plumiIdentities and inequalities: Edexcel International GCSE Further Maths questions
Edexcel International GCSE Further Maths past-paper questions on 3 Identities and inequalities, 2019–2025, with mark schemes and examiner’s reports where available.
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All 49 questions on Identities and inequalities, as a list
- June 2025 (R), Paper 1, Q5 (7 marks): Diagram NOT accurately drawn x cm (2x + 3) cm Figure 1 cm. Figure 1 shows a rectangle with width x cm and length 2 x 3 The perimeter of the rectangle is P cm an
- June 2025 (R), Paper 2, Q5 (8 marks): f x x 3 x 2 x c where c is a constant The remainder when f x is divided by x 2 is 4 times the remainder when f x is divided by x 1 (a) Show that c = 6 (4) 2 is
- June 2025, Paper 1, Q4 (8 marks): (a) On the grid opposite, draw the graph of the line with equation 0 (iii) (i) x y (ii) x 3 y 6 y = 3 x 3 2 18 (3) (b) Show, by shading on the grid, the region
- June 2025, Paper 2, Q2 (5 marks): Find the set of values for x for which (a) 9 x x 3 11 2 (1) (b) 10 x 2 x 7 12 (3) (c) both 9 x x and 10 x 2 x 3 11 2 7 12 (1)
- November 2024, Paper 1, Q1 (4 marks): (a) On the grid below, draw the line with equation (i) 3 x 4 y 24 (ii) 2 x y 5 10 0 (2) (b) Show, by shading on the grid, the region R defined by the inequaliti
- November 2024, Paper 2, Q2 (6 marks): The length of rectangle R is 2 cm greater than its width. The area of R is greater than 8 cm2 and the perimeter of R is less than 30 cm. Given that the width of
- November 2024, Paper 2, Q7 (16 marks): is a factor of x 7 (a) Use the factor theorem to show that 4 1 3 2 f ( ) x 64 x 64 x 3 (2) (b) Hence, or otherwise, find the exact roots of the equation x = 0 f
- June 2024 (R), Paper 1, Q1 (4 marks): Without using a calculator, solve the inequality x x 50 18 6 5 Give your answer in an exact form with a rationalised denominator. Show your working clearly. (4)
- June 2024 (R), Paper 2, Q4 (11 marks): f x px qx x 12 where p and q are constants. q 3 2 37 When f x is divided by x 2 the remainder is −33 5 is a factor of f x Given that x (a) (i) show that p = 2 (
- June 2024, Paper 2, Q1 (6 marks): x x x ax where a is a constant 3 2 f( ) 6 13 10 is a factor of f(x) Given that 3 x 2 (a) show that a = 21 (2) (b) Hence show algebraically that the curve y = f(
- November 2023, Paper 1, Q3 (8 marks): g' x mx 2 x 3 10 37 where m is an integer The curve y g x passes through the point with coordinates 1 20 , Given that x 5 is a factor of g(x) (a) show that g x
- November 2023, Paper 1, Q11 (8 marks): Solve the simultaneous equations x y 2 2log4 log 3 3 log x 3 8 log y 13 2 9 Show your working clearly. (8)
- November 2023, Paper 2, Q4 (8 marks): (a) On the axes opposite, draw the line with equation (i) y 1 (ii) y x 3 x 8 0 (iii) 2 y x 8 (3) (b) Show, by shading on your graph, the region R defined by the
- June 2023 (R), Paper 1, Q2 (8 marks): One solution to the following simultaneous equations y px 9 6 x 2 xy 5 1 is where p is an integer and q is a prime number. 2 , q , (a) Find the value of p and t
- June 2023 (R), Paper 1, Q5 (16 marks): f x 2 x ax 14 x b where a and b are constants. 3 2 4 the remainder is 39 When f (x) is divided by x 1 is a factor of f (x) Given that x (a) show that a 3 and fi
- June 2023 (R), Paper 2, Q2 (6 marks): Find the set of values of x for which (a) 2 x 1 5 x 2 (2) (b) 3 x 2 − x 10 (3) (c) both 2 and 3 x 1 5 x 2 x 2 − x 10 (1)
- June 2023, Paper 1, Q8 (9 marks): f′(x) = 18x 2 – 2x + 13 Given that (2x – 1) is a factor of f (x) show that the curve with equation y = f (x) has only one intersection with the x-axis. (9)
- June 2023, Paper 2, Q5 (7 marks): (a) On the grid opposite draw the line with equation (i) y = 2x + 5 (ii) 4y = x – 8 (iii) 5y + 3x = 30 (3) (b) Show, by shading, the region R defined by the ine
- January 2023 (R), Paper 1, Q5 (12 marks): f(x) = 2ax3 + x2 – bx + 3a where a and b are integers. Given that (x + 2) and (x – 1) are both factors of f(x) (a) show that a = 2 and find the value of b (5) (
- January 2023 (R), Paper 2, Q2 (6 marks): Find the set of values of x for which (a) 3x – 8 < 5x + 3 (1) (b) 4x2 – 7x + 1 > 6 – 2x2 (4) (c) both 3x – 8 < 5x + 3 and 4x2 – 7x + 1 > 6 – 2x2 (1)
- January 2023 (R), Paper 2, Q10 (13 marks): 3y 10 (a) Show that 9 (6y–5) = 3 243 (4) (b) Solve the simultaneous equations 3y 9 (x–2) = 27 243 log10 6xy = log4 2 (9)
- January 2023, Paper 1, Q4 (9 marks): f(x) = x3 + px2 + qx + 6 where p and q are constants. Given that (x – 1) is a factor of f(x) and that when f(x) is divided by (x + 1) the remainder is 8 (a) (i)
- January 2023, Paper 1, Q5 (8 marks): Given that k is a non‑zero constant curve C has equation kx2 – xy + (k + 1)x = 1 straight line l has equation y = k 2 x + 1 The point A is the only point that l
- January 2023, Paper 1, Q9 (8 marks): Giving each value in your solution to 2 decimal places, solve the simultaneous equations e2y – x + 2 = 0 ln(x + 3) – 2y – 1 = 0 (8)
- January 2023, Paper 2, Q1 (7 marks): Find the set of values of x for which (a) 2(3x – 1) < 4 – 3x (2) (b) 3x2 – 8x – 3 < 0 (4) (c) both 2(3x – 1) < 4 – 3x and 3x2 – 8x – 3 < 0 (1)
- June 2022 (R), Paper 1, Q4 (7 marks): Diagram NOT accurately drawn x metres (x + 4) metres Figure 2 Figure 2 shows a rectangle with width x metres and length (x + 4) metres. The perimeter of the rec
- June 2022 (R), Paper 2, Q4 (8 marks): f(x) = x3 + px2 + qx + 7 where p and q are integers. (x + 1) is a factor of f(x) The remainder when f(x) is divided by (x + 2) is –5 (a) Find the value of p and
- June 2022, Paper 1, Q2 (9 marks): 3x cm y cm 5x cm A E P B 3x cm 3x cm Diagram NOT accurately drawn G R F Q y cm y cm M U L T 3x cm 3x cm D C N V 3x cm y cm 5x cm Figure 1 Figure 1 shows the des
- January 2022 (R), Paper 1, Q1 (4 marks): (a) On the grid below, draw the graph of the line with equation (i) 4x + 5y = 20 (ii) 3y − 4x = −12 (2) (b) Show, by shading on the grid, the region R defined b
- January 2022 (R), Paper 2, Q1 (5 marks): Find the set of values for x for which (a) 5x − 10 > 4x − 7 (1) (b) 2x2 − 11x + 5 < 0 (3) (c) both 5x − 10 > 4x − 7 and 2x2 − 11x + 5 < 0 (1)
- January 2022, Paper 1, Q4 (8 marks): (a) On the grid opposite, draw the line with equation (i) y = 2x – 4 (ii) 2x + 3y = 12 (iii) y + 2x + 2 = 0 (3) (b) Show, by shading on the grid, the region R d
- January 2022, Paper 1, Q5 (9 marks): f(x) = ax 3 + 5bx 2 + 8ax – 4b where a and b are integers Given that (x + 2) is a factor of f(x) and that when f(x) is divided by (x + 3) the remainder is 21 (a
- November 2021, Paper 1, Q11 (12 marks): fʹ(x) = ax2 – 14x – 10 where a ∈ ¢ Given that (x – 4) is a factor of f(x) and that when f(x) is divided by (x + 1) the remainder is 25 (a) show that a = 6 (6) (
- June 2021, Paper 2, Q1 (6 marks): Find the set of values for x for which (a) 8x – 7 < 5x + 5 (2) (b) 2x2 – 5x – 3 > 0 (3) (c) both 8x – 7 < 5x + 5 and 2x2 – 5x – 3 > 0 (1)
- June 2021, Paper 2, Q6 (13 marks): f(x) = x3 + ( p + 1)x2 – 10x + q where p and q are integers. Given that (x – 3) is a factor of f(x) (a) show that 9p + q + 6 = 0 (3) Given that (x + p), where p
- June 2021, Paper 2, Q8 (8 marks): Use an algebraic method to solve the simultaneous equations log4 a + 3 log8b = 5 2 2a = 16 4 2 4 b (8)
- November 2020 (R), Paper 1, Q2 (11 marks): (a) Express x2 + 4x − 8 in the form (x + a)2 + b where a and b are constants whose values are to be found. (2) (b) Use algebra to solve the simultaneous equatio
- November 2020 (R), Paper 2, Q2 (11 marks): f(x) = x3 + px + q where p and q are constants. The remainder when f(x) is divided by (x – 1) is –12 The remainder when f(x) is divided by (x – 4) is 30 (a) Fin
- November 2020 (R), Paper 2, Q10 (14 marks): f(x) = 32x3 – 33x + 1 (a) Show that f(1) = 0 (1) (b) Hence using an algebraic method solve f(x) = 0 (4) y Diagram NOT accurately drawn C1 R B C2 a O x Figure 4
- November 2020, Paper 1, Q2 (4 marks): (a) Using the axes below sketch the line with equation (i) y = 6 (ii) y + x = 10 (iii) y = 2x − 5 Show the coordinates of any point where each line crosses the
- January 2020 (R), Paper 1, Q4 (11 marks): f(x) = 2x3 + ax2 + bx + 18 where a and b are constants When f′(x) is divided by (x – 2) the remainder is 5 Given that (x – 2) is a factor of f(x) (a) find the v
- January 2020 (R), Paper 2, Q2 (6 marks): l1 Diagram NOT y accurately drawn (0, 8) l2 R (0, 4) O (4, 0) (6, 0) x Figure 2 Figure 2 shows the shaded region R bounded by the line l1, the line l2 and the y
- January 2020, Paper 1, Q2 (4 marks): (a) On the grid below, draw the line with equation (i) 5x + 2y = 10 (ii) y = x (2) (b) Show, by shading on the grid, the region R defined by the inequalities y
- January 2020, Paper 1, Q3 (6 marks): Given that (x − 4) is a factor of px3 − 31x2 + 25x + 12 where p is a constant, (a) show that p = 6 (2) (b) Solve the equation 6x3 − 31x2 + 25x + 12 = 0 Show cle
- January 2020, Paper 2, Q2 (6 marks): Find the set of values of x for which (a) 3 + 2x x + 2 (1) (b) 8x2 + 10x < 3 (4) (c) both 3 + 2x x + 2 and 8x2 + 10x < 3 (1)
- June 2019 (R), Paper 2, Q1 (7 marks): f (x) = (x í [x2 p í x + q] 1 *LYHQ WKDW I í (a) find the value of q. (2) (b) Find the range of values of p for which the cubic equation f (x) = 0 has only one
- June 2019 (R), Paper 2, Q5 (7 marks): (a) On the grid opposite, draw the graphs of the lines with equations 2x + 3y = 24 y = 2x 3y = 2x í (3) (b) Show, by shading on the grid, the region R defined b
- June 2019, Paper 1, Q1 (4 marks): f(x)=13+2x2-5x-6 1 (a) Factorise x3 - x - 2 (1) (b) Hence, or otherwise, show that (x? - x - 2) is a factor of f(x). (3)
- June 2019, Paper 2, Q5 (6 marks): Use algebra to solve the equations xy = 36 xy + x + 2y = 53 (6)
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