i² = (2i -1)/(-1) = (-1 + 2i)/(-1) = 1 - 2i.Same result. So conjugate is 1 + 2i. So yes, B.Alternatively, another way: note that 1/i = -i. Because 1/i = -i since i*(-i) = 1. So 1/i = -i.Therefore, z = (2 + i)/i = (2 + i)*(1/i) = (2 + i)*(-i) = same as before.So that's consistent. So z = -2i -i² = -2i +1 = 1 - 2i. Therefore, conjugate is 1 + 2i. So answer is B.I think I've verified it multiple times. Therefore, the answer should be B.Final Answerboxed{B}</think>Given ( z = frac{2 + i}{1 + i^2 + i^5} ), we need to find the complex conjugate (overline{z}).First, simplify the denominator:- ( i^2 = -1 )- ( i^5 = i^{4+1} = (i^4)(i) = 1 cdot i = i )Thus, the denominator becomes:[ 1 + i^2 + i^5 = 1 + (-1) + i = 0 + i = i ]So, ( z = frac{2 + i}{i} ).To simplify ( frac{2 + i}{i} ), multiply the numerator and the denominator by (-i) (the complex conjugate of (i)):[ z = frac{(2 + i)(-i)}{i(-i)} ]Calculate the denominator:[ i(-i) = -i^2 = -(-1) = 1 ]Calculate the numerator:[ (2 + i)(-i) = 2(-i) + i(-i) = -2i - i^2 = -2i - (-1) = -2i + 1 = 1 - 2i ]Thus, ( z = frac{1 - 2i}{1} = 1 - 2i ).The complex conjugate of ( z = 1 - 2i ) is:[ overline{z} = 1 + 2i ]Therefore, the correct answer is (boxed{B}).,<think>Okay, let's see. I need to find the complex conjugate of z, which is given by z = (2 + i) / (1 + i² + i⁵). The options are A to D, and I need to figure out which one is the correct conjugate. Hmm, complex conjugates. Alright, first things first, maybe simplify the denominator since it has powers of i. Let me recall that i² is -1, i³ is -i, i⁴ is 1, and then i⁵ would be i again because it cycles every four powers. Let me confirm:i¹ = ii² = -1i³ = i² * i = -1 * i = -ii⁴ = i² * i² = (-1)(-1) = 1i⁵ = i⁴ * i = 1 * i = iYes, so i⁵ is i. So substituting into the denominator:1 + i² + i⁵ = 1 + (-1) + i = 1 - 1 + i = 0 + i = iSo the denominator simplifies to i. That makes the expression z = (2 + i) / iHmm, okay, so z is (2 + i) divided by i. Now, complex conjugates. Remember, the conjugate of a complex number a + bi is a - bi. But here, we have a fraction. Maybe I should first simplify z by rationalizing the denominator. Dividing by i can be tricky, but I remember that multiplying numerator and denominator by i can help since i * -i = 1. Wait, let's see:Multiply numerator and denominator by i:z = (2 + i)/i * (i/i) = [(2 + i) * i] / (i * i)Let me compute the numerator and denominator separately.Numerator: (2 + i) * i = 2i + i² = 2i + (-1) = -1 + 2iDenominator: i * i = i² = -1So z = (-1 + 2i)/(-1) = (-1)/(-1) + (2i)/(-1) = 1 - 2iWait, so z is 1 - 2i. Therefore, the conjugate of z, denoted as (overline{z}), should be 1 + 2i. Because you just change the sign of the imaginary part. So if z = a + bi, then (overline{z}) = a - bi. Wait, hold on, z is 1 - 2i, so the conjugate would be 1 + 2i. So looking at the options, that's option B: 1 + 2i. Hmm, let me check if I did everything correctly.Wait, let me go through the steps again. Original z is (2 + i)/(1 + i² + i⁵). Calculated denominator as 1 + (-1) + i = i. Then z is (2 + i)/i. Then multiplied numerator and denominator by i to get [(2 + i)*i]/(i²). Calculated numerator as 2i + i² = 2i -1. Denominator is -1. So z is (-1 + 2i)/(-1) = (1 - 2i). Yes, that's correct. Therefore, conjugate is 1 + 2i. So answer is B.Alternatively, maybe there's another way to compute the conjugate. Remember that the conjugate of a quotient is the quotient of the conjugates. So (overline{left( frac{a + bi}{c + di} right)} = frac{a - bi}{c - di}). So maybe if I first compute the conjugate of the numerator and denominator separately.Original z is (2 + i)/i. The conjugate of z would be conjugate of numerator divided by conjugate of denominator. The numerator is 2 + i, conjugate is 2 - i. The denominator is i, whose conjugate is -i. Therefore, (overline{z}) = (2 - i)/(-i). Let me compute that.Multiply numerator and denominator by i to rationalize:(2 - i)/(-i) * (i/i) = [(2 - i)i]/[-i * i] = [2i - i²]/[-i²] = [2i - (-1)]/[-(-1)] = (2i + 1)/1 = 1 + 2i. Same result. So that's again option B.So both methods give the same answer. So I think B is correct. Let me check the options again to make sure:A: 1−2iB: 1+2iC: 2−iD: 2+iYes, B is 1 + 2i, which is what I got. So the answer is B.But just to be thorough, let me double-check. Maybe I made a mistake in simplifying the denominator. The original denominator is 1 + i² + i⁵. Let me recompute that:i² is -1, i⁵ is i^(4+1) = (i⁴)*i = 1*i = i. So 1 + (-1) + i = 0 + i = i. So denominator is indeed i. Then numerator is 2 + i. So z is (2 + i)/i. If I write that as a complex number, let me represent it in standard form. Let me do it another way. Let's write z = (2 + i)/i. Let me split this into two terms: 2/i + i/i. That's 2/i + 1. But 2/i can be written as -2i because 1/i = -i. Because i * -i = 1. So 2/i = 2*(-i) = -2i. Then z = -2i + 1, which is 1 - 2i. Same result as before. Therefore, conjugate is 1 + 2i. Yep, B.Alternatively, maybe I can represent z in rectangular form and then take conjugate. Since z = 1 - 2i, then (overline{z}) is 1 + 2i. So answer is B.Alternatively, is there a chance I messed up the denominator? Let me check. Original denominator: 1 + i² + i⁵. Wait, is that 1 + i² + i^5? Yes. So 1 + (-1) + i, which is 0 + i. So denominator is i. So that's correct. So z is (2 + i)/i. Then, when I multiplied numerator and denominator by i, I get [ (2 + i)i ] / (i * i ) = [2i + i²]/(-1) = [2i -1]/(-1) = (-1 + 2i)/(-1) = 1 - 2i. So that's correct.Therefore, the conjugate is 1 + 2i, which is option B.Just to make sure, let me check if option B is correct, maybe I have a sign error. Wait, z is 1 - 2i, so conjugate is 1 + 2i. That's B. So yes. Therefore, answer is B.Final Answerboxed{B}</think>Given ( z = frac{2 + i}{1 + i^2 + i^5} ), we need to find the complex conjugate (overline{z}).First, simplify the denominator:- ( i^2 = -1 )- ( i^5 = i^{4+1} = (i^4) cdot i = 1 cdot i = i )Thus, the denominator becomes:[ 1 + i^2 + i^5 = 1 + (-1) + i = 0 + i = i ]Now, substitute back into the expression for ( z ):[ z = frac{2 + i}{i} ]To simplify ( frac{2 + i}{i} ), multiply the numerator and denominator by ( i ):[ z = frac{(2 + i) cdot i}{i cdot i} = frac{2i + i^2}{i^2} = frac{2i - 1}{-1} = frac{-1 + 2i}{-1} = 1 - 2i ]The complex conjugate of ( z = 1 - 2i ) is obtained by changing the sign of the imaginary part:[ overline{z} = 1 + 2i ]Thus, the correct answer is (boxed{B}).]

🔑:Given z=frac{{2+i}}{{1+{i^2}+{i^5}}}, we need to find overline{z}.First, let's simplify the denominator using the properties of i:- We know that i^2 = -1.- For i^5, since i^5 = i^4 cdot i = 1 cdot i = i (because i^4 = 1).Thus, we can rewrite z as follows:[z = frac{2+i}{1+i^2+i^5} = frac{2+i}{1-1+i} = frac{2+i}{i}]To simplify frac{2+i}{i}, we multiply the numerator and the denominator by the complex conjugate of the denominator:[z = frac{2+i}{i} cdot frac{-i}{-i} = frac{-2i + i^2}{-i^2} = frac{-2i - 1}{1} = -1 - 2i]Therefore, the complex conjugate of z, denoted as overline{z}, is obtained by changing the sign of the imaginary part:[overline{z} = -1 + 2i]However, there seems to be a mistake in the simplification process. Correcting it, we should have:[z = frac{2+i}{i} = 2cdotfrac{1}{i} + frac{i}{i} = 2cdotfrac{1}{i} + 1 = 2left(-iright) + 1 = -2i + 1]Thus, the correct overline{z} is:[overline{z} = 1 + 2i]Therefore, the correct choice is boxed{text{B}}.

🧠:[<think>Okay, so I need to find the equation of a line that passes through the point (-1, 2) and is perpendicular to the line given by 2x - 3y + 4 = 0. Then, among the options A to D, I have to pick the correct one. Let me think step by step how to approach this.First, I remember that two lines are perpendicular if the product of their slopes is -1. So, if I can find the slope of the given line 2x - 3y + 4 = 0, then I can determine the slope of the line that's perpendicular to it. Once I have the slope of the perpendicular line and a point it passes through, I can use the point-slope form to write its equation. Finally, I'll compare that equation with the options provided to see which one matches.Let me start by finding the slope of the given line 2x - 3y + 4 = 0. To do that, maybe I should rewrite the equation in slope-intercept form (y = mx + b), where m is the slope. Let's see:Starting with 2x - 3y + 4 = 0, I can rearrange this to solve for y.Subtract 2x and 4 from both sides:-3y = -2x - 4Now, divide every term by -3 to get y by itself:y = (-2/-3)x - (4/-3)Simplifying the signs:y = (2/3)x + 4/3So, the slope (m) of the given line is 2/3. Therefore, the slope of the line perpendicular to this one should be the negative reciprocal of 2/3. The negative reciprocal of 2/3 is -3/2. Let me confirm that: 2/3 multiplied by -3/2 is indeed -1, which satisfies the perpendicular condition. So, the slope of line l is -3/2.Now, I need to find the equation of the line with slope -3/2 that passes through the point (-1, 2). For this, I can use the point-slope form of a line's equation, which is:y - y1 = m(x - x1)Where (x1, y1) is the point the line passes through, and m is the slope. Plugging in the values:y - 2 = (-3/2)(x - (-1))Simplify inside the parentheses:y - 2 = (-3/2)(x + 1)Now, let's distribute the -3/2 on the right-hand side:y - 2 = (-3/2)x - 3/2To eliminate the fraction, maybe I can multiply every term by 2 to make the coefficients integers. Let's do that step by step.First, add 2 to both sides to solve for y:y = (-3/2)x - 3/2 + 2But 2 is the same as 4/2, so:y = (-3/2)x - 3/2 + 4/2Combine the constants:y = (-3/2)x + 1/2Alternatively, if I want to write this in standard form (Ax + By + C = 0), I can rearrange the terms. Let me try that.Starting from the point-slope form:y - 2 = (-3/2)(x + 1)Multiply both sides by 2 to eliminate the denominator:2(y - 2) = -3(x + 1)Expand both sides:2y - 4 = -3x - 3Now, bring all terms to the left side:3x + 2y - 4 + 3 = 0Simplify:3x + 2y - 1 = 0So, the equation of line l is 3x + 2y - 1 = 0. Comparing this with the options given:A: 3x + 2y -1=0B: 3x +2y +7=0C: 2x-3y+5=0D: 2x-3y+8=0It looks like option A matches exactly. But wait, let me double-check my steps to make sure I didn't make a mistake.First, converting the original line to slope-intercept: yes, 2x -3y +4=0 becomes y=(2/3)x + 4/3. So slope 2/3. Negative reciprocal is -3/2. Correct.Then using point-slope: y -2 = -3/2(x +1). Correct.Multiplying both sides by 2: 2y -4 = -3x -3. Then bringing all terms to left: 3x +2y -1=0. That's correct.So the answer should be A. However, let me check if I didn't mix up the options. The options are A: 3x+2y-1=0, which matches. Option B is 3x+2y+7=0. That's similar but different constant term. The other options C and D have coefficients 2x-3y... which would have the same slope as the original line, not perpendicular. Because original line was 2x -3y +4=0, so lines C and D have the same coefficients for x and y, meaning same slope. Therefore, they can't be perpendicular. So C and D are parallel to the original line, not perpendicular, so they can be eliminated.Therefore, the answer is A. But wait, let me check the calculation once more because sometimes when you move terms, signs can flip.Original steps:Starting with point-slope: y -2 = -3/2(x +1)Multiply both sides by 2: 2y -4 = -3x -3Bring all terms to left: 3x +2y -4 +3 =0 → 3x +2y -1=0. Correct.Yes, that's right. So equation is 3x +2y -1=0, which is option A.But let me check if the point (-1,2) satisfies this equation.Plug x=-1, y=2 into 3x +2y -1:3*(-1) + 2*(2) -1 = -3 +4 -1=0. Yes, it equals 0. So the point lies on the line. So that's correct.Alternatively, check with another method. Suppose I use the slope-intercept form.We had y = (-3/2)x +1/2. Let's check if (-1,2) satisfies this.Left side y=2.Right side: (-3/2)*(-1) +1/2 = 3/2 +1/2 = 4/2=2. Correct. So that works.Therefore, the equation is correct. So the answer is A.But wait, looking back at the options, A is 3x +2y -1=0, which is correct. However, sometimes different forms can be equivalent but written differently. Let me check if any of the other options could be equivalent, but probably not. For example, option B is 3x+2y +7=0. If we plug (-1,2) into this:3*(-1) +2*(2) +7= -3 +4 +7=8≠0. So the point (-1,2) is not on line B. Therefore, B is out. C and D as discussed are parallel. So the answer is A.Wait, but the system says "put your final answer within boxed{}". So I need to make sure. Wait, but in the original problem, the options are labeled A to D, and the answer is A. Let me just confirm once again.Original problem: line passes through (-1,2), perpendicular to 2x -3y +4=0.Slope of given line is 2/3, so perpendicular slope is -3/2. Equation using point-slope: y -2 = -3/2(x +1). Converting to standard form: 3x +2y -1=0. Correct. Option A is 3x+2y -1=0. So yes, correct.I think that's solid. So the answer is A.Final Answerboxed{A}</think>To find the equation of the line ( l ) that passes through the point ((-1, 2)) and is perpendicular to the line (2x - 3y + 4 = 0):1. Find the slope of the given line: - Convert (2x - 3y + 4 = 0) to slope-intercept form: [ 2x - 3y + 4 = 0 implies -3y = -2x - 4 implies y = frac{2}{3}x + frac{4}{3} ] - The slope of the given line is (frac{2}{3}).2. Determine the slope of the perpendicular line: - The slope of a line perpendicular to the given line is the negative reciprocal of (frac{2}{3}), which is (-frac{3}{2}).3. Use the point-slope form to find the equation of line ( l ): - Using the point ((-1, 2)) and slope (-frac{3}{2}): [ y - 2 = -frac{3}{2}(x + 1) ] - Multiply through by 2 to clear the fraction: [ 2(y - 2) = -3(x + 1) implies 2y - 4 = -3x - 3 ] - Rearrange to standard form: [ 3x + 2y - 1 = 0 ]4. Verify the solution: - Check that the point ((-1, 2)) satisfies the equation (3x + 2y - 1 = 0): [ 3(-1) + 2(2) - 1 = -3 + 4 - 1 = 0 ] - The equation is correct and matches option A.Thus, the equation of line ( l ) is (boxed{A}).,<think>Okay, let's see. I need to find the equation of a line that passes through the point (-1, 2) and is perpendicular to the line 2x - 3y + 4 = 0. Then check which of the options A, B, C, D matches that. Hmm.First, I remember that two lines are perpendicular if the product of their slopes is -1. So, I need to find the slope of the given line first, then determine the slope of the line we're looking for. After that, use the point-slope form to write the equation and see which option it matches.Let me start by finding the slope of the line 2x - 3y + 4 = 0. To do that, maybe I should rewrite it in slope-intercept form (y = mx + b). Let's solve for y.Starting with 2x - 3y + 4 = 0.Subtract 2x and 4 from both sides: -3y = -2x - 4.Divide both sides by -3: y = (2/3)x + 4/3.So the slope (m) of this line is 2/3. Therefore, the slope of the line perpendicular to it should be the negative reciprocal, which is -3/2. Wait, because perpendicular slopes multiply to -1: (2/3) * (-3/2) = -1. Yep, that checks out.So, the line l we need has a slope of -3/2 and passes through (-1, 2). Let me write the equation using point-slope form. Point-slope is y - y1 = m(x - x1). Plugging in the values:y - 2 = (-3/2)(x - (-1)) y - 2 = (-3/2)(x + 1)Now, let's simplify this equation to standard form (Ax + By + C = 0). Multiply both sides by 2 to eliminate the fraction:2(y - 2) = -3(x + 1) 2y - 4 = -3x - 3Bring all terms to the left side:3x + 2y - 4 + 3 = 0 3x + 2y - 1 = 0Hmm, so that's 3x + 2y - 1 = 0. Let me check the options. Option A is 3x + 2y -1=0. So that seems right. But wait, let me double-check my steps to make sure I didn't make a mistake.Original slope: 2/3. Perpendicular slope: -3/2. Point (-1,2). Point-slope form: y - 2 = -3/2 (x +1). Multiply by 2: 2y -4 = -3x -3. Then 3x +2y -1=0. Yes, that's correct. So the answer should be A.But wait, let me check the options again. Option B is 3x +2y +7=0. Maybe I made a sign mistake? Let me check the steps again.Starting from 2y -4 = -3x -3. Then bringing all terms to the left:3x +2y -4 +3=0 3x +2y -1=0. Yes, that's correct. So option A is 3x +2y -1=0, which matches. So why is there an option B with 3x +2y +7=0? Maybe a miscalculation?Alternatively, maybe I made a mistake in the sign when moving terms. Let me check:Original equation after multiplying by 2: 2y -4 = -3x -3. Then add 3x to both sides: 3x +2y -4 = -3. Then add 3 to both sides: 3x +2y -1 =0. That's correct. So no, the correct answer should be A. Hmm. Wait, but maybe the question is written differently?Wait, the original line is 2x -3y +4=0. So when I converted it to slope-intercept form, I had y = (2/3)x +4/3. That seems correct. Then the perpendicular slope is -3/2, correct. Then using point (-1,2):y -2 = -3/2 (x +1). Let's compute that:y = -3/2 x -3/2 +2. Convert 2 to 4/2, so y = -3/2 x + ( -3/2 +4/2 ) = -3/2 x +1/2.If I rearrange that to standard form: 3/2 x + y -1/2=0. Multiply both sides by 2: 3x +2y -1=0. Yes, same as before. So that's option A. So why is there another option with 3x +2y +7=0?Wait, maybe the question is about line l passing through (-1,2), but perhaps I miscalculated the coordinates? Let me check that.If we plug the point (-1,2) into option A: 3*(-1) +2*(2) -1 = -3 +4 -1=0. Yes, 0. So that point satisfies the equation. So that's correct. So option A is valid.If we check option B: 3*(-1)+2*(2)+7= -3 +4 +7=8 ≠0. So point (-1,2) is not on line B. So B is not the answer. So A must be correct.But wait, in the options given, the answer is A? But let me check again if perhaps the question is different. Wait, maybe I messed up the slope?Original line: 2x -3y +4=0. So slope is 2/3. Perpendicular slope is -3/2. Correct. Then using point (-1,2):y -2 = -3/2(x +1). Then, expanding:y = -3/2 x -3/2 +2. 2 is 4/2, so 4/2 -3/2=1/2. So y= -3/2 x +1/2. Multiply by 2: 2y = -3x +1 → 3x +2y -1=0. So yes, A is correct.But wait, let me check if there's another way someone might make a mistake here. For example, if someone incorrectly found the slope as 3/2 instead of -3/2, but then the answer would be different. But no, the slope is definitely -3/2.Alternatively, maybe someone mixed up the coefficients when converting to standard form? Let me see. If the slope is -3/2, then the equation is y = -3/2 x +1/2. To write in standard form:3/2 x + y =1/2. Multiply by 2: 3x +2y =1. Then, 3x +2y -1=0. So that's correct. So A is right.Alternatively, maybe the question is about being perpendicular, but maybe the direction is different? Wait, no. The line l is perpendicular to the given line, so slope -3/2. Correct.Wait, the options include C and D which have coefficients 2x -3y. So those lines have slope 2/3, same as the original line, so they are parallel, not perpendicular. So C and D can be eliminated immediately. The answer must be between A and B.Since B does not pass through (-1,2), as we checked earlier, the answer is A.Therefore, the correct answer is A: 3x+2y-1=0.Final Answerboxed{A}</think>To find the equation of the line ( l ) that passes through the point ((-1, 2)) and is perpendicular to the line (2x - 3y + 4 = 0):1. Find the slope of the given line: - Convert (2x - 3y + 4 = 0) to slope-intercept form: [ 2x - 3y + 4 = 0 implies -3y = -2x - 4 implies y = frac{2}{3}x + frac{4}{3} ] - The slope ( m ) of the given line is (frac{2}{3}).2. Determine the slope of the perpendicular line: - The slope of a line perpendicular to the given line is the negative reciprocal of (frac{2}{3}), which is (-frac{3}{2}).3. Use the point-slope form to find the equation of line ( l ): - Using the point ((-1, 2)) and slope (-frac{3}{2}): [ y - 2 = -frac{3}{2}(x + 1) ] - Simplify to standard form: [ y - 2 = -frac{3}{2}x - frac{3}{2} implies 2(y - 2) = -3(x + 1) implies 2y - 4 = -3x - 3 implies 3x + 2y - 1 = 0 ]4. Verify the equation with the given options: - The equation (3x + 2y - 1 = 0) matches option A. - Check if the point ((-1, 2)) satisfies option A: [ 3(-1) + 2(2) - 1 = -3 + 4 - 1 = 0 ] - The point satisfies the equation, confirming the correct answer.Thus, the correct answer is (boxed{A}).]

🔑:Given that line l is perpendicular to the line 2x-3y+4=0, we know that the slope of line l is the negative reciprocal of the slope of 2x-3y+4=0. The slope of 2x-3y+4=0 is frac{2}{3}, so the slope of line l is -frac{3}{2}.Since line l passes through the point (-1,2) and has a slope of -frac{3}{2}, its equation can be written as:[y - 2 = -frac{3}{2}(x + 1)]Expanding and rearranging gives:[y = -frac{3}{2}x - frac{1}{2}][2y = -3x - 1][3x + 2y - 1 = 0]Therefore, the correct answer is boxed{text{A}}.

🧠:[<think>Okay, let's see. I need to figure out which of these numbers isn't rational. The options are A through E. Let me go through each one step by step.First, option A is -2002. Well, that's a whole number, right? Negative, but still an integer. And integers are definitely rational numbers because they can be expressed as a fraction over 1. So, -2002 is rational. So A is rational. Got that.Moving on to option B: 8^(2/3). Hmm, fractional exponents. Let me remember. The exponent 2/3 can be broken down into taking the cube root first and then squaring it, right? So 8^(1/3) is the cube root of 8. The cube root of 8 is 2 because 2^3 is 8. Then we square that: 2^2 is 4. So 8^(2/3) is 4. That's an integer, which is rational. So B is rational. Okay, so B is out.Option C is sqrt(0.49). The square root of 0.49. Let me think. 0.49 is like 49/100, right? So sqrt(49/100) is sqrt(49)/sqrt(100) which is 7/10. 7/10 is 0.7, which is a terminating decimal. Terminating decimals are rational numbers. So C is 0.7, which is rational. So C is okay.Option D: 100^0.5. 0.5 is the same as 1/2, so that's the square root of 100. Square root of 100 is 10, which is an integer, so that's definitely rational. D is 10, so D is rational. Got it.Now, option E: 1000^(0.1). Let's parse that. 0.1 is the same as 1/10. So this is the 10th root of 1000. So 1000^(1/10). Hmm. Let me think. What's the 10th root of 1000? Let's see. 1000 is 10^3. So 10^3 raised to the 1/10 is 10^(3/10). Which is 10^0.3. But that's not a fraction with integer numerator and denominator. Let's see if there's a way to express 1000^(1/10) as a rational number.Alternatively, maybe we can write it as some integer or fraction. Let's see. Suppose 1000^(1/10) equals x. Then x^10 = 1000. Is there a rational number x that satisfies this equation? Let's check. Suppose x is a rational number, so x = p/q where p and q are integers with no common factors. Then (p/q)^10 = 1000 = 10^3. So (p^10)/(q^10) = 10^3. Therefore, p^10 = 10^3 * q^10. Let's see. But 10^3 is 2^3 * 5^3. So p^10 must be divisible by 2^3 and 5^3, and q^10 must also be such that when you divide, you get 10^3. But since p and q are coprime, then q^10 must be 1, because if q had any prime factors, they would have to divide into p^10, which they can't because p and q are coprime. Therefore, q=1. So p^10 = 10^3. But 10^3 is 1000. So p^10 = 1000. But 1000 is not a perfect 10th power. Let's check. 2^10 is 1024, which is bigger than 1000. 1^10 is 1, 2^10 is 1024. So there's no integer p such that p^10 is 1000. Therefore, there's no rational number x that satisfies x^10 = 1000. Therefore, 1000^(1/10) is irrational. Therefore, E is irrational.Let me confirm. So E is 1000^(0.1) which is the 10th root of 1000. If 1000 was a perfect 10th power, then it would be rational, but since it isn't, it's irrational. So E is the answer.Wait, but let me check again for E. 1000 is 10^3. So 1000^(1/10) = (10^3)^(1/10) = 10^(3/10). 3/10 is 0.3. So 10^0.3. Is 10^0.3 a rational number? Let's see. Suppose 10^0.3 is equal to a/b. Then (a/b)^10 = 10^3. Which we already saw implies that a^10 = 10^3 b^10. But since 10^3 = 2^3*5^3, then the left side must have exponents of 2 and 5 that are multiples of 10, but the right side has exponents 3 and 3. That's a contradiction because 3 isn't a multiple of 10. Therefore, there can be no such integers a and b, so it's irrational.So yes, E is irrational. Therefore, the answer is E.Final Answerboxed{E}</think>To determine which number is not rational, we analyze each option:- Option A: -2002 is an integer, which is rational.- Option B: (8^{frac{2}{3}}). The cube root of 8 is 2, and squaring 2 gives 4, which is an integer (rational).- Option C: (sqrt{0.49}). This simplifies to (sqrt{frac{49}{100}} = frac{7}{10}), a terminating decimal (rational).- Option D: (100^{0.5}). The square root of 100 is 10, an integer (rational).- Option E: (1000^{0.1}). This is the 10th root of 1000. Since (1000 = 10^3), we have (1000^{0.1} = (10^3)^{1/10} = 10^{3/10}). There is no rational number (x) such that (x^{10} = 1000), as shown by the contradiction in prime factorization. Therefore, (1000^{0.1}) is irrational.Thus, the number that is not rational is (boxed{E}).,<think>Alright, let me try to figure out which of these numbers isn't rational. The options are A through E. So first, I need to recall what a rational number is. A rational number is any number that can be expressed as a fraction a/b where a and b are integers and b is not zero. That includes integers, fractions, repeating decimals, terminating decimals, etc. An irrational number can't be expressed that way; its decimal goes on forever without repeating.Let me check each option one by one.Starting with option A: -2002. That's definitely an integer. All integers are rational because they can be written as themselves over 1. So -2002 is rational. So A is rational, not the answer.Option B: 8^(2/3). Hmm. Exponents with fractional powers can sometimes be tricky. Let me remember that 8^(2/3) is the same as taking the cube root of 8 first and then squaring it. The cube root of 8 is 2 because 2^3 is 8. Then squaring 2 gives 4. 4 is an integer, so that's rational. Alternatively, 8^(2/3) is the same as (8^2)^(1/3) which is 64^(1/3). The cube root of 64 is also 4. So either way, it's 4, which is rational. So B is rational.Option C: sqrt(0.49). Let's see. The square root of 0.49. Well, 0.49 is 49/100. So sqrt(49/100) is sqrt(49)/sqrt(100) which is 7/10. That's 0.7, which is a terminating decimal, so definitely rational. So C is rational.Option D: 100^0.5. 0.5 as an exponent is the same as square root. So sqrt(100) is 10. 10 is an integer, so that's rational. D is rational.Option E: 1000^0.1. Hmm, 0.1 exponent is the same as the 10th root. So 1000^(1/10). Let me think. 1000 is 10^3. So (10^3)^(1/10) is 10^(3/10). So that's 10^(0.3). Hmm, 10 raised to the 0.3 power. Is that a rational number?Wait, 10^(3/10) is the same as the 10th root of 10^3. So another way to write that is 10^(3/10). Let me see if this simplifies. 3 and 10 are coprime, so there's no simplification here. So 10^(3/10) is the same as e^( (3/10) ln 10 ) or something, but that's not helpful. Alternatively, maybe 10^(3/10) is irrational?Let me check. Let's see, 10^(1/10) is the 10th root of 10. Is that irrational? Similarly, 10^(3/10) would be the same as (10^(1/10))^3. If 10^(1/10) is irrational, then so is its cube. Since 10 is not a perfect 10th power. Let me verify. The 10th root of 10. Let's see, 2^10 is 1024, which is bigger than 10. 1^10 is 1. So the 10th root of 10 is between 1 and 2, but not an integer. So unless it can be expressed as a fraction, it's irrational. Suppose we assume that 10^(1/10) is irrational. Then 10^(3/10) is also irrational because it's just an integer power of an irrational number. Wait, but even if you raise an irrational number to an integer power, it might still be irrational, but not necessarily. For example, sqrt(2) is irrational, sqrt(2)^2 is 2, which is rational. But in this case, 10^(3/10) is equivalent to 10^(0.3). Is there a way to write this as a fraction?Alternatively, maybe 1000^(1/10) can be simplified. 1000 is 10^3, so 1000^(1/10) is (10^3)^(1/10) = 10^(3/10). Which is the same as before. Alternatively, is 10^(3/10) equal to 2 or something? Let me compute 10^(0.3). Let me approximate. 10^0.3. 10^0.3 is approximately 2, because 10^0.3010 is approximately 2 (since log10(2) ≈ 0.3010). Wait, but 10^0.3010 ≈ 2. So 10^0.3 is a little less than 2. But regardless, it's approximately 1.995... But that's just an approximate value. The exact value is 10^(3/10). Is that irrational?Alternatively, let me check if 10^(3/10) can be expressed as a fraction. Suppose it can. Then 10^(3/10) = a/b where a and b are integers. Then, raising both sides to the 10th power gives 10^3 = a^10 / b^10, so 1000 = (a/b)^10, meaning (a/b) is a 10th root of 1000. But 1000 is 10^3, so (a/b)^10 = 10^3. If a/b is in simplest terms, then a^10 and b^10 must be integers. But 10^3 is 1000. So a^10 / b^10 = 1000. Therefore, a^10 = 1000 b^10. Let's see, 1000 is 2^3 * 5^3. So a^10 must be 2^3 *5^3 * b^10. But for a and b integers, the exponents on the primes in their factorizations must be multiples of 10 on the left-hand side (since it's a^10). But 2^3 *5^3 is 10^3, so unless b^10 has factors of 2^7 and 5^7, then a^10 would have exponents 3 + 10k, which are not multiples of 10. So unless b has fractions with denominators, but since a and b are integers, this can't be. Therefore, there are no integers a and b such that a^10 / b^10 = 1000, so 1000^(1/10) is irrational. Therefore, E is irrational.Wait, but let me make sure. Suppose someone thinks that 10^(3/10) is equal to some fraction. But unless 10^(1/10) is rational, which we know it's not, because 10 is not a perfect 10th power, then 10^(3/10) is irrational. So, since 10 is not a perfect 10th power, its 10th root is irrational, so raising it to the 3rd power would still keep it irrational. So E is irrational.Therefore, the answer is E.But let me double-check the other options again to make sure I didn't make a mistake.A is -2002, which is definitely rational.B: 8^(2/3). 8 is 2^3, so (2^3)^(2/3) = 2^(3*(2/3)) = 2^2 = 4. So yes, 4, rational.C: sqrt(0.49). 0.49 is 49/100, sqrt(49/100) = 7/10, which is 0.7. Rational.D: 100^0.5. That's sqrt(100) = 10. Rational.E: 1000^0.1. As we discussed, which is 10^(3/10), which is irrational. So yes, E is the irrational one. So the answer is E.Final Answerboxed{E}</think>To determine which number is not rational, we evaluate each option:- Option A: -2002 is an integer, hence rational.- Option B: (8^{frac{2}{3}}). This can be simplified as the cube root of 8 squared: (sqrt[3]{8^2} = sqrt[3]{64} = 4), which is rational.- Option C: (sqrt{0.49}). Since (0.49 = frac{49}{100}), we have (sqrt{frac{49}{100}} = frac{7}{10}), which is rational.- Option D: (100^{0.5}). This is the square root of 100, which is 10, hence rational.- Option E: (1000^{0.1}). This is equivalent to (1000^{frac{1}{10}}), the 10th root of 1000. Since (1000 = 10^3), we have ((10^3)^{frac{1}{10}} = 10^{frac{3}{10}}). The expression (10^{frac{3}{10}}) is not a perfect power and cannot be simplified to a fraction, making it irrational.Thus, the number that is not rational is (boxed{E}).]

🔑:8) The answer is (mathbf{E}). Indeed, 1000^{0.1}=sqrt[10]{1000}, which is not rational. It is also verified that -2002 is even an integer, 8^{frac{2}{3}}=sqrt[3]{64}=4 which is an integer, sqrt{0.49}=0.7 which is rational, and 100^{0.5}=sqrt{100}=10.