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AQA Level 2 Further Maths November 2021 Paper 2 (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)
Expand and simplify 5(2x – 1) + 4(11 – x)
Give your answer in the form a(bx + c) where a, b and c are integers greater than 1(3 marks)
Question 2 (4 marks)
(a)
5m is decreased by 40%
The answer is (m + 1)Work out the value of m.
(2 marks)
(b)
Solve 3√(2w – 10) = 18
(2 marks)
Question 3 (3 marks)
The rectangle and triangle shown have equal areas.
Diagram: A rectangle with length 12de and width 2d.
Diagram: A right-angled triangle with height 9d and base 8e².
Work out the value of d/e
Give your answer in its simplest form.(3 marks)
Question 4 (3 marks)
The equations of the two circles shown are
x² + y² = 100 and x² + y² = 36
Diagram: Two circles centred at the origin O, one inside the other. The part of the ring between them in the first quadrant, between the positive x-axis and the positive y-axis, is shaded.
Work out the shaded area.
Give your answer as an integer multiple of π.(3 marks)
Question 5 (3 marks)
SQR is a right-angled triangle.
P is a point on SQ.
Angle SPR = 45°
M is the midpoint of QR.
k is a constant.Diagram: Axes with origin O. Q and P are on a vertical line, with P above Q, and S above P on the same line. R is level with S, to the right. Triangle SQR has a right angle at S. PR is drawn, and angle SPR is 45°. M is the midpoint of QR.
Work out the coordinates of M.
(3 marks)
Question 6 (3 marks)
Rearrange y = √((x + 2w)/3) to make w the subject.
(3 marks)
Question 7 (4 marks)
(a)
a is a value greater than 1
Work out the value of m for which (a^m)⁴ = (a⁵)^(2m)
(2 marks)
(b)
w³x²y⁵ = w¹³x⁷
Write y in terms of w and x.
Give your answer in its simplest form.(2 marks)
Question 8 (4 marks)
A function f is given by
beginalignedat2 f(x) &= 4x & & x < 0 \\ &= x² – 8x & & 0 ≤ x ≤ 8 \\ &= 16 – 2x & & x > 8 endalignedat
A sketch of y = f(x) is shown.
Diagram: A sketch of y = f(x): a straight line rising steeply to the origin from below the x-axis on the left, then a U-shaped curve from the origin down to its lowest point (4, –16) and back up to the x-axis at 8, then a straight line sloping down from there.
Work out all the values of x for which f(x) = -12
(4 marks)
Question 9 (4 marks)
(a)
Circle the expression that is equivalent to 1/a + 1/b
2/(a + b) ab/(b + a) 2/ab (b + a)/ab
(1 mark)
(b)
Simplify fully (6c⁴ – c³)/(36c² – 1)
(3 marks)
Question 10 (3 marks)
The radius of a sphere, in cm, is 3k/2
The volume of the sphere, in cm³, is 972π
Volume of a sphere = 4/3π r³ where r is the radius Work out the value of k.
(3 marks)
Question 11 (3 marks)
Expand and simplify fully (5x + 3y²)(4x – y²)
(3 marks)
Question 12 (5 marks)
A and B are points on the line y = 3x + 2
B, C and D (5, 0) are points on the line L.
OA : AC = 1 : 4Diagram: Axes with origin O. The line y = 3x + 2 rises from bottom left, crossing the y-axis at A. Line L slopes down from top left, crossing the y-axis at C, above A, and the x-axis at D. The two lines cross at B, to the right of the y-axis.
Work out the x-coordinate of B.
(5 marks)
Question 13 (4 marks)
P is the point on the curve y = ax³ + 10x² where x = 2
The gradient of the normal to the curve at P is -1/4Work out the value of a.
(4 marks)
Question 14 (3 marks)
(a)
mathbfA = ( 1 & 0 , 0 & -1 )
Describe geometrically the single transformation represented by mathbfA.
(1 mark)
(b)
mathbfB = ( 0 & 1 , -1 & 0 )
Describe geometrically the single transformation represented by mathbfB²
(2 marks)
Question 15 (5 marks)
A, B and C are points on a circle, centre O.
ACD is a straight line.
Angle BCD = wDiagram: A circle with centre O. A is at the bottom left of the circle, B at the right and C at the top. The straight line ACD goes from A through C to D, outside the circle. Chords AB and CB and radii OA and OB are drawn. Angle OAB is x and angle BCD is w.
Prove that w = x + 90°
(5 marks)
Question 16 (3 marks)
The coefficient of x⁴ in the expansion of (a + 2x)⁶ is 1500
Work out the two possible values of a.
(3 marks)
Question 17 (5 marks)
ABCDEFGH is a cube with side length 32 cm
M and N are points on DH and CG respectively.Diagram: A cube ABCDEFGH. The front face is ABFE with A bottom left, B bottom right, F top right and E top left; the back face is DCGH, with D behind A, C behind B, G behind F and H behind E. AD, DC and DH are hidden (dashed). M is on DH and N is on CG, with MN level. Lines MG and MB are drawn. Angle GMN is 28° and angle MNG is a right angle.
Work out the size of the angle that the line BM makes with the plane ABCD.
(5 marks)
Question 18 (5 marks)
y = 12x + 3/x
Show that y has a minimum value when x = 0.5
(5 marks)
Question 19 (3 marks)
(a)
f(x) = (x + 2)³
g is a function such that gf(x) = (x + 2)¹²
Work out an expression for g(x)
(1 mark)
(b)
h(x) = x² + 5
k is a function such that hk(x) = 4x² + 5
Work out an expression for kh(x)
(2 marks)
Question 20 (4 marks)
Show that (2sin x + cos x)/(tan x) – 1/(sin x) can be written in the form acos x + bsin x
where a and b are integers.(4 marks)
Question 21 (6 marks)
3x² + 2bx + 8a can be written in the form 3(x + a)² + b + 2
Work out the two possible pairs of values of a and b.
(6 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.