Find the equation of the tangent to the curve x2 - xy - y2 + 4 = 0 at K(2,4). The sum Sn of the first n terms of a linear sequence (A.P.) is given by Sn = n2 - 8n. If the n^th term of the sequence is

You're on a roll — here are the solutions to the math problems: Q1) Find the equation of the tangent to the curve x^2 - xy - y^2 + 4 = 0 at K(2,4). Step 1: Differentiate the equation implicitly with respect to x. (d)/(dx)(x^2) - (d)/(dx)(xy) - (d)/(dx)(y^2) + (d)/(dx)(4) = 0 2x - (1 · y + x · (dy)/(dx)) - 2y (dy)/(dx) + 0 = 0 2x - y - x (dy)/(dx) - 2y (dy)/(dx) = 0 Step 2: Rearrange the equation to solve for (dy)/(dx). 2x - y = x (dy)/(dx) + 2y (dy)/(dx) 2x - y = (x + 2y) (dy)/(dx) (dy)/(dx) = (2x - y)/(x + 2y) Step 3: Substitute the coordinates of point K(2,4) into (dy)/(dx) to find the gradient (m) of the tangent. m = (2(2) - 4)/(2 + 2(4)) = (4 - 4)/(2 + 8) = (0)/(10) = 0 Step 4: Use the point-slope form of a linear equation, y - y_1 = m(x - x_1), with m=0 and (x_1, y_1) = (2,4). y - 4 = 0(x - 2) y - 4 = 0 y = 4 The equation of the tangent is y = 4. Q2) The sum S_n of the first n terms of a linear sequence (A.P.) is given by S_n = n^2 - 8n. If the n^th term of the sequence is greater than 30, find the least possible value of n. Step 1: Find the formula for the n^th term, T_n, using the relationship T_n = S_n - S_n-1. First, find S_n-1: S_n-1 = (n-1)^2 - 8(n-1) S_n-1 = (n^2 - 2n + 1) - (8n - 8) S_n-1 = n^2 - 2n + 1 - 8n + 8 S_n-1 = n^2 - 10n + 9 Now, find T_n: T_n = (n^2 - 8n) - (n^2 - 10n + 9) T_n = n^2 - 8n - n^2 + 10n - 9 T_n = 2n - 9 Step 2: Set up the inequality given that the n^th term is greater than 30. T_n > 30 2n - 9 > 30 Step 3: Solve the inequality for n. 2n > 30 + 9 2n > 39 n > (39)/(2) n > 19.5 Step 4: Determine the least possible integer value of n. Since n must be an integer and n must be greater than 19.5, the least possible integer value for n is 20. The least possible value of n is 20. Q3) Resolve (4-x)/(x^2 - 5x - 24) into partial fractions. Step 1: Factorize the denominator. We need two numbers that multiply to -24 and add to -5. These numbers are -8 and 3. x