Algebra and functions: AQA A‑Level Maths questions

Every AQA A-Level Maths exam question on B Algebra and functions, 2018–2025, with its mark scheme and examiner’s report.

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426 questions

Q12
June 2025 · Paper 1
8 marksB Algebra and functions
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Q1
June 2025 · Paper 2
1 markB Algebra and functions
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Q3
June 2025 · Paper 2
1 markB Algebra and functions
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Q5
June 2025 · Paper 3
2 marksB Algebra and functions
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Q8
June 2025 · Paper 3
7 marksH Integration B Algebra and functions
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Q9
June 2025 · Paper 3
5 marksB Algebra and functions I Numerical methods
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Q1
June 2024 · Paper 1
1 markB Algebra and functions
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Q2
June 2024 · Paper 1
1 markB Algebra and functions F Exponentials and logarithms
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Q3
June 2024 · Paper 1
1 markB Algebra and functions
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Q7
June 2024 · Paper 1
4 marksB Algebra and functions
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Q11
June 2024 · Paper 1
5 marksB Algebra and functions
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Q13
June 2024 · Paper 1
6 marksB Algebra and functions A Proof
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Q17
June 2024 · Paper 1
6 marksB Algebra and functions
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Q3
June 2024 · Paper 2
1 markB Algebra and functions
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Q9
June 2024 · Paper 2
13 marksD Sequences and series B Algebra and functions
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Q14
June 2024 · Paper 2
3 marksQ Kinematics B Algebra and functions
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Q2
June 2024 · Paper 3
1 markB Algebra and functions
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Q7
June 2024 · Paper 3
5 marksB Algebra and functions
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Q3
June 2023 · Paper 1
1 markB Algebra and functions F Exponentials and logarithms
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Q6
June 2023 · Paper 1
5 marksF Exponentials and logarithms B Algebra and functions
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All 62 questions on Algebra and functions, as a list

June 2025, Paper 1, Q12 (8 marks): Functions f and g are defined by f(x) = x2 + 5 x ∈ ℝ g(x) = √ x x ≥ 0 12 (a) Using set notation, state the range of f [2 marks] 12 (b) Determine whether f has a
June 2025, Paper 2, Q1 (1 mark): Describe the single transformation which maps the curve with the equation y = ln x onto the curve with the equation y = 2 ln x Tick (✓) one box. [1 mark] Stretc
June 2025, Paper 2, Q3 (1 mark): The diagram shows the graph with equation y = (x + 2)(x – 7) x 7 O –2 y Solve the inequality. (x + 2)(x – 7) > 0 Tick (✓) one box. [1 mark] x ∈ (–∞, –2) ∩ (7, ∞
June 2025, Paper 3, Q5 (2 marks): The diagram shows the graphs of y = 2x and y = (x – 3)(x + 4). x y O 3 – 4 The region R is defined by the inequalities (x – 3)(x + 4) ≤ y ≤ 2x and x ≤ 0 Shade t
June 2025, Paper 3, Q8 (7 marks): (a) The expression x 2x2 + 3x + 1 can be written in the form A x + 1 + B 2x + 1 Find the value of A and the value of B [3 marks] 8 (b) Use your answer to part (
June 2025, Paper 3, Q9 (5 marks): (a) Describe a sequence of two transformations which maps the graph with equation y = 1 x onto the graph with equation y = 3 x – 4 [2 marks] 9 (b) State the equ
June 2024, Paper 1, Q1 (1 mark): Find the coefficient of x in the expansion of (4x3 – 5x2 + 3x – 2)(x5 + 4x + 1) Circle your answer. [1 mark] –5 –2 7 11
June 2024, Paper 1, Q2 (1 mark): The function f is defined by f (x) = ex + 1 for x ℝ Find an expression for f –1(x) Tick (✓) one box. [1 mark] f –1(x) = ln (x – 1) f –1(x) = ln (x) – 1 f –1(x)
June 2024, Paper 1, Q3 (1 mark): The expression 12x2 + 3x + 7 3x – 5 can be written as Ax + B + C 3x – 5 State the value of A Circle your answer. [1 mark] 3 4 7 9
June 2024, Paper 1, Q7 (4 marks): Show that 3 + √ 8n 1 + √ 2n can be written as 4n – 3 + √ 2n 2n – 1 where n is a positive integer. [4 marks]
June 2024, Paper 1, Q11 (5 marks): It is given that f (x) = x(x – a)(x – 6) where 0 < a < 6 11 (a) Sketch the graph of y = f (x) on the axes below. [3 marks] y x O 11 (b) Sketch the graph of y =
June 2024, Paper 1, Q13 (6 marks): (a) It is given that P(x) = 4x3 + 8x2 + 11x + 4 Use the factor theorem to show that (2x + 1) is a factor of P(x) [2 marks] 13 (b) Express P(x) in the form P(x)
June 2024, Paper 1, Q17 (6 marks): The function f is defined by f (x) = | x | + 1 for x ℝ The function g is defined by g(x) = ln x where g has its greatest possible domain. 17 (a) Using set notat
June 2024, Paper 2, Q3 (1 mark): Solve the inequality (1 – x)(x – 4) < 0 [1 mark] Tick (✓) one box. {x : x < 1} ∩ {x : x > 4} {x : x < 1} ∩ {x : x > 4} {x : x < 1} ∩ {x : x ≥ 4} {x : x < 1} ∩ {
June 2024, Paper 2, Q9 (13 marks): (a) (i) Find the binomial expansion of (1 + 3x)–1 up to and including the term in x2 [2 marks] 9 (a) (ii) Show that the first three terms in the binomial expans
June 2024, Paper 2, Q14 (3 marks): The displacement, r metres, of a particle at time t seconds is r = 6t – 2t 2 14 (a) Find the value of r when t = 4 [1 mark] 14 (b) Determine the range of values
June 2024, Paper 3, Q2 (1 mark): The quadratic equation 4x2 + bx + 9 = 0 has one repeated real root. Find b Circle your answer. [1 mark] b = 0 b = ±12 b = ±13 b = ±36
June 2024, Paper 3, Q7 (5 marks): The graphs with equations y = 2 + 3x – 2x2 and x + y = 1 are shown in the diagram below. The graphs intersect at the points A and B 7 (a) On the diagram above,
June 2023, Paper 1, Q3 (1 mark): The curve with equation y = ln x is transformed by a stretch parallel to the x-axis with scale factor 2 Find the equation of the transformed curve. Circle your
June 2023, Paper 1, Q6 (5 marks): Show that the equation 2 log10 x = log10 4 + log10 (x + 8) has exactly one solution. Fully justify your answer. [5 marks]
June 2023, Paper 1, Q7 (4 marks): (a) Given that n is a positive integer, express 7 3 + 5 n p − 7 5 n p − 3 as a single fraction not involving surds. [3 marks] 7 (b) Hence, deduce that 7 3 + 5 n
June 2023, Paper 1, Q16 (14 marks): (a) Given that 1 16 − 9x2 × A 4 − 3x + B 4 + 3x find the values of A and B [3 marks] 16 (b) An empty container, in the shape of a cuboid, has length 1.6 metres,
June 2023, Paper 2, Q1 (1 mark): The graph of y = ax2 + bx + c has roots x = 2 and x = 5 , as shown in the diagram below. O 5 2 y x State the set of values of x which satisfy ax2 + bx + c > 0 T
June 2023, Paper 2, Q7 (5 marks): The functions f and g are defined by f (x) = 10 − 2x p for x × 5 g(x) = 1 x for x 6= 0 The function h has maximum possible domain and is defined by h(x) = gf (x
June 2023, Paper 3, Q1 (1 mark): The graph of y = f (x) is shown below. O y x One of the four equations listed below is the equation of the graph y = f (x) Identify which one is the correct equ
June 2023, Paper 3, Q4 (2 marks): Express 5 − x 3p x2 in the form 5x p − xq where p and q are constants. [2 marks]
June 2023, Paper 3, Q6 (9 marks): (a) Sketch the curve with equation y = x2(2x + a) where a > 0 [3 marks] O y x 6 (b) The polynomial p(x) is given by p(x) = x2(2x + a) + 36 6 (b) (i) It is given
June 2022, Paper 1, Q3 (1 mark): The curve y = log4 x is transformed by a stretch, scale factor 2, parallel to the y-axis. State the equation of the curve after it has been transformed. Circle
June 2022, Paper 1, Q11 (10 marks): The polynomial p(x) is given by p(x) = x3 + (b + 2)x2 + 2(b + 2)x + 8 where b is a constant. 11 (a) Use the factor theorem to prove that (x + 2) is a factor of
June 2022, Paper 2, Q8 (5 marks): (a) Sketch the graph of y = 1 x2 [2 marks] x y 8 (b) The graph of y = 1 x2 can be transformed onto the graph of y = 9 x2 using a stretch in one direction. Beth
June 2022, Paper 2, Q10 (15 marks): A gardener has a greenhouse containing 900 tomato plants. The gardener notices that some of the tomato plants are damaged by insects. Initially there are 25 dam
June 2022, Paper 3, Q3 (1 mark): The function f is defined by f (x) = 2x + 1 Solve the equation f (x) = f −1(x) Circle your answer. [1 mark] x = −1 x = 0 x = 1 x = 2
June 2022, Paper 3, Q10 (13 marks): The function f is defined by f (x) = x2 + 10 2x + 5 where f has its maximum possible domain. The curve y = f (x) intersects the line y = x at the points P and Q
November 2021, Paper 1, Q1 (1 mark): State the set of values of x which satisfies the inequality (x − 3)(2x + 7) > 0 Tick (3) one box. [1 mark] x : − 7 2 < x < 3 x : x < −3 or x > 7 2 x : x < − 7 2
November 2021, Paper 1, Q13 (8 marks): (a) Given that P(x) = 125x3 + 150x2 + 55x + 6 use the factor theorem to prove that (5x + 1) is a factor of P(x). [2 marks] 13 (b) Factorise P(x) completely. [3
November 2021, Paper 2, Q1 (1 mark): Four possible sketches of y = ax2 + bx + c are shown below. Given b2 − 4ac = 0 and a, b and c are non-zero constants, which sketch is the only one that could po
November 2021, Paper 2, Q4 (6 marks): Figure 1 shows the graph of y = j2xj Figure 1 x O y 4 (a) On Figure 1 add a sketch of the graph of y = j3x − 6j [2 marks] 4 (b) Find the coordinates of the poin
November 2021, Paper 2, Q5 (3 marks): Express 5(x − 3) (2x − 11)(4 − 3x) in the form A (2x − 11) + B (4 − 3x) where A and B are integers. [3 marks]
November 2021, Paper 2, Q10 (11 marks): The function h is defined by h(x) = x p x − 3 where h has its maximum possible domain. 10 (a) Find the domain of h. Give your answer using set notation. [3 mark
November 2021, Paper 3, Q2 (1 mark): Simplify fully (x + 3)(6 − 2x) (x − 3)(3 + x) for x 6= ×3 Circle your answer. [1 mark] −2 2 (6 − 2x) (x − 3) (2x − 6) (x − 3)
November 2021, Paper 3, Q6 (4 marks): Given that x > 0 and x 6= 25 , fully simplify 10 + 5x − 2x 1 2 − x 3 2 5 − x p Fully justify your answer. [4 marks]
November 2020, Paper 1, Q4 (5 marks): (a) Sketch the graph of y = 4 × j2x × 6j y x O [3 marks] 4 (b) Solve the inequality 4 × j2x × 6j > 2 [2 marks]
November 2020, Paper 1, Q9 (7 marks): Chloe is attempting to write 2x2 + x (x + 1)(x + 2)2 as partial fractions, with constant numerators. Her incorrect attempt is shown below. Step 1 2x2 + x (x + 1
November 2020, Paper 1, Q13 (15 marks): The function f is defined by f (x) = 2x + 3 x × 2 x 2 R, x 6= 2 13 (a) (i) Find f ×1 [3 marks] 13 (a) (ii) Write down an expression for ff (x). [1 mark] 13 (b)
November 2020, Paper 3, Q3 (1 mark): Determine which one of these graphs does not represent y as a function of x. Tick (3) one box. [1 mark] x y x y x y x y
November 2020, Paper 3, Q4 (7 marks): p(x) = 4x3 − 15x2 − 48x − 36 4 (a) Use the factor theorem to prove that x − 6 is a factor of p(x). [2 marks] 4 (b) (i) Prove that the graph of y = p(x) intersec
November 2020, Paper 3, Q6 (7 marks): The graph of y = f (x) is shown below. (2, 6) (0, 2) O (–2, –2) (–1, 0) x y 6 (a) Sketch the graph of y = f (−x) [2 marks] x O y 6 (b) Sketch the graph of y = 2
June 2019, Paper 1, Q6 (8 marks): The function f is defined by f (x) = 1 2 (x2 + 1), x × 0 6 (a) Find the range of f. [1 mark] 6 (b) (i) Find f −1(x) [3 marks] 6 (b) (ii) State the range of f −1
June 2019, Paper 2, Q1 (1 mark): Identify the graph of y = 1 − jx + 2j from the options below. Tick (3) one box. [1 mark] A x y B x y C x y D x y
June 2019, Paper 2, Q2 (1 mark): Simplify a 2 3 × a 2 5 q Circle your answer. [1 mark] a 2 15 a 4 15 a 8 15 a 16 15
June 2019, Paper 2, Q3 (1 mark): Each of these functions has domain x 2 R Which function does not have an inverse? Circle your answer. [1 mark] f (x) = x3 f (x) = 2x + 1 f (x) = x2 f (x) = ex f
June 2019, Paper 2, Q4 (4 marks): x2 + bx + c and x2 + dx + e have a common factor (x + 2) Show that 2(d − b) = e − c Fully justify your answer. [4 marks]
June 2019, Paper 2, Q7 (10 marks): (a) Sketch the graph of any cubic function that has both three distinct real roots and a positive coefficient of x3 [2 marks] x y 7 (b) The function f (x) is de
June 2019, Paper 3, Q4 (3 marks): Sketch the region defined by the inequalities y × (1 − 2x)(x + 3) and y − x × 3 Clearly indicate your region by shading it in and labelling it R. [3 marks] x y
June 2019, Paper 3, Q7 (8 marks): (a) Express 4x + 3 (x − 1)2 in the form A x − 1 + B (x − 1)2 [3 marks] 7 (b) Show that ( 4 3 4x + 3 (x − 1)2 dx = p + ln q where p and q are rational numbers. [
June 2018, Paper 1, Q2 (1 mark): The graph of y = 5x is transformed by a stretch in the y-direction, scale factor 5 State the equation of the transformed graph. Circle your answer. [1 mark] y =
June 2018, Paper 1, Q4 (3 marks): The function f is defined by f (x) = e x−4, x 2 R Find f −1(x) and state its domain. [3 marks]
June 2018, Paper 1, Q12 (10 marks): p(x) = 30x3 − 7x2 − 7x + 2 12 (a) Prove that (2x + 1) is a factor of p(x) [2 marks] 12 (b) Factorise p(x) completely. [3 marks] 12 (c) Prove that there are no r
June 2018, Paper 2, Q4 (6 marks): A curve, C, has equation y = x2 − 6x + k , where k is a constant. The equation x2 − 6x + k = 0 has two distinct positive roots. 4 (a) Sketch C on the axes below
June 2018, Paper 2, Q8 (10 marks): (a) Determine a sequence of transformations which maps the graph of y = sinx onto the graph of y = 3 p sin x − 3 cos x + 4 Fully justify your answer. [7 marks]
June 2018, Paper 3, Q4 (3 marks): Sketch the graph of y = j2x + aj , where a is a positive constant. Show clearly where the graph intersects the axes. [3 marks] x y
June 2018, Paper 3, Q6 (13 marks): A function f is defined by f (x) = x 2x − 2 p 6 (a) State the maximum possible domain of f. [2 marks] 6 (b) Use the quotient rule to show that f 0(x) = x − 2 (2

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