AQA Level 2 Further Maths June 2025 Paper 1 (non-calculator)
23 questions · 80 marks. Every question from the paper, recreated, with the official mark scheme, an annotated worked solution and a shadow question (the same skill with new numbers). Open the paper in the Ultimate Exam Question Finder to present it, print it or build a set.
Question 1 (5 marks)
(a)
The function f is given by f(x) = 5x + 3
The range is f(x) < 23
Work out the domain of the function, giving your answer as an inequality.
(1 mark)
(b)
The function g is given by g(x) = x³ – 4
The domain is -3 < x < 2
Work out the range of the function, giving your answer as an inequality.
(2 marks)
(c)
The function h is given by h(x) = √(2x + 1)
Work out h⁻¹(x)
(2 marks)
Question 2 (2 marks)
( 5 & 0 , 1 & -6 )( 12 , 7 ) = 15( a , b )
Work out the values of a and b.
(2 marks)
Question 3 (3 marks)
A linear sequence starts 2 5 8 11
Work out the position of the term that is the first to have a value greater than 500
(3 marks)
Question 4 (4 marks)
AC is a straight line.
A is the point (–3, 2)
B is the midpoint of the line AC.
B is the point (12, 10)
(a)
Work out the coordinates of C.
(2 marks)
(b)
Work out the distance AB.
(2 marks)
Question 5 (3 marks)
Here is a sketch of y = x² + bx + c
The graph passes through (-1, 0) and (5, 0)
Diagram: A U-shaped curve crossing the x-axis at –1 and 5, crossing the y-axis below O, with its lowest point between x = 0 and x = 5.
Work out the values of b and c.
(3 marks)
Question 6 (3 marks)
Simplify fully dfrac√75 – √27√48
(3 marks)
Question 7 (2 marks)
Here is a sketch of y = tan x for 0° ≤ x ≤ 360°
Diagram: The graph of y = tan x from 0 to 360. It crosses the x-axis at 0, 180 and 360, and rises from left to right in each branch, shooting up towards x = 90 and x = 270 and coming up from below just after them.
You are given that tan 72° = k
Work out the two values of x between 0° and 360° for which tan x = -k
(2 marks)
Question 8 (1 mark)
Work out (dfracx^(9/2)x^(1/2))³ as a power of x.
Circle your answer.
x⁷ x¹² x¹⁵ x²⁷
(1 mark)
Question 9 (3 marks)
3-digit integers are made using three different digits.
The digits are chosen from 1, 2, 3, 4, 5, 6, 7, 8, 9
How many more odd integers than even integers can be made?
(3 marks)
Question 10 (4 marks)
PQ is perpendicular to QR.
Diagram: Axes with O at the origin. P is on the y-axis at (0, a), Q is at (3, 12) and R is on the x-axis at (9, 0). Lines PQ and QR meet at a right angle at Q.
Work out the value of a.
(4 marks)
Question 11 (4 marks)
Solve x³ + 10x² + 8x = 0
Where appropriate give your answers in surd form.
Simplify your answers where possible.
(4 marks)
Question 12 (4 marks)
A, B and C are points on a circle, centre O.
Diagram: A circle with centre O. A, B and C are on the circle, with B on the minor arc between A and C. Angle AOC at the centre is 4x and angle ABC is y + 3x.
Prove that y = 180 – 5x
(4 marks)
Question 13 (1 mark)
Here are four sketch graphs.
Circle the letter of the sketch graph that represents y = 5^x
A
Diagram: Graph A: a curve above the x-axis, high on the left and falling, levelling out just above the x-axis on the right.
B
Diagram: Graph B: a curve below the x-axis, just under the axis on the left and falling steeply on the right.
C
Diagram: Graph C: a curve above the x-axis, just above the axis on the left and rising steeply on the right.
D
Diagram: Graph D: a curve below the x-axis, low on the left and rising, levelling out just below the x-axis on the right.
(1 mark)
Question 14 (4 marks)
Simplify fully (2x² – 4x – 6)/(x² – 3x) ÷ (4x + 4)/(7x²)
(4 marks)
Question 15 (3 marks)
(x – 3)(x + 4)(x + p) equiv x³ + qx² + rx + 24
Work out the values of p, q and r.
(3 marks)
Question 16 (3 marks)
Diagram: Triangle ABC with A at the top, C on the left and B at the bottom right. Angle A is 45°, angle B is 30° and side CB is 10 cm.
Work out the length AC.
Give your answer in the form a√2 where a is an integer.
(3 marks)
Question 17 (3 marks)
The expansions of (4a + x)⁴ and (5 – 2x)³ have the same coefficient of x³
Work out the value of a.
(3 marks)
Question 18 (6 marks)
A 60 cm piece of wire is bent to form three sides of a rectangle.
It is then attached to a fixed rod as shown in the diagram.
Diagram: A straight rod along the top. Below it, the wire forms the other three sides of a rectangle: two sides of x cm hanging down from the rod and a bottom side of y cm.
(a)
The area of the rectangle is A cm²
Show that A = x(60 – 2x)
(2 marks)
(b)
Work out the maximum value of A as x varies.
(4 marks)
Question 19 (5 marks)
(a)
(x – 1) is a factor of 3x³ + 2x² – 3x – 2
Factorise fully 3x³ + 2x² – 3x – 2
(3 marks)
(b)
f(x) = (x – 2)(5x + 4)(3x + 2)
c is a positive integer.
The value of f(c) can be written as the product of two prime factors.
Deduce the value of f(c)
(2 marks)
Question 20 (4 marks)
y = (x³)/3 – 5x² + 28x + 11
Show that y is an increasing function for all values of x.
(4 marks)
Question 21 (2 marks)
Show that (20sin² x)/(3 + cos² x – 3sin² x) can be simplified to ktan² x where k is an integer.
(2 marks)
Question 22 (6 marks)
A circle has centre (2, 3) and radius 5
A straight line passes through (3, 7) and has a gradient of 2
Work out the coordinates of the two points where the circle and the straight line intersect.
(6 marks)
Question 23 (5 marks)
The diagram shows a trapezium.
All lengths are in centimetres.
Diagram: A trapezium with parallel sides 3x – 8 (top) and x + 5 (bottom), and a dashed perpendicular height of 2x between them.
The area of the trapezium is less than 22 cm²
Work out the greatest possible range of values of x.
Give your answer in the form m < x < n where m and n are both positive numbers.
(5 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.