Given that matrix A is [7 4p / 5p 5p]. (i) find the value of p for which the determinant of A is -2. (ii) hence find the inverse of A.

Here are the solutions to your questions: 1. Matrices Given matrix A = 2 & 4p \\ 3 & 5p . Find the value of p for which the determinant of A is -2. Step 1: Calculate the determinant of A. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. (A) = (2)(5p) - (4p)(3) = 10p - 12p = -2p Step 2: Set the determinant equal to -2 and solve for p. -2p = -2 p = (-2)/(-2) p = 1 Hence, find the inverse of A. Step 3: Substitute p=1 into matrix A. A = 2 & 4(1) \\ 3 & 5(1) = 2 & 4 \\ 3 & 5 Step 4: Calculate the inverse of A. The inverse of a 2 × 2 matrix a & b \\ c & d is (1)/(ad-bc) d & -b \\ -c & a . We already found (A) = -2. A^-1 = (1)/(-2) 5 & -4 \\ -3 & 2 A^-1 = (5)/(-2) & (-4)/(-2) \\ (-3)/(-2) & (2)/(-2) A^-1 = -2.5 & 2 \\ 1.5 & -1 1b. Probability Let P(M) be the probability of selecting Mutola and P(W) be the probability of selecting Mwambi. Given P(M) = (2)/(3) and P(W) = (3)/(5). The probabilities of not selecting them are P(M') = 1 - (2)/(3) = (1)/(3) and P(W') = 1 - (3)/(5) = (2)/(5). Assume the selections are independent events. (i) Find the probability that only one of them is selected. Step 1: Identify the events for "only one selected". This means (Mutola selected AND Mwambi NOT selected) OR (Mutola NOT selected AND Mwambi selected). P(only one) = P(M W') + P(M' W) Step 2: Calculate the probabilities for each case. Since the events are independent: P(M W') = P(M) × P(W') = (2)/(3) × (2)/(5) = (4)/(15) P(M' W) = P(M') × P(W) = (1)/(3) × (3)/(5) = (3)/(15) Step 3: Sum the probabilities. P(only one) = (4)/(15) + (3)/(15) = (7)/(15) (ii) Find the probability that none of them is selected. Step 1: Identify the event for "none selected". This means Mutola NOT selected AND Mwambi NOT selected. P(none) = P(M' W') Step 2: Calculate the probability. Since the events are independent: P(none) = P(M') × P(W') = (1)/(3) × (2)/(5) = (2)/(15) 2a. Simplify Simplify (y+1)/(y^2-1). Step 1: Factor the denominator using the difference of squares formula (a^2 - b^2 = (a-b)(a+b)). y^2 - 1 = (y-1)(y+1) Step 2: Substitute the factored denominator back into the expression. (y+1)/((y-1)(y+1)) Step 3: Cancel the common term (y+1) from the numerator and denominator (assuming y ≠ -1). (1)/(y-1) 2b. Geometric Progression The sum of n terms of a geometric progression (G.P.) is given by S_n = 10 - ((1)/(2))^n. (i) Find the sum of the first 4 terms of this G.P. Step 1: Substitute n=4 into the given formula for S_n. S_4 = 10 - ((1)/(2))^4 S_4 = 10 - (1)/(16) Step 2: Calculate the value. S_4 = (160)/(16) - (1)/(16) = (159)/(16) (ii) Find the first term. Step 1: The first term, a, is equal to the sum of the first term, S_1. Substitute n=1 into the formula for S_n. a = S_1 = 1 ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._