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AQA Level 2 Further Maths June 2022 Paper 1 (non-calculator)
21 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 (3 marks)
(x + 1) is increased by 20%
Its value is now the same as (x + 6)
Work out the value of x.
(3 marks)
Question 2 (2 marks)
The point (-6, -4) lies on a straight line with gradient 3/2
Work out the coordinates of the point where the line crosses the y-axis.
(2 marks)
Question 3 (6 marks)
(a)
beginaligned f(x) &= 4 – x && 0 ≤ x < 1 \\ &= 4x – x² && 1 ≤ x < 4 \\ &= 2x – 8 && 4 ≤ x ≤ 6 endaligned
On the grid, draw the graph of y = f(x)
Diagram: A blank grid with x from 0 to 6 and y from 0 to 6.
(4 marks)
(b)
g(x) = 6 – 3x
Work out g⁻¹(x)
(2 marks)
Question 4 (3 marks)
(a)
Circle the value of tan² 30°
1/4 1/3 1/2 3/4
(1 mark)
(b)
On the axes, sketch
y = cos x for 0° ≤ x ≤ 360°
Diagram: Blank axes: x from 0° to 360°, marked at 90°, 180°, 270° and 360°, and y from –1 to 1.
(2 marks)
Question 5 (3 marks)
(3x + a)(5x – 4) equiv 15x² – 2x + b
Work out the values of a and b.
(3 marks)
Question 6 (3 marks)
y = 2x⁴(x³ + 2 – 3/x)
Work out dy/dx
(3 marks)
Question 7 (3 marks)
ABC is a right-angled triangle with vertices A (-1, 5), B (-2, 5) and C (-1, 53/4)
Work out the length of BC.
(3 marks)
Question 8 (3 marks)
Use matrix multiplication to show that, in the x-y plane,
• a rotation, 90° anticlockwise about the origin, followed by
• a reflection in the line y = xis equivalent to a reflection in the x-axis.
(3 marks)
Question 9 (6 marks)
(a)
A quadratic sequence starts –2 –1 4 13
Work out an expression for the nth term.
(3 marks)
(b)
A different quadratic sequence has nth term n² + 10n
Use an algebraic method to work out how many terms in the sequence are less than 2000
Do not use trial and improvement.
You must show your working.(3 marks)
Question 10 (3 marks)
Rationalise and simplify fully dfrac√33 + √3
(3 marks)
Question 11 (4 marks)
Expand and simplify fully (3 + 2x)⁵
(4 marks)
Question 12 (3 marks)
The nth term of a sequence is (3n²)/(n² + 2)
(a)
One term in the sequence is 32/11
Work out the value of n.
(2 marks)
(b)
Write down the limiting value of the sequence as n to ∞
(1 mark)
Question 13 (3 marks)
Simplify fully (6x³y⁻² + 9x⁵y) ÷ 3x²y⁻³
(3 marks)
Question 14 (3 marks)
Rearrange ef = (5e + 4)/3 to make e the subject.
(3 marks)
Question 15 (5 marks)
B, C and D are points on a circle, centre P.
AB and AC are tangents to the circle.
Diagram: A circle with centre P. A is above the circle; the tangents AB and AC touch it at B (left) and C (right). D is on the circle between B and C, near the top. Chords DB and DC and radii PB and PC are drawn. The angle at A is x and angle BDC is y.
Prove that y = 90 + x/2
(5 marks)
Question 16 (6 marks)
Solve the simultaneous equations
beginaligned x – y &= 19/4 \\ xy &= -3 endaligned
Do not use trial and improvement.
You must show your working.(6 marks)
Question 17 (6 marks)
The point P lies on the circle x² + y² = 16
The line OP is at an angle of 60° to the positive x-axis.
Diagram: Axes with origin O and the circle x² + y² = 16 centred on O. P is on the circle in the first quarter; the line OP makes an angle of 60° with the positive x-axis.
(a)
Show that the coordinates of point P are (2, 2√3)
(2 marks)
(b)
Work out the equation of the tangent to the circle at P.
Write your answer in the form x + ay = b where a and b are constants.
(4 marks)
Question 18 (3 marks)
In triangle RST RS : ST = 1 : 4
Diagram: A long, thin triangle RST: T on the left, S on the right and R above, nearer to S. The angle at R is 135° and the angle at T is θ.
Work out the exact value of sin theta.
(3 marks)
Question 19 (3 marks)
Write 6x² – 24x + 17 in the form a(x + b)² + c where a, b and c are integers.
(3 marks)
Question 20 (6 marks)
The curve y = x⁴ – 18x² has three stationary points.
Work out the coordinates of the three stationary points and determine their nature.
You must show your working.(6 marks)
Question 21 (3 marks)
Show that (4cos² x + 3sin² x – 4)/(cos² x) equiv -tan² x
(3 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.