CTJan27 Online JMSS - Mathematics 20052026

1

Solve the linear equation for $x$: $5(x - 3) - 2(x + 1) = 4(x - 2)$

$x = 9$

$x = -25$

$x = 5$

$x = -9$

$x = -5$

2

Solve the following system of simultaneous linear equations for $x$ and $y$:

$3x + 2y = 12$
$5x - y = 7$

$x = 1, y = 4.5$

$x = 4, y = 0$

$x = 2, y = 3$

$x = 3, y = 2$

$x = 2, y = -3$

3

A line segment has endpoints A$(-4, 9)$ and B$(2, -1)$. What are the gradient and the midpoint of the segment AB?

Gradient: $-3/5$, Midpoint: $(-1, 4)$

Gradient: $-10$, Midpoint: $(6, 8)$

Gradient: $-5/3$, Midpoint: $(-3, 5)$

Gradient: $5/3$, Midpoint: $(-1, 4)$

Gradient: $-5/3$, Midpoint: $(-1, 4)$

4

Calculate the exact distance between the points P$(1, -3)$ and Q$(6, 9)$.

$13$

$\sqrt{119}$

$7$

$17$

$\sqrt{61}$

5

Factorise the quadratic trinomial $x^2 - 5x - 24$.

$(x - 8)(x + 3)$

$(x - 12)(x + 2)$

$(x - 6)(x + 4)$

$(x + 6)(x - 4)$

$(x + 8)(x - 3)$

6

Factorise completely $49x^2 - 81y^2$.

Does not factorise

$(7x - 9y)(7x + 9y)$

$(7x^2 - 9y^2)(7+9)$

$(49x - 81y)(x + y)$

$(7x - 9y)^2$

7

Identify the vertex of the parabola with the equation $y = 2(x - 3)^2 + 5$.

$(2, 5)$

$(-3, 5)$

$(-3, -5)$

$(3, 5)$

$(3, -5)$

8

Find the x-intercepts and y-intercept of the parabola given by the equation $y = x^2 - 8x + 15$.

x-intercepts at $(-3, 0), (-5, 0)$ and y-intercept at $(0, 15)$

x-intercepts at $(0, 3), (0, 5)$ and y-intercept at $(15, 0)$

x-intercepts at $(-8, 0), (15, 0)$ and y-intercept at $(0, 1)$

x-intercepts at $(3, 0), (5, 0)$ and y-intercept at $(0, -15)$

x-intercepts at $(3, 0), (5, 0)$ and y-intercept at $(0, 15)$

9

Simplify the expression $\frac{(a^3 b^{-2})^4}{a^5 b^{-10}}$, ensuring all indices are positive.

$a^{17} b^{-18}$

$\frac{a^7}{b^{18}}$

$a^2 b^2$

$a^7 b^2$

$a^7 b^{-18}$

10

Evaluate $27^{\frac{2}{3}} \times 16^{-\frac{1}{4}}$.

$18$

$4.5$

$-18$

$7$

$1.5$

11

Simplify $\sqrt{72} + \sqrt{50} - \sqrt{18}$.

$14\sqrt{2}$

$4\sqrt{2}$

$\sqrt{104}$

$8\sqrt{3}$

$8\sqrt{2}$

12

Simplify the expression $\frac{12}{5 - \sqrt{7}}$ by rationalising the denominator.

$\frac{10 - 2\sqrt{7}}{3}$

$\frac{15 + 3\sqrt{7}}{8}$

$\frac{10 + 2\sqrt{7}}{3}$

$\frac{60 + 12\sqrt{7}}{32}$

$2(5 + \sqrt{7})$

13

Sarah invests 8000 dollars in an account that pays 5% per annum compound interest. What is the total value of her investment after 3 years, to the nearest dollar?

9200 dollars

9261 dollars

8400 dollars

25200 dollars

9260 dollars

14

The time ($t$ hours) it takes to build a wall is inversely proportional to the number of builders ($n$) working. If it takes 3 builders 8 hours to build the wall, how long would it take 4 builders?

10.7 hours

6 hours

4 hours

12 hours

9 hours

15

A triangular prism has a length of 15 cm. Its triangular base is a right-angled triangle with side lengths 5 cm, 12 cm, and 13 cm. What is the volume of the prism?

$510 \text{ cm}^3$

$975 \text{ cm}^3$

$480 \text{ cm}^3$

$450 \text{ cm}^3$

$900 \text{ cm}^3$

16

A cylinder has a radius of 4 cm and a height of 10 cm. Calculate its total surface area, using $\pi \approx 3.14$.

$251.2 \text{ cm}^2$

$502.4 \text{ cm}^2$

$351.68 \text{ cm}^2$

$100.48 \text{ cm}^2$

$175.84 \text{ cm}^2$

17

A rectangular box has dimensions 8 cm by 6 cm by 5 cm. What is the length of the space diagonal (the longest diagonal inside the box), correct to one decimal place?

$19.0$ cm

$11.0$ cm

$10.0$ cm

$12.5$ cm

$11.2$ cm

18

Two triangles, ABC and XYZ, are similar. The side lengths of triangle ABC are 6 cm, 8 cm, and 10 cm. The longest side of triangle XYZ is 15 cm. What is the perimeter of triangle XYZ?

$29$ cm

$24$ cm

$36$ cm

$16$ cm

$360$ cm

19

From the top of a vertical cliff 80 metres high, the angle of depression to a boat at sea is $25^\circ$. How far is the boat from the base of the cliff, to the nearest metre?

$172$ m

$80$ m

$190$ m

$88$ m

$37$ m

20

Solve the following equation for $x$:

$\frac{2(x-3)}{5} - \frac{x-4}{2} = 1$

$x = 7$

$x = -42$

$x = -2$

$x = -12$

$x = \frac{14}{3}$

21

At a theatre, the cost for 3 adults and 2 children is 60 dollars, while the cost for 2 adults and 5 children is 73 dollars. What is the cost of a single adult ticket?

15 dollars

9 dollars

14 dollars

23 dollars

12 dollars

22

A line segment has endpoints at $P(-5, 9)$ and $Q(3, -7)$. What are the coordinates of the midpoint of the segment PQ?

$(8, -16)$

$(-4, 8)$

$(-1, 1)$

$(-2, 2)$

$(1, -1)$

23

What is the distance between the points $A(-2, -1)$ and $B(4, 7)$?

$100$

$10$

$2\sqrt{2}$

$14$

$\sqrt{10}$

24

Which of the following is a complete factorisation of the quadratic trinomial $3x^2 - 13x - 10$?

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

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

$(x-10)(3x+1)$

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

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

25

Factorise the expression $81x^4 - 16$ completely.

$(3x-2)^2(3x+2)^2$

$(9x^2-4)(9x^2+4)$

$(9x^2+4)(3x-2)(3x+2)$

$(81x^2+16)(x-1)(x+1)$

$(9x^2-4)^2$

26

The graph of the parabola $y = -2(x+1)^2 + 8$ is shown. What are the coordinates of its vertex?

$(-1, 8)$

$(2, 8)$

$(-1, -8)$

$(1, 8)$

$(-2, 0)$

27

Consider the parabola with the equation $y = x^2 - 3x - 10$. A graph of the parabola is shown. What are the x-coordinates of its x-intercepts?

$-10$ and $3$

$1$ and $-10$

$10$ and $-1$

$5$ and $-2$

$-5$ and $2$

28

Simplify the expression $\frac{(2a^3b^{-2})^3}{4a^5b^4}$.

$\frac{6a^4}{b^{10}}$

$2a^4b^{-10}$

$\frac{2a^4}{b^2}$

$\frac{2a^4}{b^{10}}$

$\frac{2a^{-4}}{b^{2}}$

29

If $5^{2x-1} = 125$, what is the value of $x$?

$-1$

$3$

$\frac{3}{2}$

$2$

$1$

30

Simplify the expression $\sqrt{75} - \sqrt{12} + \sqrt{48}$.

$11\sqrt{3}$

$7\sqrt{3}$

$\sqrt{111}$

$3\sqrt{3}$

$\sqrt{39}$

31

Simplify the expression $\frac{6}{\sqrt{5} - \sqrt{2}}$ by rationalising the denominator.

$2(\sqrt{5} + \sqrt{2})$

$\frac{2}{3}(\sqrt{5} + \sqrt{2})$

$6(\sqrt{5} + \sqrt{2})$

$2(\sqrt{5} - \sqrt{2})$

$\frac{6\sqrt{3}}{3}$

32

Amelia invests 8000 dollars in a bank account earning simple interest at a rate of 3.5% per annum. How much interest will she have earned after 5 years?

1472.45 dollars

1400 dollars

9400 dollars

280 dollars

140 dollars

33

Liam invests 5000 dollars in an account that pays 4% per annum compound interest, compounded annually. How much interest has he earned after 3 years? Give your answer to the nearest dollar.

625 dollars

624 dollars

725 dollars

5624 dollars

600 dollars

34

The time taken to build a house is inversely proportional to the number of builders working on it. If it takes 10 builders 120 days to build a house, how long would it take 15 builders?

1200 days

80 days

60 days

105 days

180 days

35

A triangular prism has a base which is a right-angled triangle with side lengths 5 cm, 12 cm, and 13 cm. The length of the prism is 20 cm. What is its total surface area?

$720 \text{ cm}^2$

$600 \text{ cm}^2$

$660 \text{ cm}^2$

$300 \text{ cm}^2$

$630 \text{ cm}^2$

36

A cylinder has a radius of 4 cm and a height of 10 cm. What is its volume? (Leave your answer in terms of $\pi$)

$160\pi \text{ cm}^3$

$400\pi \text{ cm}^3$

$40\pi \text{ cm}^3$

$80\pi \text{ cm}^3$

$112\pi \text{ cm}^2$

37

A 15-metre ladder leans against a vertical wall. The base of the ladder is 9 metres away from the base of the wall. How high up the wall does the ladder reach?

$\sqrt{306}$ m

24 m

6 m

12 m

144 m

38

A rectangular prism (cuboid) has side lengths of 3 cm, 4 cm, and 12 cm. What is the length of the space diagonal, which connects two opposite vertices of the prism?

13 cm

19 cm

$\sqrt{153}$ cm

169 cm

5 cm

39

Two triangles, $\triangle ABC$ and $\triangle XYZ$, are similar. The side lengths of $\triangle ABC$ are 6 cm, 8 cm, and 10 cm. The shortest side of $\triangle XYZ$ is 9 cm. What is the perimeter of $\triangle XYZ$?

36 cm

40 cm

16 cm

24 cm

27 cm

40

From the top of a 100-metre tall lighthouse, the angle of depression to a boat at sea is $30^\circ$. How far is the boat from the base of the lighthouse, to the nearest metre? You may use $\sqrt{3} \approx 1.732$.

87 m

58 m

50 m

200 m

173 m

41

A clothing store increases the price of a winter coat by 50%. After two weeks, the coat has not sold, so the store discounts the new price by 50%. The final price of the coat is what percentage of the original price?

$50%$

$100%$

$75%$

$25%$

$0%$

42

When a positive integer $N$ is divided by 8, the remainder is 5. What is the remainder when $3N + 7$ is divided by 8?

$5$

$4$

$2$

$6$

$0$

43

The diagram shows a composite shape formed from a square PQRS with side length 14 cm. A semicircle with diameter PQ is attached to the outside of the square, and a quarter-circle with centre S and radius SR is removed from the square. What is the area of the shaded region, in cm$^2$?

Diagram Description: A square PQRS is shown. The vertices are P (top-left), Q (top-right), R (bottom-right), S (bottom-left). A semicircle is drawn outside the square using PQ as its diameter. The region inside this semicircle is shaded. A quarter-circle arc is drawn with S as the centre, passing through P and R. The region of the square outside this quarter-circle is shaded. The total shaded area consists of the semicircle and the region inside the square but outside the quarter-circle.

$42$

$196 - 49\pi$

$49$

$196$

$196 - 24.5\pi$

44

The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number is 36 greater than the original number. What is the original number?

$57$

$39$

$48$

$93$

$84$

45

Consider the sequence: $3, 7, 13, 21, 31, ...$. What is the 10th term in this sequence?

$121$

$143$

$101$

$111$

$131$

46

In the diagram, ABCD is a square and ABE is an equilateral triangle constructed inside the square. The diagonal AC intersects the side BE at point F. What is the size of angle $\angle BFC$?

Diagram Description: A square ABCD is drawn. Its vertices are A (top-left), B (top-right), C (bottom-right), D (bottom-left). An equilateral triangle ABE is drawn inside the square, so vertex E is below the side AB. The diagonal from A to C is drawn. The line segment from B to E is drawn. These two lines, AC and BE, intersect at a point labelled F.

$90^\circ$

$120^\circ$

$100^\circ$

$105^\circ$

$115^\circ$

47

A point $P(3, -2)$ is reflected across the line $y=x$ to obtain point $Q$. Point $Q$ is then rotated $90^\circ$ clockwise about the origin $(0,0)$ to obtain point $R$. What are the coordinates of point $R$?

$(2, -3)$

$(-2, -3)$

$(2, 3)$

$(-3, -2)$

$(3, 2)$

48

A cyclist travels from Town A to Town B at a constant speed of 30 km/h. She immediately returns from Town B to Town A along the same route at a constant speed of 20 km/h. What is her average speed for the entire round trip?

$27.5\text{ km/h}$

$24\text{ km/h}$

$26\text{ km/h}$

$25\text{ km/h}$

$22.5\text{ km/h}$

49

Which of the following expressions is equivalent to $\frac{3^{n+2} - 3^n}{3^{n+1} + 3^n}$?

$2$

$3^n$

$3$

$\frac{1}{3}$

$\frac{8}{3}$

50

Four friends, Ali, Ben, Chloe, and Dan, ran a race. No two friends finished at the same time. They made the following statements:

Ali: "I did not finish first or last."
Ben: "I did not finish last."
Chloe: "I finished first."
Dan: "I finished last."

Exactly one of the four statements is false. Who finished in second place?

Ali

Dan

Ben

Cannot be determined

Chloe

51

A solid wooden cube has a side length of 4 cm. A smaller cube with a side length of 2 cm is removed from the very centre of the large cube. What is the total surface area of the remaining solid?

Diagram Description: A large, transparent cube is shown. Inside it, perfectly centred, is a smaller, opaque cube, representing the removed volume. The axes of the inner cube are aligned with the axes of the outer cube.

$120\text{ cm}^2$

$112\text{ cm}^2$

$88\text{ cm}^2$

$144\text{ cm}^2$

$96\text{ cm}^2$

52

The ratio of the number of apples to oranges in a box was $5:8$. After 22 apples were added to the box, the ratio of apples to oranges became $2:1$. How many oranges were in the box?

$20$

$24$

$12$

$16$

$32$

53

In the rectangle PQRS, $PQ = 20$ cm and $PS = 15$ cm. The rectangle is folded along the line QT such that the vertex P lands on the side SR at point P'. What is the length of QT?

Diagram Description: A rectangle PQRS with P at top-left, Q at top-right, R at bottom-right, S at bottom-left. A point T is on the side PS. A fold line is drawn from Q to T. The corner P is shown folded down, so that the new position P' lies on the side SR. The line segment P'T is visible. The triangle QPT is shown in its folded position as QP'T.

$17.61\text{ cm}$

$12.5\text{ cm}$

$18\text{ cm}$

$15\text{ cm}$

$\frac{50}{3}\text{ cm}$

54

The graph shows the population of a bacterial colony over time. The population follows a logistic growth model, where it initially grows exponentially, then slows down as it approaches a maximum carrying capacity. At which population size is the rate of growth of the colony the greatest?

[graph]

$\approx 25$

$\approx 50$

$\approx 10$

$\approx 75$

$\approx 100$

55

A binary operation $\otimes$ is defined on the set of real numbers as $a \otimes b = a + b - 2ab$. Find the value of $x$ such that $(x \otimes 3) \otimes 4 = -29$.

$-2/35$

$-12/35$

$1/2$

$3/5$

$0.5$

56

A school committee of 4 members is to be selected from a group of 6 teachers and 10 students. How many different committees can be formed if the committee must contain at least 2 teachers?

$675$

$1140$

$890$

$810$

$1820$

57

The seven-digit number $15A9B4C$ is divisible by 792. What is the value of $A-B+C$?

$10$

$7$

$-2$

$1$

None of the given

58

A regular hexagon and a regular octagon share a common side, with their interiors on opposite sides of the common side. What is the angle formed by extending the sides adjacent to the common side until they meet?

Diagram Description: A horizontal line segment PQ is shown. Above PQ, a regular hexagon PQRSTU is partially drawn, showing sides UP and QR. Below PQ, a regular octagon PQCDEFGH is partially drawn, showing sides HP and CD. The lines UP and CD are extended until they meet at a point X.

$90^\circ$

$80^\circ$

$75^\circ$

$120^\circ$

$105^\circ$

59

The line with equation $y = x + 2$ intersects the parabola with equation $y = x^2$ at two points, A and B. What is the area of the triangle formed by the points A, B, and the origin O(0,0)?

[graph]

$3.5$

$3$

$2$

$2.5$

$4$

60

Two trains, A and B, start at the same time from two stations 600 km apart and travel towards each other. They meet, and after meeting, train A takes 4 hours to reach train B's starting station, while train B takes 9 hours to reach train A's starting station. What is the ratio of the speed of train A to the speed of train B?

$9:4$

$1:1$

$2:3$

$4:9$

$3:2$

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