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AQA Level 2 Further Maths June 2023 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)
Solve (8d – 3)/(3d – 7) = 5/2
(3 marks)
Question 2 (4 marks)
(a)
The first four terms of a linear sequence are
15 18.5 22 25.5
Work out an expression for the nth term.
(2 marks)
(b)
A different linear sequence has nth term 318 – 9n
Work out the value of the first negative term in the sequence.
(2 marks)
Question 3 (2 marks)
( 3 & 5 , u & 2 )( 1 , 4 ) = ( t , 6 )
Work out the values of t and u.
(2 marks)
Question 4 (4 marks)
A line passes through P (1, k) and Q (r, 6) where k and r are constants.
The midpoint of PQ has x-coordinate 5
The gradient of the line is 2
Work out the value of k.
(4 marks)
Question 5 (3 marks)
y = 0.5x⁴
Work out the value of x for which the rate of change of y with respect to x is 6.75
(3 marks)
Question 6 (2 marks)
The equation of a circle is (x + 7)² + (y – 4)² = 36
Complete these statements.
The coordinates of the centre of the circle are (ldotsldots, ldotsldots)
The radius of the circle is ldotsldots
(2 marks)
Question 7 (3 marks)
Here is a sketch of the curve y = ax² + bx + c where a, b and c are constants.
The curve intersects the x-axis at (-4, 0) and (p, 0)
The turning point has x-coordinate 0.5
Diagram: A U-shaped curve crossing the x-axis at –4 and at p, to the right of the y-axis. Its lowest point is below the x-axis at x = 0.5, just to the right of the y-axis.
(a)
Work out the value of p.
(1 mark)
(b)
Solve ax² + bx + c > 0
(2 marks)
Question 8 (4 marks)
ABC is a triangle with perpendicular height AD.
Diagram: Triangle ABC with base BC horizontal and A above it. D is on BC with AD (dashed) perpendicular to BC; AD is 4 cm. BD is shorter than DC. The angle w is at C.
Area of ABC = 25 cm²
BD : DC = 2 : 3
Work out the size of angle w.
(4 marks)
Question 9 (7 marks)
The dimensions of the cuboid are given in centimetres.
Diagram: A cuboid. The front edge along the bottom is 3x, the edge going back on the right is 2x and the upright edge at the back right is x + 2y. Hidden edges are dashed.
The total length of all 12 edges is 300 cm
(a)
Show that y = (75 – 6x)/2
(2 marks)
(b)
The volume of the cuboid is V cm³
Show that V = 450x² – 30x³
(2 marks)
(c)
Use calculus to work out the maximum value of V as x varies.
(3 marks)
Question 10 (3 marks)
Line K has equation 4x – 5y = 17
Line L passes through the points (3, 6) and (-5, 16)
Tick (✓) the correct statement about lines K and L.
The lines are parallel. The lines are perpendicular. The lines are neither parallel nor perpendicular. Show working to support your answer.
(3 marks)
Question 11 (4 marks)
Expand and simplify fully (2x³ – 9)(3x² + 4) + x(x – 4)²
(4 marks)
Question 12 (3 marks)
VABCD is a pyramid.
The square horizontal base, ABCD, has side length 15 cm
V is directly above the centre, X, of the base.
VA = 28 cmDiagram: A square-based pyramid VABCD with the apex V at the top. AB is the front edge of the base, 15 cm. VA is 28 cm. X, the centre of the base, is marked with a dot. AD, DC and VD are hidden (dashed).
Work out the size of the angle that VA makes with ABCD.
(3 marks)
Question 13 (6 marks)
(a)
Circle the expression equivalent to 3x⁻⁷
-3/(x⁷) -1/(3x⁷) 1/(3x⁷) 3/(x⁷)
(1 mark)
(b)
Simplify fully (12w⁸)/((4w³)²)
(2 marks)
(c)
√y × 3√(y) = sqrt[c]y^d where c and d are positive integers.
Work out the least possible values of c and d.
(3 marks)
Question 14 (4 marks)
Simplify fully (15a²)/(a² + 6a – 16) × (8 – 4a)/3a
(4 marks)
Question 15 (4 marks)
The function g is given by g(x) = a × b^x where a and b are constants.
The domain of the function is -1 ≤ x ≤ 2
P(0, 1/2) and Q(1, 3/2) are points on the graph y = g(x)
Diagram: Axes with O at the origin and x marked at –1, 1 and 2. The curve y = g(x) is drawn from x = –1 to x = 2, just above the x-axis on the left and rising more and more steeply. It crosses the y-axis at P, and Q is on the curve at x = 1.
Work out the range of the function.
(4 marks)
Question 16 (4 marks)
(2x – 3) is a factor of 6x³ – 25x² + 28x – 6
Solve 6x³ – 25x² + 28x – 6 = 0
Give all solutions as exact values.
(4 marks)
Question 17 (4 marks)
The function h is given by h(x) = ax(3x² – 2) + 5x where a is a positive constant.
h is an increasing function for all values of x.
Work out the possible values of a.
Give your answer as an inequality.
(4 marks)
Question 18 (3 marks)
Here is a sketch of y = cos x for values of x from 0° to 360°
alpha is an obtuse angle measured in degrees.
cos alpha = -k where k is a positive constant.
Diagram: The graph of y = cos x from 0° to 360°, starting at 1, falling through 90° to –1 at 180°, then rising through 270° back to 1 at 360°. The obtuse angle α is marked on the x-axis just after 90°, with dashed lines down to the curve and across to –k on the y-axis.
(a)
Tick (✓) two boxes that show expressions for x where cos x = -k
180° – alpha 180° + alpha 270° – alpha 270° + alpha 360° – alpha 360° + alpha (2 marks)
(b)
Circle the expression for x where sin x = -k
alpha 90° + alpha 180° – alpha 180° + alpha
(1 mark)
Question 19 (3 marks)
In these simultaneous equations, k is a positive constant.
3x + 4y = k
y = 2kx
Solve the simultaneous equations.
Give the answers in their simplest form in terms of k.
(3 marks)
Question 20 (3 marks)
Show that
2sin³ x + 2sin xcos² x + 5tan xcos x simplifies to psin x where p is a constant.
(3 marks)
Question 21 (4 marks)
A, B, C, D and E are points on a circle, centre O.
Diagram: A circle with centre O, and A, B, C, D, E on it, going anticlockwise A (top left), B (lower left), C (bottom), D (right), E (top right). The pentagon ABCDE is drawn, with the radii OA, OB and OD. Angle OAB is 2x, angle AED is 7x and angle BCD is 6x.
Work out the value of x.
(4 marks)
Question 22 (3 marks)
Five-digit integers are made using
1 2 7 8 9
For each integer, all the digits are used exactly once.
The integers are
greater than 40 000 and odd.
How many different integers can be made?
You must show your working.
(3 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.