AQA Level 2 Further Maths June 2024 Paper 2 (calculator)
22 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)
A straight line
passes through the point (–1, –1)
and
has gradient 3/2Draw the line for values of x from –3 to 3
Diagram: A blank grid, x from –3 to 3 and y from –7 to 7.
(3 marks)
Question 2 (2 marks)
5n² + 2 < 38
Work out all the possible integer values of n.
(2 marks)
Question 3 (3 marks)
The equation of a curve is y = x⁴ – 3kx² where k is a constant.
When x = 2 the rate of change of y with respect to x is 23
Work out the value of k.
(3 marks)
Question 4 (4 marks)
Here is some information about three linear sequences, A, B and C.
nth term of C = nth term of A + nth term of B The nth term of C is 42 – 3n
The first four terms of B are 14 22 30 38
Work out the 20th term of A.
(4 marks)
Question 5 (4 marks)
(a)
( c & 1 , 5d & 3 )( -2 & 0 , 7 & 3 ) = ( -5 & 3 , 0 & 9 )
Work out the values of c and d.
(3 marks)
(b)
( 5 & 2 , 1 & 4 )mathbfM = ( 5 & 2 , 1 & 4 )
Write down matrix mathbfM
(1 mark)
Question 6 (2 marks)
The function f is given by f(x) = 3x² + 2 with domain -1 ≤ x ≤ 4
Work out the range of the function.
(2 marks)
Question 7 (2 marks)
The equation of a curve is y = 4 – (x – 3)²
(a)
Circle the coordinates of the point where the curve crosses the y-axis.
(–5, 0) (0, –5) (–13, 0) (0, –13)
(1 mark)
(b)
Write down the coordinates of the maximum point of the curve.
(1 mark)
Question 8 (3 marks)
y = 1/2x² + 3/(4x⁴)
Work out dfracd²ydx²
Give your answer in the form a + bxⁿ where a, b and n are integers.
(3 marks)
Question 9 (3 marks)
A set of 4-digit integers each have
a first digit greater than 6
and
a second digit less than 8What is the greatest possible number of integers in the set that are multiples of 5 ?
(3 marks)
Question 10 (2 marks)
f is a cubic function.
The curve y = f(x) has
a minimum point at (a, 0)
and
a maximum point at (1, b)Diagram: A cubic curve rising from bottom left, crossing the y-axis above O, to a maximum point at (1, b), then falling to a minimum point on the x-axis at (a, 0), then rising again.
Tick one box for each statement.
True False The tangent to the curve at (1, b) is parallel to the x‑axis There are three different values of x for which y = 0 The function is increasing for 0 < x < 1 (2 marks)
Question 11 (4 marks)
ABC and ACD are triangles.
Diagram: Triangles ABC and ACD share the side AC. In triangle ABC, angle ABC = 59°, angle BAC = 66° and BC = 18 cm. In triangle ACD, angle ACD = 70° and CD = 15 cm.
Work out the area of triangle ACD.
(4 marks)
Question 12 (6 marks)
The equation of a circle is (x – 2)² + (y + 3)² = 16
The equation of a line is y = 4 – x
The circle and line intersect at two points, A and B.
(a)
Show that the x-coordinates of A and B satisfy the equation 2x² – 18x + 37 = 0
(3 marks)
(b)
For A, the x-coordinate and y-coordinate are both positive.
Work out the coordinates of A.
Give each coordinate to 2 decimal places.(3 marks)
Question 13 (3 marks)
Expand and simplify fully (3x – 1)(4 – x)(2x + 5)
(3 marks)
Question 14 (3 marks)
P is the point on the graph y = 3^(-x) that has x-coordinate –2
Q is a point on the graph y = 1152 × (1/2)^x
y-coordinate of P = y-coordinate of Q Work out the x-coordinate of Q.
(3 marks)
Question 15 (3 marks)
Simplify fully (2x² + 9x – 18)/(12x² – 8x – 15)
(3 marks)
Question 16 (3 marks)
Simplify dfrac√(x³)(√(x³) + x³)√x
Give your answer in the form x^(a/b) + x^c where a, b and c are integers.
(3 marks)
Question 17 (6 marks)
(a)
A solid cone has base radius 7 cm, perpendicular height h cm and slant height l cm
Diagram: A cone with base radius 7 cm (dashed, from the centre of the base to its edge), perpendicular height h cm (dashed, from the apex to the centre of the base, at right angles to it) and slant height l cm.
Curved surface area of a cone = π rl The total surface area of the cone is 224π cm²
Work out the size of the angle between the slant height and the base.(4 marks)
(b)
ABCDEFGH is a cuboid.
ABCD is horizontal.
MN is vertical.
The acute angle between the planes MNB and HDCG is 36°Diagram: Cuboid ABCDEFGH with AB = 50 cm, BC = 25 cm and CG = 16 cm. ABCD is the base. M is on HG and N is on DC, with MN vertical. Dashed lines join M to N, M to B and N to B.
Work out the length CN.
(2 marks)
Question 18 (4 marks)
Rearrange m = (2k³ – 7)/(5m – 3k³) to make k the subject.
(4 marks)
Question 19 (4 marks)
In the expansion of (3 + ax)⁵ where a is a non-zero constant
8 × coefficient of x² = coefficient of x⁴ Work out the possible values of a.
(4 marks)
Question 20 (5 marks)
AB is the tangent to the circle (x – 3)² + (y – 4)² = 29 at the point (5, 9)
Diagram: Axes with origin O. A circle crosses both axes, with its centre marked by a dot. A straight line touches the circle at the point (5, 9) and crosses the y-axis at A and the x-axis at B, on the positive axes.
Work out the area of triangle AOB.
(5 marks)
Question 21 (6 marks)
f(x) = (2 – x)/3
g(x) = 18x² + 15x
f⁻¹(x) + gf(x) simplifies to ax² + bx + c Work out the values of a, b and c.
(6 marks)
Question 22 (5 marks)
A, B and C are points on a circle.
PAC is a straight line.
PB is a tangent to the circle.
AC = BCDiagram: A circle through A, B and C. The straight line PAC cuts the circle at A and C, and PB touches the circle at B. Chords AB and BC are drawn; AC and BC are marked equal. Angle APB is x and angle PBA is y.
Prove that x = 90 – 3/2y
(5 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.