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AQA Level 2 Further Maths November 2021 Paper 1 (non-calculator)
23 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)
Work out the distance between the points A (-3, 7) and B (5, 1)
(2 marks)
Question 2 (3 marks)
y = x(2x⁴ – 7x³)
Work out an expression for the rate of change of y with respect to x.
(3 marks)
Question 3 (1 mark)
Here are four sketch graphs.
Circle the letter of the sketch graph that represents 3x + 2y = 5
Diagram: Graph A: a straight line sloping down from left to right, crossing the y-axis below O.
Diagram: Graph B: a straight line sloping down from left to right, crossing the y-axis above O.
Diagram: Graph C: a straight line sloping up from left to right, crossing the y-axis above O.
Diagram: Graph D: a straight line sloping up from left to right, crossing the y-axis below O.
(1 mark)
Question 4 (5 marks)
(a)
The function f is given by f(x) = 3x – 5
The range is 13 < f(x) < 19
Work out the domain of the function.
(1 mark)
(b)
The function g is given by g(x) = x² – 4 with domain -1 < x < 3
Work out the range of the function.
(2 marks)
(c)
The function h is given by h(x) = (3 + x)/2
Work out h⁻¹(x)
(2 marks)
Question 5 (3 marks)
The nth term of a sequence is (2n + 47)/(n + 1)
(a)
A term of the sequence has a value of 5
Work out the value of n.
(2 marks)
(b)
Write down the limiting value of the sequence as n to ∞
(1 mark)
Question 6 (2 marks)
Here is a sketch of y = sin x for 0° ≤ x ≤ 360°
Diagram: The graph of y = sin x from 0° to 360°. It starts at 0, rises to 1 at 90°, falls through 0 at 180° to –1 at 270°, and rises back to 0 at 360°.
You are given that sin 220° = -k
Work out the two values of x for 0° ≤ x ≤ 360° for which y = k
(2 marks)
Question 7 (3 marks)
Solve 2x² + 4 > (2x – 3)(x + 1)
(3 marks)
Question 8 (2 marks)
Simplify √3(√75 + √48) writing your answer as an integer.
(2 marks)
Question 9 (3 marks)
Expand and simplify fully (2x – 5)(3x – 4)(x + 2)
(3 marks)
Question 10 (3 marks)
The first four terms of a quadratic sequence are
0 1 0 –3
Work out an expression for the nth term.
(3 marks)
Question 11 (4 marks)
( 2 & 1 , 0 & 3 )( a & b , 0 & 0.4 ) = kmathbfI where k is a constant and mathbfI is the identity matrix.
Work out the values of a and b.
(4 marks)
Question 12 (5 marks)
A circle, centre C (4, -2), passes through the origin and point A (8, 0) on the x-axis.
The tangent at A is shown.
Diagram: Axes with O at the origin. A circle with centre C (4, –2) passes through O and through A (8, 0) on the x-axis. The tangent to the circle at A is drawn, sloping down from left to right.
(a)
Work out the equation of the circle.
(2 marks)
(b)
Work out the equation of the tangent to the circle at A.
(3 marks)
Question 13 (3 marks)
Here is a sketch of y = k^x where k > 0
A (2, 31/16) is a point on the curve.
Diagram: A sketch of y = k to the power x: a curve above the x-axis, very close to it on the left, crossing the y-axis above O and rising more and more steeply to the right. A is marked on the curve to the right of the y-axis.
(a)
Work out the value of k.
(2 marks)
(b)
B is a point on the curve with x-coordinate –1
Work out the y-coordinate of B.
(1 mark)
Question 14 (5 marks)
Solve the simultaneous equations.
4a – b + 3c = 27
3a + 2b – c = 5
2a – 5c = -7
Do not use trial and improvement.
You must show your working.
(5 marks)
Question 15 (2 marks)
Work out the value of x where 0° ≤ x ≤ 90° for which 3tan² x = 1
(2 marks)
Question 16 (5 marks)
f(x) = 200x³ + 100x² – 18x – 9
(a)
Use the factor theorem to show that (2x + 1) is a factor of f(x).
(2 marks)
(b)
Hence solve f(x) = 0
(3 marks)
Question 17 (3 marks)
Here is the graph of y = x² – 6x + 5 for values of x between 0 and 6
Diagram: A grid with x from 0 to 6 and y from –4 to 5. The curve y = x squared – 6x + 5 is drawn from (0, 5), down through (1, 0) to its lowest point (3, –4), and up through (5, 0) to (6, 5).
By drawing a suitable linear graph on the grid, work out approximate solutions to
x² – 7x + 9 = 0
(3 marks)
Question 18 (4 marks)
Here is a triangle.
Diagram: Triangle PQR, long and thin, with P on the left, Q on the right and R below and a little to the left of Q. PQ is x cm, QR is 3 cm, PR is 7 cm and angle PQR is 60°.
Use the cosine rule to work out the value of x.
(4 marks)
Question 19 (4 marks)
y = f(x) is the graph of a cubic function.
y < 0 for x < 5
y ≥ 0 for x ≥ 5
The function is
increasing for x < -1
decreasing for -1 < x < 2
increasing for x > 2
Draw a possible sketch of y = f(x) for values of x from –2 to 6
Diagram: A blank grid with x from –2 to 6 marked on the x-axis, and the y-axis through O with no scale.
(4 marks)
Question 20 (3 marks)
Miriam’s date of birth is 14/09/2006
She makes a 4-digit number code using digits from her date of birth.
The 4-digit number she makes must
not start with 0
have all different digits.
How many codes can she make?
(3 marks)
Question 21 (5 marks)
ABC is a triangle.
The perpendicular from A meets BC at D.
BC = (6 + 2√7) cm
Diagram: Triangle ABC with BC along the bottom, B on the left and C on the right. A is above BC, and the line AD from A meets BC at D at a right angle.
Area of triangle ABC = (13 + 3√7) cm²
Work out the length, in cm, of AD.
Give your answer in the form a + b√c where a, b and c are integers.
(5 marks)
Question 22 (4 marks)
Solve 8^x = dfrac2⁵⁶ – 4²⁶ 30
(4 marks)
Question 23 (6 marks)
F, H, K and J are points on a circle.
Chords HJ and KF intersect at L.
EFG is a tangent to the circle.
FH and JK are parallel.
Diagram: A circle through H on the left, K at the top, J on the right and F at the bottom. Chords HJ and KF cross at L. FH and JK are parallel, and FJ is joined. EFG is the tangent at F, with E to the left and G to the right. Angle FHJ is 2x, angle EFH is 3y, angle HFJ is 98° and angle KLJ is 7y.
(a)
Angle FHJ = 2x
Give reasons why angle FKJ and angle HJK are also equal to 2x.
(2 marks)
(b)
Work out the values of x and y.
You must show your working.
Do not use trial and improvement.
(4 marks)
Questions © AQA, recreated by Mr Barton Maths for teaching. Each question is credited to its board, series, paper and number.