Express the angles in terms of x.

Here are the solutions to the questions:

1. (a) In a triangle PQR, angle Q is three-quarter of angle P and angle R is half of angle P. If the measure of angle P is $(x+5)^\circ$, form an equation in x and hence, calculate the angles of triangle PQR.

Step 1: Express the angles in terms of $x$.
Given:
$\angle P = (x+5)^\circ$
$\angle Q = \frac{3}{4} \angle P = \frac{3}{4}(x+5)^\circ$
$\angle R = \frac{1}{2} \angle P = \frac{1}{2}(x+5)^\circ$

Step 2: Form an equation using the sum of angles in a triangle.
The sum of angles in a triangle is $180^\circ$.

$$\angle P + \angle Q + \angle R = 180^\circ$$ $$(x+5) + \frac{3}{4}(x+5) + \frac{1}{2}(x+5) = 180$$

Step 3: Solve the equation for $x$.
Factor out $(x+5)$:

$$(x+5) \left(1 + \frac{3}{4} + \frac{1}{2}\right) = 180$$ $$(x+5) \left(\frac{4}{4} + \frac{3}{4} + \frac{2}{4}\right) = 180$$ $$(x+5) \left(\frac{9}{4}\right) = 180$$

Multiply both sides by $\frac{4}{9}$:

$$x+5 = 180 \times \frac{4}{9}$$ $$x+5 = 20 \times 4$$ $$x+5 = 80$$ $$x = 80 - 5$$ $$x = 75$$

Step 4: Calculate the measure of each angle.

$$\angle P = (x+5)^\circ = (75+5)^\circ = 80^\circ$$ $$\angle Q = \frac{3}{4}(80^\circ) = 3 \times 20^\circ = 60^\circ$$ $$\angle R = \frac{1}{2}(80^\circ) = 40^\circ$$

The equation in $x$ is $\boxed{(x+5) \left(\frac{9}{4}\right) = 180}$.
The angles of triangle PQR are $\boxed{\angle P = 80^\circ, \angle Q = 60^\circ, \angle R = 40^\circ}$.

1. (b) Find the smallest number by which 240 must be multiplied to get a perfect square.

Step 1: Find the prime factorization of 240.

$$240 = 24 \times 10 = (2^3 \times 3) \times (2 \times 5) = 2^4 \times 3^1 \times 5^1$$

Step 2: Identify prime factors with odd powers.
For a number to be a perfect square, all exponents in its prime factorization must be even. In $2^4 \times 3^1 \times 5^1$, the prime factors 3 and 5 have odd powers (1).

Step 3: Determine the smallest multiplier.
To make the powers of 3 and 5 even, we need to multiply by $3^1$ and $5^1$.
The smallest number to multiply by is $3 \times 5 = 15$.

$$240 \times 15 = (2^4 \times 3^1 \times 5^1) \times (3^1 \times 5^1) = 2^4 \times 3^2 \times 5^2 = (2^2 \times 3 \times 5)^2 = (60)^2 = 3600$$

The smallest number by which 240 must be multiplied to get a perfect square is $\boxed{15}$.

2. (a) A car travels at a distance of 60km. Construct a table of value for speed and time. Given that the time for the journey vary as: $1\frac{1}{2}\text{hrs}$, $2\text{hrs}$, $2\frac{1}{2}\text{hrs}$, and $3\text{hrs}$.

Step 1: Recall the relationship between distance, speed, and time.
Distance ($D$) = Speed ($S$) $\times$ Time ($T$)
Therefore, Speed ($S$) = $\frac{\text{Distance}}{\text{Time}}$

Step 2: Identify the given distance and times.
Distance ($D$) = $60 \text{ km}$
Times ($T$): $1\frac{1}{2}\text{ hrs} = 1.5 \text{ hrs}$, $2\text{ hrs}$, $2\frac{1}{2}\text{ hrs} = 2.5 \text{ hrs}$, $3\text{ hrs}$.

Step 3: Calculate the speed for each given time.

• For $T = 1.5 \text{ hrs}$: $S = \frac{60 \text{ km}}{1.5 \text{ hrs}} = 40 \text{ km/hr}$
• For $T = 2 \text{ hrs}$: $S = \frac{60 \text{ km}}{2 \text{ hrs}} = 30 \text{ km/hr}$
• For $T = 2.5 \text{ hrs}$: $S = \frac{60 \text{ km}}{2.5 \text{ hrs}} = 24 \text{ km/hr}$
• For $T = 3 \text{ hrs}$: $S = \frac{60 \text{ km}}{3 \text{ hrs}} = 20 \text{ km/hr}$

Step 4: Construct the table of values.

Time (hrs)	Speed (km/hr)
$1\frac{1}{2}$ (1.5)	40
2	30
$2\frac{1}{2}$ (2.5)	24
3	20

2. (b) A crate containing 24 bottles of drinks is made up of 10 bottles of Coca Cola, 8 bottles of Fanta and the rest are Sprite. If one bottle is picked at random, what is the probability of picking: (i) a coca cola drink (ii) a sprite drink

Step 1: Determine the total number of bottles and the number of each type of drink.
Total bottles = 24
Coca Cola bottles = 10
Fanta bottles = 8
Sprite bottles = Total bottles - Coca Cola bottles - Fanta bottles
Sprite bottles = $24 - 10 - 8 = 6$

Step 2: Calculate the probability for each event.
The formula for probability is $P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$.