A right-angled triangle ABC has sides AB = 12 cm and BC = 16 cm, with the right angle at B. A square is inscribed in the triangle such that one vertex is at B, and the opposite vertex lies on the hypotenuse AC. What is the area of the square?
$49\text{ cm}^2$
$(\frac{192}{28})^2\text{ cm}^2$
$(\frac{48}{7})^2\text{ cm}^2$
$64\text{ cm}^2$
$36\text{ cm}^2$
The number $N$ is the smallest integer greater than 1 such that the product $120 \times N$ is a perfect cube. What is the value of $N$?
$150$
$30$
$45$
$15$
$225$
A rectangular garden has a perimeter of 54 metres. Its length is 3 metres less than twice its width. What is the area of the garden in square metres?
$170\text{ m}^2$
$160\text{ m}^2$
$176\text{ m}^2$
$154\text{ m}^2$
$180\text{ m}^2$
A figure is formed by a square ABCD and a right-angled triangle BCE, joined at the side BC. The side length of the square is 6 cm and the length of CE is 8 cm. What is the perimeter of the composite shape ABECD?
$32\text{ cm}$
$60\text{ cm}$
$40\text{ cm}$
$42\text{ cm}$
$36\text{ cm}$
A sequence of L-shaped figures is constructed from 1x1 squares. Figure 1 has 3 squares. Figure 2 has 5 squares. Figure 3 has 7 squares. The pattern continues by adding one square to each of the two 'arms' of the L-shape. How many 1x1 squares are in Figure 10?
$23$
$19$
$100$
$21$
An electronics store advertises a laptop for 1500 dollars. The store first applies a 20% discount to the advertised price. Then, a 10% Goods and Services Tax (GST) is calculated on the discounted price and added to the bill. What is the final price paid by the customer?
$1350\text{ dollars}$
$1650\text{ dollars}$
$1200\text{ dollars}$
$1080\text{ dollars}$
$1320\text{ dollars}$
A cyclist travels from Town A to Town B at a constant speed of 30 km/h and immediately returns to Town A along the same route at a constant speed of 20 km/h. If the total journey took 5 hours, what is the distance between Town A and Town B?
$62.5\text{ km}$
$60\text{ km}$
$75\text{ km}$
$120\text{ km}$
$50\text{ km}$
If $25^x = 5^{x^2 - 3}$, what is the sum of the possible values of $x$?
$2$
$-1$
$3$
$-2$
$4$
Line $L_1$ passes through the points A$(-2, 1)$ and B$(4, 5)$. Line $L_2$ is perpendicular to $L_1$ and passes through the midpoint of the line segment AB. What is the y-intercept of Line $L_2$?
$2.5$
$5.5$
$1.5$
$4.5$
$3.5$
In a school club, the ratio of boys to girls was 5:3. After 14 boys left and 6 girls joined, the ratio of boys to girls became 2:3. How many students were in the club originally?
$56$
$32$
$64$
$48$
$40$
A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn from the bag one after the other, without replacement. What is the probability that the two marbles drawn are of different colours?
$\frac{15}{32}$
$\frac{15}{64}$
$\frac{13}{28}$
$\frac{15}{28}$
$\frac{15}{56}$
Cylinder A has a radius $r$ and height $h$. Rectangular prism B has a square base of side length $2r$ and the same height $h$. What is the ratio of the volume of Cylinder A to the volume of Prism B?
$4 : \pi$
$\pi : 4$
$\pi : 2$
$2 : \pi$
$1 : 1$
In a class of students, every student plays at least one of two sports: football or tennis. 60% of the students play football and 50% of the students play tennis. If 8 students play both sports, how many students are in the class?
$80$
$60$
$160$
A large circle contains two identical smaller circles. The two smaller circles touch each other at the center of the large circle and each smaller circle touches the circumference of the large circle. What fraction of the large circle's area is shaded (i.e., not covered by the two smaller circles)?
$\frac{1}{2}$
$\frac{1}{4}$
$\frac{\pi}{4}$
$\frac{2}{3}$
$\frac{1}{3}$
An investment of 10000 dollars increases by 10% in its first year. In the second year, its value decreases by 20%. In the third year, its value increases by 10%. What is the net percentage change in the investment over the three years?
A gain of $1.2%$
A loss of $12%$
A loss of $3.2%$
A gain of $3.2%$
$0%$
A painter mixes blue and yellow paint in the ratio 2:3 to make a shade of green paint. After using 10 litres of the green paint, he adds 6 litres of pure yellow paint to the remaining mixture. The new ratio of blue to yellow paint is now 1:2. What was the initial volume of green paint?
$50\text{ litres}$
$30\text{ litres}$
$35\text{ litres}$
$40\text{ litres}$
$25\text{ litres}$
A committee of 3 people is to be chosen from a group of 5 boys and 4 girls. How many different committees can be formed if the committee must contain at least one boy and at least one girl?
$70$
$14$
$74$
$84$
The graphs of the parabola $y = x^2 - 4x + 7$ and the line $y = x+3$ intersect at two points. What is the distance between these two intersection points?
$5$
$ \sqrt{18} $
$ \sqrt{26} $
$6$
The number of positive divisors of $N = 2^a \times 3^b$ is 12. The number of positive divisors of $2N$ is 15. What is the value of $3N$?
$16$
$144$
$216$
$72$
$108$
A four-digit number is written as $34XY$. If this number is divisible by 9 and $X-Y=2$, what is the value of $X$?
$7$
$8$
At a fruit stand, an apple and three oranges cost a total of 4.30 dollars, while three apples and an orange cost a total of 5.30 dollars. What is the cost of a single orange?
$1.60 \text{ dollars}$
$0.95 \text{ dollars}$
$1.20 \text{ dollars}$
$1.50 \text{ dollars}$
$0.85 \text{ dollars}$
In a chemical mixture, the ratio of acid to water is $3:5$. An additional 4 litres of acid is added to the 24-litre mixture. What is the new ratio of acid to water?
$13:15$
$1:1$
$7:5$
$3:7$
$5:3$
If $3^{2x-1} = \frac{27}{3^x}$, what is the value of $x$?
$\frac{4}{3}$
$\frac{3}{2}$
$1$
A store offers a 20% discount on a jacket. After the discount, a 10% Goods and Services Tax (GST) is applied to the discounted price. If the final price paid is 198 dollars, what was the original price of the jacket?
$240 \text{ dollars}$
$220 \text{ dollars}$
$250 \text{ dollars}$
$225 \text{ dollars}$
$216 \text{ dollars}$
The first four terms of a sequence are 3, 8, 15, 24. What is the 10th term in this sequence?
$110$
$99$
$120$
$121$
Point M(4, -1) is the midpoint of the line segment AB. If Point A has coordinates (-2, 5), what are the coordinates of the point that is a quarter of the way from A to B?
$(7, -4)$
$(2, 0.5)$
$(1.5, -1.5)$
$(2.5, -4.5)$
$(1, 2)$
The number $N = 12^3 \times 10^2$. How many positive integers are factors of $N$?
$24$
A square of side length 8 cm has a circle inscribed within it, and another circle circumscribed about it. What is the exact area of the region between the two circles (the annulus)?
$64\pi \text{ cm}^2$
$8\pi \text{ cm}^2$
$32\pi \text{ cm}^2$
$16\pi \text{ cm}^2$
$24\pi \text{ cm}^2$
Two cyclists, Alex and Ben, start at the same point and travel in opposite directions around a 50 km circular track. Alex cycles at a constant speed of 28 km/h and Ben cycles at 22 km/h. How long does it take for them to meet for the second time?
$2 \text{ hours}$
$2 \text{ hours, } 30 \text{ minutes}$
$1 \text{ hour}$
$5 \text{ hours}$
$1 \text{ hour, } 12 \text{ minutes}$
The graph shows the profit of two companies, A and B, over 6 months. During which month was the absolute difference in profit between Company A and Company B the greatest?
February
April
May
June
March
A solid metal cylinder with radius 6 cm and height 10 cm has a cylindrical hole of radius 3 cm drilled through its centre, for its entire height. What is the total surface area of the resulting object?
$216\pi \text{ cm}^2$
$252\pi \text{ cm}^2$
$270\pi \text{ cm}^2$
$198\pi \text{ cm}^2$
$234\pi \text{ cm}^2$
The square ABCD with vertices A(1,1), B(5,1), C(5,5), and D(1,5) is rotated 90 degrees anti-clockwise about the origin. What are the new coordinates of the vertex C?
$(-1, 5)$
$(-5, 5)$
$(1, 5)$
$(-5, -5)$
$(5, -5)$
How many distinct ways can 4 friends (Amy, Ben, Chloe, Dan) be seated in a row if Amy and Ben must sit next to each other?
$12$
$20$
The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number is 18 less than the original number. What is the original number?
$66$
$57$
$75$
The stacked bar chart shows the percentage breakdown of energy sources for two countries in a given year. If Country A's total energy production was 800 TWh and Country B's was 1200 TWh, how much more energy from Renewables (in TWh) did Country B produce than Country A?
$200 \text{ TWh}$
$280 \text{ TWh}$
$120 \text{ TWh}$
$40 \text{ TWh}$
$160 \text{ TWh}$
In the diagram provided, a regular pentagon and a regular hexagon share a common vertex and are joined side-by-side on a straight line. What is the value of the angle $x$ formed between the two shapes?
$140^\circ$
$132^\circ$
$144^\circ$
$136^\circ$
$128^\circ$
A number $X$ gives a remainder of 4 when divided by 7. What is the remainder when $3X+5$ is divided by 7?
$0$
Simplify the surd $\sqrt{48}$.
$4\sqrt{3}$
$3\sqrt{4}$
$4\sqrt{12}$
$16\sqrt{3}$
$2\sqrt{12}$
Simplify the expression $8\sqrt{7} - 3\sqrt{7} + 2\sqrt{7}$.
$3\sqrt{7}$
$7\sqrt{21}$
$13\sqrt{7}$
$7\sqrt{7}$
Simplify the expression $\sqrt{50} + \sqrt{18}$.
$2\sqrt{8}$
$15\sqrt{2}$
$28\sqrt{2}$
$8\sqrt{2}$
$\sqrt{68}$
Simplify the expression $3\sqrt{5} \times 2\sqrt{10}$.
$30\sqrt{2}$
$6\sqrt{15}$
$6\sqrt{50}$
$5\sqrt{50}$
Express the fraction $\frac{12}{\sqrt{6}}$ with a rational denominator.
$12\sqrt{6}$
$\frac{\sqrt{72}}{6}$
$\sqrt{2}$
$\frac{2}{\sqrt{6}}$
$2\sqrt{6}$
Let Set $A = {1, 3, 5, 7, 9}$ and Set $B = {2, 3, 5, 7}$. What is the intersection of A and B, denoted as $A \cap B$?
$\emptyset$
${2}$
${1, 9}$
${1, 2, 3, 5, 7, 9}$
${3, 5, 7}$
Let the universal set be $U = {x \mid x \text{ is an integer and } 1 \le x \le 10}$. Let $P = {2, 3, 5, 7}$ and $Q = {2, 4, 6, 8, 10}$. What is the complement of the union of P and Q, denoted as $(P \cup Q)'$?
${1, 3, 5, 7, 9}$
${1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$
${4, 6, 8, 10}$
A Venn diagram shows two intersecting circles for students who like Tea (T) and Coffee (C). The number in the intersection is 7. The number in the 'T only' region is 12. The number in the 'C only' region is 8. The number outside both circles is 3. How many students like Tea in total?
$27$
In a class of 30 students, 18 play basketball and 15 play soccer. If 7 students play both sports, how many students play neither sport?
$11$
What are the solutions to the quadratic equation $3x^2 + 9x - 30 = 0$?
$x = 3, x = -10$
$x = 2, x = -5$
$x = -3, x = 10$
$x = -2, x = 5$
$x = 2, x = 5$
What are the coordinates of the vertex (turning point) of the parabola? for the graph of the quadratic function $y = x^2 - 4x + 3$.
$(0, 3)$
$(2, -1)$
$(4, 3)$
$(3, 0)$
$(1, 0)$
What are the roots (x-intercepts) of the function $y = -(x+1)^2 + 4$ ?
The function has no real roots.
$x = -1$ and $x = 3$
$x = 1$ and $x = -3$
$x = 1$ and $x = 3$
$x = -1$ and $x = 4$
A quadratic equation has solutions $x=4$ and $x=-6$. Which of the following equations could represent this quadratic relationship?
$y = (x-4)(x-6)$
$y = x^2 - 10x - 24$
$y = (x+4)(x+6)$
$y = (x-4)(x+6)$
$y = (x+4)(x-6)$
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