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Question 1

Topic – 1.1

$P$ is a prime number where $60 < P < 80$.
$P$ is 2 less than a square number.

Find the value of $P$.

▶️ Answer/Explanation

Solution

Ans: 79

Find squares between 60+2 and 80+2: 8²=64, 9²=81.

P = square – 2 → 64-2=62, 81-2=79.

Check primes between 60-80: 79 is prime, 62 isn’t.

Question 2

Topic – 1.15

Hank flies from Los Angeles to Shanghai.

(a) The flight departs on Friday 22 July at 2140.
The flight takes 13 hours 35 minutes.
The local time in Shanghai is 15 hours ahead of the local time in Los Angeles.

Find the day, date and time in Shanghai when Hank’s flight arrives.

(b) The cost of the flight is \$920.
The exchange rate is \$1 = 6.87 Chinese yuan.

Find the cost of the flight in yuan.

▶️ Answer/Explanation

Solution

Ans: (a) Sunday 24 July 0215 (b) 6320.40 yuan

(a) Departure: Fri 22/7 2140 + 13h35m = Sat 23/7 1115 LA time.

Shanghai time: 1115 + 15h = Sun 24/7 0215.

(b) 920 × 6.87 = 6320.40 yuan.

Question 3

Topic – 1.6

Calculate.

$\frac{4.87 – 2.7}{-0.2 + \sqrt[3]{0.729}}$

▶️ Answer/Explanation

Solution

Ans: 3.1

Numerator: 4.87 – 2.7 = 2.17.

Denominator: -0.2 + 0.9 = 0.7 (since 0.9³=0.729).

Final calculation: 2.17 ÷ 0.7 ≈ 3.1.

Question 4

Topic – 9.3

The number of items that each of 22 people buy in a supermarket is shown in the stem-and-leaf diagram.

(a) Find the mode.

(b) Find the median.

▶️ Answer/Explanation

Solution

Ans: (a) 22 (b) 30

(a) Mode is most frequent value: 22 appears most in the data.

(b) Median is average of 11th and 12th values (for 22 people): both are 30.

Question 5

Topic – 8.2

The table shows the relative frequency of the games won by a football team.

Result of game	won	lost	drawn
Relative frequency	0.1

The number of games lost is twice the number of games drawn.

Complete the table.

▶️ Answer/Explanation

Solution

Ans: lost 0.6, drawn 0.3

Total probability = 1 → remaining = 0.9.

Let drawn = x → lost = 2x → x + 2x = 0.9 → x = 0.3.

Thus drawn = 0.3, lost = 0.6.

Question 6

Topic – 4.3

The scale drawing shows the positions of two towns, P and Q.
The scale is 1 cm represents 4 km.

(a) Find the actual distance between town P and town Q.

(b) Measure the bearing of town Q from town P.

▶️ Answer/Explanation

Solution

(a) 32.8 km

Measure distance PQ on the diagram (about 8.2 cm) and multiply by 4.

(b) 065°

Measure the angle clockwise from north at P to the line PQ.

(c) Mark X 7 cm from P at 140°

28 km ÷ 4 = 7 cm. Draw a line from P at 140° and mark X 7 cm away.

Question 7

Topic – 1.4

Without using a calculator, work out $1 \frac{5}{6} + \frac{2}{5}$.

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

▶️ Answer/Explanation

Solution

$2\frac{7}{30}$

Convert $1\frac{5}{6}$ to improper fraction: $\frac{11}{6}$

Find common denominator (30): $\frac{55}{30} + \frac{12}{30} = \frac{67}{30}$

Convert to mixed number: $2\frac{7}{30}$

Question 8

Topic – 2.5

Solve the simultaneous equations.
You must show all your working.

$4x – 2y = -13$
$-3x + 4y = 11$

▶️ Answer/Explanation

Solution

$x = -3$, $y = 0.5$

Multiply first equation by 2: $8x – 4y = -26$

Add to second equation: $5x = -15$ → $x = -3$

Substitute into first equation: $-12 – 2y = -13$ → $-2y = -1$ → $y = 0.5$

Question 9

Topic – 6.1

Calculate the value of $x$.

▶️ Answer/Explanation

Solution

$x = 54.3°$

Use cosine ratio: $\cos x = \frac{7}{12}$

Calculate inverse cosine: $x = \cos^{-1}(\frac{7}{12})$

Evaluate: $x ≈ 54.31°$ (round to 54.3°)

Question 10

Topic – 4.1

A regular polygon has an interior angle of 174°.

Find the number of sides of this polygon.

▶️ Answer/Explanation

Solution

60 sides

Use interior angle formula: $(n-2) × 180° = n × 174°$

Expand: $180n – 360 = 174n$

Solve: $6n = 360$ → $n = 60$

Question 11

Topic – 3.7

Line L has equation $y = 4 – 5x$.

Find the equation of a line that is perpendicular to line L and passes through the point (0, 6).

▶️ Answer/Explanation

Solution

Ans: $y = \frac{1}{5}x + 6$

1. Original slope is -5, so perpendicular slope is $\frac{1}{5}$ (negative reciprocal)

2. Using point-slope form with point (0,6): $y – 6 = \frac{1}{5}(x – 0)$

3. Simplify to get $y = \frac{1}{5}x + 6$

Question 12

Topic – 1.13

Chai invests some money.
By the end of the first year, the value of the investment has decreased by 35%.
By the end of the second year, the value of the investment has increased by 40% of its value at the end of the first year.

Find the overall percentage change in the value of the investment.

▶️ Answer/Explanation

Solution

Ans: -9%

1. Let initial amount be $100

2. After first year: $100 × 0.65 = $65

3. Second year increase: $65 × 0.40 = $26

4. Final amount: $65 + $26 = $91

5. Overall change: $91 – $100 = -9%

Question 13

Topic – 2.6

Solve.

$4 – 3x \geq \frac{6 – x}{5}$

▶️ Answer/Explanation

Solution

Ans: $x \leq 1$

1. Multiply both sides by 5: $20 – 15x \geq 6 – x$

2. Bring like terms together: $20 – 6 \geq 15x – x$

3. Simplify: $14 \geq 14x$

4. Divide by 14: $x \leq 1$

Question 14

Topic – 2.8

$y$ is inversely proportional to the square root of $(x – 2)$.
When $x = 4.25$, $y = 12$.

Find $x$ when $y = 3$.

▶️ Answer/Explanation

Solution

Ans: $x = 38$

1. Set up inverse proportion: $y = \frac{k}{\sqrt{x-2}}$

2. Find k using given values: $12 = \frac{k}{\sqrt{4.25-2}} ⇒ k = 12×1.5 = 18$

3. For y=3: $3 = \frac{18}{\sqrt{x-2}}$

4. Solve: $\sqrt{x-2} = 6 ⇒ x-2 = 36 ⇒ x = 38$

Question 15

Topic – 4.4

The diagram shows three shapes that are mathematically similar.
The heights of the shapes are in the ratio small : medium : large = 1 : 5 : 8.

Find the ratio shaded area : total unshaded area. Give your answer in its simplest form.

▶️ Answer/Explanation

Solution

Ans: 3:5

1. Area ratio is square of height ratio: 1:25:64

2. Shaded area = medium – small = 25-1 = 24

3. Unshaded areas = small + (large – medium) = 1 + (64-25) = 40

4. Ratio shaded:unshaded = 24:40 = 3:5

Question 16

Topic – 2.7

Find the nth term of each sequence.

(a) 8, 15, 34, 71, 132, …

(b) $\frac{2}{1}$, $\frac{3}{4}$, $\frac{4}{16}$, $\frac{5}{64}$, $\frac{6}{256}$, …

▶️ Answer/Explanation

Solution

(a) Ans: n³ + 7

Looking at differences: 7, 19, 37, 61 (second differences: 12, 18, 24). The cubic pattern suggests n³. Testing: 1³+7=8, 2³+7=15, etc.

(b) Ans: $\frac{n+1}{4^{n-1}}$

Numerator increases by 1: n+1. Denominator follows 4^(n-1) pattern: 4⁰=1, 4¹=4, 4²=16, etc.

Question 17

Topic – 2.2

$y = \frac{3x – 2}{1 – x}$

Make x the subject of the formula.

▶️ Answer/Explanation

Solution

Ans: $x = \frac{y + 2}{y + 3}$

Multiply both sides by (1-x): y(1-x) = 3x-2

Expand: y – xy = 3x – 2

Gather x terms: y + 2 = 3x + xy

Factor out x: y + 2 = x(3 + y)

Divide both sides by (3+y): x = (y+2)/(y+3)

Question 18

Topic – 5.2

The diagram shows some land in the shape of a triangle ABC.
Houses are built on this land.
Each house requires 400 m² of land.

Find the greatest number of houses that can be built on this land.

▶️ Answer/Explanation

Solution

Ans: 1150

First calculate triangle area: ½ × 800 × 2300 × sin30°

sin30° = 0.5, so area = ½ × 800 × 2300 × 0.5 = 460,000 m²

Number of houses = Total area ÷ Area per house

460,000 ÷ 400 = 1,150 houses

Question 19

Topic – 2.3

Write as a single fraction in its simplest form.

$\frac{2}{x+3} – \frac{x+2}{7}$

▶️ Answer/Explanation

Solution

Ans: $\frac{8 – 5x – x^2}{7(x+3)}$

Find common denominator: 7(x+3)

First term becomes: 14/[7(x+3)]

Second term becomes: [(x+2)(x+3)]/[7(x+3)]

Combine: [14 – (x²+5x+6)]/[7(x+3)]

Simplify numerator: 8-5x-x²

Question 20

Topic – 6.4

Solve $3(2+\cos x) = 5$ for $0° ≤ x ≤ 360°$.

▶️ Answer/Explanation

Solution

Ans: 109.5°, 250.5°

First divide both sides by 3: 2 + cosx = 5/3

Subtract 2: cosx = -1/3

Find reference angle: cos⁻¹(1/3) ≈ 70.5°

Solutions in range: 180°-70.5° ≈ 109.5°

and 180°+70.5° ≈ 250.5°

Question 21

Topic – 6.6

The diagram shows a pyramid ABCDE.
The pyramid has a square horizontal base ABCD with side 5 cm.
The vertex E is vertically above the centre O of the base.
The height OE of the pyramid is 9 cm.

Calculate the angle that EC makes with the base ABCD.

▶️ Answer/Explanation

Solution

Ans: 68.6° or 68.55° to 68.56°

1. Find half-diagonal of base: $\frac{5\sqrt{2}}{2} ≈ 3.5355$ cm

2. This forms adjacent side, height (9 cm) is opposite side

3. Angle is $\tan^{-1}\left(\frac{9}{3.5355}\right)$

4. Calculate: $\tan^{-1}(2.5456) ≈ 68.55°$

Question 22

Topic – 2.3

(a) Simplify.

$\frac{x^{\frac{2}{3}}}{x^{\frac{8}{3}}}$

(b) $16 = 64^k$

Find the value of $k$.

$3^{3x} \times \left( \frac{1}{9} \right)^{4-3x} = 3$

▶️ Answer/Explanation

Solution

(a) Ans: $\frac{1}{x^2}$ or $x^{-2}$

Using laws of indices: $x^{\frac{2}{3} – \frac{8}{3}} = x^{-2} = \frac{1}{x^2}$

(b) Ans: $\frac{2}{3}$

Express as powers of 4: $4^2 = 4^{3k}$ → $2 = 3k$ → $k = \frac{2}{3}$

(c) Ans: 1

Rewrite $\frac{1}{9}$ as $3^{-2}$: $3^{3x} \times 3^{-2(4-3x)} = 3^1$

Combine exponents: $3x – 8 + 6x = 1$ → $9x = 9$ → $x = 1$

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