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1.)Let f(x) 7 – 2x and g(x) x 3.(a)Find (g f)(x).(2)(b)–1Write down g (x).(1)(c)–1Find (f g )(5).(2)(Total 5 marks)2.)2Consider f(x) 2kx – 4kx 1, for k 0. The equation f(x) 0 has two equal roots.(a)Find the value of k.(5)(b)The line y p intersects the graph of f. Find all possible values of p.(2)(Total 7 marks)3.)–1The velocity v m s of a particle at time t seconds, is given by v 2t cos2t, for 0 t 2.(a)Write down the velocity of the particle when t 0.(1)When t k, the acceleration is zero.(b)(i)(ii)Show that k π.4Find the exact velocity when t π.4(8)(c)When t π dvπ dv, 0 and when t , 0.4 dt4 dtSketch a graph of v against t.(4)(d)Let d be the distance travelled by the particle for 0 t 1.(i)Write down an expression for d.(ii)Represent d on your sketch.(3)(Total 16 marks)

4.)3Let f(x) 3 ln x and g(x) ln 5x .(a)Express g(x) in the form f(x) ln a, where a .(4)(b)The graph of g is a transformation of the graph of f. Give a full geometric description ofthis transformation.(3)(Total 7 marks)5.)The following diagram shows part of the graph of a quadratic function f.The x-intercepts are at (–4, 0) and (6, 0) and the y-intercept is at (0, 240).(a)Write down f(x) in the form f(x) –10(x – p)(x – q).(2)(b)2Find another expression for f(x) in the form f(x) –10(x – h) k.(4)(c)2Show that f(x) can also be written in the form f(x) 240 20x – 10x .(2)–1A particle moves along a straight line so that its velocity, v m s , at time t seconds is given by2v 240 20t – 10t , for 0 t 6.(d)(i)(ii)Find the value of t when the speed of the particle is greatest.Find the acceleration of the particle when its speed is zero.(7)(Total 15 marks)

26.)Let f(x) 3x . The graph of f is translated 1 unit to the right and 2 units down.The graph of g is the image of the graph of f after this translation.(a)Write down the coordinates of the vertex of the graph of g.(2)(b)2Express g in the form g(x) 3(x – p) q.(2)The graph of h is the reflection of the graph of g in the x-axis.(c)Write down the coordinates of the vertex of the graph of h.(2)(Total 6 marks)7.)A company uses two machines, A and B, to make boxes. Machine A makes 60 % of the boxes.80 % of the boxes made by machine A pass inspection.90 % of the boxes made by machine B pass inspection.A box is selected at random.(a)Find the probability that it passes inspection.(3)(b)The company would like the probability that a box passes inspection to be 0.87.Find the percentage of boxes that should be made by machine B to achieve this.(4)(Total 7 marks)The following diagram shows the graph of f(x) e x .28.)The points A, B, C, D and E lie on the graph of f. Two of these are points of inflexion.(a)Identify the two points of inflexion.(2)(b)(i)(ii)Find f′(x).Show that f″(x) (4x – 2) e x .22(5)(c)Find the x-coordinate of each point of inflexion.

(4)(d)Use the second derivative to show that one of these points is a point of inflexion.(4)(Total 15 marks)9.)Let f(x) log3(a)x log3 16 – log3 4, for x 0.2Show that f(x) log3 2x.(2)(b)Find the value of f(0.5) and of f(4.5).(3)The function f can also be written in the form f(x) (c)(i)ln ax.ln bWrite down the value of a and of b.(ii)Hence on graph paper, sketch the graph of f, for –5 x 5, –5 y 5, using ascale of 1 cm to 1 unit on each axis.(iii)Write down the equation of the asymptote.(6)(d)–1Write down the value of f (0).(1)The point A lies on the graph of f. At A, x 4.5.(e)–1On your diagram, sketch the graph of f , noting clearly the image of point A.(4)(Total 16 marks)10.)Let f(x) 3x, g(x) 2x – 5 and h(x) (f g)(x).(a)Find h(x).(2)(b)–1Find h (x).(3)(Total 5 marks)11.)Let g(x) (a)1x sin x, for 0 x 4.2Sketch the graph of g on the following set of axes.

(4)(b)Hence find the value of x for which g(x) –1.(2)(Total 6 marks)12.)2Let f(x) 8x – 2x . Part of the graph of f is shown below.(a)Find the x-intercepts of the graph.(4)(b)(i)(ii)Write down the equation of the axis of symmetry.Find the y-coordinate of the vertex.(3)(Total 7 marks)

13.)Let f(x) log3 x , for x 0.(a)–12xShow that f (x) 3 .(2)(b)–1Write down the range of f .(1)Let g(x) log3 x, for x 0.(c)Find the value of (f–1 g)(2), giving your answer as an integer.(4)(Total 7 marks)14.)Let f(x) 1 3x x 2 3x . Part of the graph of f is shown below.3There is a maximum point at A and a minimum point at B(3, –9).(a)Find the coordinates of A.(8)(b)Write down the coordinates of(i)the image of B after reflection in the y-axis;(ii) 2 the image of B after translation by the vector ; 5 (iii)the image of B after reflection in the x-axis followed by a horizontal stretch with

scale factor1.2(6)(Total 14 marks)15.)Let f(x) p(x – q)(x – r). Part of the graph of f is shown below.The graph passes through the points (–2, 0), (0, –4) and (4, 0).(a)Write down the value of q and of r.(2)(b)Write down the equation of the axis of symmetry.(1)(c)Find the value of p.(3)(Total 6 marks)16.)2Let f(x) cos 2x and g(x) 2x – 1.(a) π Find f . 2 (2)(b) π Find (g f) . 2 (2)(c)Given that (g f)(x) can be written as cos (kx), find the value of k, k .(3)(Total 7 marks)

17.)Solve log2x log2(x – 2) 3, for x 2.(Total 7 marks)18.)Let f(x) 6 6sinx. Part of the graph of f is shown below.The shaded region is enclosed by the curve of f, the x-axis, and the y-axis.(a)Solve for 0 x 2π.(i)6 6sin x 6;(ii)6 6 sin x 0.(5)(b)Write down the exact value of the x-intercept of f, for 0 x 2.(1)(c)The area of the shaded region is k. Find the value of k, giving your answer in terms of π.(6)π Let g(x) 6 6sin x . The graph of f is transformed to the graph of g.2 (d)Give a full geometric description of this transformation.(2)(e)Given that p p3π2g ( x)dx k and 0 p 2π, write down the two values of p.(3)(Total 17 marks)19.)The diagram below shows a quadrilateral ABCD with obtuse angles AB̂C and AD̂C .

diagram not to scaleAB 5 cm, BC 4 cm, CD 4 cm, AD 4 cm, BÂC 30 , AB̂C x , AD̂C y .(a)Use the cosine rule to show that AC 41 40 cos x .(1)(b)Use the sine rule in triangle ABC to find another expression for AC.(2)(c)(i)(ii)Hence, find x, giving your answer to two decimal places.Find AC.(6)(d)(i)(ii)Find y.Hence, or otherwise, find the area of triangle ACD.(5)(Total 14 marks)20.)kxLet f(x) Ae 3. Part of the graph of f is shown below.The y-intercept is at (0, 13).(a)Show that A 10.(2)

(b)Given that f(15) 3.49 (correct to 3 significant figures), find the value of k.(3)(c)(i)Using your value of k, find f′(x).(ii)Hence, explain why f is a decreasing function.(iii)Write down the equation of the horizontal asymptote of the graph f.(5)2Let g(x) –x 12x – 24.(d)Find the area enclosed by the graphs of f and g.(6)(Total 16 marks)21.)2xConsider f(x) 2 – x , for –2 x 2 and g(x) sin e , for –2 x 2. The graph of f is given below.(a)On the diagram above, sketch the graph of g.(3)(b)Solve f(x) g(x).(2)(c)Write down the set of values of x such that f(x) g(x).(2)(Total 7 marks)22.)xLet f(x) e sin 2x 10, for 0 x 4. Part of the graph of f is given below.

There is an x-intercept at the point A, a local maximum point at M, where x p and a localminimum point at N, where x q.(a)Write down the x-coordinate of A.(1)(b)Find the value of(i)p;(ii)q.(2)(c)Find qpf ( x )dx . Explain why this is not the area of the shaded region.(3)(Total 6 marks)23.)013tThe number of bacteria, n, in a dish, after t minutes is given by n 800e(a).Find the value of n when t 0.(2)(b)Find the rate at which n is increasing when t 15.(2)(c)After k minutes, the rate of increase in n is greater than 10 000 bacteria per minute. Findthe least value of k, where k .(4)(Total 8 marks)24.)2Consider f(x) x ln(4 – x ), for –2 x 2. The graph of f is given below.

(a)Let P and Q be points on the curve of f where the tangent to the graph of f is parallel tothe x-axis.(i)Find the x-coordinate of P and of Q.(ii)Consider f(x) k. Write down all values of k for which there are exactly twosolutions.(5)32Let g(x) x ln(4 – x ), for –2 x 2.(b)Show that g′(x) 2x 44 x2 3 x 2 ln(4 x 2 ) .(4)(c)Sketch the graph of g′.(2)(d)Consider g′(x) w. Write down all values of w for which there are exactly two solutions.(3)(Total 14 marks)25.)2Let f(x) x 4 and g(x) x – 1.(a)Find (f g)(x).(2) 3 The vector translates the graph of (f g) to the graph of h. 1 (b)Find the coordinates of the vertex of the graph of h.(3)(c)2Show that h(x) x – 8x 19.(2)

(d)The line y 2x – 6 is a tangent to the graph of h at the point P. Find the x-coordinate of P.(5)(Total 12 marks)26.)32Consider the function f(x) px qx rx. Part of the graph of f is shown below.The graph passes through the origin O and the points A(–2, –8), B(1, –2) and C(2, 0).(a)Find three linear equations in p, q and r.(4)(b)Hence find the value of p, of q and of r.(3)(Total 7 marks)27.)2Let f(x) x ln(4 – x ), for –2 x 2. The graph of f is shown below.

The graph of f crosses the x-axis at x a, x 0 and x b.(a)Find the value of a and of b.(3)The graph of f has a maximum value when x c.(b)Find the value of c.(2)(c)The region under the graph of f from x 0 to x c is rotated 360 about the x-axis. Findthe volume of the solid formed.(3)(d)Let R be the region enclosed by the curve, the x-axis and the line x c, between x a andx c.Find the area of R.(4)(Total 12 marks)28.)22Let f(x) x and g(x) 2(x – 1) .(a)The graph of g can be obtained from the graph of f using two transformations.Give a full geometric description of each of the two transformations.(2)(b) 3 The graph of g is translated by the vector to give the graph of h. 2 The point (–1, 1) on the graph of f is translated to the point P on the graph of h.Find the coordinates of P.(4)(Total 6 marks)

29.)Let f(x) e(a)x 3.(i)(ii)–1Show that f (x) ln x – 3.–1Write down the domain of f .(3)(b) 1 –1Solve the equation f (x) ln . x (4)(Total 7 marks)30.)Let f(x) axx 12, –8 x 8, a . The graph of f is shown below.The region between x 3 and x 7 is shaded.(a)Show that f(–x) –f(x).(2)(b)Given that f′′(x) 2ax( x 2 3)( x 2 1) 3, find the coordinates of all points of inflexion.(7)(c)It is given thata f ( x)dx 2 ln( x2 1) C .(i)Find the area of the shaded region, giving your answer in the form p ln q.(ii)Find the value of8 2 f ( x 1)dx .4(7)

(Total 16 marks)231.)Let f(x) x and g(x) 2x – 3.(a)–1Find g (x).(2)(b)Find (f g)(4).(3)(Total 5 marks)32.)Let f(x) 2x3e 2 x sin x e cos x, for 0 x π. Given that tanπ1 , solve the equation63f(x) 0.(Total 6 marks)33.)Let f(x) (a)3x x 1, g(x) 4cos – 1. Let h(x) (g f)(x).2 3 Find an expression for h(x).(3)(b)Write down the period of h.(1)(c)Write down the range of h.(2)(Total 6 marks)34.)2Let f(x) ax bx c where a, b and c are rational numbers.(a)The point P(–4, 3) lies on the curve of f. Show that 16a –4b c 3.(2)(b)The points Q(6, 3) and R(–2, –1) also lie on the curve of f. Write down two other linearequations in a, b and c.(2)(c)These three equations may be written as a matrix equation in the form AX B, a where X b . c (i)Write down the matrices A and B.(ii)Write down A .–1

(iii)Hence or otherwise, find f(x).(8)(d)2Write f(x) in the form f(x) a(x – h) k, where a, h and k are rational numbers.(3)(Total 15 marks)35.)Consider the graph of f shown below.

(a)On the same grid sketch the graph of y f(–x).(2)The following four diagrams show images of f under different transformations.(b)Complete the following table.Description of transformationHorizontal stretch with scale factor 1.5Maps f to f(x) 1Diagram letter

(2)(c)Give a full geometric description of the transformation that gives the image inDiagram A.(2)(Total 6 marks)36.)xSolve the equation e 4 sin x, for 0 x 2π.(Total 5 marks)37.)2The quadratic equation kx (k – 3)x 1 0 has two equal real roots.(a)Find the possible values of k.(5)(b)2Write down the values of k for which x (k – 3)x k 0 has two equal real roots.(2)(Total 7 marks)38.)33xLet f(x) 2x 3 and g(x) e – 2.(a)(i)(ii)Find g(0).Find (f g)(0).(5)(b)–1Find f (x).(3)(Total 8 marks)39.)The diagram below shows the graph of a function f(x), for –2 x 4.

(a)Let h(x) f(–x). Sketch the graph of h on the grid below.(2)(b)1f(x – 1). The point A(3, 2) on the graph of f is transformed to the point P on2the graph of g. Find the coordinates of P.Let g(x)

(3)(Total 5 marks)40.)Let f(x) k log2 x.(a)–1Given that f (1) 8, find the value of k.(3)(b)–1 2 Find f . 3 (4)(Total 7 marks)41.)Let f(x) 3 20, for x 2. The graph of f is given below.x 42diagram not to scaleThe y-intercept is at the point A.(a)(i)(ii)Find the coordinates of A.Show that f′(x) 0 at A.(7)(b)The second derivative f′′(x) (i)40(3x 2 4)( x 2 4) 3. Use this tojustify that the graph of f has a local maximum at A;

(ii)explain why the graph of f does not have a point of inflexion.(6)(c)Describe the behaviour of the graph of f for large x .(1)(d)Write down the range of f.(2)(Total 16 marks)42.)Let f(x) 5 cos(a)πx and g(x) –0.5x2 5x – 8, for 0 x 9.4On the same diagram, sketch the graphs of f and g.(3)(b)Consider the graph of f. Write down(i)the x-intercept that lies between x 0 and x 3;(ii)the period;(iii)the amplitude.(4)(c)Consider the graph of g. Write down(i)the two x-intercepts;(ii)the equation of the axis of symmetry.(3)(d)Let R be the region enclosed by the graphs of f and g. Find the area of R.(5)(Total 15 marks)43.)Let f (x) ln (x 5) ln 2, for x –5.(a)Find f 1(x).(4)xLet g (x) e .(b)Find (g f) (x), giving your answer in the form ax b, where a, b, .(3)(Total 7 marks)44.)2Let f (x) 3(x 1) – 12.(a)2Show that f (x) 3x 6x – 9.(2)(b)For the graph of f

(i)write down the coordinates of the vertex;(ii)write down the equation of the axis of symmetry;(iii)write down the y-intercept;(iv)find both x-intercepts.(8)(c)Hence sketch the graph of f.(2)(d)2Let g (x) x . The graph of f may be obtained from the graph of g by the twotransformations:a stretch of scale factor t in the y-directionfollowed by p a translation of . q p Find and the value of t. q (3)(Total 15 marks)45.)2The following diagram shows part of the graph of f, where f (x) x x 2.(a)Find both x-intercepts.(4)(b)Find the x-coordinate of the vertex.(2)(Total 6 marks)

46.)Part of the graph of a function f is shown in the diagram below.y4321–2–101234x–1–2–3–4(a)On the same diagram sketch the graph of y f (x).(2)(b)Let g (x) f (x 3).(i)Find g ( 3).(ii)Describe fully the transformation that maps the graph of f to the graph of g.(4)(Total 6 marks)47.)2Let f (x) 4 tan x – 4 sin x, (a)ππ x .33On the grid below, sketch the graph of y f (x).

(3)(b)Solve the equation f (x) 1.(3)(Total 6 marks)48.)Let f (x) 3x – e(a)x–2– 4, for –1 x 5.Find the x-intercepts of the graph of f.(3)(b)On the grid below, sketch the graph of f.y321–2 –1 3)

(c)Write down the gradient of the graph of f at x 2.(1)(Total 7 marks)49.)A city is concerned about pollution, and decides to look at the number of people using taxis. At theend of the year 2000, there were 280 taxis in the city. After n years the number of taxis, T, in the city is givenbynT 280 1.12 .(a)(i)(ii)Find the number of taxis in the city at the end of 2005.Find the year in which the number of taxis is double the number of taxis there wereat the end of 2000.(6)(b)At the end of 2000 there were 25 600 people in the city who used taxis. After n years thenumber of people, P, in the city who used taxis is given byP 2 560 000.10 90e – 0.1n(i)Find the value of P at the end of 2005, giving your answer to the nearest wholenumber.(ii)After seven complete years, will the value of P be double its value at the end of2000? Justify your answer.(6)(c)Let R be the ratio of the number of people using taxis in the city to the number of taxis.The city will reduce the number of taxis if R 70.(i)Find the value of R at the end of 2000.(ii)After how many complete years will the city first reduce the number of taxis?(5)(Total 17 marks)50.)Let f be the function given by f(x) e0.5x, 0 x 3.5. The diagram shows the graph of f.

(a)–1On the same diagram, sketch the graph of f .(3)(b)–1Write down the range of f .(1)(c)–1Find f (x).(3)(Total 7 marks)51.)Let f(t) a cos b (t – c) d, t 0. Part of the graph of y f(t) is given below.When t 3, there is a maximum value of 29, at M.When t 9 , there is a minimum value of 15.(a)(i)Find the value of a.

π.6(ii)Show that b (iii)Find the value of d.(iv)Write down a value for c.(7)The transformation P is given by a horizontal stretch of a scale factor of1, followed by a2 3 .translation of 10 (b)Let M′ be the image of M under P. Find the coordinates of M′.(2)The graph of g is the image of the graph of f under P.(c)Find g(t) in the form g(t) 7 cos B(t – C) D.(4)(d)Give a full geometric description of the transformation that maps the graph of g to thegraph of f.(3)(Total 16 marks)52.)2Let f(x) 2x 4x – 6.(a)2Express f(x) in the form f(x) 2(x – h) k.(3)(b)Write down the equation of the axis of symmetry of the graph of f.(1)(c)Express f(x) in the form f(x) 2(x – p)(x – q).(2)(Total 6 marks)53.)Let f(x) x cos (x – sin x), 0 x 3.(a)Sketch the graph of f on the following set of axes.

(3)(b)The graph of f intersects the x-axis when x a, a 0. Write down the value of a.(1)(c)The graph of f is revolved 360 about the x-axis from x 0 to x a.Find the volume of the solid formed.(4)(Total 8 marks)54.)x 5 .Consider f(x) (a)Find(i)f(11);(ii)f(86);(iii)f(5).(3)(b)Find the values of x for which f is undefined.(2)(c)2Let g(x) x . Find (g f)(x).(2)(Total 7 marks)55.)2The quadratic function f is defined by f(x) 3x – 12x 11.(a)2Write f in the form f(x) 3(x – h) – k.(3)(b)The graph of f is translated 3 units in the positive x-direction and 5 units in the positivey-direction. Find the function g for the translated graph, giving your answer in the form2g(x) 3(x – p) q.(3)(Total 6 marks)

56.) 2 1 , and O Let M 3 4 0 0 . Given that M2 – 6M kI O, find k.00 (Total 6 marks)57.)Solve the following equations.(a)logx 49 2(3)(b)log2 8 x(2)(c)log25 x 12(3)(d)log2 x log2(x – 7) 3(5)(Total 13 marks)258.)Let f (x) 2x – 12x 5.(a)2Express f(x) in the form f(x) 2(x – h) – k.(3)(b)Write down the vertex of the graph of f.(2)(c)Write down the equation of the axis of symmetry of the graph of f.(1)(d)Find the y-intercept of the graph of f.(2)(e)The x-intercepts of f can be written asp q, where p, q, r .rFind the value of p, of q, and of r.(7)(Total 15 marks)59.)Let f(x) (a)1, x 0.xSketch the graph of f.(2) 2 The graph of f is transformed to the graph of g by a translation of . 3

(b)Find an expression for g(x).(2)(c)(i)Find the intercepts of g.(ii)Write down the equations of the asymptotes of g.(iii)Sketch the graph of g.(10)(Total 14 marks)60.)The function f is defined by f(x) (a)39 x2, for –3 x 3.On the grid below, sketch the graph of f.(2)(b)Write down the equation of each vertical asymptote.(2)(c)Write down the range of the function f.(2)(Total 6 marks)61.)The functions f and g are defined by f : x 3x, g : x x 2.(a)Find an expression for (f g)(x).(2)(b)–1–1Find f (18) g (18).(4)(Total 6 marks)62.)A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m, a second side, [AB],is 65 m and the angle between these two sides is 60 .

(a)Use the cosine rule to calculate the length of the third side of the field.(3)(b)Given that sin 60 3, find the area of the field in the form 3 p 3 where p is an2integer.(3)Let D be a point on [BC] such that [AD] bisects the 60 angle. The