Legacy C3 — Core Mathematics 3: Edexcel A‑Level Maths questions

Edexcel A-Level Maths past-paper questions on L.C3 Legacy C3 — Core Mathematics 3, 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 125 questions on Legacy C3 — Core Mathematics 3, as a list

June 2018, Legacy C3, Q1 (6 marks): Given y = 2 x (3 x − 1)5, (a) find d d y x , giving your answer as a single fully factorised expression. (4) (b) Hence find the set of values of x for which d d
June 2018, Legacy C3, Q2 (7 marks): The function f is defined by f ( ) , x x x x x = + + − + − 6 2 5 2 2 5 60 4 25 2 > 4 (a) Show that f ( x) = A Bx C + where A, B and C are constants to be found.
June 2018, Legacy C3, Q3 (9 marks): The value of a car is modelled by the formula V = 16 000e−k t + A, t  0, t ∈ R where V is the value of the car in pounds, t is the age of the car in years, and
June 2018, Legacy C3, Q4 (9 marks): Not to scale y B C l x A O Figure 1 Figure 1 shows a sketch of part of the curve C with equation y = e−2 x + x2 − 3 The curve C crosses the y-axis at the point
June 2018, Legacy C3, Q5 (8 marks): y y = f ( x) O x Figure 2 Figure 2 shows part of the graph with equation y = f ( x), where f ( x) = 2| 5 − x | + 3, x  0 Given that the equation f ( x) = k, wh
June 2018, Legacy C3, Q6 (11 marks): (i) Using the identity for tan (A ± B), solve, for −90° < x < 90°, tan tan tan tan 2 32 1 2 32 5 x x + ° − ° = Give your answers, in degrees, to 2 decimal place
June 2018, Legacy C3, Q7 (9 marks): The curve C has equation y x x x = + + ∈ ln( ) , 2 2 1 1 R (a) Find d d y x as a single fraction, simplifying your answer. (3) (b) Hence find the exact coordina
June 2018, Legacy C3, Q8 (7 marks): (a) By writing sec θ = 1 cosθ , show that d dθ (sec θ) = sec θ tan θ (2) (b) Given that x = esec y x > e, 0 < y < 2 π show that d d g y x x x = 1 ( ) , x > e wh
June 2018, Legacy C3, Q9 (9 marks): (a) Express sin θ − 2cos θ in the form R sin (θ − α), where R > 0 and 0 < α < 2 π Give the exact value of R and the value of α, in radians, to 3 decimal places.
June 2017, Legacy C3, Q1 (4 marks): Express 4 9 2 3 2 x x x − − + as a single fraction in its simplest form. (4)
June 2017, Legacy C3, Q2 (7 marks): Find the exact solutions, in their simplest form, to the equations (a) e3x–9 = 8 (3) (b) ln(2y + 5) = 2 + ln(4 – y) (4)
June 2017, Legacy C3, Q3 (9 marks): O x y Figure 1 Figure 1 shows a sketch of part of the graph of y = g(x), where g(x) = 3 + x + 2 , x . –2 (a) State the range of g. (1) (b) Find g–1(x) and state
June 2017, Legacy C3, Q4 (9 marks): (a) Write 5cos ș – 2sin ș in the form Rcos(ș + Į), where R and Į are constants, R > 0 and 0 -Į<2 π Give the exact value of R and give the value of Į in radia
June 2017, Legacy C3, Q5 (10 marks): Q y P C O x Figure 2 Figure 2 shows a sketch of part of the curve C with equation y = 2ln(2x + 5) – 3 2 x , x > –2.5 The point P with x coordinate –2 lies on C.
June 2017, Legacy C3, Q6 (8 marks): Given that a and b are positive constants, (a) on separate diagrams, sketch the graph with equation (i) y = 2x – a (ii) y = 2x – a + b Show, on each sketch, the
June 2017, Legacy C3, Q7 (10 marks): (i) Given y = 2x(x2 – 1)5, show that (a) d d y x = g(x)(x2 – 1)4 where g(x) is a function to be determined. (4) (b) Hence find the set of values of x for which
June 2017, Legacy C3, Q8 (9 marks): P PT T O t Figure 3 The number of rabbits on an island is modelled by the equation P = 100 1 3 0 1 0 9 e e − − + . . t t + 40, t , t . 0 where P is the number
June 2017, Legacy C3, Q9 (9 marks): (a) Prove that sin2x – tanxŁWDQx cos2x, x n + 1)90°, n ' (4) (b) Given that x DQGxVROYHIRU- x < 360°, sin2x – tanx = 3tanxsinx Give
June 2016, Legacy C3, Q1 (5 marks): The functions f and g are defined by f g : , : , , x x x x x x x → − ∈ → − ≠ ∈ 7 1 4 2 2 \ \ (a) Solve the equation fg(x) = x (4) (b) Hence, or otherwise, find
June 2016, Legacy C3, Q2 (7 marks): y x x = + 4 5 2 (a) Find d d y x , writing your answer as a single fraction in its simplest form. (4) (b) Hence find the set of values of x for which d d y x 0
June 2016, Legacy C3, Q3 (10 marks): (a) Express 2 cosș – sinș in the form Rcos(ș + Į), where R and Įare constants, R and 0 Į 9° Give the exact value of R and give the value of Įto 2 deci
June 2016, Legacy C3, Q4 (11 marks): x y O \ N A B \ g(x) Figure 1 )LJXUHVKRZVDVNHWFKRISDUWRIWKHFXUYHZLWKHTXDWLRQy = g(x), where g(x) = 4e2x – 25 , x The curve cuts the y-axis a
June 2016, Legacy C3, Q5 (10 marks): (i) Find, using calculus, the x coordinate of the turning point of the curve with equation 3e cos4 , 4 2 = x π π y x x < Give your answer to 4 decimal places.
June 2016, Legacy C3, Q6 (9 marks): f ( ) , , x x x x x x x x x = + − + − + − ∈ 4 3 2 2 3 7 6 6 2 \ (a) Given that x x x x x x x A B x 4 3 2 2 2 3 7 6 6 2 + − + − + − ≡ + + − find the values of th
June 2016, Legacy C3, Q7 (5 marks): (a) For , 2 2 −π π y VNHWFKWKHJUDSKRIy = g(x) where g(x) = arcsin [ ±- [- (2) (b) Find the exact value of x for which 3g([ ʌ = 0 (3)
June 2016, Legacy C3, Q8 (10 marks): (a) Prove that 2cot 2 tan cot , 2 + ≡ ≠ ∈ nπ x x x x n Z (4) (b) Hence, or otherwise, solve, for –ʌ-[ ʌ, 6cot 2x + 3 tan x = cosec2x – 2 Give your answers t
June 2016, Legacy C3, Q9 (8 marks): The amount of an antibiotic in the bloodstream, from a given dose, is modelled by the formula x = De–0.2t where x is the amount of the antibiotic in the bloodst
June 2015, Legacy C3, Q1 (6 marks): Given that tanș° = p, where p is a constant, p 1 use standard trigonometric identities, to find in terms of p, (a) tan2ș° (2) (b) cosș° (2) (c) cot(ș– 45)°
June 2015, Legacy C3, Q2 (10 marks): Given that f(x) = 2ex – 5, x (a) sketch, on separate diagrams, the curve with equation (i) y = f(x) (ii) y = |f(x)| On each diagram, show the coordinates of ea
June 2015, Legacy C3, Q3 (10 marks): g(ș) = 4cos2ș + 2sin2ș Given that g(ș) = R cos(2ș±Į),where 5> 0 and 0 < Į< 90°, (a) find the exact value of 5and the value of Į to 2 decimal places. (3) (
June 2015, Legacy C3, Q4 (7 marks): Water is being heated in an electric kettle. The temperature, ș °C, of the water t seconds after the kettle is switched on, is modelled by the equation ș = 120
June 2015, Legacy C3, Q5 (7 marks): The point P lies on the curve with equation x = (4y – sin2y)2 Given that P has (x, y) coordinates ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ , 2 π p , where Sis a constant, (a) find the exac
June 2015, Legacy C3, Q6 (8 marks): O x y A y = 17 – x y = 2x + 1 – 3 Figure 1 Figure 1 is a sketch showing part of the curve with equation y = 2x+1 – 3 and part of the line with equation y = 17 –
June 2015, Legacy C3, Q7 (10 marks): y O x Figure 2 Figure 2 shows a sketch of part of the curve with equation g(x) = x2(1 – x)e–2x, x . 0 (a) Show that g' ( ) x = f(x)e–2x, where f(x) is a cubic f
June 2015, Legacy C3, Q8 (9 marks): (a) Prove that sec tan cos sin cos sin 2 2 A A A A A A + ≡ + − , (2 1) 4 n π A + ≠ , n' (5) (b) Hence solve, for 0 - ș < 2ʌ, 1 sec2 tan 2 2 θ θ + = Give your a
June 2015, Legacy C3, Q9 (8 marks): Given that k is a negative constant and that the function f(x) is defined by f ( ) ( )( ) x x k x k x kx k = − − − − + 2 5 3 2 2 2 , x . 0 (a) show that f ( ) x
June 2014, Legacy C3, Q1 (6 marks): The curve C has equation y = f (x) where f ( ) , x x x x = + − 4 1 2 2 > (a) Show that f ƍ(x) = − − 9 2 2 ( ) x (3) Given that P is a point on C such that f ƍ(x
June 2014, Legacy C3, Q2 (6 marks): Find the exact solutions, in their simplest form, to the equations (a) 2 ln(2x + 1) – 10 = 0 (2) (b) 3xe4x = e7 (4)
June 2014, Legacy C3, Q3 (8 marks): The curve C has equation x = 8y tan 2y The point P has coordinates , 8 π π ⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (a) Verify that P lies on C. (1) (b) Find the equation of the tangent to
June 2014, Legacy C3, Q4 (7 marks): Figure 1 Figure 1 shows part of the graph with equation y = f (x), x . The graph consists of two line segments that meet at the point Q(6, –1). The graph cros
June 2014, Legacy C3, Q5 (10 marks): g( ) ( ) , x x x x x x x = + + + + − 3 3 2 1 6 3 2 (a) Show that g( ) , x x x x = + − 1 2 3 (4) (b) Find the range of g. (2) (c) Find the exact value of a for w
June 2014, Legacy C3, Q6 (8 marks): Figure 2 Figure 2 shows a sketch of part of the curve with equation y x x x = ⎛ ⎝⎜ ⎞ ⎠⎟+ − − 2 1 2 3 2 2 3 cos The curve crosses the x-axis at the point Q and h
June 2014, Legacy C3, Q7 (10 marks): (a) Show that cosec 2x + cot 2x = cot x, x 90n°, n ' (5) (b) Hence, or otherwise, solve, for 0 -ș < 180°, cosec (4ș + 10°) + cot (4ș + 10°) = 3 You must sh
June 2014, Legacy C3, Q8 (11 marks): A rare species of primrose is being studied. The population, P, of primroses at time t years after the study started is modelled by the equation P t t t t = + 8
June 2014, Legacy C3, Q9 (9 marks): (a) Express 2 sin ș – 4 cos ș in the form R sin(ș ± Į), where R and Į are constants, R ! 0 and 0 2 π < < α Give the value of Į to 3 decimal places. (3) H(ș) = 4
June 2014, Legacy C3 (01R), Q1 (4 marks): Express 3 2 3 1 2 3 6 4 9 2 x x x + − − + − as a single fraction in its simplest form. (4)
June 2014, Legacy C3 (01R), Q2 (12 marks): A curve C has equation y = e4x + x4 + 8x + 5 (a) Show that the x coordinate of any turning point of C satisfies the equation x3 = –2 – e4x (3) (b) On the axes g
June 2014, Legacy C3 (01R), Q3 (12 marks): (i) (a) Show that 2 tan x – cot x = 5 cosec x may be written in the form a cos2 x + b cos x + c = 0 stating the values of the constants a, b and c. (4) (b) Henc
June 2014, Legacy C3 (01R), Q4 (12 marks): (i) Given that x = sec2 2y, 0 < y < 4 π show that d d y x x x = − 1 4 1 ( ) (4) (ii) Given that y = (x2 + x3)ln 2x find the exact value of d d at e y x x = 2 ,
June 2014, Legacy C3 (01R), Q5 (8 marks): (a) Sketch the graph with equation y = |4x – 3| stating the coordinates of any points where the graph cuts or meets the axes. (2) Find the complete set of value
June 2014, Legacy C3 (01R), Q6 (12 marks): The function f is defined by f : x ĺ e2x + k2, x , k is a positive constant. (a) State the range of f. (1) (b) Find f –1 and state its domain. (3) The functio
June 2014, Legacy C3 (01R), Q7 (15 marks): Figure 1 Figure 1 shows the curve C, with equation y = 6 cos x + 2.5 sin x for 0 - x - 2ʌ (a) Express 6 cos x + 2.5 sin x in the form R cos(x – Į), where R and
June 2013, Legacy C3, Q1 (4 marks): Given that 3 2 5 4 4 4 2 4 3 2 2 2 2 x x x x ax bx c dx e x x − − − − ≡ + + + + − ≠± , find the values of the constants a, b, c, d and e. (4)
June 2013, Legacy C3, Q2 (7 marks): Given that f(x) = ln x, x > 0 sketch on separate axes the graphs of (i) y = f(x), (ii) y = |f(x)|, (iii) y = –f(x – 4). Show, on each diagram, the point where t
June 2013, Legacy C3, Q3 (8 marks): Given that 2cos(x + 50) = sin(x + 40) (a) Show, without using a calculator, that tan x = 1 3 tan (4) (b) Hence solve, for 0 < 360, 2cos(2 + 50) = s
June 2013, Legacy C3, Q4 (11 marks): f(x) = 25x2e2x – 16, x (a) Using calculus, find the exact coordinates of the turning points on the curve with equation y = f(x). (5) (b) Show that the equat
June 2013, Legacy C3, Q5 (10 marks): Given that x = sec2 3y, 0 < y < 6 π (a) find d d x y in terms of y. (2) (b) Hence show that d d y x x x = − 1 6 1 1 2 ( ) (4) (c) Find an expression for d d 2 2
June 2013, Legacy C3, Q6 (10 marks): Find algebraically the exact solutions to the equations (a) ln(4 – 2x) + ln(9 – 3x) = 2ln(x + 1), –1 < x < 2 (5) (b) 2 x e 3x+1 = 10 Give your answer to (b) in
June 2013, Legacy C3, Q7 (11 marks): The function f has domain –2 x 6 and is linear from (–2, 10) to (2, 0) and from (2, 0) to (6, 4). A sketch of the graph of y = f(x) is shown in Figure 1. Fi
June 2013, Legacy C3, Q8 (14 marks): Figure 2 Kate crosses a road, of constant width 7 m, in order to take a photograph of a marathon runner, John, approaching at 3 m s–1. Kate is 24 m ahead of Joh
June 2013, Legacy C3 (01R), Q1 (4 marks): Express 3x+5 2 xetx-12 x-3 as a single fraction in its simplest form. (4)
June 2013, Legacy C3 (01R), Q2 (5 marks): y O a I x Figure 1 Figure 1 shows a sketch of the curve with equation y = f(x), x > 0, wheref is an increasing function of x. The curve crosses the x-axis at th
June 2013, Legacy C3 (01R), Q3 (10 marks): f(x) = 7cos x + sin x Given that f(x) = Rcos(x — a), where R > 0 and 0 <a < 90°, (a) find the exact value of R and the value of a to one decimal place. GB) (b)
June 2013, Legacy C3 (01R), Q4 (11 marks): The functions f and g are defined by fix 2|x| +3, xeR, gixr>3-4y, xeR (a) State the range of f. (2) (b) Find fg(1). (2) (c) Find g", the inverse function of g.
June 2013, Legacy C3 (01R), Q5 (10 marks): (a) Differentiate cos 2x ox with respect to x. (3) (b) Show that 4 (sec? 3x) can be written in the form dx (tan 3x + tan? 3x) where yw is a constant. (3) (c) Gi
June 2013, Legacy C3 (01R), Q6 (9 marks): (i) Use an appropriate double angle formula to show that cosec2x = A cosec x sec x, and state the value of the constant 4. (3) (ii) Solve, for 0 < @< 2z, the eq
June 2013, Legacy C3 (01R), Q7 (13 marks): y var P Figure 2 Figure 2 shows a sketch of part of the curve with equation y = f(x) where f(x) = (2 + 3x +e” The curve cuts the x-axis at points A and B as sho
June 2013, Legacy C3 (01R), Q8 (13 marks): Pje Figure 3 The population of a town is being studied. The population P, at time ¢ years from the start of the study, is assumed to be 1+7e wherek is a positiv
January 2013, Legacy C3, Q1 (7 marks): The curve C has equation y x = − ( ) 2 3 5 The point P lies on C and has coordinates (w, – 32). Find (a) the value of w, (2) (b) the equation of the tangent to
January 2013, Legacy C3, Q2 (8 marks): g e ( ) x x x = + − −1 6 (a) Show that the equation g( ) x = 0 can be written as x x = − + ln( ) 6 1, x < 6 (2) The root of g( ) x = 0 is . The iterative formu
January 2013, Legacy C3, Q3 (9 marks): y x y =f(x) Q(0,2) P(–3,0) O Figure 1 Figure 1 shows part of the curve with equation y x = f ( ), x ∈ . The curve passes through the points Q( , ) 0 2 and P( ,
January 2013, Legacy C3, Q4 (8 marks): (a) Express 6 cos+8sin in the form Rcos(), where R > 0 and 0 2 < < π α . Give the value of to 3 decimal places. (4) (b) p( ) cos sin = + +
January 2013, Legacy C3, Q5 (11 marks): (i) Differentiate with respect to x (a) y x x = 3 2 ln (b) y x x = + ( sin ) 2 3 (6) Given that x y = cot , (ii) show that d d y x x = − + 1 1 2 (5)
January 2013, Legacy C3, Q6 (11 marks): (i) Without using a calculator, find the exact value of (sin 22.5° + cos 22.5°)2 You must show each stage of your working. (5) (ii) (a) Show that cos 2 + sin
January 2013, Legacy C3, Q7 (12 marks): h( ) ( )( ) x x x x x = + + + − + + 2 2 4 5 18 5 2 2 2 , x 0 (a) Show that h( ) x x x = + 2 5 2 (4) (b) Hence, or otherwise, find ′ h ( ) x in its simplest fo
January 2013, Legacy C3, Q8 (9 marks): The value of Bob’s car can be calculated from the formula V t t = + + − − 17000 2000 500 0 25 0 5 e e . . where V is the value of the car in pounds (£) and t is
June 2012, Legacy C3, Q1 (4 marks): Express 2 3 2 9 4 2 3 1 2 ( ) x x x + − − + as a single fraction in its simplest form. (4)
June 2012, Legacy C3, Q2 (9 marks): f ( ) x x x x = + + − 3 2 3 4 12 (a) Show that the equation f ( ) x = 0 can be written as x x x = − + ⎛ ⎝⎜ ⎞ ⎠⎟ √ 4 3 3 ( ) ( ) , x ≠−3 (3) The equation x x x 3
June 2012, Legacy C3, Q3 (9 marks): Figure 1 Figure 1 shows a sketch of the curve C which has equation y x x = e ˲3 3 sin , −3 3 x π π (a) Find the x coordinate of the turning point P on C, fo
June 2012, Legacy C3, Q4 (7 marks): Figure 2 Figure 2 shows part of the curve with equation y x = f ( ) The curve passes through the points P( . , ) −1 5 0 and Q( , ) 0 5 as shown. On separate dia
June 2012, Legacy C3, Q5 (9 marks): (a) Express 4 2 2 cosec cosec 2 θ θ − in terms of sin and cos . (2) (b) Hence show that 4 2 2 cosec cosec 2 θ θ − 2 sec = θ (4) (c) Hence or otherwise solve,
June 2012, Legacy C3, Q6 (14 marks): The functions f and g are defined by f e : x x + 2, x ∈ g : ln x x , x 0 (a) State the range of f. (1) (b) Find fg( ) x , giving your answer in its simpl
June 2012, Legacy C3, Q7 (11 marks): (a) Differentiate with respect to x, (i) x x 1 2 3 ln( ) (ii) 1 10 2 1 5 − − x x ( ) , giving your answer in its simplest form. (6) (b) Given that x y = 3 2 tan
June 2012, Legacy C3, Q8 (12 marks): f ( ) cos sin x x x = − 7 2 24 2 Given that f ( ) cos( ) x R x = + 2 α , where R 0 and 0 90 < < α , (a) find the value of R and the value of . (3) (b) Hen
January 2012, Legacy C3, Q1 (9 marks): Differentiate with respect to x, giving your answer in its simplest form, (a) x x 2 3 ln( ) (4) (b) sin 4 3 x x (5)
January 2012, Legacy C3, Q2 (6 marks): P(–3, 0) Q(2, –4) y x O Figure 1 Figure 1 shows the graph of equation y = f (x). The points P (– 3, 0) and Q (2, – 4) are stationary points on the graph. Sketch
January 2012, Legacy C3, Q3 (6 marks): The area, A mm2, of a bacterial culture growing in milk, t hours after midday, is given by A t = 20 1 5 e . , t . 0 (a) Write down the area of the culture at mi
January 2012, Legacy C3, Q4 (7 marks): The point P is the point on the curve x y = + ⎛ ⎝⎜ ⎞ ⎠⎟ 2 12 tan π with y-coordinate 4 π . Find an equation of the normal to the curve at P. (7)
January 2012, Legacy C3, Q5 (10 marks): Solve, for 0 - R - 180°, 2cot 2 3R 7 cosec 3R – 5 Give your answers in degrees to 1 decimal place. (10)
January 2012, Legacy C3, Q6 (12 marks): f cos x x x x ( ) = − + ( ) 2 1 2 3 2 , 0 - x - ʌ (a) Show that the equation f (x) = 0 has a solution in the interval 0.8 < x < 0.9 (2) The curve with equation
January 2012, Legacy C3, Q7 (12 marks): The function f is defined by f : , , x x x x x x x 3 1 2 7 4 1 4 1 2 2 + ( ) + − − + ∈R > (a) Show that f x x ( ) = − 1 2 1 (4) (b) Find f −( ) 1 x (3) (c) Fi
January 2012, Legacy C3, Q8 (13 marks): (a) Starting from the formulae for sin ( A + B ) and cos ( A + B ), prove that tan tan tan tan tan A B A B A B + ( ) = + − 1 (4) (b) Deduce that 1 3tan tan 6 3
June 2011, Legacy C3, Q1 (5 marks): Differentiate with respect to x (a) ln x x 2 3 5 + + ( ) (2) (b) cos x x2 (3)
June 2011, Legacy C3, Q2 (8 marks): f ( ) sin x = 2 n( ) , x x + −2 2 0 - x 2ʌ (a) Show that f (x) = 0 has a root Į between x = 0.75 and x = 0.85 (2) The equation f (x) = 0 can be written as x x
June 2011, Legacy C3, Q3 (6 marks): Figure 1 Figure 1 shows part of the graph of y = f (x), x \ . The graph consists of two line segments that meet at the point R (4, – 3 ), as shown in Figure 1
June 2011, Legacy C3, Q4 (8 marks): The function f is defined by f : ln x 6 4 − ( ), x 2 + x \ , x . –1 (a) Find f −1( ). x (3) (b) Find the domain of f −1. (1) The function g is defined by g e
June 2011, Legacy C3, Q5 (11 marks): The mass, m grams, of a leaf t days after it has been picked from a tree is given by m p kt = − e where k and p are positive constants. When the leaf is picked
June 2011, Legacy C3, Q6 (12 marks): (a) Prove that 1 2 2 2 90 sin cos sin tan , , θ θ θ θ θ − = ≠ ° n n ∈ (4) (b) Hence, or otherwise, (i) show that tan 15° = 2 – ¥ 3, (3) (ii) solve, for 0 x 360
June 2011, Legacy C3, Q7 (13 marks): f ( ) ( )( ) , , x x x x x x x x = − + − − − ≠± ≠− 4 5 2 1 3 2 9 3 1 2 2 (a) Show that f ( ) ( )( ) x x x = + + 5 2 1 3 (5) The curve C has equation y = f (x).
June 2011, Legacy C3, Q8 (12 marks): (a) Express 2cos 3x – 3sin 3x in the form R cos (3x + Į), where R and Į are constants, R 0 and 0 2 < < π α . Give your answers to 3 significant figures. (4) f e
January 2011, Legacy C3, Q1 (9 marks): (a) Express 7cos 24sin x x − in the form R cos (x + Į) where 0 R and 0 Į 2 π . Give the value of Į to 3 decimal places. (3) (b) Hence write down the minimum val
January 2011, Legacy C3, Q2 (9 marks): (a) Express 4 1 3 2( 1) 2( 1)(2 1) x x x x − − − − − as a single fraction in its simplest form. (4) Given that 4 1 3 f ( ) 2( 1) 2( 1)(2 1) x x x x x − = − − −
January 2011, Legacy C3, Q3 (6 marks): Find all the solutions of 2 cos 2ș = 1 – 2 sin ș in the interval 0 - ș 360°. (6)
January 2011, Legacy C3, Q4 (8 marks): Joan brings a cup of hot tea into a room and places the cup on a table. At time t minutes after Joan places the cup on the table, the temperature, ș °C, of the
January 2011, Legacy C3, Q5 (13 marks): Figure 1 Figure 1 shows a sketch of part of the curve with equation f ( ) y x = , where f ( ) (8 )ln , 0 x x x x = − The curve cuts the x-axis at the points A a
January 2011, Legacy C3, Q6 (13 marks): The function f is defined by f: x ѻ 3 2 5 x x − − , x \ , 5 x ≠ (a) Find 1 f ( ). x − (3) Figure 2 The function g has domain –1 -x -8, and is linear from (–
January 2011, Legacy C3, Q7 (8 marks): The curve C has equation 3 sin 2 2 cos2 x y x + = + (a) Show that ( ) 2 d 6sin 2 4cos2 2 d 2 cos2 y x x x x + + = + (4) (b) Find an equation of the tangent to C
January 2011, Legacy C3, Q8 (9 marks): (a) Given that ( ) d cos sin d x x x = − show that ( ) d sec sec tan . d x x x x = (3) Given that sec2 x y = (b) find d d x y in terms of y. (2) (c) Hence find
June 2010, Legacy C3, Q1 (5 marks): (a) Show that sin cos tan 2 1 2 θ θ θ + = (2) (b) Hence find, for –180° - θ < 180°, all the solutions of sin cos 2 1 2 1 θ θ + = 2 Give your answers to 1 decima
June 2010, Legacy C3, Q2 (7 marks): A curve C has equation 2) 3 5 ( 3 x y − = , 3 5 ≠ x The point P on C has x-coordinate 2. Find an equation of the normal to C at P in the form 0, ax by c + + = w
June 2010, Legacy C3, Q3 (9 marks): f ( ) 4cosec 4 1 x x x = − + , where x is in radians. (a) Show that there is a root α of f ( ) 0 x = in the interval [1.2, 1.3]. (2) (b) Show that the equation
June 2010, Legacy C3, Q4 (10 marks): The function f is defined by f : 2 5 x x − a , (a) Sketch the graph with equation f ( ), y x = showing the coordinates of the points where the graph cuts or mee
June 2010, Legacy C3, Q5 (12 marks): Figure 1 Figure 1 shows a sketch of the curve C with the equation 2 (2 5 2)e . x y x x − = − + (a) Find the coordinates of the point where C crosses the y-axis.
June 2010, Legacy C3, Q6 (10 marks): Figure 2 Figure 2 shows a sketch of the curve with the equation f ( ), y x = . The curve has a turning point at A(3, 4) − and also passes through the point (0,
June 2010, Legacy C3, Q7 (15 marks): (a) Express 2 sin θ – 1.5 cos θ in the form R sin (θ – α), where R > 0 and 0 < < α π 2 . Give the value of α to 4 decimal places. (3) (b) (i) Find the maximum v
June 2010, Legacy C3, Q8 (7 marks): (a) Simplify fully 2 9 5 2 15 2 2 x x x x + − + − (3) Given that ln( ) ln( ) 2 9 5 1 2 15 2 2 x x x x + − = + + − , 5, x ≠− (b) find x in terms of e. (4)
January 2010, Legacy C3, Q1 (4 marks): Express x x x + − − + 1 3 3 1 3 1 2 as a single fraction in its simplest form. (4)
January 2010, Legacy C3, Q2 (8 marks): f( ) x x x x = + − − 3 2 2 3 11 (a) Show that f(x) = 0 can be rearranged as x x x x = + + ⎛ ⎝⎜ ⎞ ⎠⎟ ≠− 3 11 2 2 , . (2) The equation f(x) = 0 has one positive r
January 2010, Legacy C3, Q3 (9 marks): (a) Express 5 cos x – 3 sin x in the form R cos(x + α), where R > 0 and 0 < α < π 1 2 . (4) (b) Hence, or otherwise, solve the equation 5 cos x – 3 sin x = 4 fo
January 2010, Legacy C3, Q4 (9 marks): (i) Given that y x x = + ln ( ) 2 1 , find d d y x . (4) (ii) Given that x = tan y, show that d d y x x = + 1 1 2 . (5)
January 2010, Legacy C3, Q5 (3 marks): Sketch the graph of y x = ln , stating the coordinates of any points of intersection with the axes. (3)
January 2010, Legacy C3, Q6 (9 marks): Figure 1 Figure 1 shows a sketch of the graph of y = f (x). The graph intersects the y-axis at the point (0, 1) and the point A(2, 3) is the maximum turning poi
January 2010, Legacy C3, Q7 (11 marks): (a) By writing sec cos x x as 1 , show that d x dx x x (sec ) sec tan = . (3) Given that y = e 2x sec 3x, (b) find d d y x . (4) The curve with equation y = e 2
January 2010, Legacy C3, Q8 (7 marks): Solve cosec2 2x – cot 2x = 1 for 0 - x - 180°. (7)
January 2010, Legacy C3, Q9 (15 marks): (i) Find the exact solutions to the equations (a) ln (3x – 7) = 5 (3) (b) 3 xe 7x + 2 = 15 (5) (ii) The functions f and g are defined by f (x) = e 2x + 3, x ∈ \

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