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Question 1

Topic – E2.7

These are the first eight terms of a sequence.

c -3 -9 -15 -21 -27 -33 k

Find the value of c and the value of k.

▶️ Answer/Explanation

Solution

Ans: c = 3, k = -39

The sequence decreases by 6 each time (arithmetic sequence).

Working backwards: -3 + 6 = 3 → c = 3

Working forwards: -33 – 6 = -39 → k = -39

Question 2

Topic – E4.6

The diagram shows an isosceles triangle.

Find the value of x.

▶️ Answer/Explanation

Solution

Ans: x = 94

In an isosceles triangle, base angles are equal (both 43°).

Sum of angles in triangle = 180°

x + 43° + 43° = 180°

x = 180° – 86° = 94°

Question 3

Topic – E1.1

Complete each statement with a number from the list:

…… is a natural number.

…… is an irrational number.

…… is the reciprocal of 4.

▶️ Answer/Explanation

Solution

Ans: 24, √3, 0.25

24 is a natural number (positive integer).

√3 is irrational (cannot be expressed as a fraction).

0.25 is the reciprocal of 4 (1/4 = 0.25).

Question 4

Topic – E9.4

The scatter diagram shows the number of rooms and the number of people in each of eight buildings.

(a) One building has 67 rooms. Write down the number of people in this building.

(b) In another building there are 42 people and 33 rooms. Plot this point.

(ii) Estimate the number of rooms for 45 people.

(d) What type of correlation is shown?

▶️ Answer/Explanation

Solution

Ans:

(a) 76 people

(b) Point plotted at (42,33)

(c)(i) Straight line through middle of data points

(c)(ii) 27-33 rooms (depending on line)

(d) Positive correlation (as rooms increase, people increase)

Question 5

Topic – E5.1

Convert 7.51 m² into cm².

▶️ Answer/Explanation

Solution

Ans: 75100 cm²

1 m = 100 cm

1 m² = 100 × 100 = 10,000 cm²

7.51 m² = 7.51 × 10,000 = 75,100 cm²

Question 6

Topic – E5.2

The diagram shows a trapezium.

The area of the trapezium is $42cm^{2}$

Calculate the value of x.

▶️ Answer/Explanation

Solution

Ans: 4.5

Area of trapezium = ½ × (sum of parallel sides) × height

42 = ½ × (9.5 + x) × 6

42 = 3 × (9.5 + x)

14 = 9.5 + x

x = 4.5 cm

Question 7

Topic – E1.4

Without using a calculator, work out $\frac{2}{7} ÷ \frac{6}{11}$.

Give your answer as a fraction in its simplest form.

▶️ Answer/Explanation

Solution

Ans: $\frac{11}{21}$

Dividing by a fraction is same as multiplying by its reciprocal

$\frac{2}{7} ÷ \frac{6}{11} = \frac{2}{7} × \frac{11}{6}$

Multiply numerators and denominators: $\frac{22}{42}$

Simplify by dividing numerator and denominator by 2: $\frac{11}{21}$

Question 8

Topic – E4.6

The diagram shows a parallelogram.

Work out the size of the smallest interior angle of the parallelogram.

▶️ Answer/Explanation

Solution

Ans: 70°

Adjacent angles in parallelogram add to 180°

(132-2x) + (15+5x) = 180

147 + 3x = 180 → 3x = 33 → x = 11

First angle: 132 – 2(11) = 110°

Second angle: 15 + 5(11) = 70°

Question 9

Topic – E5.3

Points A, B, C and D lie on a circle.
ABCD is a square with area 72 cm².

Calculate the area of the circle. Give your answer as a multiple of π.

▶️ Answer/Explanation

Solution

Ans: 36π cm²

Square area = side² = 72 → side = √72 cm

Diagonal of square = diameter of circle = √72 × √2 = √144 = 12 cm

Radius = 6 cm

Area of circle = πr² = π × 6² = 36π cm²

Question 10

Topic – E1.14

Calculate $\sqrt[3]{1 + 10.9 \times 0.4^{2}}$.

▶️ Answer/Explanation

Solution

Ans: 1.4

First calculate $0.4^2 = 0.16$.

Then multiply: $10.9 \times 0.16 = 1.744$.

Add 1: $1 + 1.744 = 2.744$.

Finally take cube root: $\sqrt[3]{2.744} = 1.4$.

Question 11

Topic – E2.2

Factorise fully.

(a) 24x² – 9xy

(b) 63x² – 28y²

▶️ Answer/Explanation

Solution

(a) Ans: 3x(8x – 3y)

Take out common factor 3x: 3x(8x) – 3x(3y) = 3x(8x – 3y)

(b) Ans: 7(3x + 2y)(3x – 2y)

First take out 7: 7(9x² – 4y²). Then recognize difference of squares: 7(3x + 2y)(3x – 2y)

Question 12

Topic – E2.8

y is directly proportional to the square root of (x + 1).
y = 10.5 when x = 8.

Find y when x = 1.56.

▶️ Answer/Explanation

Solution

Ans: 5.6

First write equation: y = k√(x+1). Substitute y=10.5, x=8 to find k=3.5.

Then use k to find y when x=1.56: y = 3.5√(1.56+1) = 3.5×1.6 = 5.6

Question 13

Topic – E2.6

The region R satisfies these inequalities:

-3 < y ≤ 2 y ≤ x – 1

By drawing suitable straight lines and shading unwanted regions, find and label the region R.

▶️ Answer/Explanation

Solution

Ans:

1. Draw dashed line y=-3 (not included) and shade above it

2. Draw solid line y=2 (included) and shade below it

3. Draw solid line y=x-1 (included) and shade below it

The region R is where all three conditions overlap

Question 14

Topic – E2.9

The diagram shows the speed-time graph for 17 seconds of a car journey.

(a) Find the acceleration of the car during the first 10 seconds.

(b) Calculate the total distance travelled by the car during the 17 seconds.

▶️ Answer/Explanation

Solution

(a) Ans: 2 m/s²

Acceleration = gradient = (20-0)/10 = 2 m/s²

(b) Ans: 240 m

Distance = area under graph = (½×10×20) + (7×20) = 100 + 140 = 240 m

Question 15

Topic – E1.17

At the start of an experiment there are 40,000 bacteria.
The number of bacteria increases at a rate of 15% per hour.

Calculate the number of bacteria after 3 hours.

▶️ Answer/Explanation

Solution

Ans: 60,835

Use compound growth formula: 40,000 × (1.15)³

First hour: 40,000 × 1.15 = 46,000

Second hour: 46,000 × 1.15 = 52,900

Third hour: 52,900 × 1.15 = 60,835

Question 16

Topic – E8.4

75 people are asked if they have a car, C, and if they have a job, J.
The Venn diagram shows the results.

A person is chosen at random from those who have a car.

Find the probability that this person also has a job.

▶️ Answer/Explanation

Solution

Ans: 17/20

From the Venn diagram:

People with cars (C) = 9 (C∩J) + 51 (C only) = 60

People with jobs among car owners = 9 (C∩J)

The Venn diagram shows:

• People with both car and job (C∩J) = 51

• People with car only = 9

• Therefore probability = 51/(51+9) = 51/60 = 17/20

Question 17

Topic – E4.8

A, B and C are points on the circumference of a circle with center O.
DA and DC are tangents to the circle.
Angle ABC = 64°.

Work out the value of x.

▶️ Answer/Explanation

Solution

Ans: 52

Angle at center AOC = 2 × angle ABC = 128°.

Tangents meet radius at 90°, so angles OAD and OCD are 90°.

Quadrilateral AOCD has angles summing to 360°: 90 + 90 + 128 + x = 360.

Therefore x = 360 – 308 = 52°.

Question 18

Topic – E1.2

(a) ξ = {8×10⁻¹, 0.8̇, 8%, 0.08}

A = {a: 0.08 ≤ a ≤ 1}
B = {b: b > 0.8}

Complete the Venn diagram.

(b) Shade the region (A∪C)∩B’ in the Venn diagram.

▶️ Answer/Explanation

Solution

18(a) Answer:

First convert all numbers to decimal form:

8×10⁻¹ = 0.8

0.8̇ (recurring) = 0.888…

8% = 0.08

0.08 remains 0.08

Set A (0.08 ≤ a ≤ 1) contains: 0.8, 0.8̇, 0.08

Set B (b > 0.8) contains: 0.8̇ (only)

Therefore:

• Only in A: 0.8, 0.08

• A∩B: 0.8̇

• Outside both: none (all elements are in A)

18(b) Answer:

To shade (A∪C)∩B’:

1. A∪C means everything in A or C (but C isn’t defined here)

2. B’ means everything not in B

3. The intersection would be all elements that are in (A or C) AND not in B

Assuming C is empty, this would be all elements of A not in B: 0.8 and 0.08

Question 19

Topic – E5.4

A solid is made from a cylinder and a hemisphere, both of radius 4.3 cm.
The cylinder has length 11.9 cm.

a) Calculate the volume of the solid.
[The volume, V, of a sphere with radius r is $V = \frac{4}{3} \Pi r^{3}$]

b) Calculate the total surface area of the solid.
[The surface area, A, of a sphere with radius r is $A = 4 \Pi r^{2}$]

▶️ Answer/Explanation

Solution

Ans: a) 858 cm³ b) 496 cm²

a) Hemisphere volume = ⅔πr³ = ⅔π(4.3)³ ≈ 166.5 cm³.

Cylinder volume = πr²h = π(4.3)²(11.9) ≈ 691.2 cm³.

Total volume ≈ 166.5 + 691.2 = 857.7 cm³.

b) Hemisphere area = 2πr² = 2π(4.3)² ≈ 116.2 cm².

Cylinder curved area = 2πrh = 2π(4.3)(11.9) ≈ 321.7 cm².

Total area ≈ 116.2 + 321.7 + π(4.3)² ≈ 496 cm².

Question 20

Topic – E2.7

Find an expression for the nth term of this sequence.

$\frac{1}{7}$, 1, 7, 49, 343, 2401, …

▶️ Answer/Explanation

Solution

Ans: 7ⁿ⁻²

Observe the pattern: 7⁻¹, 7⁰, 7¹, 7², 7³, 7⁴,…

The first term (n=1) is 7⁻¹ = 1/7.

Second term (n=2) is 7⁰ = 1.

Third term (n=3) is 7¹ = 7.

Therefore, nth term = 7ⁿ⁻².

Question 21

Topic – E2.2

Expand and simplify.

$(x+3)(x+5)(2x+1)$

▶️ Answer/Explanation

Solution

Ans: $2x^3 + 17x^2 + 38x + 15$

First expand $(x+3)(x+5)$ to get $x^2 + 8x + 15$.

Then multiply by $(2x+1)$: $2x^3 + x^2 + 16x^2 + 8x + 30x + 15$.

Combine like terms: $2x^3 + 17x^2 + 38x + 15$.

Question 22

Topic – E3.5

$A$ is the point (17, 9) and $B$ is the point (23, 39).

Find the equation of the perpendicular bisector of line $AB$. Give your answer in the form $y = mx + c$.

▶️ Answer/Explanation

Solution

Ans: $y = -\frac{1}{5}x + 28$

1. Find midpoint: $\left(\frac{17+23}{2}, \frac{9+39}{2}\right) = (20, 24)$

2. Gradient of AB: $\frac{39-9}{23-17} = 5$

3. Perpendicular gradient: $-\frac{1}{5}$ (negative reciprocal)

4. Equation using point-slope form: $y – 24 = -\frac{1}{5}(x – 20)$

5. Simplify: $y = -\frac{1}{5}x + 4 + 24$ → $y = -\frac{1}{5}x + 28$

Question 23

Topic – E4.4

The small box is mathematically similar to the large box.
The volume of the large box is 72.8% greater than the volume of the small box.
The small box has length 3.5 cm and the large box has length $x$ cm.

Calculate the value of $x$.

▶️ Answer/Explanation

Solution

Ans: 4.2

Volume scale factor = 1 + 0.728 = 1.728.

Length scale factor = $\sqrt[3]{1.728} = 1.2$.

Large box length = 3.5 cm × 1.2 = 4.2 cm.

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