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AQA Level 2 Further Maths Specimen Paper 1 (non-calculator)
20 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 (4 marks)
(a)
(y⁶ × y)/(y^m) = y⁴
Circle the value of m.
–2 1.5 2 3
(1 mark)
(b)
aⁿ × a⁵ = a⁵
Work out the value of n.
(1 mark)
(c)
(c⁵)^p = (c²)⁶
Work out the value of p.
(2 marks)
Question 2 (2 marks)
Solve 3√(7x – 13) = 2
(2 marks)
Question 3 (4 marks)
3a(2x – 1) + 4(ax + 5) equiv 60x + b
Work out the values of a and b.
(4 marks)
Question 4 (4 marks)
ABC is a straight line with AB : BC = 5 : 2
Diagram: A straight line from A (3, 7) down to the right through B (5, 5.5) to C, a little further on beyond B.
Work out the coordinates of C.
(4 marks)
Question 5 (3 marks)
y = 2x¹⁰ – 3/(x²)
Work out dy/dx
(3 marks)
Question 6 (3 marks)
Simplify fully (15x²y – 5xy²)/(12x – 4y)
(3 marks)
Question 7 (3 marks)
ABCD is a rhombus with side length 8 cm
Angle ABC = 60°
Diagram: A rhombus ABCD with BC along the bottom, 8 cm long. Angle ABC, at the bottom left, is 60°.
Work out the area of the rhombus.
Give your answer in the form a√b cm² where a and b are integers.
(3 marks)
Question 8 (3 marks)
The curve y = 2x³ – 3x² – 12x + 6
has a maximum point at L (-1, 13)
has a minimum point at M (2, -14)
intersects the y-axis at N.
The curve crosses the x-axis at three distinct points.
On the axes below, sketch the curve.
Label the points L, M and N on your sketch.
Diagram: Blank axes: an x-axis and a y-axis crossing at O, with no scale.
(3 marks)
Question 9 (4 marks)
A, B, C and D are points on a circle.
angle BCA = x angle ACD = 2x angle CAD = 3x angle CAB = 4x
Diagram: A circle through C on the left, A on the right, B above the line CA and D below it. Angle BCA is x and angle ACD is 2x at C; angle CAB is 4x and angle CAD is 3x at A.
Prove that AC is a diameter.
(4 marks)
Question 10 (5 marks)
f(x) = (9x/2)⁻¹
g(x) = √(1 – px³) where p is a constant.
Given that f(1/3) = g(1/3) work out the value of p.
(5 marks)
Question 11 (3 marks)
A circle, centre C, touches the y-axis at the point (0, 2)
The line y = k intersects the circle at the points (1, k) and (5, k)
Diagram: Axes with origin O. A circle with centre C touches the y-axis at (0, 2) and crosses the x-axis. A horizontal line y = k above C cuts the circle at (1, k) and (5, k).
Work out the equation of the circle.
(3 marks)
Question 12 (3 marks)
AB = 4 cm
AC = 7 cm
cos x = -2/7
Diagram: Triangle ABC with A at the top. AB is 4 cm and AC is 7 cm. The angle at A, between AB and AC, is x and is obtuse.
Work out the length of BC.
(3 marks)
Question 13 (5 marks)
Rearrange t = (3w³ + a)/(w³ – 2) to make w the subject.
(5 marks)
Question 14 (4 marks)
Rationalise and simplify dfrac√3 – 7√3 + 1
Give your answer in the form a + b√3 where a and b are integers.
(4 marks)
Question 15 (9 marks)
Point A lies on the curve y = x² + 5x + 8
The x-coordinate of A is –4
(a)
Show that the equation of the normal to the curve at A is 3y = x + 16
(5 marks)
(b)
The normal at A also intersects the curve at B.
Work out the x-coordinate of B.
(4 marks)
Question 16 (4 marks)
The coefficient of the x⁴ term in the expansion of (2x + a)⁶ is 60
Work out the possible values of a.
(4 marks)
Question 17 (5 marks)
Solve the simultaneous equations
beginaligned 2a + b – c &= 8 \\ 4a – 3b – 2c &= -9 \\ 6a + 3b + c &= 0 endaligned
(5 marks)
Question 18 (3 marks)
Solve x^(-2/3) = 121/4
(3 marks)
Question 19 (4 marks)
f(x) = 2x³ – 12x² + 25x – 11
Use differentiation to show that f(x) is an increasing function for all values of x.
(4 marks)
Question 20 (5 marks)
(a)
Show that 2cos²theta equiv 2 – 2sin²theta
(1 mark)
(b)
Hence, solve 2cos²theta + 3sintheta = 3 for 0 < theta < 180°
(4 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.