Year 10 Mathematics End of Year Practice Exam
Section A: Non-Linear Graphs (20 marks)
1) Sketch the graph of y = x² – x – 12, showing all intercepts and the vertex. (5)
2) Find the equation of the parabola with vertex (2, -3) and a y intercept of 5. (4)
3) Sketch the graph of y = x³ + x² – 4x – 4. (4)
4) Sketch the hyperbola 𝑦 =
3
𝑥+2
− 1, showing all intercepts and asymptotes. (5)
5) Does the point (3, 1) lie outside the circle with centre (2, -3) and radius 4? (2)
Section A Mark
/20
Section B: Rates of Change (11 marks)
6) A truck can carry 3.6 tonnes of Chunky Chicken each day to hungry cadets who must
eat it. How many kilograms does it carry each minute, on average? (1)
[A] 2kg/min
[B] 2.25kg/min
[C] 2.5kg/min
[D] 2.75kg/min
7) The weight ‘M’ of an astronaut varies in direct linear proportion to their weight ‘E’ on
Earth. A 72kg person on Earth weighs 27.4kg on Mars.
7a) Calculate how much a 60kg person weighs on Mars. (2)
7b) How much does an astronaut who weighs 30kg on Mars weigh on Earth? (2)
8) The intensity of light (L) on a screen varies inversely with the square of the distance
(D) between the screen and the light source. If the screen has 24 units of illumination
when the screen is 4 metres away, determine the illumination when the screen is 6 metres
away. (3)
9) A runner sprinting a 100m race accelerates throughout the whole race.
9a) What type of variation (direct, indirect/inverse or unrelated) does this constitute? (1)
9b) Draw the graph of the runner, with the speed of the runner throughout the race on
the x-axis and the time taken to finish the race on the y-axis. Note that you only need to
draw the shape of the graph, no values need to be drawn since none have been given. (1)
10) Which car is travelling faster? (1)
Section B Mark
/11
Section C: Simultaneous Equations (18 marks)
11) To what extent does William Shakespeare’s Jacobean tragedy Macbeth (1606)
represe- oops wrong subject Solve y = 9 – x and 2x + 3y = 21 simultaneously. (2)
12) Solve xy = 12 and x + 3y = -20 simultaneously. (3)
13) In 6 years, Declan will be a third of his dad Oscar’s age. In 21 years (separate to the 6
years) Declan will be a half of Oscar’s age. How old are Declan and Oscar now? (4)
14) In my paddock I only have meese riding unicycles (5 legs) and geese riding bicycles
(4 legs). I can count 15 heads and 73 legs in my paddock. How many geese riding bicycles
are there? (4)
15) Oh no! There was a storm after Question 14, and my paddock has flooded.
Now
all the geese have lost their bicycles. However, the meese are (somehow) still on their
unicycles. Additionally, a flock of geese and a small herd of meese have decided to enter
the paddock from the nearby river. I can now count 20 heads and 85 legs. How many
animals decided to enter after the storm? (5)
Section C Mark
/18
Section D: Right-Angled and Non-Right Angled Trigonometry (25 marks)
16) State the sine rule (1) and the cosine rule for an angle (1) and for a side (1). (3)
17) Find, correct to two decimal places, the length of theta. (2)
18) Find the largest angle in a triangle with sides 18.3 cm, 29.8 cm and 24.4 cm. (3)
19) Extension – likely a set paper level question (5) It is known that sin (a) = 1/5.
19a) Find all possible values of cos (a). Hint – draw a triangle! (3)
19b) Find cos (a) if it is also known that tan (a) is negative. (2)
20) Extension – likely a set paper level question (12)
Mark distribution: a) 2 marks, b) 2 marks, c) 2 marks, d) 3 marks, e) 3 marks Total 12 marks.
Section D Mark
/25
Section E: Logarithms (15 marks)
21) Solve
(2)
22) Solve log 3 7 to 2 decimal places. (2)
23) Knowing that log 𝑎 5 is equal to 0.46 and log 𝑎 2, determine the value of log 𝑎 100. (3)
24) The cow population on Bovine Paradise Island is doubling every year. A few years
ago, a study estimated that there were 500 cows there. How many years later will the
cow population be 10 000? Answer correct to the nearest tenth of a year. (4)
25) Use the substitution u = 2ˣ to solve the equation 4ˣ − 7 × 2ˣ + 12 = 0. (4)
Section E Mark
/15
Section F: Polynomials (15 marks)
26) Determine the degree, leading coefficient and the constant term of the polynomial
(2x² + 1) (3x³ − 2) (4x⁴ + 3) (5x⁵ − 4) (1 mark per correct answer, 3 marks total)
27) Divide P(x) = x³ + 3x² - 4x – 12 by (x+2). Hence, fully factorise P(x). (4)
28) Find a and b in f(x) = x³ + ax² - bx – 6, given that:
a) (x+3) is a factor AND b) when f(x) is divided by (x-2), the remainder is 20. (5)
29) Divide P(x) = x³ - 7x – 6 by (x-3). Hence, fully factorise P(x). (4)
30a) Show that 2 and 5 are zeroes of P(x) = x⁴ − 3x³ − 15x² + 19x + 30. (2)
30b) Hence explain why (x − 2) (x − 5) is a factor of P(x). (1)
30c) Divide P(x) by (x − 2) (x − 5) and hence express P(x) as the product of four linear
factors. (4)
Section F Mark
/23
Results
Section A:
/
Section B:
/
Section C:
/
Section D:
/
Section E:
/
Section F:
/23
Total Mark:
/112
General comments
110+ = Stop flexing – we get it. But congratulations, you’re definitely ready
for the test.
100+ = Congratulations! You’re ready for the test – many of the questions
are from next year’s Extension 1 textbook, so keep it up.
90+ = Congratulations! You’re ready for the test! Make sure to sharpen up
on the extension questions if you got them wrong: they can appear in the Set
Paper (although they are questions from Extension 1 Maths).
80+ = Some more revision might be useful, especially if you struggled in a
particular section.
70+ = You need to revise a bit more – especially if you struggled in more
than one section.