The graph shows the lines for the equations $y = -2x + 4$ and $y = \frac{1}{2}x - 1$. Which of the following systems of inequalities correctly represents the shaded region?
$y \le -2x + 4$ and $y \ge \frac{1}{2}x - 1$
$y \le -2x + 4$ and $y \le \frac{1}{2}x - 1$
$y \ge -2x + 4$ and $y \le \frac{1}{2}x - 1$
$y > -2x + 4$ and $y > \frac{1}{2}x - 1$
A store increases the price of a jacket by 25%. After a month, the jacket is put on sale and its new price is decreased by a certain percentage, bringing it back to its original price. What was the percentage decrease?
20%
22.5%
15%
25%
Let $K$ be the smallest positive integer which is a multiple of every integer from 1 to 7 inclusive. What is the sum of the digits of the smallest prime number strictly greater than $K$?
7
5
6
8
The bar chart below shows a company's quarterly profits in 2022 and 2023. What was the approximate percentage increase in total profit from the year 2022 to the year 2023?
17%
73%
31%
24%
A regular pentagon and a regular hexagon share a common side, with the polygons positioned on opposite sides of this side. What is the measure of the reflex angle formed inside the combined shape at a vertex where the polygons meet?
$216^\circ$
$228^\circ$
$132^\circ$
$240^\circ$
The $n$-th term of a sequence is given by the formula $T_n = n^2 - 11n + 35$, where $n$ is a positive integer. What is the minimum value of a term in this sequence?
0
35
11
A shape is formed by a central square of side length 8 cm, with a semi-circle drawn on each side, external to the square. What is the perimeter of this entire shape?
$32\pi$ cm
$8\pi$ cm
$16\pi + 32$ cm
$16\pi$ cm
$y = x^2 - 4x + 7$ and $y = x+3$. For how many integer values of $x$ is the value of the quadratic function less than the value of the linear function?
1
2
4
3
A sealed rectangular container with dimensions 6 cm, 8 cm, and 10 cm is resting on its smallest face (6 cm by 8 cm) and is filled with water to half of its height. The container is then tilted to rest on its largest face (8 cm by 10 cm). What will be the new depth of the water?
4 cm
3 cm
5 cm
2.5 cm
Two runners, Alex and Ben, start a race at the same time. Alex runs at a constant speed of 4 m/s. Ben starts slower but accelerates, such that his distance from the start, $d$ metres, after $t$ seconds is given by $d = 2t + 0.1t^2$. The graph shows their distances over time. At what time, $t > 0$, does Ben overtake Alex?
10 s
25 s
15 s
20 s
A password consists of 4 distinct digits chosen from the set {1, 2, 3, 4, 5}. The password must be an even number and the first digit must be a prime number. How many such passwords can be formed?
60
24
30
36
A scientific calculator is priced at 80 dollars. It is first discounted by 15%. A 5% shipping fee is then added to the discounted price. Finally, a 10% Goods and Services Tax (GST) is calculated on this new total. What is the final cost?
75.02 dollars
74.12 dollars
76.00 dollars
78.54 dollars
The integer $N$ is given by the prime factorisation $N = 2^a \times 3^4 \times 7^2$. If $N$ has exactly 60 positive divisors, what is the value of the exponent $a$?
A farmer has only sheep and chickens. In total, she counts 60 heads and 174 legs among her animals. How many more chickens than sheep does the farmer have?
33
27
The diagram shows a square with side length 12 cm. A semicircle is drawn with its diameter as one of the sides of the square, located inside the square. What is the area of the region inside the square but outside the semicircle?
$144 - 72\pi$ cm$^2$
$144 - 36\pi$ cm$^2$
$144 - 18\pi$ cm$^2$
$122\pi$ cm$^2$
A sequence begins with the term $T_1 = 13$. To get the next term, you square the previous term and sum its digits. What is the 2024th term of this sequence?
10
16
13
Which of the following expressions is equivalent to $\frac{(2x^{-3}y^2)^{-4}}{(4x^2y^{-5})^2}$?
$\frac{x^8}{256y^{18}}$
$\frac{x^{16}y^2}{256}$
$\frac{x^8y^2}{64}$
$\frac{x^8y^2}{256}$
Driver A and Driver B are 450 km apart, driving towards each other on a straight highway. Driver A travels at a constant 80 km/h and Driver B travels at a constant 100 km/h. If they both start at 9:00 am, at what time will they meet?
11:00 am
11:30 am
1:30 pm
12:00 pm
A regular pentagon ABCDE is drawn. A point F is located inside the pentagon such that triangle ABF is equilateral. What is the size of angle $\angle FCD$?
$48^\circ$
$60^\circ$
$66^\circ$
$54^\circ$
A point $K(-3, 4)$ is rotated $180^\circ$ about the origin, and then reflected across the line $y = -x$. What are the coordinates of the final image of point $K$?
$(-4, 3)$
$(4, -3)$
$(4, 3)$
$(3, -4)$
A committee of 3 people is to be chosen from a group of 4 boys and 5 girls. What is the probability that the committee consists of exactly 2 girls and 1 boy?
$\frac{20}{84}$
$\frac{5}{14}$
$\frac{1}{2}$
$\frac{10}{21}$
The graph shows the trajectory of a projectile, with its height $h$ in metres as a function of time $t$ in seconds, modelled by a quadratic equation. The projectile reaches its maximum height of 80 metres at $t=4$ seconds. It was launched from a height of 32 metres. At what time does the projectile hit the ground ($h=0$)? Rounded to the seconds.
8 seconds
12 seconds
9 seconds
10 seconds
The sum of five consecutive positive integers is a perfect cube. The sum of the three middle integers is a perfect square. What is the smallest possible value of the middle integer?
225
2025
75
675
The bar chart shows the number of units sold by Team Alpha and Team Bravo over four months. In which month did Team Bravo have the greatest percentage increase in sales compared to its own sales in the previous month?
April
March
February
January
What is the completely factorised form of the expression $4(x+1)^2 - 9(x-1)^2$?
$(-x-1)(5x+5)$
$(x-5)(5x-1)$
$(5-x)(5x-1)$
$(5x-5)(-x-1)$
What is the value of the expression $\frac{\sqrt{48} + \sqrt{12}}{5\sqrt{3} - \sqrt{75}}$?
Undefined
$6\sqrt{3}$
$6$
$3$
Evaluate the expression: $5 \times [ (3-7)^2 \div (\frac{1}{2})^{-3} ] - \sqrt{9+16}$.
$-15$
$635$
$5$
Simplify the expression $\frac{(2x^3y^{-2})^4}{4x^{-2}y} \div \frac{x^5}{y^{-3}}$. Assume $x, y e 0$.
$\frac{4x^{19}}{y^6}$
$4x^9y^{-6}$
$\frac{4x^9}{y^6}$
$\frac{4x^9}{y^{12}}$
A cuboid has dimensions $3$ cm, $4$ cm, and $12$ cm. What is the length of the space diagonal, connecting opposite corners of the cuboid?
$\sqrt{153}$ cm
$19$ cm
$13$ cm
$5$ cm
A concert sold two types of tickets: adult and child. An adult ticket costs 15 dollars more than a child ticket. A group of 4 adults and 6 children paid a total of 210 dollars. What is the cost of one adult ticket?
$45$ dollars
$30$ dollars
$15$ dollars
$25$ dollars
The graph of the quadratic function $y = 2x^2 + bx + c$ is shown. Based on its x-intercepts, what is the completely factorised form of the expression $2x^2 + bx + c$?
$(x+3)(2x-1)$
$2(x+3)(x-0.5)$
$(x-3)(2x+1)$
$2(x-3)(x+0.5)$
What is the result of simplifying $\frac{10}{3\sqrt{2} - 2\sqrt{3}}$ by rationalising the denominator?
$5\sqrt{2} + \frac{10\sqrt{3}}{3}$
$\frac{15\sqrt{2} - 10\sqrt{3}}{3}$
$\sqrt{2} + \frac{2\sqrt{3}}{3}$
$\frac{15\sqrt{2} + 10\sqrt{3}}{3}$
A triangle has side lengths $2\sqrt{5}$ cm, $3\sqrt{5}$ cm, and $5\sqrt{2}$ cm. Which statement accurately describes the triangle?
It is a right-angled triangle.
It is an obtuse-angled triangle.
Such a triangle cannot exist.
It is an acute-angled triangle.
If $27^{2x-1} = \frac{\sqrt{3}}{9^{1-x}}$, what is the value of $x$?
$x = \frac{3}{8}$
$x = \frac{3}{4}$
$x = \frac{9}{8}$
$x = -\frac{3}{8}$
One of the factors of $6x^2 - xy - 12y^2$ is $(2x-3y)$. What is the other factor?
$(4x-3y)$
$(3x-4y)$
$(3x+4y)$
$(4x+3y)$
Find the largest integer value of $x$ that satisfies the inequality $\frac{2x-1}{3} - \frac{3x-5}{4} > \frac{1}{2}$.
$-6$
$4$
There is no largest integer.
What is the value of $(\frac{2}{3} - \frac{1}{2})^2 \div \frac{5}{6} + \frac{1}{3} \times \sqrt{\frac{9}{4}}$?
$\frac{8}{15}$
$\frac{23}{60}$
$\frac{17}{30}$
$\frac{1}{5}$
Two cyclists, Alice and Bob, start at the same point. Alice cycles due North and Bob cycles due East. The bar chart shows their constant speeds. After 2 hours, how far apart are they, 'as the crow flies'?
$28$ km
$56$ km
$20$ km
$40$ km
Given that $(a - \sqrt{b})^2 = 21 - 8\sqrt{5}$, where $a$ is a positive integer and $b$ is a prime number, find the value of $a^2-b$.
$11$
$-1$
$21$
Simplify the expression $\frac{2x^2+5x-3}{x^2-9} \div \frac{2x^2-x}{x^2+3x}$.
$\frac{x-3}{x+3}$
$\frac{2x-1}{x-3}$
$\frac{x+3}{x-3}$
$1$
A chord of length 16 cm is drawn in a circle of radius 10 cm. What is the perpendicular distance from the center of the circle to the chord?
$\sqrt{164}$ cm
$6$ cm
$2$ cm
$8$ cm
Consider the system of linear equations: $3x - 2y = 7$ and $kx - 4y = 14$. For what value of $k$ does this system have infinitely many solutions?
Any value except 6
Evaluate $(-\frac{8}{27})^{\frac{2}{3}} + (16)^{-\frac{3}{4}}$.
$-\frac{23}{72}$
$\frac{41}{72}$
$\frac{23}{72}$
$\frac{5}{17}$
The price of a jacket is reduced by 20%. A 10% sales tax is then applied to the discounted price. If the final price paid is 88 dollars, what was the original price of the jacket?
80 dollars
96 dollars
110 dollars
100 dollars
A team of 5 identical machines can complete a job in 12 hours. The team starts work. After 4 hours, 2 machines break down. How many additional hours are needed for the remaining machines to complete the job?
20 hours
12 hours
$13\frac{1}{3}$ hours
8 hours
A sequence of figures is built from squares. Figure 1 is 1 square. Figure 2 has 5 squares. Figure 3 has 13 squares, and Figure 4 has 25 squares. The pattern is formed by adding a new larger layer of squares around the previous figure. If the pattern continues, how many squares are in Figure 10?
199
221
161
181
In the diagram, $ABCD$ is a square and $\triangle ADE$ is an equilateral triangle with point $E$ inside the square. What is the size of $\angle AEC$?
$150^\circ$
$120^\circ$
$135^\circ$
$75^\circ$
The bar chart shows Maya's scores on four of her five science tests for the semester. Her average score for all five tests was exactly 80. What was her score on Test 5?
86
79
84
80
Four identical circles are inscribed within a large square, touching each other and the sides of the square. If the total area of the four circles is $36\pi$ cm$^2$, what is the area of the small region in the centre, bounded by the arcs of the four circles?
$36\pi - 36$ cm$^2$
$36 - 9\pi$ cm$^2$
36 cm$^2$
A bag contains 24 marbles, all of which are either red or blue. The ratio of red marbles to blue marbles is 1:2. If two marbles are drawn from the bag at random without replacement, what is the probability that they are of different colours?
$\frac{32}{69}$
$\frac{16}{69}$
$\frac{37}{69}$
$\frac{4}{9}$
The graph shows the percentage of a large construction project completed over 12 weeks. During which two-week period was the project's completion rate the greatest?
Weeks 4-6
Weeks 8-10
Weeks 10-12
Weeks 2-4
Great effort! Check the detailed feedback above.
Incorrect password
