CTJan27 Online JMSS - Mathematical Reasoning 15042026

A shape is formed by a square with a side length of 10 cm and a semicircle attached to one of its sides, as shown in the diagram. The diameter of the semicircle is a side of the square. What is the perimeter of the composite shape in cm?

Diagram Description: A square ABCD with side length 10. A semicircle is drawn with diameter CD, outside the square.

The number $N$ is the product of the first three distinct prime numbers. How many factors does $N^2$ have, including 1 and itself?

Liam is three times as old as his sister, Sarah. In 4 years' time, he will be twice as old as Sarah. What is the sum of their current ages?

A sequence is generated by the rule: 'each term is 2 less than double the previous term'. If the first term is 5, what is the fourth term?

In a box, there are only red, blue, and green pens. The ratio of red pens to blue pens is $3:5$. The ratio of blue pens to green pens is $2:3$. If there are 30 green pens, how many red pens are there?

Two cars, Car A and Car B, start at the same point and travel in opposite directions along a straight road. Car A travels at a constant speed of 70 km/h and Car B travels at a constant speed of 90 km/h. How many minutes will it take for them to be 400 km apart?

A triangle PQR has vertices at $P(1, 2)$, $Q(7, 2)$, and $R(4, 6)$. What is the area of the triangle in square units?

A bag contains 5 red marbles and 3 blue marbles. A marble is drawn at random and not replaced. Then, a second marble is drawn. What is the probability that both marbles drawn are red?

A three-digit number is formed using the digits 1, 2, and 3, with each digit used exactly once. How many of these three-digit numbers are prime?

A solid metal cylinder has a radius of 4 cm and a height of 10 cm. A cylindrical hole of radius 2 cm is drilled through the centre of the cylinder, parallel to its axis. What is the volume of the remaining metal in cm$^3$?

The graph shows the height, $h$ (in metres), of the water level in a container over time, $t$ (in seconds), as it is filled at a constant rate. Which of the following containers matches the graph?

In a group of 80 students, 45 play soccer and 35 play basketball. If 10 students play neither sport, how many students play both soccer and basketball?

A figure is made from a large semicircle with radius 10 cm. A smaller semicircle with radius 5 cm is removed from it, and another semicircle with radius 5 cm is added to it, as shown. What is the area of the shaded region in cm$^2$?

Diagram Description: A large semicircle with diameter AB of length 20, centre O. The region is the area of this large semicircle. From this, a smaller semicircle with diameter AO (length 10) is removed. To this, a semicircle with diameter OB (length 10) is added on top. Effectively, it's a shape known as a lune, but described as a composite shape. The final shaded area is equivalent to the area of the large semicircle.

A bag contains only red and blue marbles in the ratio $7:5$. After 2 red marbles are added and 2 blue marbles are removed, the new ratio of red to blue marbles is $3:2$. How many marbles were in the bag originally?

The sum of five consecutive integers is $S$. In terms of $S$, what is the largest of these five integers?

A solid cube with a side length of 4 cm is painted red on all its faces. It is then cut into 64 smaller cubes, each with a side length of 1 cm. A small cube is selected at random. What is the probability that the selected cube has exactly two red faces?

The vertices of a quadrilateral ABCD are $A(-2, 3)$, $B(3, 3)$, $C(5, -1)$, and $D(0, -1)$. Which of the following most accurately describes this quadrilateral?

A company's monthly revenue is modelled by the function $R(x) = 20x$ and its monthly cost is modelled by $C(x) = x^2 - 10x + 150$, where $x$ is the number of units sold. The company makes a profit when revenue exceeds cost. For which integer values of $x$ does the company make a profit?

An ant is at one corner (A) of a 2 cm x 3 cm x 4 cm rectangular prism and wishes to travel to the diagonally opposite corner (G) by only moving along the edges. How many distinct paths of the shortest possible length can the ant take?

The numbers 1 to 9 are placed in a 3x3 grid such that each number is used exactly once. The numbers in each row and each column add up to the same total, $T$. In the grid shown, what is the value of the number in the centre square (labelled with an 'X')?

A train travelling at a constant speed of 90 km/h takes 15 seconds to completely pass a stationary pole. It then takes 45 seconds to completely cross a bridge. What is the length of the bridge in metres?

A shape is constructed by taking a regular hexagon with side length 6 and attaching a square to each of its six sides. Then, an equilateral triangle is attached to the outward-facing side of each of these squares. What is the perimeter of the final composite shape?

At a school play, student tickets cost 8 dollars each and adult tickets cost 12 dollars each. A total of 150 tickets were sold, raising 1520 dollars. How many student tickets were sold?

A positive integer $N$ is the product of three distinct prime numbers. The sum of these three prime numbers is 20. What is the largest possible value of $N$?

The ratio of red marbles to blue marbles in a bag was initially 2:3. After 10 blue marbles were added to the bag, the ratio of red to blue marbles became 2:5. How many marbles were in the bag originally?

Cyclist A starts from Town P and cyclist B starts from Town Q, riding towards each other. The towns are 190 km apart. Cyclist A's speed is 20 km/h and Cyclist B's speed is 30 km/h. If Cyclist A starts one hour before Cyclist B, at what distance from Town P do they meet?

The graph shows the volume of water in a tank over a 25-minute period. The process involves a filling phase, a pause, and an emptying phase. What is the positive difference between the rate of filling and the rate of emptying, in litres per minute?

A student has 3 distinct maths books, 2 distinct science books, and 1 English book. In how many ways can these 6 books be arranged on a shelf if the two science books must be kept together?

In a sequence of five positive integers, the mean is 8, the median is 7, and the mode is 4. The range of the sequence is 11. What is the largest integer in the sequence?

A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn from the bag one after another without replacement. What is the probability that both marbles drawn are the same colour?

A solid metal cylinder has a radius of 3 cm and a height of 10 cm. A cone with the same radius and a height of 4 cm is drilled out from one end of the cylinder. What is the volume of the remaining solid?

Three years ago, Amelia's age was one-third of her father's current age. In five years' time, her father's age will be 11 years more than twice Amelia's current age. What is the sum of their current ages?

A positive integer $N$ is less than 100 and has exactly three factors. How many possible values of $N$ are there?

Alice, Ben, and Chloe share a sum of money. The ratio of Alice's share to Ben's share is 2:3. The ratio of Ben's share to Chloe's share is 4:5. If Chloe gives 42 dollars to Alice, Alice will have the same amount of money as Ben. How much money did Ben have originally?

A car travels from City A to City B, a distance of 360 km. For the first third of the distance, the car travels at an average speed of 60 km/h. For the remaining journey, it travels at a constant speed of V km/h. If the total journey time was 4.5 hours, what is the value of V?

The monthly cost of two mobile phone plans. Plan A is represented by the equation $C = 10 + 0.20D$ and Plan B by $C = 30 + 0.10D$, where $C$ is the cost in dollars and $D$ is the data usage in gigabytes (GB). For what amount of data usage is the monthly cost of both plans exactly the same?

A robot starts at point A (top-left vertex) and must reach point B (bottom-right vertex) on a grid of wires that is 4 units wide and 3 units high. The robot can only move right or down along the wires. How many distinct paths are there from A to B?

I am thinking of a three-digit number, $N$. The sum of its digits is 14. The hundreds digit is one more than the tens digit. If the digits are reversed, the new number is 297 less than the original number. What is the number $N$?

In a class of 30 students, 18 study Physics, 15 study Chemistry, and 5 study neither subject. A student is chosen at random from the class. Given that the student studies Chemistry, what is the probability that they also study Physics?

Five identical cubes, each with side length 2 cm, are glued together to form a shape resembling a plus sign ('+'). One cube is in the centre, and the other four are attached to four of its faces. What is the total surface area of this new solid?

The number $N = 12^3 \times 15^2$ is written as a product of its prime factors in the form $p_1^{a_1} p_2^{a_2} \dots$. What is the sum of the exponents ($a_1 + a_2 + \dots$)?

A large cube with a side length of 10 cm is painted green on all its faces. It is then cut into 1000 identical small cubes, each with a side length of 1 cm. What percentage of the total surface area of all the small cubes is NOT painted green?

Evaluate the following expression: $10 - (20 \div (8-3) \times 2^2) + (-1)^5$

A straight line passes through the points $A(-2, 7)$ and $B(4, -5)$. What is the y-intercept of this line?

Simplify the expression $\sqrt{72} + \sqrt{50} - \sqrt{128}$.

Which of the following number lines correctly represents the solution set for the inequality $7 - 3x \geq 19$?

What is the simplified expansion of $(3x - 4)(2x + 5)$?

Simplify the expression $\frac{(8x^{-6}y^{3/2})^{-1/3}}{(16x^4y^{-1/3})^{1/2}}$ completely. Assume $x, y > 0$. Express your answer using only positive exponents.

A line $L_1$ has the equation $3x + 2y = 6$. A second line, $L_2$, is perpendicular to $L_1$ and passes through the point $(3, -1)$. What is the equation of $L_2$?

Which expression is equivalent to $\frac{12}{3 - \sqrt{5}}$?

What is the fully factorised form of $6x^2 - 13x + 5$?

A mobile phone plan costs 30 dollars per month, which includes 2 gigabytes (GB) of data. Any additional data used is charged at a rate of 12 dollars per GB. If a customer wants their monthly bill to be no more than 80 dollars, what is the maximum total amount of data, in GB, they can use?

If $(x-2)^{3/2} = 64$, what is the value of $x$?

What are the solutions to the quadratic equation $2x^2 - 8x + 3 = 0$?

A line is defined by the equation $2x + 3y = 12$. The line forms a triangle with the positive x-axis and the positive y-axis. What is the area of this triangle?

In a right-angled triangle, the two shorter sides have lengths of $(\sqrt{6} + \sqrt{2})$ cm and $(\sqrt{6} - \sqrt{2})$ cm. What is the exact length of the hypotenuse?

A rectangular field has an area of 104 square metres. Its length is 5 metres longer than its width. What is the perimeter of the field?

The line $L$ has the equation $y = 2x + 1$. The point $P$ has coordinates $(k, 9)$. For which values of $k$ is the point $P$ located below the line $L$?

If $9^n = 5$ and $5^m = 81$, what is the value of the product $nm$?

Let the value of $k$ be defined by the expression $k = \frac{(3^2)^3 \times 3^{-4}}{3^0 + 8}$. A quadratic equation is given by $2x^2 - kx - 5 = 0$. Using the quadratic formula, what is the value of the positive solution for $x$?