Given that K = 1, 2, 3, ..., 11: (i) List the prime numbers in K. (ii) Find the probability that a number selected at random from the set K is not a prime number. 1. (b) Factorize completely: (x+y)(2m

The prime numbers in $K$ are $2, 3, 5, 7, 11$.
*   **Prime numbers: $\{2, 3, 5, 7, 11\}$**

• (ii) Find the probability that a number selected at random from the set $K$ is not a prime number.

$$
P(\text{not prime}) = \frac{\text{Number of non-prime numbers}}{\text{Total number of elements}}
$$
$$
P(\text{not prime}) = \frac{6}{11}
$$
*   **Probability: $\boxed{\frac{6}{11}}$**

1. (b) Factorize completely: $(x+y)(2m-n) - m(x+y)$

The common factor is $(x+y)$.

• Factorized expression: $\boxed{(x+y)(m-n)}$

1. (c) A typist charges GH¢ 68.00 for the first 8 sheets typed, and GH¢ 11.00 for each additional sheet. Calculate the amount earned if the typist typed a total of 35 sheets.

Additional sheets = Total sheets - First 8 sheets
Additional sheets = $35 - 8 = 27$ sheets.

Cost for additional sheets = Number of additional sheets $\times$ Cost per additional sheet
Cost for additional sheets = $27 \times \text{GH¢ } 11.00 = \text{GH¢ } 297.00$.

Total amount earned = Cost for first 8 sheets + Cost for additional sheets
Total amount earned = $\text{GH¢ } 68.00 + \text{GH¢ } 297.00 = \text{GH¢ } 365.00$.

• Amount earned: $\boxed{\text{GH¢ } 365.00}$

2. (a) The image of $P(2, 5)$ when translated by the vector $r$ is $(-3, 8)$. Find:

• (i) $r$;

Let the translation vector be $r = (a, b)$.
The translation formula is $P' = P + r$. $(x', y') = (x+a, y+b)$ $(-3, 8) = (2+a, 5+b)$

For the x-coordinate: $-3 = 2+a \implies a = -3 - 2 \implies a = -5$.  
For the y-coordinate: $8 = 5+b \implies b = 8 - 5 \implies b = 3$.
*   **Translation vector $r$: $\boxed{\begin{pmatrix} -5 \\ 3 \end{pmatrix}}$**

• (ii) the image $Q'$ of $Q(-4, -6)$ when translated by $r$.

$$
Q' = (-4 + (-5), -6 + 3)
$$
$$
Q' = (-4 - 5, -6 + 3)
$$
$$
Q' = (-9, -3)
$$
*   **Image $Q'$: $\boxed{(-9, -3)}$**

2. (b) Joyce and Richard contributed GH¢ 10,000.00 and GH¢ 5,000.00 respectively to start a business. They agreed that Richard will be paid one-third of the profit as a manager, and the rest of the profit will be shared in the ratio of their contributions. If the profit of GH¢ 9,000.00 was made, how much did:

• (i) Joyce receive;
• (ii) Richard receive in total?

Richard's manager's share = $\frac{1}{3} \times \text{GH¢ } 9,000.00 = \text{GH¢ } 3,000.00$.

Remaining profit = Total profit - Richard's manager's share
Remaining profit = $\text{GH¢ } 9,000.00 - \text{GH¢ } 3,000.00 = \text{GH¢ } 6,000.00$.

Joyce's contribution : Richard's contribution = GH¢ 10,000.00 : GH¢ 5,000.00
Ratio = $10,000 : 5,000 = 2 : 1$.
Sum of ratios = $2 + 1 = 3$.

Joyce's share = $\frac{2}{3} \times \text{GH¢ } 6,000.00 = \text{GH¢ } 4,000.00$.

• (i) Joyce receive: $\boxed{\text{GH¢ } 4,000.00}$

Richard's share = $\frac{1}{3} \times \text{GH¢ } 6,000.00 = \text{GH¢ } 2,000.00$.

Richard's total receipt = Richard's manager's share + Richard's share from remaining profit
Richard's total receipt = $\text{GH¢ } 3,000.00 + \text{GH¢ } 2,000.00 = \text{GH¢ } 5,000.00$.

• (ii) Richard receive in total: $\boxed{\text{GH¢ } 5,000.00}$