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AQA Level 2 Further Maths June 2023 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)
The function f is given by f(x) = 2x + 1
(a)
Work out x when f(x) = -5
(2 marks)
(b)
The function g is given by g(x) = x²
Work out fg(3)
(2 marks)
Question 2 (2 marks)
Factorise fully 6x²y + 21xy
(2 marks)
Question 3 (3 marks)
(a)
Circle the transformation matrix that represents a reflection in the line y = -x
( 0 & -1 , -1 & 0 ) ( 0 & 1 , 1 & 0 ) ( 0 & 1 , -1 & 0 ) ( 0 & -1 , 1 & 0 )
(1 mark)
(b)
Show that ( 2 & 4 , -1 & -3 )( -3 & -4 , 1 & 2 ) = kmathbfI where k is an integer.
(2 marks)
Question 4 (5 marks)
S (7, 2) and T (5, -4) are points on a straight line.
(a)
Work out the gradient of the line.
(2 marks)
(b)
Work out the distance between S and T.
Give your answer in the form a√b where a and b are both integers greater than 1
(3 marks)
Question 5 (3 marks)
X_n and Y_n are the nth terms of two sequences.
X_n = (n – 1)(n + 1)
Y_n = (n + 1)(n + 2)
Prove that every term of the sequence with nth term Y_n – X_n is a multiple of 3
(3 marks)
Question 6 (4 marks)
The equation of a curve is y = x⁶ + 4x² – 7
Work out the equation of the tangent to the curve at the point (1, -2)
Give your answer in the form y = mx + c
(4 marks)
Question 7 (4 marks)
The diagram below shows a cone and a prism.
All measurements are in cmThe cone has base radius r and perpendicular height x.
The prism has a triangular cross section with base y and perpendicular height y.
The length of the prism is x.Diagram: A cone. Its perpendicular height, from the centre of the base to the top, is x. Its base radius is r.
Diagram: A triangular prism. The triangle at the front has base y and perpendicular height y. The length of the prism is x.
Volume of a cone = 1/3 × area of base × perpendicular height The volume of the cone is four times the volume of the prism.
Express r in terms of y.
(4 marks)
Question 8 (6 marks)
A circle has centre (0, 0) and radius 5
A straight line has equation 2y = x + 5
Work out the coordinates of the two points where the circle and straight line intersect.
Do not use trial and improvement.
You must show your working.(6 marks)
Question 9 (4 marks)
Rearrange w = (y² + 5)/(y² – 2) to make y the subject.
(4 marks)
Question 10 (4 marks)
Rationalise the denominator and simplify fully
dfrac1 + √53 – √5
(4 marks)
Question 11 (3 marks)
y = 1/12x⁴ + 3x² + 4
Work out the positive value of x for which dfracd²ydx² = 55
(3 marks)
Question 12 (4 marks)
(a)
Write down the value of x for 0° ≤ x ≤ 360° when sin x = -1
(1 mark)
(b)
Work out the values of y for 0° ≤ y ≤ 360° when √3tan y = 1
(3 marks)
Question 13 (3 marks)
Write (2x – 3)/x – 1/3x + 1 as a single fraction.
Give your answer in its simplest form.
(3 marks)
Question 14 (4 marks)
Solve 8/x + 3x ≤ 10 where x is positive.
(4 marks)
Question 15 (4 marks)
Solve (x^(1/2) – x^(3/2))² = x² + x
(4 marks)
Question 16 (4 marks)
The expansions of (1 + 12x)⁴ and (a + 4x)³ have the same coefficient of x²
Work out the value of a.
(4 marks)
Question 17 (5 marks)
The curve y = ax³ + bx² + 7 has a stationary point at (-2, 11)
Work out the values of a and b.
(5 marks)
Question 18 (5 marks)
Solve the simultaneous equations
2x + y = 13
x + 3z = 2
z – 2y = -7
Do not use trial and improvement.
You must show your working.(5 marks)
Question 19 (3 marks)
8x² + 20x + n equiv c(x + d)² + 3 where c, d and n are constants.
Work out the values of c, d and n.
(3 marks)
Question 20 (6 marks)
P, Q and R are points on a circle, centre O.
Angle POR = 120° PQ = 4 cm QR = 5 cm
Diagram: A circle with centre O. P, Q and R are on the circle: Q at the top left, P at the bottom and R on the right. Chords PQ (4 cm), QR (5 cm) and PR are drawn, and radii OP and OR. Angle POR is 120°.
Work out the radius of the circle.
Give your answer in the form √k where k is an integer.
(6 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.