0580_w22_qp

Question 1

Topic – E1.1

Write down a common multiple of 18 and 24.

▶️ Answer/Explanation

Solution

Ans: Any multiple of 72

Find the LCM of 18 and 24:

Prime factors of 18: $2 × 3^2$

Prime factors of 24: $2^3 × 3$

LCM = $2^3 × 3^2 = 72$

Any multiple of 72 (e.g., 72, 144, 216) is correct.

Question 2

Topic – E1.15

A train journey starts at 23.40 and finishes at 06.50.

Work out the time taken for this journey.

▶️ Answer/Explanation

Solution

Ans: 7 h 10 min

From 23:40 to midnight: 20 minutes

From midnight to 06:50: 6 hours 50 minutes

Total time = 20 min + 6 h 50 min = 7 h 10 min

Question 3

Topic – E1.4

Write 32 cm as a fraction of 2 m.

Give your answer in its simplest form.

▶️ Answer/Explanation

Solution

Ans: $\frac{4}{25}$

Convert 2 m to cm: 2 m = 200 cm

Fraction = $\frac{32}{200}$

Simplify by dividing numerator and denominator by 8: $\frac{4}{25}$

Question 4

Topic – E1.11

Divide $200 in the ratio 7:3.

▶️ Answer/Explanation

Solution

Ans: $140, $60

Total parts = 7 + 3 = 10

Value per part = $200 ÷ 10 = $20

First part = 7 × $20 = $140

Second part = 3 × $20 = $60

Question 5

Topic – E4.6

The diagram shows two straight lines intersecting two parallel lines.

Find the value of $x$.

▶️ Answer/Explanation

Solution

Ans: $x = 54$

Using the fact that angles in a triangle sum to $180^\circ$:

First find the angle adjacent to $71^\circ$: $180^\circ – 71^\circ – 55^\circ = 54^\circ$

This angle is equal to $x$ due to alternate angles being equal for parallel lines.

Question 6

Topic – E1.13

The price of a computer is $520.
This price is reduced by 15% in a sale.

Work out the sale price.

▶️ Answer/Explanation

Solution

Ans: $442

Method 1: Calculate 15% of $520 = $78, then subtract from original price: $520 – $78 = $442

Method 2: Calculate 85% of $520 directly: 0.85 × 520 = $442

Question 7

Topic – E1.2

The Venn diagram shows the elements of the sets E, P and Q.
Complete the statements.

(a) P = {……}

(b) n(P ∪ Q) = ……

▶️ Answer/Explanation

Solution

Ans:

(a) P = {a, b, c, d}

(b) n(P ∪ Q) = 6

For (a), list all elements in circle P.

For (b), count all unique elements in P and Q combined.

Question 8

Topic – E2.7

(a) 3, 9, 27, 81, …

Write down the next term in this sequence.

(b) 13, 17, 21, 25, …

Find the nth term of this sequence.

▶️ Answer/Explanation

Solution

Ans:

(a) 243 (each term is multiplied by 3)

(b) 4n + 9

For (a), recognize the geometric sequence pattern (×3).

For (b), find common difference (4) and adjust for first term: 4×1 + 9 = 13.

Question 9

Topic – E1.4

Without using a calculator, work out $\frac{1}{3} + \frac{5}{6}$.

You must show all your working and give your answer as a mixed number in its simplest form.

▶️ Answer/Explanation

Solution

Ans: $1\frac{1}{6}$

Convert $\frac{1}{3}$ to $\frac{2}{6}$ to get common denominator.

$\frac{2}{6} + \frac{5}{6} = \frac{7}{6} = 1\frac{1}{6}$

Question 10

Topic – E2.4

Simplify $18x^{18} ÷ 9x^9$.

▶️ Answer/Explanation

Solution

Ans: $2x^9$

Divide coefficients: 18 ÷ 9 = 2

Subtract exponents: $x^{18} ÷ x^9 = x^{18-9} = x^9$

Combine results: $2x^9$

Question 11

Topic – E2.5

Solve the simultaneous equations.

$x – 3y = 7$
$2x – 3y = 11$

▶️ Answer/Explanation

Solution

Ans: x = 4, y = -1

Subtract first equation from second: (2x – 3y) – (x – 3y) = 11 – 7

This gives x = 4

Substitute x = 4 into first equation: 4 – 3y = 7 → -3y = 3 → y = -1

Question 12

Topic – E4.4

Triangle PQR is similar to triangle ABC with $\frac{PR}{AC} = \frac{2}{3}$.

AB = 9 cm and the area of triangle ABC is 18 cm².

(a) Find the length of PQ.

(b) Find the area of triangle PQR.

▶️ Answer/Explanation

Solution

Ans: (a) 6 cm (b) 8 cm²

(a) Since PR/AC = 2/3, all corresponding sides have ratio 2:3. PQ/AB = 2/3 → PQ = (2/3)×9 = 6 cm

(b) Area ratio is square of side ratio: (2/3)² = 4/9. Area PQR = (4/9)×18 = 8 cm²

Question 13

Topic – E2.9

The diagram shows the speed-time graph of the first 15 seconds of a car journey.

(a) Find the acceleration of the car during the first 5 seconds.

(b) Find the distance travelled during the 15 seconds.

▶️ Answer/Explanation

Solution

Ans: (a) 2.8 m/s² (b) 175 m

(a) Acceleration = gradient = 14 m/s ÷ 5 s = 2.8 m/s²

(b) Distance = area under graph = triangle (½×5×14) + rectangle (10×14) = 35 + 140 = 175 m

Question 14

Topic – E7.1

Describe fully the single transformation that maps triangle A onto triangle B.

▶️ Answer/Explanation

Solution

Ans: Rotation, 90° clockwise about (5,3)

From the diagram, triangle B is a 90° clockwise rotation of triangle A. The center of rotation is where the perpendicular bisectors of corresponding points meet, which is at (5,3).

Question 15

Topic – E5.3

The perimeter of a sector of a circle with radius 8 cm is 26 cm.

Calculate the angle of this sector.

▶️ Answer/Explanation

Solution

Ans: 71.6°

Perimeter = 2 radii + arc length: 26 = 8 + 8 + (θ/360)×2π×8

Arc length = 10 cm → θ = (10×360)/(16π) ≈ 71.6°

Question 16

Topic – E4.7

The diagram shows a circle and eight chords.

Calculate the values of $ u $, $ v $, $ w $ and $ x $.

▶️ Answer/Explanation

Solution

Ans: $ u = 20 $, $ v = 52 $, $ w = 108 $, $ x = 36 $

1. $ u = 20° $ (given angle)

2. $ v = 180 – 72 – 56 = 52° $ (angles in triangle)

3. $ w = 180 – 72 = 108° $ (angles on straight line)

4. $ x = 180 – 108 – 36 = 36° $ (angles in triangle)

Question 17

Topic – E2.4

Simplify $ \left(3125x^{3125}\right)^{\frac{1}{5}} $.

▶️ Answer/Explanation

Solution

Ans: $ 5x^{625} $

1. Recognize 3125 is $ 5^5 $

2. Apply exponent rules: $ (5^5)^{\frac{1}{5}} = 5 $

3. For $ x $ term: $ 3125 \times \frac{1}{5} = 625 $

4. Combine results: $ 5x^{625} $

Question 18

Topic – E6.2

Calculate the length $ BC $.

▶️ Answer/Explanation

Solution

Ans: 12.7 cm (12.68 to 12.69)

1. Find angle C: $ 180 – 115 – 35 = 30° $

2. Apply sine rule: $ \frac{BC}{\sin 115} = \frac{7}{\sin 30} $

3. Calculate: $ BC = \frac{7 \times \sin 115}{\sin 30} $

4. Evaluate: $ \sin 115 ≈ 0.9063 $, $ \sin 30 = 0.5 $

5. Final calculation: $ BC ≈ 12.688 $ cm

Question 19

Topic – E2.2

Expand and simplify $ (2x + 3)(x – 2)^2 $.

▶️ Answer/Explanation

Solution

Ans: $ 2x^3 – 5x^2 – 4x + 12 $

1. First expand $ (x – 2)^2 = x^2 – 4x + 4 $

2. Multiply by $ (2x + 3) $: $ 2x(x^2 – 4x + 4) + 3(x^2 – 4x + 4) $

3. Expand: $ 2x^3 – 8x^2 + 8x + 3x^2 – 12x + 12 $

4. Combine like terms: $ 2x^3 – 5x^2 – 4x + 12 $

Question 20

Topic – E2.2

Factorise completely:

(a) $ 1 + x – y – xy $

(b) $ 2x^3 – 18xy^2 $

▶️ Answer/Explanation

Solution

Ans:

(a) $ (1 + x)(1 – y) $

1. Group terms: $ (1 + x) – y(1 + x) $

2. Factor out common $ (1 + x) $ term

(b) $ 2x(x + 3y)(x – 3y) $

1. Factor out 2x: $ 2x(x^2 – 9y^2) $

2. Recognize difference of squares: $ x^2 – 9y^2 = (x + 3y)(x – 3y) $

3. Combine factors

Question 21

Topic – E2.11

The graph of a cubic function has two turning points.
When $x < 0$ and when $x > 4$ the gradient of the graph is positive.
When $0 < x < 4$ the gradient of the graph is negative.
The graph passes through the origin.

Sketch the graph.

▶️ Answer/Explanation

Solution

Sketch should show:

1. A cubic curve with maximum at origin (0,0)

2. A minimum point somewhere in the range 0 < x < 4

3. Positive gradient for x < 0 and x > 4

4. Negative gradient between 0 < x < 4

Question 22

Topic – E2.10

(a) On the diagram, sketch the graph of $y = \cos x$ for $0^\circ \leq x \leq 360^\circ$.

(b) Solve the equation $\cos x = -\frac{1}{2}$ for $0^\circ \leq x \leq 360^\circ$.

▶️ Answer/Explanation

Solution

(a)

The cosine graph should start at (0°,1), decrease to (180°,-1), and return to (360°,1), forming a smooth wave.

(b) $x = 120^\circ$ or $x = 240^\circ$

These are the angles in the range where cosine equals -1/2. 120° is in the second quadrant and 240° is in the third quadrant.

Question 23

Topic – E1.11

$y$ is inversely proportional to $\sqrt{x}$ and $x$ is directly proportional to $w^2$.
When $w = 12$, $y = 12$.

Find $y$ in terms of $w$.

▶️ Answer/Explanation

Solution

$y = \frac{144}{w}$

1. First relationship: $y = \frac{k}{\sqrt{x}}$

2. Second relationship: $x = mw^2$

3. Combine: $y = \frac{k}{\sqrt{mw^2}} = \frac{k}{\sqrt{m}w}$

4. Substitute w=12, y=12 to find k/√m = 144

Question 24

Topic – E1.10

Violet and Wilfred recorded their times to run 200 m, correct to the nearest second.
Violet took 36 seconds and Wilfred took 39 seconds.

Work out the upper bound of the difference between their times.

▶️ Answer/Explanation

Solution

4 seconds

1. Violet’s time range: 35.5 ≤ t < 36.5

2. Wilfred’s time range: 38.5 ≤ t < 39.5

3. Maximum difference occurs when Violet’s time is minimum (35.5) and Wilfred’s is maximum (39.5)

4. Difference: 39.5 – 35.5 = 4 seconds

Question 25

Topic – E8.1

A bag contains 5 red balls, 4 blue balls and 3 green balls.

(a) (i) Megan picks a ball at random. Write down the probability that the ball is red or blue.

(ii) Megan replaces the ball.
She picks a ball at random, notes the colour and replaces the ball.
She repeats this 60 times.

Calculate the number of times the ball is expected to be red or blue.

(b) Mick picks 2 of the 12 balls at random, without replacement. Calculate the probability that the balls are different colours.

The probability that she picks a green ball on pick $n$ is $\frac{21}{220}$.

Find the value of $n$.

▶️ Answer/Explanation

Solution

(a)(i) $\frac{3}{4}$ (9 balls out of 12 are red or blue)

(a)(ii) 45 (60 × 3/4)

(b) $\frac{47}{66}$

1. Total ways to pick 2 balls: 12C2 = 66

2. Calculate 1 – P(same color) = 1 – (5C2 + 4C2 + 3C2)/66

(c) n = 5

This is found by calculating when P(green on nth pick without replacement) = 21/220

Resources

Members

Company