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AQA Level 2 Further Maths June 2019 Paper 2 (calculator) · old spec 8360
27 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 (5 marks)
(a)
a( 3 , 5 ) = 4( 2a + 3 , b )
Work out the values of a and b.
(3 marks)
(b)
( m & -1 , 1 & 1 )( 2 & 2 , -2 & -1 ) = mathbfI where mathbfI is the identity matrix.
Work out the value of m.
(2 marks)
Question 2 (3 marks)
Here is a sketch of quadrilateral PQRS.
M is the midpoint of PS.Diagram: Axes with the quadrilateral PQRS: P (4, 1), Q (10, –3), R (14, 0) and S (6, 11). M is marked at the midpoint of PS.
Use gradients to show that MR is parallel to PQ.
(3 marks)
Question 3 (4 marks)
-2 < a < 0 and -1 < b < 1
Tick the correct box for each statement.
Always true Sometimes true Never true a² < 0 -1 < b³ < 1 b/a < 0 a – b > 0 (4 marks)
Question 4 (4 marks)
P is a point on a curve.
The curve has gradient function (x⁵ – 17)/10
The tangent to the curve at P is parallel to the line 3x – 2y = 9Work out the x-coordinate of P.
(4 marks)
Question 5 (5 marks)
(a)
Write sqrt[4]a × a⁻⁹ as an integer power of a.
(2 marks)
(b)
Simplify fully ((4cd²)³)/(2cd⁴)
(3 marks)
Question 6 (3 marks)
Here is a sketch of the curve y = (2x + 3)(x – 2)
The curve intersects the x-axis at A and B.Diagram: A U-shaped curve crossing the x-axis at A, left of the origin, and at B, right of the origin. Its lowest point is below the x-axis, just right of the y-axis.
(a)
Complete the coordinates of A and B.
(2 marks)
(b)
Write down the range of values for x for which (2x + 3)(x – 2) < 0
(1 mark)
Question 7 (2 marks)
(a)
On the grid, sketch a graph for which
the rate of change of y with respect to x is always zero.Diagram: A blank grid with x from 0 to 6 and y from 0 to 6.
(1 mark)
(b)
On the grid, sketch a graph for which
the rate of change of y with respect to x is always a positive constant.Diagram: A blank grid with x from 0 to 6 and y from 0 to 6.
(1 mark)
Question 8 (7 marks)
(a)
A linear sequence has first term 7 + 12√5
The term-to-term rule is
add 9 – 2√5 One term of the sequence is an integer.
Work out the value of this integer.
(2 marks)
(b)
The nth term of a different sequence is (3n² – 1)/(n² + 1)
Work out the sum of the first three terms.
(2 marks)
(c)
The first four terms of a quadratic sequence are
-3 3 13 27
Work out an expression for the nth term.
(3 marks)
Question 9 (2 marks)
Factorise fully (p + 6)¹¹ – (p + 6)¹⁰
(2 marks)
Question 10 (4 marks)
(a)
f(x) = x³ – 2
The domain of f(x) is x ≤ 3
Work out the range of f(x).
(2 marks)
(b)
g(x) = 5 – x²
The domain of g(x) is -2 ≤ x ≤ 1
Work out the range of g(x).
(2 marks)
Question 11 (1 mark)
Here is a sketch of a quadratic curve which has a maximum point at (-2, 5)
Diagram: A sketch of a curve shaped like an upside-down U, crossing the x-axis twice, with its maximum point left of the y-axis and above the x-axis.
What is the equation of the normal to the curve at the maximum point?
Circle your answer.
x = -2 y = 5 x = 5 y = -2 (1 mark)
Question 12 (3 marks)
The diagram shows a solid hemisphere.
The diameter is 12a cm
The volume is 486π cm³Diagram: A solid hemisphere resting on its flat circular face. An arrow under it across the full width is labelled 12a cm.
Work out the value of a.
(3 marks)
Question 13 (4 marks)
Simplify fully (x – x³)/(2x + 2x²)
You must show your working.
(4 marks)
Question 14 (3 marks)
Here is a triangle.
Diagram: A triangle. Two sides, a cm and 3a cm, meet at an angle of 120°. The third side, opposite this angle, is b cm.
Use the cosine rule to work out the ratio b² : a²
(3 marks)
Question 15 (4 marks)
Rearrange m = (2p + 1)/p + (p + 5)/3p to make p the subject.
(4 marks)
Question 16 (3 marks)
The curve y = 2√(x – a) + 5 passes through the point (1, 8)
Work out the value of a.
(3 marks)
Question 17 (5 marks)
Show that (x + 1)(x + 3)(x + 4) – x(x² + 7x + 11)
can be written in the form (x + a)(x + b) where a and b are positive integers.(5 marks)
Question 18 (3 marks)
Solve 4(x – 5)² = k² where k is a constant.
Give your answers in their simplest form in terms of k.
(3 marks)
Question 19 (6 marks)
ABC is a right-angled triangle.
ACD is an isosceles triangle.
All dimensions are in centimetres.Diagram: Quadrilateral ABCD split by the line AC. Triangle ABC has a right angle at B, with AB = 3x (vertical) and BC = 4x (horizontal). Triangle ACD is above and to the right of AC, with AD = CD = 6.5x.
(a)
Show that AC = 5x
(1 mark)
(b)
Work out an expression, in cm², for the area of quadrilateral ABCD.
Give your answer in the form px² where p is an integer.(5 marks)
Question 20 (5 marks)
A, B, C and D are points on a circle.
D, E and F are points on a different circle, centre C.
DCE, ADF and BCF are straight lines.
angle DEF = xDiagram: Two overlapping circles. The first passes through A, B, C and D. The second has centre C and passes through D, E and F. DCE is a diameter of the second circle; A, D and F lie on one straight line and B, C and F on another. AB is joined, and FE. Angle DEF is marked x.
(a)
Prove that angle BAD = 2x
(3 marks)
(b)
In the case when AB is parallel to DE, work out the size of angle x.
(2 marks)
Question 21 (4 marks)
ABCDEFGH is a cuboid.
BC = 15 cm CD = 12 cm DH = 8 cm
Diagram: A cuboid ABCDEFGH. The front face is BCGF, with B bottom left, C bottom right, G top right and F top left. A, D, H and E are behind B, C, G and F. The edges BA, AD and AE are hidden (dashed). The line CE is drawn from the bottom front right corner to the top back left corner.
Work out the size of the angle between the line CE and the plane CDHG.
(4 marks)
Question 22 (5 marks)
(a)
Show that (2sin² x – 1 + cos² x)/(sin x cos x) is equivalent to tan x
(3 marks)
(b)
Hence solve (2sin² x – 1 + cos² x)/(sin x cos x) = -1 for 0° ≤ x ≤ 360°
(2 marks)
Question 23 (6 marks)
A circle has centre C and equation (x – 1)² + (y + 3)² = 25
P(4, -7) and Q are points on the circle.The tangent at Q is parallel to the x-axis.
The tangents at P and Q intersect at point R.Diagram: Axes with a circle centre C. Q is the highest point of the circle; the tangent at Q is a horizontal line. P (4, –7) is on the circle, lower right; the tangent at P slopes up to the right and meets the tangent at Q at R.
(a)
Write down the coordinates of C.
(1 mark)
(b)
Show that the equation of the tangent at Q is y = 2
(1 mark)
(c)
Work out the x-coordinate of R.
(4 marks)
Question 24 (4 marks)
Show that the curve y = 3/5x⁵ + x⁴ has exactly two stationary points.
(4 marks)
Question 25 (3 marks)
f(x) = x³ – 10x – c where c is a positive integer.
(x + c) is a factor of f(x).Use the factor theorem to work out the value of c.
(3 marks)
Question 26 (4 marks)
f(x) is a function with domain all values of x.
f(x) = √(x² + 6x – a) where a is a constant.
Work out the possible values of a.
Give your answer as an inequality.(4 marks)
Question 27 (3 marks)
The curve y = f(x) has dy/dx = (x + 2)⁶ + (x + 2)⁴
The curve has exactly one stationary point at P where x = -2
Use the expression for dy/dx to show that P is a point of inflection.
(3 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.