Calculate the determinant of the 2 x 2 matrix.

Here are the solutions to questions 17, 18, and 19 based on the image provided: 17. Find the value of x if x & 3 \\ 4 & 7 = 9 Step 1: Calculate the determinant of the 2 × 2 matrix. The determinant of a matrix a & b \\ c & d is given by ad - bc. For the given matrix x & 3 \\ 4 & 7 , the determinant is (x)(7) - (3)(4). 7x - 12 Step 2: Set the determinant equal to 9 as given in the problem. 7x - 12 = 9 Step 3: Solve the linear equation for x. Add 12 to both sides: 7x = 9 + 12 7x = 21 Divide by 7: x = (21)/(7) x = 3 The correct answer is (b) 3. 3 18. Rationalise 2sqrt(3) - sqrt(5)sqrt(5) - sqrt(3) Step 1: Identify the conjugate of the denominator. The denominator is sqrt(5) - sqrt(3). Its conjugate is sqrt(5) + sqrt(3). Step 2: Multiply the numerator and the denominator by the conjugate. 2sqrt(3) - sqrt(5)sqrt(5) - sqrt(3) × sqrt(5) + sqrt(3)sqrt(5) + sqrt(3) Step 3: Expand the numerator: (2sqrt(3) - sqrt(5))(sqrt(5) + sqrt(3)). = (2sqrt(3))(sqrt(5)) + (2sqrt(3))(sqrt(3)) - (sqrt(5))(sqrt(5)) - (sqrt(5))(sqrt(3)) = 2sqrt(15) + 2(3) - 5 - sqrt(15) = 2sqrt(15) + 6 - 5 - sqrt(15) = (2sqrt(15) - sqrt(15)) + (6 - 5) = sqrt(15) + 1 Step 4: Expand the denominator using the difference of squares formula (a-b)(a+b) = a^2 - b^2: (sqrt(5) - sqrt(3))(sqrt(5) + sqrt(3)). = (sqrt(5))^2 - (sqrt(3))^2 = 5 - 3 = 2 Step 5: Combine the simplified numerator and denominator. sqrt(15) + 12 Upon reviewing the provided options, none of them match the calculated answer sqrt(15) + 12. There might be a typo in the question or the options. sqrt(15) + 12 19. If y=x, find dy/dx. Step 1: The given function is y=x. Step 2: Differentiate y with respect to x. Using the power rule for differentiation, if y = x^n, then (dy)/(dx) = nx^n-1. For y=x, we have n=1. (dy)/(dx) = 1 · x^1-1 (dy)/(dx) = 1 · x^0 Since any non-zero number raised to the power of 0 is 1 (x^0=1 for x ≠ 0), (dy)/(dx) = 1 · 1 (dy)/(dx) = 1 The correct answer is (a) 1. 1