A set K=1, 2, ..., 11. Find (a) the prime numbers in K and the probability that a number selected at random from K is not prime, and (b) factorize (x+y)(2m-n)-m(x+y), and (c) calculate cost for 35 she

The set $K = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}$.
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.
The prime numbers in $K$ are $2, 3, 5, 7, 11$.

$$\text{Prime numbers in } K = \boxed{\{2, 3, 5, 7, 11\}}$$

Total number of elements in $K$ is 11.
Number of prime numbers in $K$ is 5.
Number of non-prime numbers in $K = \text{Total numbers} - \text{Prime numbers} = 11 - 5 = 6$.
The probability that a number selected at random from $K$ is not a prime number is:

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

To factorize $(x+y)(2m-n) - m(x+y)$, we identify the common factor $(x+y)$.

$$(x+y)(2m-n) - m(x+y) = (x+y)[(2m-n) - m]$$ $$= (x+y)(2m-n-m)$$ $$= (x+y)(m-n)$$ $$\text{Factorized expression} = \boxed{(x+y)(m-n)}$$

Cost for the first 8 sheets = GH¢ 68.00.
Total sheets typed = 35 sheets.
Number of additional sheets = $35 - 8 = 27$ sheets.
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

$$= \text{GH¢ } 68.00 + \text{GH¢ } 297.00 = \text{GH¢ } 365.00$$ $$\text{Total amount earned} = \boxed{\text{GH¢ } 365.00}$$

The translation vector is $r = (-3, 8)$.

(i) To find the image $P'$ of $P(2, 5)$ translated by $r$:

$$P' = (2 + (-3), 5 + 8) = (2 - 3, 5 + 8) = (-1, 13)$$ $$P' = \boxed{(-1, 13)}$$

(ii) To find the image $Q'$ of $Q(-4, -6)$ translated by $r$:

$$Q' = (-4 + (-3), -6 + 8) = (-4 - 3, -6 + 8) = (-7, 2)$$ $$Q' = \boxed{(-7, 2)}$$

Total profit = GH¢ 9,000.00.
Richard's manager's share = $\frac{1}{3} \times \text{GH¢ } 9,000.00 = \text{GH¢ } 3,000.00$.
Remaining profit = $\text{GH¢ } 9,000.00 - \text{GH¢ } 3,000.00 = \text{GH¢ } 6,000.00$.
Contribution ratio (Joyce : Richard) = $10,000 : 5,000 = 2 : 1$.
Total ratio parts = $2 + 1 = 3$.

(i) Joyce's share from remaining profit = $\frac{2}{3} \times \text{GH¢ } 6,000.00 = \text{GH¢ } 4,000.00$.
Joyce receives:

$$\text{Joyce's total} = \boxed{\text{GH¢ } 4,000.00}$$

(ii) Richard's share from remaining profit = $\frac{1}{3} \times \text{GH¢ } 6,000.00 = \text{GH¢ } 2,000.00$.
Richard's total receipt = Richard's manager's share + Richard's share from remaining profit

$$= \text{GH¢ } 3,000.00 + \text{GH¢ } 2,000.00 = \text{GH¢ } 5,000.00$$ $$\text{Richard's total} = \boxed{\text{GH¢ } 5,000.00}$$

Original daily wage = GH¢ 25.00.
Increase in daily wage = 10% of GH¢ 25.00 = $0.10 \times \text{GH¢ } 25.00 = \text{GH¢ } 2.50$.
New daily wage = GH¢ 25.00 + GH¢ 2.50 = GH¢ 27.50.
Number of days worked = 30 days.
Total amount paid = New daily wage $\times$ Number of days worked

$$= \text{GH¢ } 27.50 \times 30 = \text{GH¢ } 825.00$$ $$\text{Total amount paid} = \boxed{\text{GH¢ } 825.00}$$

The diagram shows an L-shaped room.
Let's determine the lengths of all sides:
The top horizontal side is 8 m.
The right vertical side is 3 m.
The bottom horizontal side is 5 m.
The left vertical side is 5 m.
The missing horizontal segment (part of the top side of the "cut-out") is $8 \text{ m} - 5 \text{ m} = 3 \text{ m}$.
The missing vertical segment (part of the left side of the "cut-out") is $5 \text{ m} - 3 \text{ m} = 2 \text{ m}$.

(i) Perimeter: Sum of all outer sides.

$$\text{Perimeter} = 8 \text{ m} + 3 \text{ m} + 5 \text{ m} + 2 \text{ m} + 3 \text{ m} + 5 \text{ m}$$ $$= 26 \text{ m}$$ $$\text{Perimeter} = \boxed{26 \text{ m}}$$

(ii) Area: Divide the L-shape into two rectangles.
We can consider it as a large rectangle of $8 \text{ m} \times 5 \text{ m}$ with a smaller rectangle of $3 \text{ m} \times 2 \text{ m}$ removed from the top right corner.
Area of large rectangle = $8 \text{ m} \times 5 \text{ m} = 40 \text{ m}^2$.
Area of removed rectangle = $(8-5) \text{ m} \times (5-3) \text{ m} = 3 \text{ m} \times 2 \text{ m} = 6 \text{ m}^2$.

$$\text{Area} = 40 \text{ m}^2 - 6 \text{ m}^2 = 34 \text{ m}^2$$ $$\text{Area} = \boxed{34 \text{ m}^2}$$

(iii) Cost of carpeting the floor:
Cost per square metre = GH¢ 20.00.
Total area = $34 \text{ m}^2$.
Total cost = Area $\times$ Cost per square metre

$$= 34 \text{ m}^2 \times \text{GH¢ } 20.00/\text{m}^2 = \text{GH¢ } 680.00$$ $$\text{Cost of carpeting} = \boxed{\text{GH¢ } 680.00}$$