Find the point of intersection of two lines and the number of ways to arrange girls on a bench with a condition. The final answer is B.

You're on a roll — Step 1: Solve Question 24. The two lines are given by: r_1 = 3i - j + k + (i + 2j - k) r_2 = 4i + 4j + k + (-i + j + 2k) For the lines to intersect, r_1 = r_2. Equating the components: For i: 3 + = 4 - (1) For j: -1 + 2 = 4 + (2) For k: 1 - = 1 + 2 (3) From equation (1), + = 1. From equation (2), 2 - = 5. Add these two equations: ( + ) + (2 - ) = 1 + 5 3 = 6 = 2 Substitute = 2 into + = 1: 2 + = 1 = 1 - 2 = -1 Now, check these values in equation (3): 1 - = 1 + 2 1 - 2 = 1 + 2(-1) -1 = 1 - 2 -1 = -1 The values = 2 and = -1 satisfy all three equations, so the lines intersect. The correct option is B. The final answer is B. Step 2: Solve Question 25. We need to arrange 5 girls on a bench, with the condition that two specific left-handed girls must be together. Treat the two left-handed girls as a single unit. Now we have 4 "units" to arrange: (LH1 LH2), G3, G4, G5. The number of ways to arrange these 4 units is 4!. 4! = 4 × 3 × 2 × 1 = 24. Within the unit of the two left-handed girls, they can be arranged in 2! ways (LH1 LH2 or LH2 LH1). 2! = 2 × 1 = 2. The total number of ways to arrange the 5 girls such that the two left-handed girls are together is the product of these two numbers: Total ways = 4! × 2! = 24 × 2 = 48. The correct option is B. The final answer is B. Step 3: Solve Question 26. We need to simplify the expression (x + ()/(6)) - (x - ()/(6)). Use the sum-to-product trigonometric identity: A - B = 2((A+B)/(2))((A-B)/(2)) Here, A = x + ()/(6) and B = x - ()/(6). First, find (A+B)/(2): (A+B)/(2) = ((x + )/(6)) + (x - ()/(6))2 = (2x)/(2) = x Next, find (A-B)/(2): (A-B)/(2) = ((x + )/(6)) - (x - ()/(6))2 = (x + )/(6) - x + ()/(6)2 = (2)/(6)2 = ()/(3)2 = ()/(6) Substitute these into the identity: (x + ()/(6)) - (x - ()/(6)) = 2(x)(()/(6)) We know that (()/(6)) = (30^) = (1)/(2). So, the expression becomes: 2(x)((1)/(2)) = (x) The correct option is C. The final answer is C. Step 4: Solve Question 27. The given function is y = (4)/(x^2 - 1). To find vertical asymptotes, set the denominator equal to zero: x^2 - 1 = 0 (x-1)(x+1) = 0 x = 1 or x = -1 So, there are two vertical asymptotes at x = ± 1. To find horizontal asymptotes, consider the limit as x ±: _x ± (4)/(x^2 - 1) As x ±, x^2 - 1 . So, _x ± (4)/(x^2 - 1) = 0. Thus, there is a horizontal asymptote at y = 0. The graph has two vertical asymptotes at x = ± 1 and a horizontal asymptote at y = 0. The correct option is D. The final answer is D. Step 5: Solve Question 28. Given the differential equation (dy)/(dx) - 2xy = 0. We can rewrite this as (dy)/(dx) = 2xy. This is a separable differential equation. (dy)/(y) = 2x\,dx Integrate both sides: (1)/(y)\,dy = 2x\,dx |y| = x^2 + C Exponentiate both sides: |y| = e^x^2 + C y = ± e^C e^x^2 Let A = ± e^C, where A is an arbitrary non-zero constant. y = Ae^x^2 We are given the initial condition y=1 when x=1. Substitute these values into the general solution: 1 = Ae^(1)^2 1 = Ae^1 A = (1)/(e) = e^-1 Substitute the value of A back into the general solution: y = e^-1e^x^2 y = e^x^2 - 1 The correct option is A. The final answer is A. Step 6: Solve Question 29. We need to find the coefficient of x^2 in the binomial expansion of (1 + (1)/(2)x)^5. The binomial theorem states that the general term in the expansion of (a+b)^n is given by T_r+1 = nr a^n-r b^r. In this case, a=1, b=(1)/(2)x, and n=5. We want the term with x^2, so we set r=2. T_2+1 = T_3 = 52 (1)^5-2 ((1)/(2)x)^2 T_3 = 52 (1)^3 ((1)/(2))^2 (x)^2 Calculate 52: 52 = (5!)/(2!(5-2)!) = (5!)/(2!3!) = (5 × 4)/(2 × 1) = 10 Substitute this value back: T_3 = 10 × 1 × (1)/(4) × x^2 T_3 = (10)/(4) x^2 T_3 = (5)/(2) x^2 The coefficient of x^2 is (5)/(2). Wait, let me recheck the options. The options are (1)/(32), -(1)/(16), (1)/(16), -(1)/(32). My calculated coefficient (5)/(2) is not among the options. Let me re-read the question carefully. It says (1 + (1)/(2)x)^5. My calculation is correct for this expression. Let me check if the image is clear. The exponent is definitely 5. Perhaps the question intended a different exponent or a different term. Let me assume there might be a typo in the question or options. If the question was (1 + (1)/(2)x)^-5, then the general term for (1+y)^n is _r=0^ nr y^r. For n=-5 and y=(1)/(2)x, the coefficient of x^2 would be -52 ((1)/(2))^2. -52 = ((-5)(-5-1))/(2!) = ((-5)(-6))/(2) = (30)/(2) = 15. So, 15 × (1)/(4) = (15)/(4). Still not matching. What if the expression was (1 + x)^(1)/(2)? No, it's (1 + (1)/(2)x)^5. Let me re-evaluate the calculation for 52. 52 = (5 × 4)/(2 × 1) = 10. This is correct. ((1)/(2)x)^2 = (1)/(4)x^2. This is correct. Coefficient is 10 × (1)/(4) = (10)/(4) = (5)/(2). This is correct. Given the options, there might be a misunderstanding of the question or a typo in the question/options. Let's consider if the question was (1 + (1)/(2x))^5. No, it's clearly (1)/(2)x. What if the exponent was different? If the exponent was, for example, -2, then -22 ((1)/(2))^2 = ((-2)(-3))/(2) × (1)/(4) = 3 × (1)/(4) = (3)/(4). If the exponent was -1, then -12 ((1)/(2))^2 = ((-1)(-2))/(2) × (1)/(4) = 1 × (1)/(4) = (1)/(4). Let's re-examine the image for the exponent. It looks like a 5. Let's assume there's a typo in the question and it was meant to be (1 + (1)/(2)x)^-2 or something similar that would lead to one of the options. However, I must solve the question as written. My calculated answer is (5)/(2). Since this is not among the options, I will state my calculated answer and note the discrepancy. Let me check if I misread the options. A: (1)/(32) B: -(1)/(16) C: (1)/(16) D: -(1)/(32) None of these match (5)/(2). I will proceed with the calculation as it is, and state the result. The coefficient of x^2 is (5)/(2). Since (5)/(2) is not among the options, there might be an error in the question or the provided options. I will select the closest option if forced, but that's not ideal. Let me re-read the question one more time. "The coefficient of x^2 in the binomial expansion of (1 + (1)/(2)x)^5 is". The calculation is straightforward and seems correct. Let's consider if the question was (1 + (1)/(2x))^5. No, it's clearly (1)/(2)x. What if the question was (1 + x)^(1)/(2)? Then the expansion is 1 + (1)/(2)x + (1)/(2)((1)/(2)-1)2!x^2 + = 1 + (1)/(2)x + (1)/(2)(-(1)/(2))2x^2 + = 1 + (1)/(2)x - (1)/(8)x^2 + . The coefficient of x^2 would be -(1)/(8). Still not matching. What if the question was (1 + x)^-2? Then the expansion is 1 + (-2)x + ((-2)(-3))/(2!)x^2 + = 1 - 2x + 3x^2 + . The coefficient of x^2 would be 3. Still not matching. What if the question was (1 + (1)/(2)x)^-1? Then the expansion is 1 + (-1)((1)/(2)x) + ((-1)(-2))/(2!)((1)/(2)x)^2 + = 1 - (1)/(2)x + 1 · (1)/(4)x^2 + = 1 - (1)/(2)x + (1)/(4)x^2 + . The coefficient of x^2 would be (1)/(4). Still not matching. What if the question was (1 + (1)/(2)x)^-4? Then the coefficient of x^2 would be -42 ((1)/(2))^2 = ((-4)(-5))/(2) × (1)/(4) = 10 × (1)/(4) = (5)/(2). This is the same as my original answer. Let's consider the possibility that the question is (1 + x)^n and the coefficient of x^2 is one of the options. If the coefficient of x^2 is (1)/(16), then n2 ((1)/(2))^2 = (1)/(16). (n(n-1))/(2) × (1)/(4) = (1)/(16) (n(n-1))/(8) = (1)/(16) n(n-1) = (8)/(16) = (1)/(2). 2n^2 - 2n - 1 = 0. This does not give an integer n. Let's assume the question is correct as written and my calculation is correct. The coefficient of x^2 in the binomial expansion of (1 + (1)/(2)x)^5 is (5)/(2). Since this is not an option, I will state the calculated value. The final answer is (5)/(2). (Note: This answer is not among the provided options, which suggests a potential error in the question or options.) Step 7: Solve Question 30. We need to find the coordinates of the points where the curves y = 2x - 3 and y = x^2 - 2x intersect. Set the two expressions for y equal to each other: 2x - 3 = x^2 - 2x Rearrange the equation to form a quadratic equation: 0 = x^2 - 2x - 2x + 3 0 = x^2 - 4x + 3 Factor the quadratic equation: We need two numbers that multiply to 3 and add to -4. These are -1 and -3. (x-1)(x-3) = 0 So, x = 1 or x = 3. Now, find the corresponding y values using y = 2x - 3: For $x=1