Calculus: Edexcel International GCSE Further Maths questions

Edexcel International GCSE Further Maths past-paper questions on 9 Calculus, 2019–2025, with mark schemes and examiner’s reports where available.

All 159 questions on Calculus, as a list

June 2025 (R), Paper 1, Q3 (10 marks): A particle P is moving along a straight line, from a fixed origin O , the velocity, v m/s, of P is given by At time t seconds t 0 v 2 t 2 19 t 35 At time t seco
June 2025 (R), Paper 1, Q8 (15 marks): f x x 2 x 2 4 9 The curve C has equation y f x The line l with equation y 8 x 9 0 is a tangent to C The line k is perpendicular to l and is also the tangent to
June 2025 (R), Paper 1, Q9 (9 marks): 4 9 Given that f x x 2 1 3 (a) show that f ' x x x 2 8 1 (2) x sin3 The curve C has equation y 4 1 x 2 The point A on C has x coordinate 2 π 3 (b) Show that the
June 2025 (R), Paper 1, Q10 (15 marks): (a) Using a formula from page 2, show that A 1 cos 2 (i) sin 2 A 2 A cos 2 1 (ii) cos 2 A 2 (5) Diagram NOT y accurately drawn C R A O x D Figure 2 Figure 2 sh
June 2025 (R), Paper 1, Q11 (8 marks): 3 m Diagram NOT accurately drawn 4 m h m A Figure 3 Figure 3 shows a hollow right circular cone with radius 3 metres and height 4 metres above the vertex A The
June 2025 (R), Paper 2, Q7 (11 marks): Diagram NOT accurately drawn x cm y cm 2x cm Figure 3 Figure 3 shows a solid cuboid with sides of length x cm, 2x cm and y cm The volume of the cuboid is 72 cm3
June 2025 (R), Paper 2, Q8 (5 marks): Use calculus to find the gradient of the curve with equation 1 y e3 x x 9 2 3 at the point on the curve where x =1 Give your answer to 3 significant figures. (5
June 2025 (R), Paper 2, Q11 (12 marks): y C Diagram NOT accurately drawn A n R B O x Figure 4 x Figure 4 shows part of a curve C with equation y e = 3 3 lies on C The point A with coordinates 2, p (a)
June 2025, Paper 1, Q1 (7 marks): A particle P moves in a straight line. At time t seconds, the velocity, v m/s, of P is given by 2 v 3 t 9 t 7 (a) Show that P never comes to rest. (2) (b) Find
June 2025, Paper 1, Q7 (6 marks): Diagram NOT accurately drawn 20 cm h A Figure 2 Figure 2 shows a hollow hemisphere with radius 20 cm The hemisphere contains liquid, which is dripping out of a
June 2025, Paper 1, Q9 (13 marks): y Diagram NOT accurately drawn A x P M O N B Figure 4 Figure 4 shows a sketch of part of the curve C with equation y f x where f x x 3 ax 2 bx c 2 1 100 The cur
June 2025, Paper 1, Q11 (7 marks): y Diagram NOT accurately drawn l C O x Figure 5 Figure 5 shows part of the curve C with equation y 8 and the 4 x line l with equation x b where b > 0 = The fini
June 2025, Paper 2, Q6 (10 marks): r cm Diagram NOT accurately drawn h cm Figure 1 Figure 1 shows a solid right circular cylinder with radius r cm and height h cm The total surface area of the cy
June 2025, Paper 2, Q9 (8 marks): e 4 t y cos 2 t d y (a) Show that 2 4 t t y e sin 2 4 t d (3) 2 y y Given that d d M Ny 0 where M and N are integers t 2 t d d (b) find the value of M and the v
June 2025, Paper 2, Q10 (17 marks): A curve C has equation x 7 x 3 y x 2 3 2 (a) Write down an equation of the asymptote to C that is (i) parallel to the y-axis (ii) parallel to the x-axis (2) (b)
June 2025, Paper 2, Q11 (15 marks): (a) Show that cos 4 4 sin cos 2 (4) (b) Hence, or otherwise, solve the equation 0 2 4 4 for 8 cos 2 3 2 cos 2 sin 4 8 8 Give your solutions to 2 decimal places.
November 2024, Paper 1, Q5 (10 marks): A particle P is moving along a straight line. , the velocity, v m/s, of P is given by At time t seconds t 0 v 3 t 2 16 t 5 (a) Find the values of t when P is in
November 2024, Paper 1, Q7 (11 marks): C D Diagram NOT accurately drawn B r cm A O E x cm G F Figure 2 Figure 2 shows a shape ABCDEOFG ABCO is a quarter circle with radius r cm CDEO and AOFG are cong
November 2024, Paper 1, Q9 (10 marks): Diagram NOT accurately drawn y C l R x O Figure 3 Figure 3 shows part of the curve C with equation y 2 x and part of the line l with 1 equation 2 y x 4 0 The re
November 2024, Paper 1, Q10 (18 marks): x x x 2 f ( ) 3 4 9 Given that f ( ) x can be expressed in the form A B x ( C )2 where A, B and C are positive constants (a) find the value of A, the value of B
November 2024, Paper 2, Q3 (5 marks): 4 3 The curve C has equation y e3 x x 2 1 Using calculus, find the exact value of the gradient of the tangent to C when x = 1 (5)
November 2024, Paper 2, Q5 (5 marks): The height of liquid in a vessel P is h The volume, V, of the liquid in P is given by V h 3 = 6 Liquid is leaking from P at a constant rate of 36 cm3/s Find the
November 2024, Paper 2, Q9 (14 marks): (a) Using a formula given on page 2, show that cos 2 2 cos 2 1 (2) (b) Hence show that 3 a b 2 2 cos 1 d 4 c 3 where a, b and c are integers to be found. (4) Di
November 2024, Paper 2, Q10 (18 marks): A curve C has equation x 5 2 2 y x x 3 2 3 (a) Find the coordinates of the point where C intersects the (i) x-axis (ii) y-axis (2) (b) Write down an equation of
June 2024 (R), Paper 1, Q6 (6 marks): A particle P is moving in a straight line. The displacement s of P, in metres, at time t seconds, t 0, is given by s e2 t sin 3 t 2 At time t = 0, P is at the p
June 2024 (R), Paper 1, Q11 (11 marks): f θ 2 cos θ sin θ 2 sin θ cos θ 3 (a) Show that f θ sin 2 θ 2 cos 2 θ 2 (3) Diagram NOT y accurately drawn S O a θ Figure 4 Figure 4 shows part of the curve S w
June 2024 (R), Paper 2, Q5 (6 marks): The force F newtons between two magnetic poles is given by the formula 3 F = 20 2 r where r is the distance, in centimetres, between the poles. The distance bet
June 2024 (R), Paper 2, Q7 (14 marks): y Diagram NOT accurately drawn C O x Figure 2 Figure 2 shows a sketch of part of the curve C with equation x 1 where x 5 2 y 4 x 5 4 (a) Write down the equation
June 2024 (R), Paper 2, Q9 (8 marks): y Diagram NOT S A accurately drawn B L O x Figure 3 Figure 3 shows a sketch of part of the curve S with equation y 2 3e x 4 and the line L The curve S has inter
June 2024 (R), Paper 2, Q11 (12 marks): (a) Using a formula on page 2 show that cos A A 2 2 cos 2 1 (2) cos 4 A 1 2 (b) Hence show that 2 cos 2 A 1 2 (3) 2 2 cos x 1 2 sin 2 x 1 The curve with equatio
June 2024, Paper 1, Q4 (8 marks): The surface area of a sphere with radius r cm is increasing at a constant rate of 50π cm2/s Find, in cm3, the exact volume of the sphere at the instant when the
June 2024, Paper 1, Q5 (9 marks): A particle P is moving along the x-axis. the acceleration, a m/s2, of P is given by a At time t seconds t 0 t 3 4 When t = 0, P is at rest. (a) Find the velocit
June 2024, Paper 1, Q7 (13 marks): y Diagram NOT accurately drawn O x Figure 1 Figure 1 shows a sketch of part of the curve C with equation x 2 y x 3 8 4 The point P lies on C and has coordinates
June 2024, Paper 2, Q5 (7 marks): y x x x e2 2 5 y y 2 d d Show that 2 y x 2 e 4 4 x x 2 d d (7)
June 2024, Paper 2, Q6 (10 marks): y C S Diagram NOT accurately drawn 4 R A O x Figure 3 1 Figure 3 shows part of the curve C with equation y and part of the x , 0 x 4 curve S with equation y x 2
June 2024, Paper 2, Q8 (10 marks): x cm A Diagram NOT B accurately drawn y cm C F D E Figure 4 Figure 4 shows a solid right triangular prism ABCDEF The cross section of the prism is an isosceles
June 2024, Paper 2, Q11 (16 marks): Using formulae from page 2, show that (a) (i) cos A A 2 2 2 cos 1 (3) (ii) sin A = A A 2 2 sin cos (1) A A cos 3 3 cos (b) Show that cos A 3 4 (4) Hence, or oth
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, Q5 (8 marks): y 5 C O x Figure 2 Figure 2 shows the graph of part of the curve C with equation y x 6 2 The finite region enclosed by the curve C and the straight line with eq
November 2023, Paper 1, Q8 (8 marks): 3 x 1 2 e 8 y x 2 5 (a) Find d y d x where A, B and C are prime numbers Give your answer in the form A e3 x 1 Bx A Cx 3 to be found. (5) The value of x increase
November 2023, Paper 2, Q3 (7 marks): A particle P moves along the x-axis. the acceleration, a m/s2, of P is given by a 0 At time t seconds t 6 t 16 When t = 0, P is at the origin and is moving with
November 2023, Paper 2, Q5 (15 marks): y Diagram NOT S accurately drawn O x B A P l Figure 1 2 Figure 1 shows part of the curve S with equation y px qx r where p, q and r are constants. The points A,
November 2023, Paper 2, Q6 (7 marks): The volume of oil in a container is V cm3 when the height of the oil is h cm. Oil is pouring into the container at a constant rate of 12 cm3/s. Given that V 3 =
November 2023, Paper 2, Q7 (9 marks): Two numbers x and y are such that 3 x y 4 S x 3 y 2 5 (a) Show that S x 3 x 2 x 5 9 24 16 (2) Given that x can vary, (b) use calculus to find the value of x for
June 2023 (R), Paper 1, Q6 (11 marks): Diagram NOT accurately drawn A cm2 h cm 60° Figure 4 Figure 4 shows a container in the shape of a right circular cone. The container is fixed with its axis of s
June 2023 (R), Paper 1, Q7 (8 marks): The curve with equation y mx 64 x 39 has a stationary point with 2 where m and n are integers. coordinates 4, n Using calculus (a) find the value of m and the v
June 2023 (R), Paper 1, Q9 (14 marks): 1 9 (a) Expand 1 2 x in ascending powers of x up to and including the term in x3 3 expressing each coefficient as a fraction in its lowest terms. (3) (b) Find t
June 2023 (R), Paper 2, Q3 (10 marks): Diagram NOT accurately drawn A B r cm θ rad O Figure 1 Figure 1 shows the sector OAB of a circle with centre O. The radius of the circle is r cm and the angle A
June 2023 (R), Paper 2, Q5 (7 marks): A particle P is moving along the x-axis. 0, the velocity, v m/s, of P is given by At time t seconds, t v t t 2 2 19 35 (a) Find the acceleration of P when t = 5
June 2023 (R), Paper 2, Q7 (7 marks): y Diagram NOT accurately drawn S R m O n θ Figure 2 1 Figure 2 shows part of the curve S with equation y cos 3 θ 3 sin 3 θ 2 where m θ n The curve S meets the x
June 2023 (R), Paper 2, Q9 (15 marks): 3 2 x 9 A curve C has equation y where x 6 x 6 (a) Write down an equation of the asymptote to C that is parallel to the (i) x-axis (ii) y-axis (2) (b) Find the
June 2023, Paper 1, Q2 (5 marks): y (sin 2 x ) 3 2 x sin 2 x A Bx cos 2 x Show that d y where A and B are integers to be found. d x 3 2 x (5)
June 2023, Paper 1, Q4 (10 marks): A particle P is moving along the x-axis. At time t seconds, t 0 , the velocity, v m / s, of P is given by v = 2t 2 – 16t + 30 2, of P when t = 5 (a) Find the ac
June 2023, Paper 1, Q5 (11 marks): A solid cuboid has width x cm, length 4x cm and height h cm. 3 and the surface area of the cuboid is S cm 2 The volume of the cuboid is 75 cm 2 + 375 (a) Show t
June 2023, Paper 1, Q7 (10 marks): e4 x 7 The equation of a curve is y = x − 2 3 When x is increased to (x + δx) , y increases to (y + δy) where δx and δy are small. x (a) Show that δy ≈ e2 x 4 7
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 1, Q11 (14 marks): f (x) = 10 + 6x – x 2 Given that f (x) can be written in the form A(x + B) 2 + C where A, B and C are constants, (a) find the value of A, the value of B and the
June 2023, Paper 2, Q4 (8 marks): Diagram NOT y accurately drawn l B S A O x Figure 2 The curve S with equation y = x2 4 + 2 where x 0 and the line l with equation 2y – x – 4 = 0 where x 0 inter
June 2023, Paper 2, Q6 (6 marks): A Diagram NOT r cm π rad accurately drawn B 5 cm 3 r cm 5 cm E r cm π rad F 3 r cm C 5 cm D Figure 3 Figure 3 shows a right triangular prism ABCDEF. A cross sec
June 2023, Paper 2, Q7 (11 marks): 3 x 7 (a) Expand 1 in ascending powers of x up to and including the term in x3 3 Where appropriate express each coefficient as an exact fraction in its lowest t
June 2023, Paper 2, Q10 (16 marks): The curve C with equation y = 6 − 3 x where x ≠ 4 , crosses the x-axis at the point P x − 4 and the y-axis at the point Q (a) Find the coordinates of (i) P (ii)
January 2023 (R), Paper 1, Q6 (10 marks): 2 1 6 The curve C has equation y = e( x + ) x + 2 1 2 y Kx 3 ( x + 1 ) (a) Show that d e = where K is a constant whose value is to be found. x x 2 2 d ( + 1 ) (
January 2023 (R), Paper 1, Q7 (11 marks): A particle P is moving along the x-axis. At time t seconds (t 0) the velocity of P is v m/s where v = t2 – 10t + 28 (a) Find the velocity of P when t = 1 (1) Gi
January 2023 (R), Paper 1, Q8 (15 marks): y Diagram NOT C accurately drawn l R (0, c) (a, 0) (–1, 0) O (b, 0) x Figure 1 Figure 1 shows part of the curve C with equation y = f(x) The curve C passes thro
January 2023 (R), Paper 2, Q3 (8 marks): Given that y = e3x sin 2x show that 13y + d 2 x = 6 d y y d 2 d x (8)
January 2023 (R), Paper 2, Q6 (6 marks): O Diagram NOT accurately drawn r cm π rad 6 A B Figure 2 Figure 2 shows the sector OAB of a circle with centre O and radius r cm. π radians OA = OB = r cm ∠AOB
January 2023 (R), Paper 2, Q7 (8 marks): y S y = a Diagram NOT R accurately drawn y = x2 + 2 O x Figure 3 Figure 3 shows part of the curve S with equation y = x2 + 2 The finite region R, shown shaded i
January 2023 (R), Paper 2, Q11 (10 marks): Diagram NOT r cm accurately drawn h cm Figure 4 Figure 4 shows an open container in the shape of a cylinder with radius r cm and height h cm. Given that the tot
January 2023, Paper 1, Q8 (13 marks): A E D Diagram NOT accurately drawn x cm 0.5 rad F B y cm C G Figure 1 Figure 1 shows a badge, shown shaded, made from two identical rectangles, ABCD and DEFG, a
January 2023, Paper 1, Q10 (9 marks): y Diagram NOT y = x2 + 1 accurately drawn A B R x2 + y2 = 11 O x Figure 2 The region R, shown shaded in Figure 2, is bounded by the curve with equation y = x2 +
January 2023, Paper 2, Q3 (13 marks): f(x) = 8x2 + 10x – 3 Given that f(x) can be written in the form A(x + B)2 + C where A, B and C are constants, (a) find the value of A, the value of B and the va
January 2023, Paper 2, Q4 (7 marks): The equation of a curve is y = x 3 sin x 1 Find an equation of the tangent to the curve at the point on the curve where x = π 2 Give your answer in the form y =
January 2023, Paper 2, Q7 (9 marks): y = e2x cos 2x (a) Show that y d 2 sin 2 y 2 e 2 x x x d (4) (b) Hence show that d y d y 2 y 4 8 d x d x 2 (5)
January 2023, Paper 2, Q9 (7 marks): A cube has edges of length x cm. The total surface area, A cm2, of the cube is increasing at a constant rate of 0.45 cm2/s Find the rate of increase, in cm3/s,
January 2023, Paper 2, Q11 (14 marks): y Diagram NOT accurately drawn A O x B y = 4 – e 2x Figure 3 Figure 3 shows part of the curve C with equation y = 4 – e 2x The curve C crosses the y-axis at the
June 2022 (R), Paper 1, Q2 (7 marks): A particle P is moving along a straight line, which passes through the fixed point O. At time t seconds (t 0), the velocity, v m / s, of P is given by v = t2 –
June 2022 (R), Paper 1, Q5 (8 marks): Differentiate with respect to x 3 ( ) (a) e4 x + 6 2 x 2 ( ) ( ) where A and B are integers. Give your answer in the form e4 x Ax B 6 + 2 + x (5) sin3 x (b) ( )
June 2022 (R), Paper 1, Q10 (11 marks): Using suitable results for sin (A + B) and sin (A – B) from the formulae page, (a) show that 2 sin 4 x cos x = sin 5 x + sin 3 x (3) y Diagram NOT accurately dr
June 2022 (R), Paper 2, Q2 (6 marks): When poured from a pipe, concrete is formed into the shape of a cuboid with a square base of side x and with a height of 3x The volume of the cuboid increases a
June 2022 (R), Paper 2, Q8 (11 marks): y A y = 9 – 9e–3x D y = e3x – 1 C O x Figure 2 Figure 2 shows part of the curve C with equation y = e3x – 1 and part of the curve D with equation y = 9 – 9e–3x
June 2022 (R), Paper 2, Q10 (18 marks): A curve C has equation x − 7 2 3 y = x ≠ x − 2 3 2 (a) Write down an equation of the asymptote to C that is (i) parallel to the y-axis, (ii) parallel to the x-a
June 2022 (R), Paper 2, Q11 (12 marks): Diagram NOT accurately drawn h cm r cm Figure 3 Figure 3 shows a solid metal right circular cylinder of radius r cm and height h cm. The total surface area of t
June 2022, Paper 1, Q5 (11 marks): –1 can be expressed in the form p (1 + qx) –1 where p and q 5 Given that (2 + 3x) are constants, (a) find the value of p and the value of q (2) –1 in ascending
June 2022, Paper 1, Q7 (15 marks): The point with coordinates (4, –104) lies on the curve C with equation y = f (x) Given that fʹ (x) = 4x3 – 12x2 – 19x + 12 (a) (i) show that C passes through th
June 2022, Paper 1, Q8 (7 marks): The volume of a sphere is 500 cm3 (a) Calculate the radius, in cm to 3 significant figures, of the sphere. (2) The surface area of the sphere is increased by 20
June 2022, Paper 1, Q9 (16 marks): f (x) = 3x4 + 4x3 – 36x2 + 64 2 (ax2 + bx + c) Given that f (x) can be written in the form (x – 2) (a) find the value of a, the value of b and the value of c (4
June 2022, Paper 2, Q2 (6 marks): A particle P moves along the x‑axis. At time t seconds, the displacement, x metres, of P from the origin O is given by x = t4 – 13.5t + 12 (a) Find the velocity
June 2022, Paper 2, Q6 (8 marks): (a) Using a formula from page 2, show that cos 2 A = 1 – 2 sin2A (2) The finite region R is bounded by the curve with equation y = 3 + 2 sin x , the x‑axis, π t
June 2022, Paper 2, Q8 (12 marks): E F Diagram NOT H accurately drawn G D C A B Figure 2 Figure 2 shows a waste paper basket in the shape of a right prism with 5 faces and a cross section that is
June 2022, Paper 2, Q11 (17 marks): A curve C has equation ( ) y = 2 a − 1 x + 1 where a is a constant and x ≠ 6 ax − a 6 (a) Find d y d x (3) The curve crosses the y‑axis at the point A. The norm
January 2022 (R), Paper 1, Q7 (11 marks): y cm Diagram NOT accurately drawn 3x cm 5x cm 4x cm Figure 2 Figure 2 shows a block of wood in the shape of a right triangular prism. The cross section of the p
January 2022 (R), Paper 1, Q8 (8 marks): The curve C has equation y = 2x2 − sin x The point A on C has x coordinate π Show that an equation of the normal to C at the point A is x + (4π + 1)y − π (8π2 +
January 2022 (R), Paper 1, Q11 (9 marks): y Diagram NOT accurately drawn b O x R a Figure 3 The finite region R, shown shaded in Figure 3, is bounded by the curve with equation y = e2x − 9 and the coord
January 2022 (R), Paper 2, Q3 (6 marks): Differentiate with respect to x (a) e2x 5 x − 3 (3) x 3 (b) cos 3 x (3)
January 2022 (R), Paper 2, Q6 (11 marks): The curve C has equation y = 4 x The point A on C has coordinates (9, 12) The tangent to C at the point A meets the x-axis at the point T (a) Find the coordinat
January 2022 (R), Paper 2, Q8 (7 marks): Liquid drips onto a large horizontal flat cloth, forming a circular stain. The liquid starts to drip onto the cloth at time t = 0 The area of the stain increase
January 2022 (R), Paper 2, Q10 (12 marks): y Diagram NOT accurately drawn C2 A R1 B C1 R2 O π 3 π x 2 Figure 2 Figure 2 shows part of the curve C1 with equation y = sin x + 1 and part of the curve C2 wit
January 2022, Paper 1, Q1 (4 marks): π 4∫ 3 1 Using calculus find the exact value of π cos 4θ dθ a Give your answer in the form − where a is a prime number and b is an integer. b (4)
January 2022, Paper 1, Q8 (10 marks): Diagram NOT accurately drawn h cm r cm Figure 1 A solid right circular cylinder has base radius r cm and height h cm as shown in Figure 1. 3 and a total surface
January 2022, Paper 1, Q11 (9 marks): Given that y = e4 x 2 – 4x + 1) (8x 32 (a) show that d y x = x 2e 4x d (5) y Diagram NOT accurately drawn –2 O x R Figure 2 Figure 2 shows part of the curve C w
January 2022, Paper 2, Q3 (5 marks): The curve C has equation y = 9 – x 2 Use algebraic integration to find the area of the finite region bounded by C and the x-axis. (5)
January 2022, Paper 2, Q5 (9 marks): A particle P is moving along the x-axis. At time t seconds (t 0) the velocity, v m/s, of P is given by v = 3t 2 – 23t + 30 (a) Find the values of t when P is in
January 2022, Paper 2, Q10 (18 marks): The curve C has equation y = 2 x − 1 x ≠ –4 x + 4 (a) Write down an equation of the asymptote to C that is (i) parallel to the x-axis, (ii) parallel to the y-ax
November 2021, Paper 1, Q4 (5 marks): x y = sin 2 | x| > 3 4 x − 2 9 y x − x − x x 2 Show that d 2 ( 9 )cos 2 sin 2 = x ( ) d 3 x − 2 9 (5)
November 2021, Paper 1, Q6 (6 marks): The volume of a sphere with radius r cm is increasing at a constant rate of 3 cm3/s. Find the rate, in cm2/s, at which the surface area of the sphere is increas
November 2021, Paper 1, Q10 (15 marks): y Diagram NOT accurately drawn A C O M N x Figure 4 2 + sin 3x where 0 x < 2 π Figure 4 shows the curve C with equation y = 1 3 The curve C crosses the x‑axis 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) (
November 2021, Paper 2, Q2 (6 marks): The finite region enclosed by the curve with equation y = 4 – x2 and the line with equation y = x + 2 is rotated through 360° about the x‑axis. Use algebraic in
November 2021, Paper 2, Q4 (11 marks): The curve C has equation y = 2x2 + px + q where p and q are integers. The curve C has a stationary point at (3, –5) (a) Show that p = –12 and find the value of
November 2021, Paper 2, Q5 (6 marks): y and x vary so that y = xe–2x Given that the value of x increases by 3%, use calculus to find, in terms of x, an estimate for the percentage change in y Give y
November 2021, Paper 2, Q6 (8 marks): A particle P is moving along the x‑axis. At time t seconds (t 0) the displacement, s metres, of P from the origin O, is given by s = t3 – 4t2 – 16t – 8 (a) Find
November 2021, Paper 2, Q10 (13 marks): 1 3 x 2 10 (a) Show that 1 3 9 3 x (2) 3 (b) Hence expand - x in ascending powers of x up to and including the term in x3 9 3 expressing each coefficient as an
June 2021, Paper 1, Q3 (10 marks): A curve C has equation y = ax − 3 5 where a is a constant and x ≠ –5 x + The gradient of C at the point on the curve where x = 2 is 18 49 (a) Show that a = 3 (3
June 2021, Paper 1, Q7 (11 marks): x 7 The curve C has equation y = x2 + 4 (a) Using calculus, find the coordinates of the stationary points on C. (5) y x x (b) Show that d 2 2 ( 2 − 12 ) = x x +
June 2021, Paper 1, Q9 (5 marks): Find an equation of the normal to the curve with equation y = (x3 – 2x)e(1 – x) at the point on the curve with coordinates (1, –1) (5)
June 2021, Paper 1, Q11 (15 marks): (a) Using a formula from page 2, show that cos 2x = 1 – 2 sin2 x (3) y Diagram NOT accurately drawn y = sin x + 2 R2 A B R1 y = cos 2x + 2 C O x Figure 2 Figure
June 2021, Paper 2, Q3 (9 marks): Diagram NOT accurately drawn P Q 2 π rad 18 cm 18 cm 3 O Figure 1 Figure 1 shows a sector OPQ of a circle with centre O. The radius of the circle is 18 cm and t
June 2021, Paper 2, Q9 (8 marks): Diagram NOT accurately drawn 4x cm x cm Figure 3 Figure 3 shows a metal solid S. The solid is a right triangular prism. The cross section of S is an equilateral
June 2021, Paper 2, Q11 (8 marks): y y = ex Diagram NOT accurately drawn R y = 4e–x O a x Figure 4 The region R, shown shaded in Figure 4, is bounded by the curve with equation y = ex, the curve
November 2020 (R), Paper 1, Q4 (6 marks): The curve C has equation y = x3 − 3x2 − 24x + 6 (a) Use calculus to find the coordinates of each of the stationary points on C. (4) (b) Determine the nature of
November 2020 (R), Paper 1, Q7 (9 marks): The length of each side of a cube S1 is increasing at a constant rate of 0.1 m/s. (a) Find, in m3/s, the rate of increase of the volume of the cube S1 when the
November 2020 (R), Paper 1, Q12 (7 marks): y S C B Diagram NOT accurately drawn R O A x Figure 2 The region R, shown shaded in Figure 2, is bounded by the x-axis, the curve S with equation y = 2 sin x an
November 2020 (R), Paper 2, Q6 (15 marks): y Diagram NOT accurately drawn C O A B x Figure 3 The curve C with equation y = x2 – 5x + 4 crosses the x‑axis at the points A and B, as shown in Figure 3 (a) F
November 2020 (R), Paper 2, Q9 (8 marks): Given that x = e–t sin 2t show that 2 + 2 d d x x t + 5x = 0 2 d t d (8)
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, Q1 (3 marks): Differentiate with respect to x e x cos 2x 6 3 2 (3)
November 2020, Paper 1, Q7 (13 marks): f(x) = x2 − 9x + 14 Given that f(x) can be written in the form (x + a)2 + b, where a and b are constants, (a) find the value of a and the value of b. (2) (b) He
November 2020, Paper 1, Q8 (10 marks): Given that 2xy + 5y = ex ( ) y y 2 x + 3 (a) show that d = x x + d 2 5 (5) (b) find the value of d y x when x = 0 d (2) (c) find an equation of the normal to th
November 2020, Paper 1, Q10 (12 marks): Diagram NOT accurately drawn R cm l cm r cm L cm Figure 4 Figure 4 shows a right circular cone with base radius r cm and slant height l cm. Figure 4 also shows
November 2020, Paper 2, Q1 (4 marks): A particle P is moving in a straight line. At time t seconds, t 0, the displacement, s metres, of P from a fixed point O of the line is given by 1 s = 3 + 8 t +
November 2020, Paper 2, Q2 (4 marks): The region enclosed by the curve with equation y = e3x, the x-axis, the y-axis and the line with equation x = 3 is rotated through 360° about the x-axis. Use al
November 2020, Paper 2, Q8 (15 marks): The curve C1 has equation y = 5e–2x + 4 The curve C2 has equation y = e2x The curves C1 and C2 intersect at the point A. (a) Find the exact coordinates of A. (4
November 2020, Paper 2, Q9 (18 marks): A curve C has equation 2 + 4 x − x 2 1 y = x ≠− 2 x + 1 2 (a) Write the equation of C in the form ax2 + (by – 4)x + (y – c) = 0, where a, b and c are integers w
January 2020 (R), Paper 1, Q3 (6 marks): Given that y = e2x(x2 + 1) (a) find d y d x (3) The straight line l is the tangent to the curve with equation y = e2x(x2 + 1) at the point on the curve where x
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 1, Q8 (8 marks): A particle P moves along the positive x‑axis. At time t seconds (t 0) the velocity, v m/s, of P is given by v = 3 + 5t – 2t2 At time t seconds, P is at the poin
January 2020 (R), Paper 1, Q11 (11 marks): Diagram NOT accurately drawn y C R P (0, b) l O x (16, 0) Figure 2 Figure 2 shows part of the curve C with equation y = x − 2 Figure 2 also shows the straight l
January 2020 (R), Paper 2, Q6 (6 marks): Diagram NOT accurately drawn r cm h cm 60° A Figure 4 Figure 4 shows a hollow right circular cone fixed with its axis of symmetry vertical. The cone is inverted
January 2020 (R), Paper 2, Q10 (12 marks): The equation of a curve C is y = f (x) where f ¢(x) = 3x2 − 4x − p and p ≠ 0 The points with coordinates (2, 0) and (−1, 9) lie on C. (a) Show that C has equati
January 2020, Paper 1, Q5 (11 marks): y Diagram NOT 25 accurately drawn A 13 B R x −5 O 5 Figure 2 Figure 2 shows part of the curve with equation y = 25 − x2 and part of the line with equation y + x
January 2020, Paper 1, Q11 (13 marks): Diagram NOT accurately drawn x cm Figure 4 A company manufactures chocolate bars that are inside packaging that is in the shape of a right triangular prism. The
January 2020, Paper 2, Q1 (4 marks): A particle P is moving along a straight line that passes through the fixed point O. At time t seconds, t 0, the displacement, s metres, of P from O is given by
January 2020, Paper 2, Q6 (8 marks): Diagram NOT y accurately drawn R A 2 x O Figure 2 The region R, shown shaded in Figure 2, is bounded by the curve with equation y = 2x – x2 and the line with eq
January 2020, Paper 2, Q8 (8 marks): Given that y = e3x sin 2x show that d y d y 2 − 6 + 13y = 0 d x d x 2 (8)
January 2020, Paper 2, Q9 (16 marks): A curve C has equation 2 x ≠ p y = qx − x − p The curve crosses the y‑axis at the point A. The line l with equation y = x + 2 is the normal to C at A. (a) (i) S
January 2020, Paper 2, Q10 (9 marks): The volume of a sphere is increasing at a constant rate of 40 cm3/s. Find the rate of increase, in cm2/s, of the surface area of the sphere at the instant when
June 2019 (R), Paper 1, Q4 (10 marks): f (x) = e3x 1 x + 2 3e x x 2 ( 2 + 3 ) D 6KRZ WKDW I௘މ x) = x 1 + 2 (4) (b) Find an equation of the normal to the curve with equation y = f (x) at the point on
June 2019 (R), Paper 1, Q5 (5 marks): A circle has radius 3U cm and area A cm2 Given that the value of U increases by 0.05% use calculus to find an estimate for the percentage increase in the value
June 2019 (R), Paper 1, Q9 (10 marks): E Diagram NOT accurately drawn F H G D h cm C A 3x cm x cm B Figure 3 Figure 3 shows a solid cuboid ABCDEFGH AB = x cm BC = 3x cm AH = h cm The volume of the cu
June 2019 (R), Paper 2, Q3 (6 marks): Diagram NOT accurately drawn x cm 10x cm Figure 2 Figure 2 shows a solid right circular cylindrical metal rod. The diameter of the rod is x cm and the length of
June 2019 (R), Paper 2, Q4 (8 marks): A particle P moves along the x-axis. At time t seconds (t . 0) the acceleration, a m/s2 , of P is given by a = 6t í When t = 0, P is at rest at the origin. (a)
June 2019 (R), Paper 2, Q9 (9 marks): The finite region R enclosed by the y-axis, the straight line with equation y + 2x = 13 and the curve with equation y = x2 í LV GHILQHG IRU SRLQWV ZLWK FRRUGLQD
June 2019, Paper 1, Q6 (11 marks): B Diagram NOT y metres accurately drawn C x metres E D Figure 2 Figure 2 shows a lawn ABCDEF, where ABDE is a rectangle of length y metres and width 2x metres.
June 2019, Paper 1, Q10 (14 marks): (a) Expand (1+ 2x2)≥ in ascending powers of x up to and including the term in r*, expressing each coefficient as an exact fraction in its lowest terms. (3) (b)
June 2019, Paper 1, Q11 (17 marks): The curve C' has equation 3y = x- + 2 The point P lies on C' and has x coordinate 4 The line k is the tangent to C at P. (a) Find an equation for k, giving your
June 2019, Paper 2, Q2 (6 marks): Oil is leaking from a pipe and forms a circular pool on a horizontal surface. The area of the surface of the pool is increasing at a constant rate of 8 cm2/s. F
June 2019, Paper 2, Q3 (6 marks): A particle P moves in a straight line. At time t seconds, the velocity, v m/s, of P is given by v = t2 – 4t + 7 (a) Find the acceleration of P, in m/s2, when t
June 2019, Paper 2, Q6 (8 marks): (a) Given that y = (4x – 3)e2x y (i) find d x d (3) y d x x y (ii) show that ( − = − 4 3 ) ( 8 2 ) x d (2) x (b) Differentiate sin 5 with respect to x x − 3 2 (
June 2019, Paper 2, Q9 (14 marks): 28 ⎛ ⎞ 9 The curve C, with equation y = f(x), passes through the point with coordinates − 2 , − ⎝⎜ ⎠⎟ 3 *LYHQ WKDW Iމ x) = x3 – x2 – 4x + 4 (a) show that C pass

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