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AQA Level 2 Further Maths June 2022 Paper 2 (calculator)
22 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 (2 marks)
Factorise fully 12w + 18w²
(2 marks)
Question 2 (2 marks)
M is the midpoint of PQ.
Diagram: A straight line from P at the bottom left to Q at the top right, with its midpoint M marked. P (–4, a), M (2, –1), Q (8, 6).
Work out the value of a.
(2 marks)
Question 3 (6 marks)
(a)
Work out 3( 4 & 2 , 1 & 0 )( 2 & 0 , -1 & 5 )
Give your answer as a single matrix.
(3 marks)
(b)
( 7 & a² , b & -5 )( 2 , a ) = ( 78 , 12 )
Work out the values of a and b.
(3 marks)
Question 4 (4 marks)
Line A has equation y + 4x = 6
Line B is parallel to line A and passes through the point (2, 1)
The point (d, 2d) lies on line B.Work out the value of d.
(4 marks)
Question 5 (3 marks)
Work out all the negative integer values of x for which 3x² < 48
(3 marks)
Question 6 (3 marks)
Prove algebraically that when n is an integer
((2n + 1)² – (2n – 1)²)/4 is always even.
(3 marks)
Question 7 (2 marks)
How many integers between 200 000 and 400 000 can be formed using only the digits
1 2 3 5 8 9
with no repetition of any digit?
(2 marks)
Question 8 (4 marks)
A curve has equation y = x³ – 5x²
At two points on the curve, the rate of change of y with respect to x is 4
(a)
Work out an equation, in terms of x, to represent this information.
Give your answer in the form ax² + bx + c = 0 where a, b and c are integers.(2 marks)
(b)
Hence, work out the two possible values of x.
Give your answers to 3 significant figures.(2 marks)
Question 9 (4 marks)
The first three terms of a linear sequence are
30 30 + 4k 30 + 8k
where k is a constant.
(a)
Work out an expression, in terms of k, for the 4th term.
Give your answer in its simplest form.(1 mark)
(b)
The 100th term of the sequence is 525
Work out the value of k.(3 marks)
Question 10 (1 mark)
Here are four sketch graphs.
Circle the letter of the sketch graph that represents y = 3 × 2^x
Diagram: Graph A: a curve above the x-axis, through the point (0, 6) on the y-axis, high on the left and falling, levelling out just above the x-axis on the right.
Diagram: Graph B: a curve above the x-axis, through the point (0, 3) on the y-axis, high on the left and falling, levelling out just above the x-axis on the right.
Diagram: Graph C: a curve above the x-axis, through the point (0, 6) on the y-axis, just above the x-axis on the left and rising steeply on the right.
Diagram: Graph D: a curve above the x-axis, through the point (0, 3) on the y-axis, just above the x-axis on the left and rising steeply on the right.
(1 mark)
Question 11 (2 marks)
Here is a right-angled triangle.
Diagram: A right-angled triangle with the right angle at the bottom left. The sloping side is 4 cm and the bottom side is (a – 3) cm. The angle x is at the bottom right, between these two sides.
You are given that a > 5
Use trigonometry to work out the range of values of x.
(2 marks)
Question 12 (5 marks)
Work out the gradient of the curve y = (12x³ – 8x + 3)/(4x²)
at the point where x = -1
You must show your working.(5 marks)
Question 13 (3 marks)
A (-2, 5) and B (4, 13) are points on a circle.
AB is a diameter.Work out the equation of the circle.
Give your answer in the form (x – a)² + (y – b)² = c where a, b and c are integers.(3 marks)
Question 14 (3 marks)
PQRS is a cyclic quadrilateral.
Diagram: A circle through P, Q, R and S, with the quadrilateral PQRS drawn inside it. P is at the top left, S at the top right, R on the right and Q at the bottom left. Angle PSR is marked 4(x + 15°).
Angle PSR = 4(x + 15°)
Angle PQR is 40° smaller than angle PSR.Work out the value of x.
(3 marks)
Question 15 (5 marks)
Simplify fully (x/2 + 3x/5) ÷ √((x⁶)/4)
(5 marks)
Question 16 (4 marks)
Here is an isosceles triangle.
All the angles are acute.Diagram: An isosceles triangle. The bottom side and the left-hand sloping side are both 16 cm. The angle y is at the bottom right.
The area of the triangle is 120 cm²
Work out the size of angle y.
(4 marks)
Question 17 (5 marks)
Solve the simultaneous equations
beginaligned a + 3b – 2c &= 4 \\ 4a – 3b + 5c &= -5 \\ 2a + b + 3c &= 9 endaligned
Do not use trial and improvement.
You must show your working.(5 marks)
Question 18 (5 marks)
ABCDEFGH is a cuboid.
AB = 40 cm BC = 9 cm CG = 20 cm
P is a point on HG such that HP : PG = 3 : 7
AP = 25 cmDiagram: A cuboid ABCDEFGH. The base ABCD has AB = 40 cm along the front and BC = 9 cm going back; E, F, G and H are above A, B, C and D, and CG = 20 cm. P is a point on the top edge HG. The lines AP, PC and AC are drawn and angle APC is marked. AD, DC and DH are hidden (dashed).
Work out the size of angle APC.
(5 marks)
Question 19 (3 marks)
Expand and simplify fully (3x + 4)(2x – 3)(5x – 2)
(3 marks)
Question 20 (6 marks)
f(x) = 2x³ + 11x² + 12x – 9
(a)
Use the factor theorem to show that (2x – 1) is a factor of f(x).
(2 marks)
(b)
Show that f(x) = 0 has exactly two solutions.
(4 marks)
Question 21 (4 marks)
Work out the values of x between 0° and 360° for which
2tan² x = 3
Give your answers to 1 decimal place.
You must show your working.(4 marks)
Question 22 (4 marks)
Using powers of 2 or otherwise, work out the non-zero value of x for which
(16^x)^x = dfrac12^(3x)
You must show your working.
(4 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.