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AQA Level 2 Further Maths Specimen 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)
A sketch of the lines y = 2x and y = 6 is shown.
Diagram: A sketch of the line y = 2x through the origin O, and the horizontal line y = 6. P is where y = 6 crosses the y-axis and Q is where the two lines meet.
Work out the area of triangle OPQ.
(3 marks)
Question 2 (2 marks)
A circle, centre (0, 0) has circumference 20π
Work out the equation of the circle.
(2 marks)
Question 3 (2 marks)
M is the midpoint of the line AB.
Diagram: A straight line from A, at the top left, down to B, at the bottom right, with its midpoint M marked.
Work out the values of p and r.
(2 marks)
Question 4 (2 marks)
(a)
Circle the solution of -3x < -18
x > -6 x < -6 x > 6 x < 6 (1 mark)
(b)
Circle the solution of x² ≥ 16
x ≥ -4 or x ≤ 4 x ≤ -4 or x ≥ 4 x ≥ -4 or x ≥ 4 x ≤ -4 or x ≤ 4 (1 mark)
Question 5 (2 marks)
Here is a sketch of y = f(x) where f(x) is a quadratic function.
The graph
intersects the x-axis at A (-1, 0) and B
has a maximum point at (0.5, 6)
Diagram: A sketch of an n-shaped parabola. It crosses the x-axis at A, left of the origin O, and at B, right of O, with its highest point just right of the y-axis.
(a)
Work out the coordinates of B.
(1 mark)
(b)
The equation f(x) = k has exactly one solution.
Write down the value of k.
(1 mark)
Question 6 (4 marks)
mathbfA = ( 4 & -1 , -7 & 2 ) mathbfB = ( s , -5 ) mathbfC = ( -1 , t ) mathbfD = ( 2 & 1 , 7 & u )
s, t and u are constants.
(a)
mathbfAB = mathbfC
Work out the values of s and t.
(3 marks)
(b)
mathbfAD = mathbfI
Work out the value of u.
(1 mark)
Question 7 (3 marks)
Work out the equation of the straight line that is
parallel to the line 2y = x
and
intersects the x-axis at (4, 0)
(3 marks)
Question 8 (4 marks)
(a)
Work out ab/cd ÷ bc/ad
Give your answer as a single fraction in its simplest form.
(2 marks)
(b)
Work out 7/(2x²) + 4/3x
Give your answer as a single fraction in its simplest form.
(2 marks)
Question 9 (5 marks)
A, B and C are points on a circle, centre O.
Diagram: A circle with centre O. A is at the top of the circle and B and C are at the bottom left and bottom right, with the triangle ABC drawn and the radii OB and OC. Angle BAC is 2x – 50°, angle BOC is x + 62° and angle OCB is y.
Work out the size of angle y.
(5 marks)
Question 10 (5 marks)
y = (6x⁹ + x⁸)/(2x⁴)
Work out the value of dfracd²ydx² when x = 0.5
(5 marks)
Question 11 (4 marks)
For sequence A, nth term = n/(14n + 30)
For sequence B, nth term = 2/n
The kth term of sequence A equals the kth term of sequence B.
Work out the value of k.
You must show your working.(4 marks)
Question 12 (7 marks)
This shape is made from two rectangles.
All dimensions are in centimetres.Diagram: An L-shaped shape made from two rectangles. The top edge is 30x and the right-hand edge goes down 15x, then a step goes in 10x, then down y to the bottom edge, which is 20x.
(a)
The perimeter of the shape is 252 cm
Show that y = 126 – 45x
(2 marks)
(b)
The area of the shape is A cm²
Show that A = 2520x – 450x²
(2 marks)
(c)
Use differentiation to work out the maximum value of A as x varies.
(3 marks)
Question 13 (4 marks)
f(x) = 3x² + 6 for all x
g(x) = √(x – 5) x ≥ 5
(a)
Work out the value of gf(4)
(2 marks)
(b)
Show that fg(x) can be written in the form a(x – a) where a is an integer.
(2 marks)
Question 14 (3 marks)
Use the sine rule to work out the size of obtuse angle x.
Diagram: A triangle. The short side on the left is y and the long side across the top is 2y. The obtuse angle x is at the bottom left, between y and the base, and the angle 18° is at the bottom right, opposite y.
(3 marks)
Question 15 (4 marks)
Here is a sketch of the curve y = ab^(-x) where a and b are positive constants.
(0, 3) and (2, 0.48) lie on the curve.
Diagram: A sketch of a curve that falls from left to right, getting closer to the x-axis without meeting it. It crosses the y-axis at (0, 3) and passes through (2, 0.48).
Work out the values of a and b.
(4 marks)
Question 16 (5 marks)
Simplify (8x³ – 50x)/(2x(6x² – x – 35))
Give your answer in the form (ax + b)/(cx + d) where a, b, c and d are integers.
(5 marks)
Question 17 (4 marks)
By multiplying both sides of the equation by x^(1/2)
Solve 2x^(3/2) – 3x^(1/2) = 7x^(-1/2) for x > 0
Give your answer to 3 significant figures.You must show your working.
(4 marks)
Question 18 (3 marks)
How many odd numbers greater than 30 000 can be formed from these digits
2 4 6 7 8
with no repetition of any digit?
(3 marks)
Question 19 (5 marks)
f(x) = 3x³ – 2x² – 7x – 2
(a)
Use the factor theorem to show that (3x + 1) is a factor of f(x).
(2 marks)
(b)
Factorise f(x) fully.
(3 marks)
Question 20 (5 marks)
VABCD is a pyramid with a horizontal rectangular base ABCD.
V is directly above the centre of the base.VA = VB = VC = VD = 10 cm
AB = 8 cm BC = 6 cm
M is the midpoint of BC.Diagram: A pyramid VABCD with a rectangular base. AB is the front edge of the base and BC goes back on the right; V is at the top, above the centre of the base. M is the midpoint of BC. The lines DC, DM and VM are dashed.
Work out the size of angle VMD.
(5 marks)
Question 21 (4 marks)
Show that (2n + 3)³ + n³ is divisible by 9 for all integer values of n.
(4 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.