Legacy C4 — Core Mathematics 4: Edexcel A‑Level Maths questions

Edexcel A-Level Maths past-paper questions on L.C4 Legacy C4 — Core Mathematics 4, 2010–2025, with mark schemes and examiner’s reports where available. Add MadAsMaths questions to your practice, or browse them separately by topic and difficulty.

All 119 questions on Legacy C4 — Core Mathematics 4, as a list

June 2018, Legacy C4, Q1 (8 marks): (a) Find the binomial series expansion of 4 9 4 9 − x x , < in ascending powers of x, up to and including the term in x 2 Give each coefficient in its simplest
June 2018, Legacy C4, Q2 (10 marks): The curve C has equation x2 + xy + y 2 − 4x − 5y + 1 = 0 (a) Use implicit differentiation to find d d y x in terms of x and y. (5) (b) Find the x coordinates of
June 2018, Legacy C4, Q3 (14 marks): (i) Given that 13 4 2 1 3 2 1 2 1 3 2 2 − + + ≡ + + + + + x x x A x B x C x ( ) ( ) ( ) ( ) ( ) (a) find the values of the constants A, B and C. (4) (b) Hence f
June 2018, Legacy C4, Q4 (6 marks): Diagram not drawn to scale 50 cm r cm h cm 30 ° Figure 1 A water container is made in the shape of a hollow inverted right circular cone with semi-vertical angl
June 2018, Legacy C4, Q5 (7 marks): x y O A (k, 2) Diagram not drawn to scale C Figure 2 Figure 2 shows a sketch of the curve C with parametric equations x = 1 + t − 5 sin t , y = 2 − 4 cos t , −
June 2018, Legacy C4, Q6 (6 marks): Given that y = 2 when x = − 8 π , solve the differential equation d d y x y x x = − 2 2 3 2 1 2 1 2 cos < < giving your answer in the form y = f (x). (6)
June 2018, Legacy C4, Q7 (15 marks): The point A with coordinates (−3, 7, 2) lies on a line l1 The point B also lies on the line l1 Given that AB → = 4 6 2 −        , (a) find the coordinat
June 2018, Legacy C4, Q8 (9 marks): x y O Diagram not drawn to scale 4 π R Figure 3 (a) Find x x x cos4 d ∫ (3) Figure 3 shows part of the curve with equation y x x x = sin , 2 0  The finite regi
June 2017, Legacy C4, Q1 (8 marks): The curve C has parametric equations x t y t = − = − 3 4 5 6 , , t > 0 (a) Find d d y x in terms of t (2) The point P lies on C where t = 1 2 (b) Find the equat
June 2017, Legacy C4, Q2 (6 marks): f(x) = (2 + kx)–3, kx < 2, where k is a positive constant The binomial expansion of f(x), in ascending powers of x, up to and including the term in x2 is A Bx x
June 2017, Legacy C4, Q3 (12 marks): O 1 x R y Figure 1 Figure 1 shows a sketch of part of the curve with equation y x = + 6 2 ( ) , e x The finite region R, shown shaded in Figure 1, is bounded b
June 2017, Legacy C4, Q4 (9 marks): The curve C has equation 4x 2 – y3 – 4xy + 2 y = 0 The point P with coordinates (–2, 4) lies on C. (a) Find the exact value of d d y x at the point P. (6) The n
June 2017, Legacy C4, Q5 (7 marks): S O ln4 y x Figure 2 The finite region S, shown shaded in Figure 2, is bounded by the y-axis, the x-axis, the line with equation x = ln4 and the curve with equa
June 2017, Legacy C4, Q6 (13 marks): With respect to a fixed origin O, the lines l1 and l2 are given by the equations 1 4 1 : 28 5 4 1 l λ − ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = + − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ r , 2 5 3 : 3 0 1
June 2017, Legacy C4, Q7 (8 marks): 200cm hcm P Figure 3 Figure 3 shows a vertical cylindrical tank of height 200 cm containing water. Water is leaking from a hole P on the side of the tank. At ti
June 2017, Legacy C4, Q8 (12 marks): y C R k x O P(k, 8) Figure 4 Figure 4 shows a sketch of part of the curve C with parametric equations x = 3șsinș, y = sec3ș, 0 -ș <2 π The point P(k, 8) lies
June 2016, Legacy C4, Q1 (6 marks): Use the binomial series to find the expansion of 1 2 5 2 5 3 ( ) , + x x < in ascending powers of x, up to and including the term in x3. Give each coefficient a
June 2016, Legacy C4, Q2 (9 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation y = x2ln x, x . 1 The finite region R, shown shaded in Figure 1, is bounded by the curve, th
June 2016, Legacy C4, Q3 (9 marks): The curve C has equation 2x2y + 2x + 4y – cos (ʌ\) = 17 (a) Use implicit differentiation to find d d y x in terms of x and y. (5) The point P with coordinates 3
June 2016, Legacy C4, Q4 (7 marks): The rate of decay of the mass of a particular substance is modelled by the differential equation d d x t = −5 2 x, t . 0 where x is the mass of the substance me
June 2016, Legacy C4, Q5 (6 marks): Figure 2 Figure 2 shows a sketch of the curve C with parametric equations 4tan , 5 3sin 2 , 0 2 π x t y t t = = < The point P lies on C and has coordinates 4
June 2016, Legacy C4, Q6 (15 marks): (i) Given that y > 0, find 3 4 3 2 y y y y − + ∫ ( ) d (6) (ii) (a) Use the substitution x = 4sin2ș to show that 3 3 2 0 0 d sin d 4 π x x λ θ θ x ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠
June 2016, Legacy C4, Q7 (8 marks): (a) Find ( ) 2 1 3 2 x x − ∫ d giving your answer in its simplest form. (2) y C O x S P(k, 8) 8 1 2 Figure 3 Figure 3 shows a sketch of part of the curve C with
June 2016, Legacy C4, Q8 (15 marks): With respect to a fixed origin O, the line l1 is given by the equation 8 5 1 4 3 3 μ − ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = + ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠ r where ȝ is a scalar parameter. Th
June 2015, Legacy C4, Q1 (8 marks): (a) Find the binomial expansion of ( ) , 4 5 4 5 1 2 + < x x in ascending powers of x, up to and including the term in x2. Give each coefficient in its simplest
June 2015, Legacy C4, Q2 (11 marks): The curve C has equation x2 – 3xy – 4y2 + 64 = 0 (a) Find d d y x in terms of x and y. (5) (b) Find the coordinates of the points on C where d d y x = 0 (Soluti
June 2015, Legacy C4, Q3 (8 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation y x x x = − 4 1 2 e , x .0 The curve meets the x-axis at the origin O and cuts the x-axis a
June 2015, Legacy C4, Q4 (11 marks): With respect to a fixed origin O, the lines l1 and l2 are given by the equations l1: r = 5 3 − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ p + Ȝ 0 1 3 − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ , l2: r = 8 5 2 − ⎛
June 2015, Legacy C4, Q5 (6 marks): A curve C has parametric equations x = 4t + 3, y = 4t + 8 + 5 2t , t 0 (a) Find the value of d d y x at the point on C where t = 2, giving your answer as a fr
June 2015, Legacy C4, Q6 (8 marks): y O R 3 x Diagram not to scale Figure 2 Figure 2 shows a sketch of the curve with equation y x x = − + ( )( ) 3 1 , 0 -x -3 The finite region R, shown shaded
June 2015, Legacy C4, Q7 (13 marks): (a) Express 2 2 P P ( ) − in partial fractions. (3) A team of biologists is studying a population of a particular species of animal. The population is modelled
June 2015, Legacy C4, Q8 (10 marks): y x l P Q R O C Diagram not to scale Figure 3 Figure 3 shows a sketch of part of the curve C with equation y = 3x The point P lies on C and has coordinates (2,
June 2014, Legacy C4, Q1 (7 marks): A curve C has the equation x3 + 2xy – x – y3 – 20 = 0 (a) Find d d y x in terms of x and y. (5) (b) Find an equation of the tangent to C at the point (3, –2), g
June 2014, Legacy C4, Q2 (5 marks): Given that the binomial expansion of (1 + kx)–4, ¨kx ¨< 1, is 1 – 6x + Ax2 + … (a) find the value of the constant k, (2) (b) find the value of the constant A, g
June 2014, Legacy C4, Q3 (11 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation y x x = + 10 2 5√ , x > 0 The finite region R, shown shaded in Figure 1, is bounded by the c
June 2014, Legacy C4, Q4 (5 marks): hcm Figure 2 A vase with a circular cross-section is shown in Figure 2. Water is flowing into the vase. When the depth of the water is cm, h the volume of water
June 2014, Legacy C4, Q5 (5 marks): y O x C Figure 3 Figure 3 shows a sketch of the curve C with parametric equations x = 4cos 6 π t⎛ ⎞ + ⎜ ⎟ ⎝ ⎠, y = 2sint, 0 - t 2ʌ (a) Show that x + y = 2 3 cos
June 2014, Legacy C4, Q6 (12 marks): (i) Find xe4xdx (3) (ii) Find 8 2 1 3 ( ) x − dx, x 1 2 (2) (iii) Given that y = 6 π at x = 0, solve the differential equation d d y x = ex cosec2y cosecy
June 2014, Legacy C4, Q7 (15 marks): Figure 4 Figure 4 shows a sketch of part of the curve C with parametric equations x = 3tanș, y = 4cos2ș, 0 - ș < 2 π The point P lies on C and has coordinates (
June 2014, Legacy C4, Q8 (15 marks): Relative to a fixed origin O, the point A has position vector − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 2 4 7 and the point B has position vector − ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ 1 3 8 The line l1 pa
June 2014, Legacy C4 (01R), Q1 (8 marks): (a) Find the binomial expansion of 1 9 10 √( ) − x , x < 9 10 in ascending powers of x up to and including the term in x2. Give each coefficient as a simplified
June 2014, Legacy C4 (01R), Q2 (9 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation y = (2 – x)e2x, x ∈ The finite region R, shown shaded in Figure 1, is bounded by the curve,
June 2014, Legacy C4 (01R), Q3 (10 marks): x2 + y2 + 10x + 2y – 4xy = 10 (a) Find d d y x in terms of x and y, fully simplifying your answer. (5) (b) Find the values of y for which d d y x = 0 (5)
June 2014, Legacy C4 (01R), Q4 (10 marks): (a) Express 25 2 1 2x x ( ) + in partial fractions. (4) Figure 2 Figure 2 shows a sketch of part of the curve C with equation y x x = + 5 2 1 √( ) , x > 0 The f
June 2014, Legacy C4 (01R), Q5 (6 marks): At time t seconds the radius of a sphere is r cm, its volume is V cm3 and its surface area is S cm2. [You are given that V = 4 3ʌr3 and that S = 4ʌr2] The volum
June 2014, Legacy C4 (01R), Q6 (10 marks): With respect to a fixed origin, the point A with position vector i + 2j + 3k lies on the line l1 with equation 1 0 2 2 3 1 λ ⎛⎞ ⎛ ⎞ ⎜⎟ ⎜ ⎟ = + ⎜⎟ ⎜ ⎟ − ⎝⎠ ⎝ ⎠ r
June 2014, Legacy C4 (01R), Q7 (10 marks): The rate of increase of the number, N, of fish in a lake is modelled by the differential equation d d ( ) N t kt N t = − − ( )1 5000 t > 0, 0 < N < 5000 In the
June 2014, Legacy C4 (01R), Q8 (12 marks): Figure 3 The curve shown in Figure 3 has parametric equations x = t – 4 sin t, y = 1 – 2 cos t, 2 2 3 3 π π t − The point A, with coordinates (k, 1), lies o
June 2013, Legacy C4, Q1 (7 marks): (a) Find x x x 2e d . ∫ (5) (b) Hence find the exact value of x x x 2 0 1 e d . ∫ (2)
June 2013, Legacy C4, Q2 (9 marks): (a) Use the binomial expansion to show that 2 1 1 1 , | | 1 1 2 x x x x x + ⎛ ⎞≈ + + < ⎜ ⎟ ⎝ ⎠ − (6) (b) Substitute x = 1 26 into 2 1 x x = + + x x + ⎛ ⎞ ⎜ ⎟ ⎝
June 2013, Legacy C4, Q3 (8 marks): Figure 1 Figure 1 shows the finite region R bounded by the x-axis, the y-axis, the line 2 π x = and the curve with equation 1 sec , 0 2 2 π y x x ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠
June 2013, Legacy C4, Q4 (9 marks): A curve C has parametric equations x = 2sin t, y = 1 – cos 2t, – 2 π t 2 π (a) Find d d y x at the point where t = 6 π (4) (b) Find a cartesian equation for
June 2013, Legacy C4, Q5 (10 marks): (a) Use the substitution x = u2, u > 0, to show that 2 d = (2 1) du x u u − 1 (2 x x −1) ∫ ∫ (3) (b) Hence show that 9 1 1 d = 2ln a x b ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ ∫ (2 x x −1
June 2013, Legacy C4, Q6 (11 marks): Water is being heated in a kettle. At time t seconds, the temperature of the water is °C. The rate of increase of the temperature of the water at any time t i
June 2013, Legacy C4, Q7 (12 marks): A curve is described by the equation x2 + 4xy + y2 + 27 = 0 (a) Find d d y x in terms of x and y. (5) A point Q lies on the curve. The tangent to the curve at Q
June 2013, Legacy C4, Q8 (9 marks): With respect to a fixed origin O, the line l has equation 13 2 8 2 1 1 λ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = + ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠ r 13 2 8 2 1 1 λ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = + ⎜ ⎟ ⎜ ⎟ − ⎝
June 2013, Legacy C4 (01R), Q1 (4 marks): Express in partial fractions 5x +3 (2x + 1)(x +1)? (4)
June 2013, Legacy C4 (01R), Q2 (7 marks): The curve C has equation 314+ xy-y?+5=0 Show that 2 at the point (1, 3) on the curve C can be written in the form + In(ue’), x Aa where / and yu are integers to
June 2013, Legacy C4 (01R), Q3 (8 marks): Using the substitution w = 2 + \(2x + 1), or other suitable substitutions, find the exact value of 4 1 J ————~ dx 924+ (2x +1) giving your answer in the form A
June 2013, Legacy C4 (01R), Q4 (9 marks): (a) Find the binomial expansion of V(8 — 9x), |x|< 89 in ascending powers ofx, up to and including the term in x°. Give each coefficient as a simplified fractio
June 2013, Legacy C4 (01R), Q5 (11 marks): x|° t=8 ‘ Figure 1 Figure 1 shows part of the curve with equation x = 4te +3. The finite region R shown shaded in Figure | is bounded by the curve, the x-axis,
June 2013, Legacy C4 (01R), Q6 (12 marks): Relative to a fixed origin O, the pointA has position vector 21i— 17j + 6k and the point B has position vector 25i — 14j + 18k. The line / has vector equation a
June 2013, Legacy C4 (01R), Q7 (12 marks): y Cc O x Figure 2 Figure 2 shows a sketch of the curve C with parametric equations x =27 sect, y=3 tant, 0<1<t (a) Find the gradient ofthe curve C at the point
June 2013, Legacy C4 (01R), Q8 (12 marks): In an experiment testing solid rocket fuel, some fuel is burned and the waste products are collected. Throughout the experiment the sum of the masses of the unb
January 2013, Legacy C4, Q1 (5 marks): Given f(x) = (2 + 3x)–3, |x| < 2 3 find the binomial expansion of f(x), in ascending powers of x, up to and including the term in x3. Give each coefficient as a
January 2013, Legacy C4, Q2 (7 marks): (a) Use integration to find 1 3x∫ ln x dx (5) (b) Hence calculate 1 3 1 2 x ∫ ln x dx (2)
January 2013, Legacy C4, Q3 (4 marks): Express 9 20 10 2 3 1 2x x x x + − + − ( )( ) in partial fractions. (4)
January 2013, Legacy C4, Q4 (12 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation y x x = + 1 √. The finite region R, shown shaded in Figure 1, is bounded by the curve, the x
January 2013, Legacy C4, Q5 (15 marks): Figure 2 Figure 2 shows a sketch of part of the curve C with parametric equations x t = − 1 1 2 , y = 2t – 1 The curve crosses the y-axis at the point A and cro
January 2013, Legacy C4, Q6 (9 marks): Figure 3 Figure 3 shows a sketch of part of the curve with equation y = 1 – 2cos x, where x is measured in radians. The curve crosses the x-axis at the point A
January 2013, Legacy C4, Q7 (14 marks): With respect to a fixed origin O, the lines l1 and l2 are given by the equations l1 : r = (9i + 13j – 3k) + (i + 4j – 2k) l2 : r = (2i – j + k) + (2i + j + k)
January 2013, Legacy C4, Q8 (9 marks): A bottle of water is put into a refrigerator. The temperature inside the refrigerator remains constant at 3 °C and t minutes after the bottle is placed in the r
June 2012, Legacy C4, Q1 (10 marks): f ( ) ( ) ( ) ( ) x x x A x B x C x = − = + − + − 1 3 1 3 1 3 1 2 2 (a) Find the values of the constants A, B and C. (4) (b) (i) Hence find f d ( ) . x x ∫ (ii)
June 2012, Legacy C4, Q2 (6 marks): x x x Figure 1 Figure 1 shows a metal cube which is expanding uniformly as it is heated. At time t seconds, the length of each edge of the cube is x cm, and the
June 2012, Legacy C4, Q3 (9 marks): f ( ) , x x = − ( ) 6 9 4 √ x < 9 4 (a) Find the binomial expansion of f ( ) x in ascending powers of x, up to and including the term in x3. Give each coefficie
June 2012, Legacy C4, Q4 (5 marks): Given that y = 2 at x = π 4 , solve the differential equation d d y x y x = 3 2 cos (5)
June 2012, Legacy C4, Q5 (12 marks): The curve C has equation 16 9 54 0 3 2 y x y x + − = (a) Find d d y x in terms of x and y. (5) (b) Find the coordinates of the points on C where d d y x = 0. (7
June 2012, Legacy C4, Q6 (12 marks): x O y C Figure 2 Figure 2 shows a sketch of the curve C with parametric equations x = t , y = 4 cos2 t , 0 t (a) Show that d d y x = k t,
June 2012, Legacy C4, Q7 (11 marks): O x 4 1 y R Figure 3 Figure 3 shows a sketch of part of the curve with equation y x x = 1 2 2 ln . The finite region R, shown shaded in Figure 3, is bounded by
June 2012, Legacy C4, Q8 (10 marks): Relative to a fixed origin O, the point A has position vector 10 2 3 i j k + + ( ) , and the point B has position vector 8 3 4 i j k + + ( ) . The line l passes
January 2012, Legacy C4, Q1 (8 marks): The curve C has the equation 2x + 3y 2 + 3x 2 y = 4x 2. The point P on the curve has coordinates (–1, 1). (a) Find the gradient of the curve at P. (5) (b) Hence
January 2012, Legacy C4, Q2 (6 marks): (a) Use integration by parts to find x x x sin . 3 d ∫ (3) (b) Using your answer to part (a), find x x x 2 3 cos . d ∫ (3)
January 2012, Legacy C4, Q3 (9 marks): (a) Expand 1 2 5 2 ( ) −x , x < 2 5 in ascending powers of x, up to and including the term in x2, giving each term as a simplified fraction. (5) Given that the
January 2012, Legacy C4, Q4 (5 marks): O 2 y x S Figure 1 Figure 1 shows the curve with equation y x x x = + ⎛ ⎝⎜ ⎞ ⎠⎟ 2 3 4 0 2 , The finite region S, shown shaded in Figure 1, is bounded by the c
January 2012, Legacy C4, Q5 (8 marks): C O x y Figure 2 Figure 2 shows a sketch of the curve C with parametric equations x t y t t = + ⎛ ⎝⎜ ⎞ ⎠⎟ = 4 6 3 2 0 2 sin , cos , π π < (a) Find an expressi
January 2012, Legacy C4, Q6 (12 marks): O R x y 2 π Figure 3 Figure 3 shows a sketch of the curve with equation y x x x = + 2 2 1 0 2 sin ( cos ) , . π The finite region R, shown shaded in Figure
January 2012, Legacy C4, Q7 (15 marks): Relative to a fixed origin O, the point A has position vector ( 2i – j + 5k ), the point B has position vector ( 5i + 2j + 10k ), and the point D has position v
January 2012, Legacy C4, Q8 (12 marks): (a) Express 1 5 P P ( ) − in partial fractions. (3) A team of conservationists is studying the population of meerkats on a nature reserve. The population is mod
June 2011, Legacy C4, Q1 (4 marks): Find the values of the constants A, B and C. (4) 9 1 2 1 1 1 2 1 2 2 2 x x x A x B x C x ( ) ( ) ( ) ( ) ( ) − + = − + − + +
June 2011, Legacy C4, Q2 (6 marks): Find the first three non-zero terms of the binomial expansion of f ( ) x in ascending powers of x. Give each coefficient as a simplified fraction. (6) f ( ) x x
June 2011, Legacy C4, Q3 (6 marks): h Figure 1 A hollow hemispherical bowl is shown in Figure 1. Water is flowing into the bowl. When the depth of the water is h m, the volume V m3 is given by 1 1
June 2011, Legacy C4, Q4 (15 marks): O x √2 y R Figure 2 Figure 2 shows a sketch of the curve with equation y = x3 ln (x2 + 2), x .0. The finite region R, shown shaded in Figure 2, is bounded by t
June 2011, Legacy C4, Q5 (7 marks): Find the gradient of the curve with equation ln y = 2x ln x, x 0, y 0 at the point on the curve where x = 2. Give your answer as an exact value. (7)
June 2011, Legacy C4, Q6 (14 marks): With respect to a fixed origin O, the lines l1 and l2 are given by the equations l1: 6 1 3 2 2 3 − ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = − + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ ⎝ ⎠ λ r , l2: r =
June 2011, Legacy C4, Q7 (15 marks): y x O l C P S Q √3 Figure 3 Figure 3 shows part of the curve C with parametric equations x = tan ș , y = sin ș , 0 -ș < 2 ʌ The point P lies on C and has coord
June 2011, Legacy C4, Q8 (8 marks): (a) Find (2) (b) Given that 1.5 y = at x = – 2, solve the differential equation ( ) 2 4 3 d d y y x x + = ˲ giving your answer in the form f ( ). y x = (6) ( )
January 2011, Legacy C4, Q1 (6 marks): Use integration to find the exact value of x x x sin 2 0 2 d π ∫ (6)
January 2011, Legacy C4, Q2 (5 marks): The current, I amps, in an electric circuit at time t seconds is given by I = 16 – 16(0.5)t , t 0 Use differentiation to find the value of d d I t when 3 t =
January 2011, Legacy C4, Q3 (12 marks): (a) Express ( )( ) 5 1 3 2 x x − + in partial fractions. (3) (b) Hence find ( )( ) 5 d 1 3 2 x x x − + ∫ , where 1 x . (3) (c) Find the particular solution of t
January 2011, Legacy C4, Q4 (10 marks): Relative to a fixed origin O, the point A has position vector 3 2 − + i j k and the point B has position vector 2 2 − + − i j k . The points A and B lie on a st
January 2011, Legacy C4, Q5 (13 marks): (a) Use the binomial theorem to expand ( ) 2 2 2 3 , , 3 − −x x in ascending powers of x, up to and including the term in 3x . Give each coefficient as a simpli
January 2011, Legacy C4, Q6 (15 marks): The curve C has parametric equations ln x t = , 2 2 y t = − , 0 t Find (a) an equation of the normal to C at the point where 3 t = , (6) (b) a cartesian equatio
January 2011, Legacy C4, Q7 (14 marks): (a) Given that y x = + − 1 4 1 G ( ) , complete the table below with values of y corresponding to 3 x = and 5 x = . Give your values to 4 decimal places. x 2 3
June 2010, Legacy C4, Q1 (8 marks): Figure 1 Figure 1 shows part of the curve with equation 2 (0.75 cos ). y x = + √ The finite region R, shown shaded in Figure 1, is bounded by the curve, the y-a
June 2010, Legacy C4, Q2 (6 marks): Using the substitution cos 1, u x = + or otherwise, show that 2 cos 1 0 e sin d e(e 1) x x x π + = − ∫ (6)
June 2010, Legacy C4, Q3 (7 marks): A curve C has equation 2 2 2 x y xy + = Find the exact value of d d y x at the point on C with coordinates (3, 2). (7)
June 2010, Legacy C4, Q4 (10 marks): A curve C has parametric equations 2 sin , 2tan , 0 2 x t y t t π = = - < (a) Find d d y x in terms of t. (4) The tangent to C at the point where 3 t π = cuts t
June 2010, Legacy C4, Q5 (11 marks): 2 2 5 10 ( 1)( 2) 1 2 x x B C A x x x x + − ≡ + + − + − + (a) Find the values of the constants A, B and C. (4) (b) Hence, or otherwise, expand 2 2 5 10 ( 1)( 2)
June 2010, Legacy C4, Q6 (10 marks): f(θ) = 4 2 cos θ – 3 2 sin θ (a) Show that f(θ) 1 7 cos 2 2 = + 2θ. (3) (b) Hence, using calculus, find the exact value of 2 0 π ∫ θ f(θ) dθ. (7)
June 2010, Legacy C4, Q7 (12 marks): The line 1l has equation = + ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ λ r 1 2 1 2 3 −4 , where λ is a scalar parameter. The line 2l has equation = + ⎛ ⎜ ⎜ ⎝ ⎞ ⎟ ⎟ ⎠ ⎛ ⎜
June 2010, Legacy C4, Q8 (11 marks): Figure 2 Figure 2 shows a cylindrical water tank. The diameter of a circular cross-section of the tank is 6 m. Water is flowing into the tank at a constant rate
January 2010, Legacy C4, Q1 (9 marks): (a) Find the binomial expansion of √( ), , 1 8 1 8 − < x x in ascending powers of x up to and including the term in x3, simplifying each term. (4) (b) Show that
January 2010, Legacy C4, Q2 (13 marks): Figure 1 Figure 1 shows a sketch of the curve with equation y = x 1n x, x . 1. The finite region R, shown shaded in Figure 1, is bounded by the curve, the x-axi
January 2010, Legacy C4, Q3 (9 marks): The curve C has the equation cos cos , 2 3 1 + = x y −π π π 4 4 0 6 - - - - x y , (a) Find d d y x in terms of x and y. (3) The point P lies on C where x = π 6
January 2010, Legacy C4, Q4 (12 marks): The line l1 has vector equation and the line l2 has vector equation where λ and μ are parameters. The lines l1 and l2 intersect at the point A and the acute ang
January 2010, Legacy C4, Q5 (8 marks): (a) Find (2) (b) Given that 8 = y at 1, = x solve the differential equation d d y x x y x = + ( ) 9 6 1 3 giving your answer in the form 2 g( ). y x = (6) 9 6 d
January 2010, Legacy C4, Q6 (5 marks): The area A of a circle is increasing at a constant rate of 1.5 cm2 s–1. Find, to 3 significant figures, the rate at which the radius r of the circle is increasi
January 2010, Legacy C4, Q7 (9 marks): Figure 2 Figure 2 shows a sketch of the curve C with parametric equations x t = − 5 4 2 , y t t = − ( ) 9 2 The curve C cuts the x-axis at the points A and B. (
January 2010, Legacy C4, Q8 (10 marks): (a) Using the substitution x = 2 cos u, or otherwise, find the exact value of 1 4 2 2 1 x x x √ √ ( ) − ∫ d 2 (7) Figure 3 Figure 3 shows a sketch of part of th

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