Calculate the area of the region bounded by y = 16 - x2, the x-axis, and the y-axis in the first quadrant. Also, find the value of x for which the matrix [[1, 4], [x, x-3]] is singular.

Step 1: Identify the function and the limits of integration. The shaded region is bounded by the curve y = 16 - x^2, the x-axis (y=0), and the y-axis (x=0). To find the x-intercept, set y=0: 0 = 16 - x^2 x^2 = 16 x = ± 4 Since the region is in the first quadrant, the relevant x-intercept is x=4. The area is given by the definite integral of the function from x=0 to x=4. Area = _0^4 (16 - x^2)\,dx Step 2: Evaluate the integral. Area = [ 16x - (x^3)/(3) ]_0^4 Area = ( 16(4) - (4^3)/(3) ) - ( 16(0) - (0^3)/(3) ) Area = ( 64 - (64)/(3) ) - (0 - 0) Area = (192)/(3) - (64)/(3) Area = (128)/(3) The exact area of the shaded region is (128)/(3) square units. --- Step 1: Calculate the determinant of the matrix. A matrix a & b \\ c & d is singular if its determinant ad - bc = 0. For the given matrix 1 & 4 \\ x & x-3 , the determinant is: det = (1)(x-3) - (4)(x) det = x - 3 - 4x det = -3x - 3 Step 2: Set the determinant to zero and solve for x. For the matrix to be singular: -3x - 3 = 0 -3x = 3 x = (3)/(-3) x = -1 The value of x for which the matrix is singular is -1. --- Step 1: Rewrite the equation in terms of using the identity ^2 = 1 - ^2 . 2 ^2 - 5 + 1 = 0 2(1 - ^2 ) - 5 + 1 = 0 2 - 2 ^2 - 5 + 1 = 0 -2 ^2 - 5 + 3 = 0 Multiply by -1 to make the leading coefficient positive: 2 ^2 + 5 - 3 = 0 Step 2: Solve the quadratic equation for . Let y = . The equation becomes 2y^2 + 5y - 3 = 0. Factor the quadratic equation: (2y - 1)(y + 3) = 0 So, 2y - 1 = 0 or y + 3 = 0. 2y = 1 y = (1)/(2) y = -3 Substitute back y = : = (1)/(2) or = -3 Since the range of is [-1, 1], = -3 has no solution. Therefore, we only consider = (1)/(2). Step 3: Find the values of for = (1)/(2) within the range 0^ ≤ ≤ 360^. The basic angle for = (1)/(2) is 60^. Since cosine is positive, is in the first and fourth quadrants. • First quadrant: = 60^. • Fourth quadrant: = 360^ - 60^ = 300^. The values of are 60^, 300^. --- Step 1: Determine the value of one ratio unit using Kamau's share. The total amount to be shared is Sh. 12 000. The ratio for Kamau : Njoroge : Mwangi is 5 : 6 : x. Kamau's share is Sh. 4 000. Let k be the value of one ratio unit. Kamau's share = 5k = 4000. k = (4000)/(5) k = 800 So, one ratio unit is Sh. 800. Step 2: Calculate the total number of ratio units. The total amount is Sh. 12 000. Total ratio units = Sum of ratio parts = 5 + 6 + x = 11 + x. Total amount = (Total ratio units) × (Value of one ratio unit) 12000 = (11 + x) × 800 Step 3: Solve for x. (12000)/(800) = 11 + x 15 = 11 + x x = 15 - 11 x = 4 The value of x is 4. --- Step 1: Let A = xy to simplify the expression. The expression is 2x^2y^2 - 5xy - 12. Substitute A = xy: 2A^2 - 5A - 12 Step 2: Factor the quadratic expression 2A^2 - 5A - 12. We look for two numbers that multiply to 2 × (-12) = -24 and add up to -5. These numbers are 3 and -8. Rewrite the middle term: 2A^2 + 3A - 8A - 12 Factor by grouping: A(2A + 3) - 4(2A + 3) (A - 4)(2A + 3) Step 3: Substitute xy back for A. (xy - 4)(2xy + 3) The factorised expression is (xy - 4)(2xy + 3). That's 2 down. 3 left today — send the next one.