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Question 1

Topic – 4.5

(a) Write down the order of rotational symmetry of this diagram.

(b) On the diagram, draw all the lines of symmetry.

▶️ Answer/Explanation

Solution

(a) Ans: 4

The diagram has rotational symmetry of order 4, meaning it looks identical after rotations of 90°.

(b) Ans:

The diagram has 4 lines of symmetry that divide it into identical mirror-image halves.

Question 2

Topic – 8.1

The probability that a train is late is 0.15.

Write down the probability that the train is not late.

▶️ Answer/Explanation

Solution

Ans: 0.85

Since P(late) = 0.15, P(not late) = 1 – 0.15 = 0.85.

This uses the rule that probabilities of complementary events sum to 1.

Question 3

Topic – 9.3

The stem-and-leaf diagram shows the number of hours that each of 16 students studied last week.

Find:

(a) the median,

(b) the mode,

▶️ Answer/Explanation

Solution

(a) Ans: 28

Median is the average of the 8th and 9th values (both 28) in the ordered data set.

(b) Ans: 21

Mode is the most frequent value, which appears twice (21 hours).

(c) Ans: 35

Range = maximum (47) – minimum (12) = 35 hours.

Question 4

Topic – 4.6

The diagram shows two parallel lines intersected by two straight lines.

Find the values of a, b and c.

▶️ Answer/Explanation

Solution

Ans: a = 59, b = 37, c = 84

a = 59° (corresponding angles)

b = 180° – 143° = 37° (angles on straight line)

c = 180° – a – b = 180° – 59° – 37° = 84° (angles in triangle)

Question 5

Topic – 7.2

Work out:

(a) $\begin{pmatrix} 6 \\ -5 \end{pmatrix} + \begin{pmatrix} 8 \\ -1 \end{pmatrix}$

(b) $3 \begin{pmatrix} -4 \\ 7 \end{pmatrix}$

▶️ Answer/Explanation

Solution

(a) Ans: $\begin{pmatrix} 14 \\ -6 \end{pmatrix}$

Add corresponding components: 6+8=14 and -5+(-1)=-6.

(b) Ans: $\begin{pmatrix} -12 \\ 21 \end{pmatrix}$

Multiply each component by 3: 3×(-4)=-12 and 3×7=21.

Question 6

Topic – 2.7

(a) The nth term of a sequence is n² + 3n. Find the first three terms of this sequence.

(b) These are the first five terms of a different sequence.

25, 18, 11, 4, -3

Find the nth term of this sequence.

▶️ Answer/Explanation

Solution

(a) 4, 10, 18

Substitute n=1: 1² + 3×1 = 4

n=2: 2² + 3×2 = 10

n=3: 3² + 3×3 = 18

(b) 32 – 7n

Difference is -7 each time. First term is 25 when n=1: 25 = a + (-7×1) → a = 32

Question 7

Topic – 2.5

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

$2x + y = 3$
$x – 5y = 40$

▶️ Answer/Explanation

Solution

x = 5, y = -7

From first equation: y = 3 – 2x

Substitute into second equation: x – 5(3 – 2x) = 40

Simplify: x – 15 + 10x = 40 → 11x = 55 → x = 5

Substitute x=5 into y=3-2x → y = -7

Question 8

Topic – 1.4

Without using a calculator, work out $ 1\frac{3}{8} – \frac{5}{6}$.

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

▶️ Answer/Explanation

Solution

13/24

Convert 1⅜ to improper fraction: 11/8

Find common denominator (24): 33/24 – 20/24

Subtract numerators: 13/24

This is already in simplest form.

Question 9

Topic – 3.4

A is the point (5, -5) and B is the point (9, 3).

(a) Find the coordinates of the midpoint of AB.

(b) Find the length of AB.

▶️ Answer/Explanation

Solution

(a) (7, -1)

Midpoint formula: ((5+9)/2, (-5+3)/2) = (7, -1)

(b) 8.94 or √80

Distance formula: √[(9-5)² + (3-(-5))²] = √[16 + 64] = √80 ≈ 8.94

Question 10

Topic – 7.1

(a) Describe fully the single transformation that maps

(i) triangle A onto triangle B
(ii) triangle A onto triangle C

(b) Draw the image of triangle A after a translation by the vector $\begin{pmatrix}
2 \\10
\end{pmatrix}$.

▶️ Answer/Explanation

Solution

(a)(i) Rotation 90° anticlockwise about (0, -1)

(a)(ii) Enlargement with scale factor 1/3, center (6, 6)

(b) Triangle at (-4,7), (-4,1), (-1,1)

For (b), each vertex of triangle A moves right 2 units and up 10 units.

Question 11

Topic – 2.4

(a) Simplify fully $(4ab^5)^4$

(b) $2p^{\frac{1}{3}} = 6$. Find the value of $p$.

▶️ Answer/Explanation

Solution

(a) 256a⁴b²⁰

Apply power to each term: 4⁴ = 256, a⁴, (b⁵)⁴ = b²⁰

(b) 27

Divide both sides by 2: p^1/3 = 3

Cube both sides: p = 3³ = 27

(c) 6

Express all terms as powers of 3: 81 = 3⁴, 9 = 3²

Equation becomes (3⁴)² ÷ 3^t = 3²

Simplify: 3^8-t = 3² ⇒ 8-t = 2 ⇒ t = 6

Question 12

Topic – 1.17

The profit a company makes decreases exponentially at a rate of 0.9% per year.
In 2014, the profit was $9500.

Calculate the profit in 2019.

▶️ Answer/Explanation

Solution

$9080 or 9080.13

Use exponential decay formula: $9500 × (1 – 0.009)^5$.

Calculate 0.991^5 ≈ 0.9558, then multiply by 9500.

5 years from 2014 to 2019.

Question 13

Topic – 4.3

On a map, a lake has an area of 32 cm².
The scale of the map is 1 : 24 000.

Calculate the actual area of the lake. Give your answer in km².

▶️ Answer/Explanation

Solution

1.8432 km²

First find linear scale factor: 24,000.

Area scale factor is (24,000)² = 576,000,000.

Actual area = 32 × 576,000,000 = 18,432,000,000 cm².

Convert to km²: ÷100,000² = 1.8432 km².

Question 14

Topic – 2.8

$y$ is directly proportional to the square root of $(x-3)$.
When $x = 28$, $y = 20$.

Find $y$ when $x = 39$.

▶️ Answer/Explanation

Solution

First find constant of proportionality: $y = k\sqrt{x-3}$ → $20 = k\sqrt{25}$ → $k = 4$.

Now find y when x=39: $y = 4\sqrt{39-3} = 4×6 = 24$.

Question 15

Topic – 2.2

Make $h$ the subject of the formula $2mh = g(1-h)$.

▶️ Answer/Explanation

Solution

$h = \frac{g}{2m+g}$

Expand brackets: $2mh = g – gh$.

Collect h terms: $2mh + gh = g$.

Factor out h: $h(2m + g) = g$.

Divide both sides: $h = \frac{g}{2m + g}$.

Question 16

Topic – 3.5

(a) Find the gradient of line l.

(b) Find the equation of line l in the form y = mx + c.

(c) Find the equation of the line that is perpendicular to line l and passes through the point (12, -7). Give your answer in the form y = mx + c.

▶️ Answer/Explanation

Solution

(a) -3/4 or -0.75

Gradient = rise/run = -3/4

(b) y = -3/4x + 2

Using y-intercept at (0,2) and gradient from (a)

(c) y = 4/3x – 23

Perpendicular gradient = 4/3 (negative reciprocal). Substituted (12,-7) into y=4/3x+c to find c.

Question 17

Topic – 8.3

A bag contains 3 blue buttons, 8 white buttons and 5 red buttons.
Two buttons are picked at random from the bag, without replacement.

Work out the probability that the two buttons are either both red or both white.

▶️ Answer/Explanation

Solution

19/60

P(both white) = (8/16)×(7/15) = 56/240

P(both red) = (5/16)×(4/15) = 20/240

Total probability = (56+20)/240 = 76/240 = 19/60

Question 18

Topic – 7.4

S is a point on PQ such that PS : SQ = 4 : 5.

Find $\overrightarrow{OS}$, in terms of a and b, in its simplest form.

▶️ Answer/Explanation

Solution

(5/9)a + (4/9)b

PS:SQ = 4:5 means S divides PQ in ratio 4:5

OS = OP + (4/9)PQ = a + (4/9)(b-a)

Simplify: a + (4/9)b – (4/9)a = (5/9)a + (4/9)b

Question 19

Topic – 6.4

(a) Sketch the graph of y = tan x for 0° ≤ x ≤ 360°.

(b) Solve the equation 5tan x = 1 for 0° ≤ x ≤ 360°.

▶️ Answer/Explanation

Solution

(a) Graph with correct shape: asymptotes at 90° and 270°, passing through (0,0), (180,0), (360,0)

(b) 11.3° and 191.3°

tan x = 1/5 ⇒ x = tan⁻¹(0.2) ≈ 11.3°

Second solution: 11.3° + 180° = 191.3°

Question 20

Topic – 1.10

The distance between two towns is 600 km, correct to the nearest 10 km.
A car takes 8 hours 40 minutes, correct to the nearest 10 minutes, to travel this distance.

Calculate the lower bound for the average speed of the car in km/h.

▶️ Answer/Explanation

Solution

68 km/h

Lower bound distance = 595 km (600-5)

Upper bound time = 8h 45m = 8.75 hours

Lower bound speed = 595 ÷ 8.75 = 68 km/h

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