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AQA Level 2 Further Maths June 2017 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)
(a)
The nth term of a sequence is (3 – 5n)/2
Work out the difference between the 20th term and the 8th term.
(2 marks)
(b)
The nth term of another sequence is 3n/(1 – 2n)
Write down the limiting value of the sequence as n to ∞
(1 mark)
Question 2 (5 marks)
mathbfA = ( 4 & -1 , 3 & -2 ) mathbfB = ( 5 , 2 )
(a)
Work out mathbfA²
(2 marks)
(b)
kmathbfB = ( 11 – 3k , 11 – 6k ) where k is a constant.
Work out the value of k.
(2 marks)
(c)
Give a reason why it is not possible to work out mathbfBA
(1 mark)
Question 3 (4 marks)
(a)
p, q and r are all integers greater than 1
pqr = 1365
Work out one possible set of values for p, q and r.
(2 marks)
(b)
a and b are both square numbers greater than 1
ab – 11b is also a square number.
By factorising ab – 11b, work out one possible pair of values for a and b.
You must show your working.(2 marks)
Question 4 (2 marks)
Solve dfrac563√(x) = 4
(2 marks)
Question 5 (3 marks)
M is the midpoint of PQ.
Diagram: A straight line segment from P (a, 6) at the bottom left up to Q (4, 10) at the top right, with M (3a, 8) marked in the middle.
Work out the value of a.
(3 marks)
Question 6 (3 marks)
A cone has vertex V.
C is the centre of the base.
The slant height, VA, is 20 cm
The angle between VA and VC is 38°Diagram: A cone standing on its circular base. V is the vertex at the top, C is the centre of the base, directly below V (the dashed line VC meets the base at a right angle), and A is on the edge of the base, to the right of C. The slant height VA is 20 cm.
Work out the radius of the base.
(3 marks)
Question 7 (4 marks)
The equation of the line through B, P and A is 4x + 5y = 40
BP : PA = 2 : 3
Diagram: Axes with a straight line sloping down from left to right. It crosses the y-axis at B and the x-axis at A, and P is a point on it between B and A. O, the origin, is joined to P.
Work out the area of triangle OBP.
(4 marks)
Question 8 (4 marks)
The perimeter of a triangular flower bed, ABC, is marked out using 27 metres of rope.
Diagram: A triangle ABC with BC along the bottom and A at the top. AB = 7 m and AC = 8 m; BC is not labelled. The angle at A is marked.
Work out the size of angle BAC.
(4 marks)
Question 9 (5 marks)
-11 < 5x ≤ 5 and 6x + 7 ≤ 4x + 4
Show that there is exactly one integer that x can be.
(5 marks)
Question 10 (3 marks)
ABC is an isosceles triangle with AB = AC
The area of ABC is 57.76 cm²Diagram: Triangle ABC with A at the bottom left, B to its right and C above and to the left of A. Angle BAC is marked 150°.
Work out the length of AB.
(3 marks)
Question 11 (6 marks)
A function f(x) is defined as
beginarrayrll f(x) = & 3 – 2x & -2 ≤ x < 0 \\ = & (1 + x)(3 – x) & 0 ≤ x < 4 \\ = & 5x – 25 & 4 ≤ x ≤ 5 endarray
(a)
Draw the graph of y = f(x) on the axes below.
Diagram: A blank grid with x from –3 to 6 and y from –6 to 9.
(4 marks)
(b)
State the range of f(x)
(2 marks)
Question 12 (4 marks)
(a)
Factorise fully 75 – 3x²
(2 marks)
(b)
Simplify fully (3n + 1)² – (3n – 1)²
(2 marks)
Question 13 (3 marks)
Simplify fully 8a/(3a + 6) × (5a + 10)/(3a²) ÷ 4/(15a³)
(3 marks)
Question 14 (5 marks)
The line y = ax + b is perpendicular to the line x + 4y = 74
The lines intersect at the point where x = 2
Diagram: Axes with two straight lines that cross at right angles. The line y = ax + b slopes steeply upwards from left to right. The line x + 4y = 74 slopes gently downwards. A dashed vertical line runs down from the point where they cross to the x-axis at 2.
Work out the values of a and b.
(5 marks)
Question 15 (3 marks)
Rearrange w = (8x – y)/y to make y the subject.
(3 marks)
Question 16 (3 marks)
(a)
a = 3^(2b)
Circle the correct expression for 1/a3^(2b – 1) 3^(-2b) -3^(2b) (1/3)^(-2b) (1 mark)
(b)
y = 5^x
Circle the correct expression for 25y5^(x + 2) 25^x 5^(2x) 125^x (1 mark)
(c)
w = 2^m
Circle the correct expression for w³8^(3m) 6^m 2^(m + 3) 2^(3m) (1 mark)
Question 17 (10 marks)
Here is a sketch of y = x³ – 6x² + 7
Diagram: A sketch of a cubic curve, rising from the bottom left. It crosses the x-axis to the left of O, has a maximum point on the y-axis, comes down through the x-axis to a minimum point below the axis on the right, then rises through the x-axis again.
(a)
Use differentiation to work out the coordinates of the stationary point that is a minimum.
You must show your working.(4 marks)
(b)
The three roots of x³ – 6x² + 7 = 0 are the x-coordinates of the points where the graph intersects the x-axis.
Show that x = -1 is one root of x³ – 6x² + 7 = 0
(1 mark)
(c)
Hence, work out the other two roots of x³ – 6x² + 7 = 0
Give your answers to 2 decimal places.
You must show your working.(5 marks)
Question 18 (4 marks)
The diagram shows a rectangle with a diagonal drawn.
The given expressions for the measurements are in centimetres.Diagram: A rectangle with a right angle marked at the bottom left. Its short side is x and its long side is 2x. A dashed diagonal runs from the top left corner to the bottom right corner and is labelled 4y.
Work out an expression for the area of the rectangle, in cm²
Give your answer in its simplest form, in terms of y.(4 marks)
Question 19 (4 marks)
Here is a sketch of y = sin x for 0° ≤ x ≤ 360°
Diagram: The graph of y = sin x from 0° to 360°. It starts at 0, rises to 1 at 90°, comes down through 0 at 180° to –1 at 270°, and returns to 0 at 360°.
alpha is an acute angle measured in degrees.
sin alpha = k where k is a constant.Write the answers to each of the following in terms of k, without involving trigonometric functions.
(a)
sin(180° – alpha)
(1 mark)
(b)
sin(360° – alpha)
(1 mark)
(c)
cos alpha
(2 marks)
Question 20 (5 marks)
Two circles overlap.
A, B and E lie on the circle, centre O.
B, C, D and E lie on the other circle.
AOBC and AED are straight lines.
CD = CE
angle BAE = xDiagram: Two overlapping circles. The circle on the right has centre O; A is on its right-hand side, and B and E are on its left-hand side, where it crosses the larger circle on the left, which also passes through C (at the top) and D (at the bottom left). The straight line AOBC runs from A through O and B to C, and the straight line AED runs from A through E to D. CD, CE and BE are joined; CD and CE are marked equal. Angle BAE is marked x.
(a)
Give a reason why angle BEA = 90°
(1 mark)
(b)
Prove that angle DCE = 2x
(4 marks)
Question 21 (9 marks)
Here is a sketch of y = (x + 2)(4 – x)
The graph intersects the axes at A (-2, 0), B (4, 0) and C.
Diagram: A sketch of an upside-down U-shaped curve. It crosses the x-axis at A (–2, 0) and B (4, 0), and the y-axis at C, above O. Its maximum point is just to the right of the y-axis.
(a)
Work out the coordinates of C.
(1 mark)
(b)
Work out the gradient function of the curve.
(3 marks)
(c)
The normal to the curve at C intersects the x-axis at D.
Show that length BD = 2 × length AB
(5 marks)
Question 22 (5 marks)
The equation of a circle is (x – 2)² + (y – 1)² = 16
The equation of a line is y = 2x + 1
The circle and the line intersect at two points.
Work out the coordinates of the two points.
You must show your working.
Do not use trial and improvement.(5 marks)
Question 23 (3 marks)
In this question, tan x neq 0 and sin x neq 0
Show that 1/(tan² x) – 1/(sin² x) is a constant.
(3 marks)
Question 24 (5 marks)
Write 12x² – 60x + 5 in the form a(bx + c)² + d where a, b, c and d are integers.
(5 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.