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 $A = \begin{bmatrix} 2 & 4p \\ 3 & 5p \end{bmatrix}$.

• Find the value of $p$ for which the determinant of $A$ is $-2$.

Step 1: Calculate the determinant of $A$.
For a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, the determinant is $ad - bc$.

$$\det(A) = (2)(5p) - (4p)(3) = 10p - 12p = -2p$$

Step 2: Set the determinant equal to $-2$ and solve for $p$.

$$-2p = -2$$ $$p = \frac{-2}{-2}$$ $$p = \boxed{1}$$

• Hence, find the inverse of $A$.

Step 3: Substitute $p=1$ into matrix $A$.

$$A = \begin{bmatrix} 2 & 4(1) \\ 3 & 5(1) \end{bmatrix} = \begin{bmatrix} 2 & 4 \\ 3 & 5 \end{bmatrix}$$

Step 4: Calculate the inverse of $A$.
The inverse of a $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $\frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.
We already found $\det(A) = -2$.

$$A^{-1} = \frac{1}{-2} \begin{bmatrix} 5 & -4 \\ -3 & 2 \end{bmatrix}$$ $$A^{-1} = \begin{bmatrix} \frac{5}{-2} & \frac{-4}{-2} \\ \frac{-3}{-2} & \frac{2}{-2} \end{bmatrix}$$ $$A^{-1} = \boxed{\begin{bmatrix} -2.5 & 2 \\ 1.5 & -1 \end{bmatrix}}$$

1b. Probability
Let $P(M)$ be the probability of selecting Mutola and $P(W)$ be the probability of selecting Mwambi.
Given $P(M) = \frac{2}{3}$ and $P(W) = \frac{3}{5}$.
The probabilities of not selecting them are $P(M') = 1 - \frac{2}{3} = \frac{1}{3}$ and $P(W') = 1 - \frac{3}{5} = \frac{2}{5}$.
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).

$$P(\text{only one}) = P(M \cap W') + P(M' \cap W)$$

Step 2: Calculate the probabilities for each case.
Since the events are independent:

$$P(M \cap W') = P(M) \times P(W') = \frac{2}{3} \times \frac{2}{5} = \frac{4}{15}$$ $$P(M' \cap W) = P(M') \times P(W) = \frac{1}{3} \times \frac{3}{5} = \frac{3}{15}$$

Step 3: Sum the probabilities.

$$P(\text{only one}) = \frac{4}{15} + \frac{3}{15} = \boxed{\frac{7}{15}}$$

• (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.

$$P(\text{none}) = P(M' \cap W')$$

Step 2: Calculate the probability.
Since the events are independent:

$$P(\text{none}) = P(M') \times P(W') = \frac{1}{3} \times \frac{2}{5} = \boxed{\frac{2}{15}}$$

2a. Simplify
Simplify $\frac{y+1}{y^2-1}$.

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.

$$\frac{y+1}{(y-1)(y+1)}$$

Step 3: Cancel the common term $(y+1)$ from the numerator and denominator (assuming $y \neq -1$).

$$\boxed{\frac{1}{y-1}}$$

2b. Geometric Progression
The sum of $n$ terms of a geometric progression (G.P.) is given by $S_n = 10 - \left(\frac{1}{2}\right)^n$.

• (i) Find the sum of the first 4 terms of this G.P.

Step 1: Substitute $n=4$ into the given formula for $S_n$.

$$S_4 = 10 - \left(\frac{1}{2}\right)^4$$ $$S_4 = 10 - \frac{1}{16}$$

Step 2: Calculate the value.

$$S_4 = \frac{160}{16} - \frac{1}{16} = \boxed{\frac{159}{16}}$$

• (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 $S_n$.

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