AQA Level 2 Further Maths June 2025 Paper 2 (calculator)
21 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 (3 marks)
Sequence nth term A 6n/(7n + 25) B 20/(n + 4) The 5th term in sequence A is also a term in sequence B.
Work out the position of the term in sequence B.
(3 marks)
Question 2 (4 marks)
C and E are points on the circle with equation x² + y² = 144
CD is a straight line.
OD : DE = 5 : 1Diagram: A circle centred at the origin O. C is where it meets the positive y-axis and E where it meets the positive x-axis. D is on the x-axis between O and E. The region between the arc CE, the line ED and the line DC is shaded.
Work out the shaded area.
Give your answer as a decimal.(4 marks)
Question 3 (3 marks)
The equation of a straight line is 2ax + y = 25 where a is a constant.
The line passes through the point (3, 1)Work out the value of the gradient of the line.
(3 marks)
Question 4 (2 marks)
-5 ≤ n ≤ 5
Work out the range of possible values of n²
(2 marks)
Question 5 (3 marks)
The equation of a curve is y = x⁵ + px⁴ where p is a constant.
When x = -1 the gradient of the curve is –3Work out the value of p.
(3 marks)
Question 6 (3 marks)
C is a point on the straight line AB.
Diagram: Axes with the straight line from A to B. C is a point on the line between A and B.
AC : CB = 3 : 2
Work out the equation of the straight line passing through O and C.
(3 marks)
Question 7 (5 marks)
(a)
2^(5a + 1) = 2^a
Work out the value of a.
(2 marks)
(b)
3⁻⁷ ÷ 3^c = (3⁻³)^(2c)
Work out the value of c.
(3 marks)
Question 8 (3 marks)
k is a positive integer.
Prove that 4k(k + 8) – (4k + 9)(k – 1) is a multiple of 9
(3 marks)
Question 9 (2 marks)
beginaligned f(x) &= x² + px && x ≤ 1 \\ &= 2p – 1/2x && x > 1 endaligned
A sketch of y = f(x) is shown.
Diagram: A sketch of y = f(x): a parabola through the origin with its lowest point just left of the y-axis and below the x-axis, rising to x = 1, then a straight line sloping down from there and crossing the x-axis.
Work out the value of p.
(2 marks)
Question 10 (2 marks)
The first four terms of a quadratic sequence are
–1 11 31 59
Work out an expression for the nth term.
Give your answer in the form an² + b where a and b are constants.(2 marks)
Question 11 (6 marks)
(a)
Simplify fully 3/4w + 1/6w
(2 marks)
(b)
Factorise fully (x – 1)⁹ + (x – 1)⁸(x + 4)
(2 marks)
(c)
Factorise fully 8y² – 50
(2 marks)
Question 12 (4 marks)
(a)
Circle the matrix that represents an enlargement, scale factor 3, centre O.
( 3 & 0 , 0 & 3 ) ( 3 & 3 , 0 & 0 ) ( 0 & 3 , 3 & 0 ) ( 3 & 0 , 3 & 0 )
(1 mark)
(b)
OABC is the unit square.
O (0, 0) A (1, 0) B (1, 1) C (0, 1)
K is the transformation represented by ( 1 & 0 , 0 & -1 )
L is the transformation represented by ( 0 & 1 , -1 & 0 )OABC is mapped to OA’B’C’ under a combined transformation of K followed by L.
Work out the coordinates of A’, B’ and C’.
(3 marks)
Question 13 (4 marks)
Rearrange x = (3k²)/(5k² – 2) to make k the subject.
(4 marks)
Question 14 (5 marks)
All lengths shown are in centimetres.
Diagram: A right-angled triangle. The vertical side is w, the horizontal side is 2w + 5 and the sloping side is 3w + 2. The angle x is at the bottom right.
(a)
Use Pythagoras’ theorem to show that 4w² – 8w – 21 = 0
(2 marks)
(b)
Work out the size of angle x.
(3 marks)
Question 15 (3 marks)
For a curve y = f(x)
dy/dx = 1 – 9/(x²)
(a)
Show that the curve has a turning point where x = 3
(1 mark)
(b)
Use (d²y)/(dx²) to work out the nature of the turning point.
You must show your working.(2 marks)
Question 16 (4 marks)
A, B, C and D are points on a circle.
EBF is a tangent.Diagram: A circle through A, B, C and D, with the quadrilateral ABCD and its diagonal BD. EBF is the tangent to the circle at B. Angle BAD is 4x – 15°, angle DBC is x + 20° and angle CBF is ½x + 60°.
Work out the value of x.
(4 marks)
Question 17 (5 marks)
Solve
beginaligned 2a + 4b + c &= 12 \\ 3a – b + 2c &= -2 \\ a – 5b – 3c &= 6 endaligned
Do not use trial and improvement.
You must show your working.(5 marks)
Question 18 (5 marks)
ABCDEF is a triangular prism.
ABCD is a rectangle and angle BCF is 90°AB = 14 cm BC = 8 cm CF = 6 cm
M is the midpoint of EF.Diagram: A triangular prism ABCDEF. The base ABCD is a rectangle with AB = 14 cm along the front and BC = 8 cm going back. CF = 6 cm is vertical, with angle BCF a right angle. EF is the top edge, above DC, and M is the midpoint of EF. AD, DC and DE are hidden (dashed).
Work out the size of angle BMD.
(5 marks)
Question 19 (4 marks)
f(x) = 4x³ – 7 g(x) = 6x + 8
Work out the value of x for which f(2x) = gf(x)
(4 marks)
Question 20 (5 marks)
Here is a sketch of the curve with equation y = 4 + 2x – x²
The tangent to the curve at the turning point P is shown.
The normal to the curve at Q meets this tangent at R.Diagram: A sketch of the n-shaped curve y = 4 + 2x − x² with its turning point P (1, 5). The horizontal tangent at P is drawn. Q (3, 1) is on the curve to the right of P, and R is on the tangent, far to the right.
Work out the x-coordinate of R.
(5 marks)
Question 21 (5 marks)
(a)
theta is an angle between 270° and 360°
cos theta = 1/5k and tan theta = -3k where k is a positive constant.
Work out the value of theta.
(2 marks)
(b)
Work out the values of x between 0° and 360° for which
8cos² x – cos x = 0
(3 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.