1.	Let f(x) = 7 – 2x and g(x) = x + 3.
(a)	Find (g ° f)(x).
(2)
 
(b)	Write down g–1(x).
(1)
 
(c)	Find (f ° g–1)(5).
(2)
(Total 5 marks)

2.	Consider f(x) = 2kx2 – 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.	Let f(x) = 3 ln x and g(x) = ln 5x3.
(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 of this transformation.
(3)
(Total 7 marks)

4.	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)	Find another expression for f(x) in the form f(x) = –10(x – h)2 + k.
(4)
 
(c)	Show that f(x) can also be written in the form f(x) = 240 + 20x – 10x2.
(2)
 
	A particle moves along a straight line so that its velocity, v m s–1, at time t seconds is given by
v = 240 + 20t – 10t2, for 0 ≤ t ≤ 6.
(d)	(i)	Find the value of t when the speed of the particle is greatest.
(ii)	Find the acceleration of the particle when its speed is zero.
(7)
(Total 15 marks)

5.	Let f(x) = 3x, g(x) = 2x – 5 and h(x) = (f ° g)(x).
(a)	Find h(x).
(2)
 
(b)	Find h–1(x).
(3)
(Total 5 marks)

6.	Let g(x) = x sin x, for 0 ≤ x ≤ 4.
(a)	Sketch 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)

7.	Let f(x) = 8x – 2x2. Part of the graph of f is shown below.

(a)	Find the x-intercepts of the graph.
(4)
 
(b)	(i)	Write down the equation of the axis of symmetry.
(ii)	Find the y-coordinate of the vertex.
(3)
(Total 7 marks)

8.	Let f(x) = log3, for x > 0.
(a)	Show that f–1(x) = 32x.
(2)
 
(b)	Write down the range of f–1.
(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)

9.	Let f(x) = . Part of the graph of f is shown below.
		
	There 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)	the image of B after translation by the vector ;
(iii)	the image of B after reflection in the x-axis followed by a horizontal stretch with scale factor .
(6)
(Total 14 marks)

10.	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)

11.	Let f(x) = cos 2x and g(x) = 2x2 – 1.
(a)	Find .
(2)
 
(b)	Find (g ° f).
(2)
 
(c)	Given that (g ° f)(x) can be written as cos (kx), find the value of k, k  .
(3)
(Total 7 marks)

12.	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)
 
	Let g(x) = 6 + 6sin. The graph of f is transformed to the graph of g.
(d)	Give a full geometric description of this transformation.
(2)
 
(e)	Given that  = k and 0 ≤ p < 2π, write down the two values of p.
(3)
(Total 17 marks)

13.	Let f(x) = x2 + 4 and g(x) = x – 1.
(a)	Find (f ° g)(x).
(2)
 
	The vector  translates the graph of (f ° g) to the graph of h.
(b)	Find the coordinates of the vertex of the graph of h.
(3)

(c)	Show that h(x) = x2 – 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)

14.	Let f(x) = x2 and g(x) = 2(x – 1)2.
(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)	The graph of g is translated by the vector  to give the graph of h.
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)

15.	Let f(x) = ex+3.
(a)	(i)	Show that f–1(x) = ln x – 3.
(ii)	Write down the domain of f–1.
(3)
 
(b)	Solve the equation f–1(x) = .
(4)
(Total 7 marks)

16.	Let f(x) = , –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) = , find the coordinates of all points of inflexion.
(7)

(c)	It is given that .
(i)	Find the area of the shaded region, giving your answer in the form p ln q.
(ii)	Find the value of .
(7)
(Total 16 marks)

17.	Let f(x) = x2 and g(x) = 2x – 3.
(a)	Find g–1(x).
(2)
 
(b)	Find (f ° g)(4).
(3)
(Total 5 marks)

18.	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 transformation	Diagram letter	
Horizontal stretch with scale factor 1.5		
Maps f to f(x) + 1		
(2)
 
(c)	Give a full geometric description of the transformation that gives the image in Diagram A.
(2)
(Total 6 marks)

19.	Let f(x) = 2x3 + 3 and g(x) = e3x – 2.
(a)	(i)	Find g(0).
(ii)	Find (f ° g)(0).
(5)
 
(b)	Find f–1(x).
(3)
(Total 8 marks)

20.	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)	Let g(x) = f(x – 1). The point A(3, 2) on the graph of f is transformed to the point P on the graph of g. Find the coordinates of P.
(3)
(Total 5 marks)

21.	Let f(x) = k log2 x.
(a)	Given that f–1(1) = 8, find the value of k.
(3)

(b)	Find f–1.
(4)
(Total 7 marks)

22.	Let f (x) = ln (x + 5) + ln 2, for x > –5.
(a)	Find f −1(x).
(4)
Let g (x) = ex.
(b)	Find (g ◦ f) (x), giving your answer in the form ax + b, where a, b,Î .
(3)
(Total 7 marks)

23.	Let f (x) = 3(x + 1)2 – 12.
(a)	Show that f (x) = 3x2 + 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)	Let g (x) = x2. The graph of f may be obtained from the graph of g by the two transformations:
		a stretch of scale factor t in the y-direction
		followed by
		a translation of 
Find  and the value of t.
(3)
(Total 15 marks)

24.	The following diagram shows part of the graph of f, where f (x) = x2 − x − 2.

(a)	Find both x-intercepts.
(4)
(b)	Find the x-coordinate of the vertex.
(2)
(Total 6 marks)

25.	Part of the graph of a function f is shown in the diagram below.

(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)

26.	Let f be the function given by f(x) = e0.5x, 0 ≤ x ≤ 3.5. The diagram shows the graph of f.

(a)	On the same diagram, sketch the graph of f–1.
(3)
 
(b)	Write down the range of f–1.
(1)
 
(c)	Find f–1(x).
(3)
(Total 7 marks)

27.	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.
(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 of , followed by a translation of .
(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 the graph of f.
(3)
(Total 16 marks)

28.	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)	Let g(x) = x2. Find (g ° f)(x).
(2)
(Total 7 marks)

29.	The quadratic function f is defined by f(x) = 3x2 – 12x + 11.
(a)	Write f in the form f(x) = 3(x – h)2 – k.
(3)
 
(b)	The graph of f is translated 3 units in the positive x-direction and 5 units in the positive
y-direction. Find the function g for the translated graph, giving your answer in the form
g(x) = 3(x – p)2 + q.
(3)
(Total 6 marks)

30.	Let M = , and O = . Given that M2 – 6M + kI = O, find k.
(Total 6 marks)

31.	Let f (x) = 2x2 – 12x + 5.
(a)	Express f(x) in the form f(x) = 2(x – h)2 – 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 as , where p, q, r  .
Find the value of p, of q, and of r.
(7)
(Total 15 marks)

32.	Let f(x) = , x ≠ 0.
(a)	Sketch the graph of f.
(2)

	The graph of f is transformed to the graph of g by a translation of .
(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)

33.	The functions f (x) and g (x) are defined by f (x) = ex and g (x) = ln (1+ 2x).
(a)	Write down f −1(x).
(b)	(i)	Find ( f ◦ g) (x).
(ii)	Find ( f ◦ g)−1 (x).
(Total 6 marks)

34.	The graph of the function y = f (x), 0 £ x £ 4, is shown below.

(a)	Write down the value of
(i)	f ′ (1);
(ii)	f ′ (3).

(b)	On the diagram below, draw the graph of y = 3 f (−x).

(c)	On the diagram below, draw the graph of y = f (2x).

(Total 6 marks)

35.	Let f (x) = , x ³ − 4 and g (x) = x2, x Î .
(a)	Find (g ◦ f ) (3).
(b)	Find f −1(x).
(c)	Write down the domain of f −1.
(Total 6 marks)

36.	Consider two different quadratic functions of the form f (x) = 4x2 − qx + 25. The graph of each function has its vertex on the x-axis.
(a)	Find both values of q.
(b)	For the greater value of q, solve f (x) = 0.
(c)	Find the coordinates of the point of intersection of the two graphs.
(Total 6 marks)

37.	The graph of a function f is shown in the diagram below. The point A (–1, 1) is on the graph, and y = −1 is a horizontal asymptote.

(a)	Let g (x) = f (x −1) + 2. On the diagram, sketch the graph of g.
(b)	Write down the equation of the horizontal asymptote of g.
(c)	Let A′ be the point on the graph of g corresponding to point A. Write down the coordinates of A′.
(Total 6 marks)

38.	The following diagram shows part of the graph of f (x).

Consider the five graphs in the diagrams labelled A, B, C, D, E below.

(a)	Which diagram is the graph of f (x + 2) ?
 
(b)	Which diagram is the graph of – f (x) ?
 
(c)	Which diagram is the graph of f (–x)
(Total 6 marks)

39.	Let f (x) = x3 − 4 and g (x) = 2x.
(a)	Find (g ◦ f ) (−2).
(b)	Find f −1 (x).
(Total 6 marks)

40.	Part of the graph of the function y = d (x −m)2 + p is given in the diagram below.
The x-intercepts are (1, 0) and (5, 0). The vertex is V(m, 2).

(a)	Write down the value of
(i)	m;
(ii)	p.
(b)	Find d.
(Total 6 marks)

41.	Let g (x) = 3x – 2, h (x) = , x  4.
(a)	Find an expression for (h ◦ g) (x). Simplify your answer.

(b)	Solve the equation (h ◦ g) (x) = 0.
(Total 6 marks)

42.	The functions f and g are defined by .
(a)	Find an expression for (f ° g) (x).
(b)	Show that f –l (18) + g–l (18) = 22.
(Total 6 marks)

43.	Consider the line L with equation y + 2x = 3. The line L1 is parallel to L and passes through the point (6, –4).
(a)	Find the gradient of L1.
(b)	Find the equation of L1 in the form y = mx + b.
(c)	Find the x-coordinate of the point where line L1 crosses the x-axis.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................
(c)	..................................................................	
(Total 6 marks)

44.	The function f is given by f (x) = e(x–11) –8.
(a)	Find f –1(x).
(b)	Write down the domain of f –l(x).
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

45.	The graph of y = f (x) is shown in the diagram.

(a)	On each of the following diagrams draw the required graph,
(i)	y = 2 f (x);

(ii)	y = f (x – 3).

(b)	The point A (3, –1) is on the graph of f. The point A¢ is the corresponding point on the graph of y = –f (x) + 1. Find the coordinates of A¢.
Working:		
	Answer:
(b)	..................................................................	
(Total 6 marks)

46.	Part of the graph of f (x) = (x – p) (x – q) is shown below.

The vertex is at C. The graph crosses the y-axis at B.
(a)	Write down the value of p and of q.
(b)	Find the coordinates of C.
(c)	Write down the y-coordinate of B.

(Total 6 marks)

47.	Consider the functions f (x) = 2x and g (x) = , x  3.
(a)	Calculate (f ◦ g) (4).
(b)	Find g−1(x).
(c)	Write down the domain of g−1.

(Total 6 marks)

48.	The equation x2 – 2kx + 1 = 0 has two distinct real roots. Find the set of all possible values of k.
Working:		
	Answer:
…………………………………………..	
(Total 6 marks)

49.	Let f (x) = 2x + 1 and g (x) = 3x2 – 4.
	Find
(a)	f –1(x);
(b)	(g ° f ) (–2);
(c)	( f ° g) (x).
Working:		
	Answers:
(a)	…………………………………………..
(b)	…………………………………………..
(c)	…………………………………………..	
(Total 6 marks)

50.	The function f (x) is defined as f (x) = –(x – h)2 + k. The diagram below shows part of the graph of f (x). The maximum point on the curve is P (3, 2).

(a)	Write down the value of
(i)	h;
(ii)	k.
(2)
 
(b)	Show that f (x) can be written as f (x) = –x2 + 6x – 7.
(1)
 
(c)	Find f ¢ (x).
(2)
 
	The point Q lies on the curve and has coordinates (4, 1). A straight line L, through Q, is perpendicular to the tangent at Q.
(d)	(i)	Calculate the gradient of L.
(ii)	Find the equation of L.
(iii)	The line L intersects the curve again at R. Find the x-coordinate of R.
(8)
(Total 13 marks)

51.	Let f (x) = e–x, and g (x) = , x ¹  –1. Find
(a)	f –1 (x);
(b)	(g ° f ) (x).
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

52.	Consider the following relations between two variables x and y.
A.	y = sin x
B.	y is directly proportional to x
C.	y = 1 + tan x
D.	speed y as a function of time x, constant acceleration
E.	y = 2x
F.	distance y as a function of time x, velocity decreasing
	Each sketch below could represent exactly two of the above relations on a certain interval.

	Complete the table below, by writing the letter for the two relations that each sketch could represent.
sketch	relation letters	
(i)			
(ii)			
(iii)			

53.	The equation kx2 + 3x + 1 = 0 has exactly one solution. Find the value of k.
Working:		
	Answer:
..................................................................	
(Total 6 marks)

54.	The function f is given by f (x) = x2 – 6x + 13, for x ³ 3.
(a)	Write f (x) in the form (x – a)2 + b.
(b)	Find the inverse function f –1.
(c)	State the domain of f –1.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................
(c)	..................................................................	
(Total 6 marks)

55.	The diagram shows part of the graph of y = a (x – h)2 + k. The graph has its vertex at P, and passes through the point A with coordinates (1, 0).

(a)	Write down the value of
(i)	h;
(ii)	k.
(b)	Calculate the value of a.
Working:		
	Answers:
(a)	(i)	..........................................................
	(ii)	..........................................................
(b)	.................................................................	
(Total 6 marks)

56.	Consider functions of the form y = e–kx
(a)	Show that  =  (1 – e–k).
(3)
 
(b)	Let k = 0.5
(i)	Sketch the graph of y = e–0.5x, for –1 £ x £ 3, indicating the coordinates of the
y-intercept.
(ii)	Shade the region enclosed by this graph, the x-axis, y-axis and the line x = 1.
(iii)	Find the area of this region.
(5)
 
(c)	(i)	Find  in terms of k, where y = e–kx.
	The point P(1, 0.8) lies on the graph of the function y = e–kx.
(ii)	Find the value of k in this case.
(iii)	Find the gradient of the tangent to the curve at P.
(5)
(Total 13 marks)

57.	Consider the function f (x) = 2x2 – 8x + 5.
(a)	Express f (x) in the form a (x – p)2 + q, where a, p, q Î .
(b)	Find the minimum value of f (x).
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

58.	Consider the functions f : x  4(x – 1) and g : x  .
(a)	Find g–1.
(b)	Solve the equation ( f ° g–1) (x) = 4.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

59.	The sketch shows part of the graph of y = f (x) which passes through the points A(–1, 3), B(0, 2), C(l, 0), D(2, 1) and E(3, 5).

	A second function is defined by g (x) = 2f (x – 1).
(a)	Calculate g (0), g (1), g (2) and g (3).
(b)	On the same axes, sketch the graph of the function g (x).
Working:		
	Answers:
(a)	..................................................................
	..................................................................	
(Total 6 marks)

60.	The diagram shows part of the graph with equation y = x2 + px + q. The graph cuts the x-axis at –2 and 3.

	Find the value of
(a)	p;
(b)	q.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

61.	The function f is deﬁned by

	Evaluate f –1(5).
Working:		
	Answer:
.......................................................................	
(Total 4 marks)

62.	The following diagram shows the graph of y = f (x). It has minimum and maximum points at 
(0, 0) and ().

(a)	On the same diagram, draw the graph of .
(b)	What are the coordinates of the minimum and maximum points of
?
Working:		
	Answer:
(b)  ................................................................	
(Total 4 marks)

63.	The diagram shows parts of the graphs of y = x2 and y = 5 – 3(x – 4)2.

	The graph of y = x2 may be transformed into the graph of y = 5 – 3(x – 4)2 by these transformations.
	A reflection in the line y = 0	followed by
a vertical stretch with scale factor k	followed by
a horizontal translation of p units	followed by
a vertical translation of q units.
	Write down the value of
(a)	k;
(b)	p;
(c)	q.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................
(c)	..................................................................	
(Total 4 marks)

64.	Given that f (x) = 2e3x, find the inverse function f –1(x).
Working:		
	Answer:
.......................................................................	
(Total 4 marks)

65.	The diagram shows the graph of y = f (x), with the x-axis as an asymptote.

(a)	On the same axes, draw the graph of y =f (x + 2) – 3, indicating the coordinates of the images of the points A and B.

(b)	Write down the equation of the asymptote to the graph of y = f (x + 2) – 3.
Working:		
	Answer:
(b)  ....................................................................	
(Total 4 marks)

66.	Two functions f, g are defined as follows:
	f : x ® 3x + 5
g : x ® 2(1 – x)
	Find
(a)	f –1(2);
(b)	(g  f )(–4).
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

67.	The quadratic equation 4x2 + 4kx + 9 = 0, k > 0 has exactly one solution for x. 
Find the value of k.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

68.	The diagram shows three graphs.

	A is part of the graph of y = x.
	B is part of the graph of y = 2x.
	C is the reflection of graph B in line A.

	Write down
(a)	the equation of C in the form y =f (x);
(b)	the coordinates of the point where C cuts the x-axis.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

69.	The diagrams show how the graph of y = x2 is transformed to the graph of y = f (x) in three steps.
	For each diagram give the equation of the curve.

Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................
(c)	..................................................................	
(Total 4 marks)

70.	The diagram shows the graph of the function y = ax2 + bx + c.

	Complete the table below to show whether each expression is positive, negative or zero.
Expression	positive	negative	zero	
a				
c				
b2 – 4ac				
b				

Working:		
(Total 4 marks)

71.	(a)	Factorize x2 – 3x – 10.
(b)	Solve the equation x2 – 3x – 10 = 0.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

72.	The diagram represents the graph of the function
f : x  (x – p)(x – q).

(a)	Write down the values of p and q.

(b)	The function has a minimum value at the point C. Find the x-coordinate of C.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

73.	The graph represents the function
f : x  p cos x, p Î .

	Find
(a)	the value of p;
(b)	the area of the shaded region.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

74.	Two functions f and g are defined as follows:
f (x) = cos x,	0 £ x £ 2p;
g (x) = 2x + 1,	x Î .
	Solve the equation (g  f)(x) = 0.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

75.	The function f is given by
f (x) = , x Î , x ¹ 3.
(a)	(i)	Show that y = 2 is an asymptote of the graph of y = f (x).
(2)
(ii)	Find the vertical asymptote of the graph.
(1)
(iii)	Write down the coordinates of the point P at which the asymptotes intersect.
(1)
 
(b)	Find the points of intersection of the graph and the axes.
(4)
(c)	Hence sketch the graph of y = f (x), showing the asymptotes by dotted lines.
(4)

(d)	Show that f¢ (x) =  and hence find the equation of the tangent at 
the point S where x = 4.
(6)
 
(e)	The tangent at the point T on the graph is parallel to the tangent at S.
	Find the coordinates of T.
(5)
(f)	Show that P is the midpoint of [ST].
(l)
(Total 24 marks)

76.	The diagram shows the parabola y = (7 – x)(l + x). The points A and C are the x-intercepts and the point B is the maximum point.

	Find the coordinates of A, B and C.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

77.	Three of the following diagrams I, II, III, IV represent the graphs of
(a)	y = 3 + cos 2x
(b)	y = 3 cos (x + 2)
(c)	y = 2 cos x + 3.

	Identify which diagram represents which graph.

Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................
(c)	..................................................................	
(Total 4 marks)

78.	The function f is given by f (x) =  Find the domain of the function.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

79.	A population of bacteria is growing at the rate of 2.3% per minute. How long will it take for the size of the population to double? Give your answer to the nearest minute.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

80.	Let f (x) = , and g (x) = 2x. Solve the equation
(f –1  g)(x) = 0.25.
Working:		
	Answer:
......................................................................	
(Total 4 marks)