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AQA Level 2 Further Maths June 2018 Paper 2 (calculator) · old spec 8360
24 questions · 105 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)
The nth term of a sequence is (1420 – 5n)/(1420 + 5n)
(a)
Work out the position of the term that has the value zero.
(2 marks)
(b)
Write down the limiting value of the sequence as n to ∞
(1 mark)
Question 2 (2 marks)
P (-3, -10) and Q (a, b) are points on a straight line with gradient 12
Work out one possible pair of integer values for a and b.
(2 marks)
Question 3 (4 marks)
p = (m + 2)/(m² + 1)
(a)
Work out the value of p when m = -5.5
(1 mark)
(b)
Work out the values of m when p = 2
(3 marks)
Question 4 (1 mark)
A, B, C and D are points on a circle, centre O.
Diagram: A circle with centre O. A is at the top, B on the left, C at the bottom and D on the right. The lines AB, AD, BC and CD form a quadrilateral, and BC is extended beyond C. O is joined to B and to D. Angle x is at A between AB and AD, angle w is at O between OB and OD on the side towards C, and angle y is at C between CD and the extension of BC.
Which statement is correct?
Tick one box.x + y = 180° and w = 2x x + y = 180° and x = 2w x = y and w = 2x x = y and x = 2w (1 mark)
Question 5 (4 marks)
On the grid, draw the graph of y = f(x)
beginarrayrll f(x) = & x + 4 & -4 ≤ x < 0 \\ = & 4 – 3x & 0 ≤ x < 2 \\ = & -2 & 2 ≤ x ≤ 5 endarray
Diagram: A blank grid with x from –5 to 6 and y from –3 to 5.
(4 marks)
Question 6 (7 marks)
f(x) = x² – 7 for all values of x
g(x) = 1 – 3x for -4 ≤ x ≤ 4
(a)
Work out the range of f(x).
Give your answer as an inequality.(1 mark)
(b)
Work out the range of g(x).
Give your answer as an inequality.(2 marks)
(c)
Solve 2f(x) = g(x)
You must show your working.
Give your answers to 3 decimal places.(4 marks)
Question 7 (2 marks)
PQRS is a trapezium.
Diagram: Axes with the trapezium PQRS. P (2, 0) and Q (10, 0) are on the x-axis. S (5, a) and R (9, a) are level with each other above them, so SR is parallel to PQ.
The area of the trapezium is 63 square units.
Work out the value of a.
(2 marks)
Question 8 (4 marks)
Here is a sketch of triangle ABC.
P is a point on AB.Diagram: Axes with the triangle ABC: A (1, 6), B (7, 1) and C (9, 12). P is a point on AB, nearer to B than to A.
AP : PB is 3 : 1
Work out the length PC.
Give your answer to 4 significant figures.(4 marks)
Question 9 (4 marks)
y = (2x⁷ + 15x²)/3x
Work out the value of x when dy/dx = 133
(4 marks)
Question 10 (5 marks)
The transformation matrix ( a & b , 2a & 3b ) maps the point (1, -3) onto the point (1, 4)
Work out the values of a and b.
You must show your working.(5 marks)
Question 11 (3 marks)
Expand and simplify fully (x + 2)(x + 3)(x + 4)
(3 marks)
Question 12 (7 marks)
(a)
Write 7/9x + 2/(3x²) as a single fraction in its simplest form.
(3 marks)
(b)
Show that (x⁴)/(x + 4) × (x + 2)/x ÷ (x²)/(3x + 12)
simplifies to the form ax² + bx where a and b are integers.
(4 marks)
Question 13 (4 marks)
(a)
Here is a sketch of y = tan x for 0° ≤ x ≤ 360°
Diagram: The graph of y = tan x from 0° to 360°. It crosses the x-axis at 0°, 180° and 360°, and rises from left to right in each branch, shooting up towards the dashed lines x = 90° and x = 270° and coming up from below just after them.
How many solutions of tan x = k where k > 0 are between 90° and 360°?
(1 mark)
(b)
0 < p < 1
How many solutions of sin x = p – 1 are between 0° and 180°?
You may use a sketch graph to help you.(1 mark)
(c)
State the coordinates of each point where the graph
y = cos x for 0° ≤ x ≤ 360°
meets or intersects an axis.
(2 marks)
Question 14 (7 marks)
(a)
Factorise fully 12pq³r – 18pq²r² + 24pq²r
(2 marks)
(b)
Factorise fully 6(y + 3)⁵ + 4(y + 3)⁴
Give your answer in its simplest form.
Do not attempt to expand (y + 3)⁵ or (y + 3)⁴(3 marks)
(c)
Factorise fully 48 – 75x²
(2 marks)
Question 15 (4 marks)
Work out the rate of change of y with respect to x at the point on the curve
y = x²(x² – 9) where x = -2
You must show your working.
(4 marks)
Question 16 (4 marks)
A = 2 – 5x
B = 3x – 1
C = x²Show that (2A + 3B)² equiv A + B + C
(4 marks)
Question 17 (5 marks)
A circle has equation x² + y² = 29
P is the point (-5, 2)
(a)
Show that P is on the circle.
(1 mark)
(b)
The tangent to the circle at P intersects the x-axis at point Q.
Work out the x-coordinate of Q.
You must show your working.(4 marks)
Question 18 (6 marks)
(a)
Work out all the integer values of x for which
-5 < 4x + 3 ≤ 13
(3 marks)
(b)
Work out the range of values of x for which
x² – 11x + 28 > 0
You must show your working.
(3 marks)
Question 19 (5 marks)
Use matrix multiplication to show that, in the x–y plane,
• a reflection in the line y = -x, followed by
• a rotation, 90° anticlockwise about the origin, followed by
• a reflection in the x-axisis equivalent to a transformation by the identity matrix.
(5 marks)
Question 20 (6 marks)
PQRSTU is a triangular prism.
PQRS is a rectangle and angle QRU = 90°PQ = 10 cm
QR = 12 cm
UR = 7 cm
M is the midpoint of PQ.Diagram: A triangular prism PQRSTU. The base PQRS is a rectangle with PQ = 10 cm along the front and QR = 12 cm going back. RU = 7 cm is vertical, with angle QRU a right angle, and T is above S. TU is the top edge. M is the midpoint of PQ. PS, SR and ST are hidden (dashed).
(a)
Calculate the size of the angle between the line UM and the plane PQRS.
(4 marks)
(b)
Calculate the size of the angle between the planes UMR and UQR.
(2 marks)
Question 21 (3 marks)
The continuous curve y = f(x) has exactly two stationary points.
Here is some information about the curve.x dy/dx is x < -1 positive x = -1 zero -1 < x < 2 negative x = 2 zero x > 2 negative f(-1) = 3 and f(2) = 1
State the coordinates and the nature of each of the stationary points.
(3 marks)
Question 22 (5 marks)
(a)
8 cos x + 5 sin x = 0 where 90° < x < 180°
Work out the size of angle x.
(3 marks)
(b)
6 sin² x + 4 cos² x equiv A + B cos² x where A and B are integers.
Work out the values of A and B.
You must show your working.(2 marks)
Question 23 (6 marks)
For each of these two function machines, when the input is a the output is b.
k > 0 and k neq 1 and a > 0
Diagram: Two function machines. The first: input a, square, add 3, multiply by k, output b. The second: input a, multiply by k, square, add 3, output b.
Work out an expression for a in terms of k.
Give your answer in its simplest form.(6 marks)
Question 24 (4 marks)
Work out the value of p when
9^(0.5p) × 81 = 27^(2p – 1)
(4 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.