plumiAlgebra and functions: WJEC A‑Level Maths questions
Every WJEC A-Level Maths exam question on P.2 Algebra and functions, 2018–2025, with its mark scheme and examiner’s report.
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![Summer 2025 Unit 3, question 8: 3 2 x 5 x 7 x 29 x 30 By algebraic division, or otherwise, simplify the expression . [5] 8. 3( 5)( 3)( 2) x x x ](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun25-08-qp.webp)
![Summer 2025 Unit 3, question 11: 5 2 x d x 11. Find , giving your answer in the form lna + b, where a, b are constants. 2 1 2 1 x x [10] 1](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun25-11-qp.webp)
![Summer 2025 Unit 3, question 15: f x , where x ` 3. 15. A function f is defined by 2 x 3 (a) Determine the range of f. [2] f 1( ) x (b) Find an expression for . [5]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun25-15-qp.webp)
![Summer 2025 AS Unit 1, question 1: Solve the inequality 2 x 1 x 6 9 2 x x 1. . [5]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun25-01-qp.webp)
![Summer 2025 AS Unit 1, question 3: 3. Showing all your working, simplify the following expressions. 8 4 2 a a 3 (a) [3] 1 a 3](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun25-03-qp.webp)
![Summer 2025 AS Unit 1, question 5: x kx k 3 0 5. Find the range of values of k for which the quadratic equation has two distinct real roots. [6]](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun25-05-qp.webp)
![Summer 2025 AS Unit 1, question 6: 2 f x ( ) = x 3 x 6 x . 8 6. The function f is defined by f x ( ) = 0 (a) Use an algebraic method to solve the equation . Show all your working. [6] y f](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun25-06-qp.webp)
![Summer 2025 AS Unit 1, question 11: f x ( ) x 3 9 x 2 28 x 4 . 11. The function f is defined by f x is an increasing function. ( ) (a) Show that [4] y = f x . ( ) (b) Two straight lines, b](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun25-11-qp.webp)
![Summer 2024 Unit 3, question 1: 1. The function f is given by 25 32 x . f x 2 5 1 2 x x x Express f x in terms of partial fractions. (a) [4] 2 f x d x ln P (b](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun24-01-qp.webp)
![Summer 2024 Unit 3, question 10: 10. The function f has domain 4, and is defined by 2 3 x 1 x f x . x 1 2 x 2 x 3 2 x f x (a) Show that . [4] x 3 Determine the](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun24-10-qp.webp)
![Summer 2024 AS Unit 1, question 6: 6. (a) Find the exact value of x that satisfies the equation 5 7 x 4 147 . [4] 1 x 2 Show that (8 x 18) 9 x , may be written as 2(2 x 3) (b) , where . [](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun24-06-qp.webp)
![Summer 2024 AS Unit 1, question 8: x 10 x 5 x 8. Prove that for all real values of x. [4] !](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun24-08-qp.webp)
![Summer 2024 AS Unit 1, question 11: 3x y 11. (a) Sketch the graph of . Clearly label the coordinates of the point where the graph crosses the y-axis. [2] 3 x ( 1) y (b) On the same set of ax](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun24-11-qp.webp)
![Summer 2024 AS Unit 1, question 16: x kx 4 0 16. (a) Find the range of values of k for which the quadratic equation has no real roots. [4] y x 2 3 x 4 (b) Determine the coordinates of](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/2300u10-jun24-16-qp.webp)
![Summer 2023 Unit 3, question 3: 0 3 a) Express )2 in terms of partial fractions. [4] ( ) 1+ 2x ( 1− x 9 b) Using your answer from part (a), find the expansion of )2 in ( ) 1+ 2x ( 1− x ascendi](https://pub-e8a9dbab8e2a4d089f52b2a45e0d0b50.r2.dev/bank/1300u30-jun23-03-qp.webp)
All 59 questions on Algebra and functions, as a list
- Summer 2025, Unit 3, Q8 (5 marks): 3 2 x 5 x 7 x 29 x 30 By algebraic division, or otherwise, simplify the expression . [5] 8. 3( 5)( 3)( 2) x x x
- Summer 2025, Unit 3, Q11 (10 marks): 5 2 x d x 11. Find , giving your answer in the form lna + b, where a, b are constants. 2 1 2 1 x x [10] 1
- Summer 2025, Unit 3, Q15 (7 marks): f x , where x ` 3. 15. A function f is defined by 2 x 3 (a) Determine the range of f. [2] f 1( ) x (b) Find an expression for . [5]
- Summer 2025, AS Unit 1, Q1 (5 marks): Solve the inequality 2 x 1 x 6 9 2 x x 1. . [5]
- Summer 2025, AS Unit 1, Q3 (6 marks): 3. Showing all your working, simplify the following expressions. 8 4 2 a a 3 (a) [3] 1 a 3
- Summer 2025, AS Unit 1, Q4 (10 marks): y x 4 x 6 4. The diagram below shows a sketch of the curve C with equation and the line L y x+ 16 with equation . The curve and the line intersect at t
- Summer 2025, AS Unit 1, Q5 (6 marks): x kx k 3 0 5. Find the range of values of k for which the quadratic equation has two distinct real roots. [6]
- Summer 2025, AS Unit 1, Q6 (16 marks): 2 f x ( ) = x 3 x 6 x . 8 6. The function f is defined by f x ( ) = 0 (a) Use an algebraic method to solve the equation . Show all your working. [6] y f
- Summer 2025, AS Unit 1, Q11 (9 marks): f x ( ) x 3 9 x 2 28 x 4 . 11. The function f is defined by f x is an increasing function. ( ) (a) Show that [4] y = f x . ( ) (b) Two straight lines, b
- Summer 2025, AS Unit 1, Q12 (4 marks): y = f x . ( ) 12. The diagram below shows the graph of y x O0 1 Draw a sketch of each of the following transformations on the diagrams provided. Clearly label t
- Summer 2024, Unit 3, Q1 (11 marks): 1. The function f is given by 25 32 x . f x 2 5 1 2 x x x Express f x in terms of partial fractions. (a) [4] 2 f x d x ln P (b
- Summer 2024, Unit 3, Q4 (6 marks): f x 3 x 4 4. A function f is given by . y f x (a) Sketch the graph of . Clearly label the coordinates of the point A, where the graph meets the x-ax
- Summer 2024, Unit 3, Q10 (14 marks): 10. The function f has domain 4, and is defined by 2 3 x 1 x f x . x 1 2 x 2 x 3 2 x f x (a) Show that . [4] x 3 Determine the
- Summer 2024, AS Unit 1, Q6 (7 marks): 6. (a) Find the exact value of x that satisfies the equation 5 7 x 4 147 . [4] 1 x 2 Show that (8 x 18) 9 x , may be written as 2(2 x 3) (b) , where . [
- Summer 2024, AS Unit 1, Q8 (4 marks): x 10 x 5 x 8. Prove that for all real values of x. [4] !
- Summer 2024, AS Unit 1, Q10 (6 marks): 10. Water is being emptied out of a sink. The depth of water, y cm, at time t seconds, may be modelled by y t 2 14 t 49 0 X t X 7. (a) Find the value of t
- Summer 2024, AS Unit 1, Q11 (4 marks): 3x y 11. (a) Sketch the graph of . Clearly label the coordinates of the point where the graph crosses the y-axis. [2] 3 x ( 1) y (b) On the same set of ax
- Summer 2024, AS Unit 1, Q16 (10 marks): x kx 4 0 16. (a) Find the range of values of k for which the quadratic equation has no real roots. [4] y x 2 3 x 4 (b) Determine the coordinates of
- Summer 2024, AS Unit 1, Q17 (7 marks): f x ( ) log (2 x ) 17. A function f is defined by . Another function g is defined by 10 y f x ( ) g x ( ) log (5 x ) . The diagram below shows a sketc
- Summer 2023, Unit 3, Q3 (11 marks): 0 3 a) Express )2 in terms of partial fractions. [4] ( ) 1+ 2x ( 1− x 9 b) Using your answer from part (a), find the expansion of )2 in ( ) 1+ 2x ( 1− x ascendi
- Summer 2023, Unit 3, Q4 (8 marks): ( ) is defined by 3 2 f x 6 x 35 x 7 x 6 −∞,∞ 0 4 A function f with domain . ( ) = 0 in the interval [–1, 1]. [2] a) Determine the number of roots of
- Summer 2023, Unit 3, Q7 (10 marks): y 5 x 3 y 2 x intersect at the point A. Show that the 3 0 7 a) The graphs of and coordinates of A are (2, 7). [2] y 5 x 3 y 2 x 3 b) On the same
- Summer 2023, Unit 3, Q8 (7 marks): 4 x 12 x 9 The function f is defined by 0 8 f x , where x 1 1. 2 2 x x – 3 5 ( ) can be written as 2 + Show that f x a) x −1 . [3] 7∫ ( ) b) Hence fi
- Summer 2023, Unit 3, Q10 (8 marks): 0 Two real functions are defined as 8 ( ) = f x 4 4 for x x , x − 4 ( ) = x − 2 ( )2 g x x for . ( ). Find an expression for fg x a)
- Summer 2023, AS Unit 1, Q4 (10 marks): Find the remainder when the polynomial 3x3 + 2x2 + x −1 is divided by 3 x 0 4 a) . [3] ( ) = 2x3 − 3x2 + ax + 6 is divisible by x + 2 ( ), where a is a re
- Summer 2023, AS Unit 1, Q5 (4 marks): × a − a 3 0 5 3 512a2 Simplify the expression . [4] 1 a 6
- Summer 2023, AS Unit 1, Q7 (13 marks): y 2 x 5 x 12 y mx 14 The curve C has equation and the line L has equation , 0 7 where m is a real constant. a) Given that L is a tangent to C, i) show
- Summer 2023, AS Unit 1, Q11 (7 marks): ( ) = 8 1 1 The function f is defined by f x x2 . ( ) . Sketch the graph of y = f x a) [2] ( ). Indicate the vertical On a separate set of axes, sketch the grap
- Summer 2022, Unit 3, Q5 (9 marks): 5 The function f (x) is defined by 6x + 4 ( ) = ) . f x ( ) x +1 ( ) 2x + 3 ( x −1 a) Express f (x) in terms of partial fractions. [4] ∫ 3x + 2 dx b) Find , giv
- Summer 2022, Unit 3, Q10 (5 marks): Solve the equation 1 0 6x5 −17x4 − 5x3 + 6x2 [5] = 0 . ( ) 3x + 2
- Summer 2022, Unit 3, Q12 (13 marks): 2 The diagram below shows a sketch of the graph of y = f (x), where ( ) = 2x2 +12x +10. f x The graph intersects the x-axis at the points (p, 0), (q, 0) and the
- Summer 2022, AS Unit 1, Q1 (3 marks): 1 Write down the inverse function of y = ex. On the same set of axes, sketch the graphs of y = ex and its inverse function, clearly labelling the coordinates of
- Summer 2022, AS Unit 1, Q2 (6 marks): Showing all your working, simplify the following expression. [6] 0 2 5 48 + 2 + 5 3 ( ) 3 5 + 3 3 − 2 3
- Summer 2022, AS Unit 1, Q4 (4 marks): 4 Solve the inequality x2 + 3x – 6 > 4x – 4. [4]
- Summer 2022, AS Unit 1, Q5 (9 marks): 5 The curve C1 has equation y = –x2 + 2x + 3 and the curve C2 has equation y = x2 – x – 6. The two curves intersect at the points A and B. a) Determine the coor
- Summer 2022, AS Unit 1, Q8 (7 marks): 8 a) The graph G shows the relationship between the variables y and x, where y∝x. Sketch the graph G. [1] b) Mary and Jeff work for a company which pays its emp
- Summer 2022, AS Unit 1, Q9 (4 marks): Find the range of values of k for which the quadratic equation x2 + 2kx + 8k = 0 has 0 9 no real roots. [4]
- Summer 2022, AS Unit 1, Q12 (9 marks): 2 a) Solve the equation 2x3 – x2 – 5x – 2 = 0. [6] Find all values of θ in the range 0° < θ < 180° satisfying b) cos(2θ– 51°) = 0.891. [3]
- Summer 2022, AS Unit 1, Q14 (12 marks): ( ) = 3x3 − 5x2 + x − 6 1 4 A curve C has equation . f x a) Find the coordinates of the stationary points of C and determine their nature. [8] b) Without solvin
- Summer 2019, Unit 3, Q1 (7 marks): 0 1 a) Express in terms of partial fractions. [4] ( ) x + 2 ( )2 x −1 ∫ 9 dx b) Find . [3] ( ) x + 2 ( )2 x −1
- Summer 2019, Unit 3, Q5 (7 marks): 5 a) Find the range of values of x for which |1 – 3x| > 7. [3] b) Sketch the graph of y = |1 – 3x| – 7. Clearly label the minimum point and the points where the
- Summer 2019, Unit 3, Q7 (5 marks): 7 The diagram below shows a sketch of the graph of y = f (x). The graph crosses the y-axis at the point (0, –2), and the x-axis at the point (8, 0). y x O (8, 0
- Summer 2019, Unit 3, Q11 (7 marks): 1 The function f (x) is defined by x2 −1 ( ) = f x x with domain x x 1. Find an expression for f –1(x). State the domain for f –1 and sketch both f (x) and a) f
- Summer 2019, AS Unit 1, Q2 (7 marks): 2 Find all the values of k for which the equation x2 + 2kx + 9k = – 4x has two distinct real roots. [7]
- Summer 2019, AS Unit 1, Q3 (6 marks): Use an algebraic method to solve the equation 12x3 – 29x2 + 7x + 6 = 0. 0 3 Show all your working. [6]
- Summer 2019, AS Unit 1, Q7 (6 marks): 7 Given that a, b are integers, simplify the following. Show all your working. 2 3 + a a) [3] 3 −1 2 6b2 b) [3] − 27 + 192 2
- Summer 2019, AS Unit 1, Q10 (13 marks): 0 a) Solve the following simultaneous equations. 33x × 9 y = 27 2−3x × 8−y = 1 [6] 64 b) Find the value of x satisfying the equation loga3 + 2logax – loga(x – 1
- Summer 2018, Unit 3, Q1 (4 marks): 1 Solve the equation |2x + 1| = 3|x – 2|. [4]
- Summer 2018, Unit 3, Q3 (6 marks): The diagram below shows a sketch of the graph of y = f (x). The graph passes through 0 3 the points (– 2, 0), (0, 8), (4, 0) and has a maximum point at (1, 9).
- Summer 2018, Unit 3, Q5 (8 marks): 5 a) Show that 3x A B C , 2 ≡ ) + ) + ( ( x −1 x − 4 ( ) x − 4 ( ) ( ) x −1 x − 4 2 where A, B and C are constants to be found. [3] 3x 7∫ dx b) Evaluate , givin
- Summer 2018, Unit 3, Q12 (10 marks): Given that f is a function, 1 2 a) state the condition for f –1 to exist, i) find ff –1(x). ii) [2] The functions g and h, are given by b) g(x) = x2 – 1, h(x) =
- Summer 2018, Unit 3, Q17 (6 marks): By drawing suitable graphs, show that x – 1 = cosx has only one root. Starting with 1 7 x0 = 1, use the Newton-Raphson method to find the value of this root cor
- Summer 2018, AS Unit 1, Q1 (7 marks): 1 Showing all your working, simplify 24 a , a) [3] ( ) ( ) 2 2 a + 3 a − 3 − 3 7 + 5 3 . b) [4] 7 + 3
- Summer 2018, AS Unit 1, Q5 (4 marks): 5 The diagram below shows a sketch of y = f (x). y x O Sketch the graph of y = 4 + f (x), clearly indicating any asymptotes. a) [2] Sketch the graph of y = f (x
- Summer 2018, AS Unit 1, Q6 (10 marks): 6 y L B C A x O The sketch shows the curve C with equation y = 14 + 5x – x2 and line L with equation y = x + 2. The line intersects the curve at the points A an
- Summer 2018, AS Unit 1, Q8 (5 marks): 8 Given that (x – 2) and (x + 2) are factors of the polynomial 2x3 + px2 + qx – 12, a) find the values of p and q, [4] b) determine the other factor of the poly
- Summer 2018, AS Unit 1, Q11 (6 marks): 1 a) The vectors u and v are defined by u = 9i – 40j and v = 3i – 4j. Determine the range of values for μ such that μ | v | > | u |. [3] b) The point A has posi
- Summer 2018, AS Unit 1, Q12 (5 marks): 2 Find the values of m for which the equation 4x2 + 8x – 8 = m(4x – 3) has real roots. [5]
- Summer 2018, AS Unit 1, Q17 (11 marks): 7 a) Solve 2log10x = 1 + log105 – log102. [4] Solve 3 = 2e0·5x. b) [2] Express 4x – 10 × 2x in terms of y, where y = 2x. c) Hence solve the equation 4x – 10 × 2
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