1.	Let f(x) = log3  + log3 16 – log3 4, for x > 0.
(a)	Show 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)	Write 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 a scale of 1 cm to 1 unit on each axis.
 
(iii)	Write down the equation of the asymptote.
(6)
 
(d)	Write down the value of f–1(0).
(1)
 
	The point A lies on the graph of f. At A, x = 4.5.
(e)	On your diagram, sketch the graph of f–1, noting clearly the image of point A.
(4)
(Total 16 marks)

2.	Consider f(x) = 2 – x2, for –2 ≤ x ≤ 2 and g(x) = sin ex, 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)

3.	Let f(x) = ex 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 local minimum 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 . Explain why this is not the area of the shaded region.
(3)
(Total 6 marks)

4.	Consider f(x) = x ln(4 – x2), 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 to the 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 two solutions.
(5)
 
	Let g(x) = x3 ln(4 – x2), for –2 < x < 2.
(b)	Show that g′(x) = .
(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)

5.	Consider the function f(x) = px3 + qx2 + 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)

6.	Let f(x) = ax2 + 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 linear equations in a, b and c.
(2)

(c)	These three equations may be written as a matrix equation in the form AX = B,
where X = .
(i)	Write down the matrices A and B.
(ii)	Write down A–1.
(iii)	Hence or otherwise, find f(x).
(8)
 
(d)	Write f(x) in the form f(x) = a(x – h)2 + k, where a, h and k are rational numbers.
(3)
(Total 15 marks)

7.	Solve the equation ex = 4 sin x, for 0 ≤ x ≤ 2π.
(Total 5 marks)

8.	Let f(x) =  and g(x) = –0.5x2 + 5x – 8, for 0 ≤ x ≤ 9.
(a)	On 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)

9.	A city is concerned about pollution, and decides to look at the number of people using taxis. At the end of the year 2000, there were 280 taxis in the city. After n years the number of taxis, T, in the city is given by
		T = 280 ´ 1.12n.
(a)	(i)	Find the number of taxis in the city at the end of 2005.
(ii)	Find the year in which the number of taxis is double the number of taxis there were at 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 the number of people, P, in the city who used taxis is given by
		P = 
(i)	Find the value of P at the end of 2005, giving your answer to the nearest whole number.
(ii)	After seven complete years, will the value of P be double its value at the end of 2000? 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)

10.	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)

11.	The function f is defined by f(x) = , for –3 < x < 3.
(a)	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)

12.	Let f (x) = ln (x + 2), x > −2 and g (x) = e(x−4), x > 0.
(a)	Write down the x-intercept of the graph of f.
(b)	(i)	Write down f (−1.999).
(ii)	Find the range of f.
(c)	Find the coordinates of the point of intersection of the graphs of f and g.
(Total 6 marks)

13.	The function f (x) is defined as f (x) = 3 + , x  .
(a)	Sketch the curve of f for −5 £ x £ 5, showing the asymptotes.
(3)
 
(b)	Using your sketch, write down
(i)	the equation of each asymptote;
(ii)	the value of the x-intercept;
(iii)	the value of the y-intercept.
(4)
 
(c)	The region enclosed by the curve of f, the x-axis, and the lines x = 3 and x = a, is revolved through 360° about the x-axis. Let V be the volume of the solid formed.
(i)	Find dx.
(ii)	Hence, given that V = , find the value of a.
(10)
(Total 17 marks)

14.	Let f (x) = , where p, qÎ +.
Part of the graph of f, including the asymptotes, is shown below.

(a)	The equations of the asymptotes are x =1, x = −1, y = 2. Write down the value of
(i)	p;
(ii)	q.
(2)
 
(b)	Let R be the region bounded by the graph of f, the x-axis, and the y-axis.
(i)	Find the negative x-intercept of f.
(ii)	Hence find the volume obtained when R is revolved through 360° about the x-axis.
(7)
 
(c)	(i)	Show that f ′ (x) = .
(ii)	Hence, show that there are no maximum or minimum points on the graph of f.
(8)
 
(d)	Let g (x) = f ′ (x). Let A be the area of the region enclosed by the graph of g and the x-axis, between x = 0 and x = a, where a > 0. Given that A = 2, find the value of a.
(7)
(Total 24 marks)

15.	Consider the function f (x) e(2x–1) + , x  .
(a)	Sketch the curve of f for −2 £ x £ 2, including any asymptotes.
(3)
 
(b)	(i)	Write down the equation of the vertical asymptote of f.
(ii)	Write down which one of the following expressions does not represent an area between the curve of f and the x-axis.
		 f (x)dx
		 f (x)dx
(iii)	Justify your answer.
(3)
 
(c)	The region between the curve and the x-axis between x = 1 and x = 1.5 is rotated through 360° about the x-axis. Let V be the volume formed.
(i)	Write down an expression to represent V.
(ii)	Hence write down the value of V.
(4)
 
(d)	Find f ′ (x).
(4)

(e)	(i)	Write down the value of x at the minimum point on the curve of f.
(ii)	The equation f (x) = k has no solutions for p £ k < q. Write down the value of p and of q.
(3)
(Total 17 marks)

16.	Let ƒ (x) = 3 sin 2x, for 1 £ x £ 4 and g (x) = −5x2 + 27x − 35 for 1 £ x £ 4. The graph of ƒ is shown below.

(a)	On the same diagram, sketch the graph of g.
 
(b)	One solution of ƒ (x) = g (x) is 1.89. Write down the other solution.
 
(c)	Let h (x) = g (x) − ƒ (x). Given that h (x) > 0 for p < x < q, write down the value of p and of q.
(Total 6 marks)

17.	The function f is given by f (x) = mx3 + nx2 + px + q, where m, n, p, q are integers.
The graph of f passes through the point (0, 0).
(a)	Write down the value of q.
(1)
 
The graph of f also passes through the point (3, 18).
(b)	Show that 27 m+ 9n + 3p =18.
The graph of f also passes through the points (1, 0) and (–1, –10).
(2)
 
(c)	Write down the other two linear equations in m, n and p.
(2)
 
(d)	(i)	Write down these three equations as a matrix equation.
(ii)	Solve this matrix equation.
(6)
 
(e)	The function f can also be written f (x) = x (x −1)(rx − s) where r and s are integers. Find r and s.
(3)
(Total 14 marks)

18.	The function f is defined as f (x) = (2x +1) e−x, 0 £ x £ 3. The point P(0, 1) lies on the graph of f (x), and there is a maximum point at Q.
(a)	Sketch the graph of y = f (x), labelling the points P and Q.
(3)
 
(b)	(i)	Show that f ′ (x) = (1− 2x) e−x.
(ii)	Find the exact coordinates of Q.
(7)
 
(c)	The equation f (x) = k, where k Î , has two solutions. Write down the range of values of k.
(2)
 
(d)	Given that f ²(x) = e−x (−3 + 2x), show that the curve of f has only one point of inflexion.
(2)
 
(e)	Let R be the point on the curve of f with x-coordinate 3. Find the area of the region enclosed by the curve and the line (PR).
(7)
(Total 21 marks)

19.	The function f is defined by  for –3 < x < 3.
(a)	On the grid below, sketch the graph of f.

(b)	Write down the equation of each vertical asymptote.
(c)	Write down the range of the function f.
(Total 6 marks)

20.	Let f (x) = 6 sin px , and g (x) = 6e–x – 3 , for 0 £ x £ 2. The graph of f is shown on the diagram below. There is a maximum value at B (0.5, b).

(a)	Write down the value of b.
(b)	On the same diagram, sketch the graph of g.
(c)	Solve f (x) = g (x) , 0.5 £ x £ 1.5.

(Total 6 marks)

21.	Consider the graph of the function, f , defined by
f (x) = 3x4 – 4x3 – 30x2 – 36x + 112, – 2 £ x £ 4.5.
(a)	Given that f (x) = 0 has one solution at x = 4, find the other solution.
(2)
 
(b)	The tangent to the graph of f is horizontal at x = 3 and at one other value of x. Find this other value.
(3)
 
(c)	Find the x-coordinates of both points of inflexion on the graph of f .
(4)
 
(d)	Write down both coordinates of the point of inflexion on the graph of f where the tangent is horizontal.
(2)
A sketch of the graph of  is given below.

(e)	Write down the equations of the two vertical asymptotes.
(2)

(f)	The tangent to the graph of  is horizontal at P. Write down the x-coordinate of P.
(2)
(Total 15 marks)

22.	Let f (x) = 2 + cos (2x) – 2 sin (0.5x) for 0 £ x £ 3, where x is in radians.
(a)	On the grid below, sketch the curve of y = f (x), indicating clearly the point P on the curve where the derivative is zero.

(b)	Write down the solutions of f (x) = 0.
Working:		
	Answer:
(b)	…………………………………………..	
(Total 6 marks)

23.	Let h (x) = (x – 2) sin (x – 1) for –5 £ x £ 5. The curve of h (x) is shown below. There is a minimum point at R and a maximum point at S. The curve intersects the x-axis at the points (a, 0) (1, 0) (2, 0) and (b, 0).

(a)	Find the exact value of
(i)	a;
(ii)	b.
(2)
 
	The regions between the curve and the x-axis are shaded for a £ x £ 2 as shown.
(b)	(i)	Write down an expression which represents the total area of the shaded regions.
(ii)	Calculate this total area.
(5)
 
(c)	(i)	The y-coordinate of R is –0.240. Find the y-coordinate of S.
(ii)	Hence or otherwise, find the range of values of k for which the equation
(x – 2) sin (x – 1) = k has four distinct solutions.
(4)
(Total 11 marks)

24.	Solve the equation ex = 5 – 2x, giving your answer correct to four significant figures.
Working:		
	Answer:
......................................................................	
(Total 6 marks)

25.	The diagram below shows a sketch of the graph of the function y = sin (ex) where –1 £ x £ 2, and x is in radians. The graph cuts the y-axis at A, and the x-axis at C and D. It has a maximum point at B.

(a)	Find the coordinates of A.
(2)
 
(b)	The coordinates of C may be written as (ln k, 0). Find the exact value of k.
(2)

(c)	(i)	Write down the y-coordinate of B.
(ii)	Find .
(iii)	Hence, show that at B, x = ln .
(6)
 
(d)	(i)	Write down the integral which represents the shaded area.
(ii)	Evaluate this integral.
(5)
 
(e)	(i)	Copy the above diagram into your answer booklet. (There is no need to copy the shading.) On your diagram, sketch the graph of y = x3.
(ii)	The two graphs intersect at the point P. Find the x-coordinate of P.
(3)
(Total 18 marks)

26.	Let g (x) = x4 – 2x3 + x2 – 2.
(a)	Solve g (x) = 0.
(2)
	Let f (x) = . A part of the graph of f (x) is shown below.

(b)	The graph has vertical asymptotes with equations x = a and x = b where a < b. Write down the values of
(i)	a;
(ii)	b.
(2)
 
(c)	The graph has a horizontal asymptote with equation y = l. Explain why the value of f (x) approaches 1 as x becomes very large.
(2)
 
(d)	The graph intersects the x-axis at the points A and B. Write down the exact value of the x-coordinate at
(i)	A;
(ii)	B.
(2)

(e)	The curve intersects the y-axis at C. Use the graph to explain why the values of f¢ (x) and f¢¢ (x) are zero at C.
(2)
(Total 10 marks)

27.	(a)	On the following diagram, sketch the graphs of y = ex and y = cos x
for –2 £ x £ 1.

(b)	The equation ex = cos x has a solution between –2 and –1.
	Find this solution.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

28.	Note: Radians are used throughout this question.
	Let f (x) = sin (1 + sin x).
(a)	(i)	Sketch the graph of y = f (x), for 0 £ x £ 6.
(ii)	Write down the x-coordinates of all minimum and maximum points of f, for 
0 £ x £ 6. Give your answers correct to four signiﬁcant ﬁgures.
(9)
 
(b)	Let S be the region in the ﬁrst quadrant completely enclosed by the graph of f and both coordinate axes.
(i)	Shade S on your diagram.
(ii)	Write down the integral which represents the area of S.
(iii)	Evaluate the area of S to four signiﬁcant ﬁgures.
(5)
 
(c)	Give reasons why f (x) ≥ 0 for all values of x.
(2)
(Total 16 marks)

29.	The diagram below shows the graph of y = x sin , for 0 £ x < m, and 0 £ y < n, where x is in radians and m and n are integers.

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

30.	(a)	Sketch, on the given axes, the graphs of y = x2 and y – sin x for –1 £ x £ 2.

(b)	Find the positive solution of the equation
x2 = sin x,
giving your answer correct to 6 significant figures.
Working:		
	Answer:
(b)  ....................................................................	
(Total 4 marks)

31.	Let f(x) = 3x2. 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)	Express g in the form g(x) = 3(x – p)2 + 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)

32.	Let f(x) = Aekx + 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)
 
	Let g(x) = –x2 + 12x – 24.
(d)	Find the area enclosed by the graphs of f and g.
(6)
(Total 16 marks)

33.	The number of bacteria, n, in a dish, after t minutes is given by n = 800e013t.
(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. Find the least value of k, where k  .
(4)
(Total 8 marks)

 for –3 < x < 3.
(a)	On the grid below, sketch the graph of f.
34.	Let f(x) =  +1, g(x) = 4cos – 1. Let h(x) = (g ° f)(x).
(a)	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)

35.	The quadratic equation kx2 + (k – 3)x + 1 = 0 has two equal real roots.
(a)	Find the possible values of k.
(5)
 
(b)	Write down the values of k for which x2 + (k – 3)x + k = 0 has two equal real roots.
(2)
(Total 7 marks)

 
(b)	Write down the equation of each vertical asymptote.
(c)	Write down the range of the function f.
(Total 6 marks)

36.	Let f (x) = 3x – ex–2 – 4, for –1 £ x £ 5.
(a)	Find the x-intercepts of the graph of f.
(3)
 
(b)	On the grid below, sketch the graph of f.

(3)

(c)	Write down the gradient of the graph of f at x = 2.
(1)
(Total 7 marks)

37.	Let f(x) = 2x2 + 4x – 6.
(a)	Express f(x) in the form f(x) = 2(x – h)2 + 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)

38.	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)	Find f–1(18) + g–1(18).
(4)
(Total 6 marks)

39.	(a)	Given that (2x)2 + (2x) −12 can be written as (2x + a)(2x + b), where a, b Î , find the value of a and of b.
(b)	Hence find the exact solution of the equation (2x)2 + (2x) −12 = 0, and explain why there is only one solution.
(Total 6 marks)

40.	The population of a city at the end of 1972 was 250 000. The population increases by 1.3% per year.
(a)	Write down the population at the end of 1973.
(b)	Find the population at the end of 2002.
(Total 6 marks)

41.	The following diagram shows part of the graph of a quadratic function, with equation in the form y = (x − p)(x − q), where p, q Î .

(a)	Write down
(i)	the value of p and of q;
(ii)	the equation of the axis of symmetry of the curve.
(3)
 
(b)	Find the equation of the function in the form y = (x − h)2 + k, where h, k Î .
(3)
 
(c)	Find .
(2)

(d)	Let T be the tangent to the curve at the point (0, 5). Find the equation of T.
(2)
(Total 10 marks)

42.	Let f (x) = loga x, x > 0.
(a)	Write down the value of
(i)	f (a);
(ii)	f (1);
(iii)	f (a4 ).
(3)
 
(b)	The diagram below shows part of the graph of f.

On the same diagram, sketch the graph of f−1.
(3)
(Total 6 marks)

43.	The following diagram shows part of the graph of f (x) = 5 − x2 with vertex V (0, 5).
Its image y = g (x) after a translation with vector  has vertex T (3, 6).

(a)	Write down the value of
(i)	h;
(ii)	k.
(2)
 
(b)	Write down an expression for g (x).
(2)
 
(c)	On the same diagram, sketch the graph of y = g (−x).
(2)
(Total 6 marks)

44.	The area A km2 affected by a forest fire at time t hours is given by A = A0 ekt.
When t = 5, the area affected is 1 km2 and the rate of change of the area is 0.2 km2 h−1.
(a)	Show that k = 0.2.
(4)
(b)	Given that A0 = , find the value of t when 100 km2 are affected.
(2)
(Total 6 marks)

45.	(a)	Consider the equation 4x2 + kx + 1 = 0. For what values of k does this equation have two equal roots?
(3)
 
Let f be the function f (q ) = 2 cos 2q + 4 cos q + 3, for −360° £ q £ 360°.
(b)	Show that this function may be written as f (q ) = 4 cos2 q + 4 cos q + 1.
(1)
 
(c)	Consider the equation f (q ) = 0, for −360° £ q £ 360°.
(i)	How many distinct values of cos q satisfy this equation?
(ii)	Find all values of q which satisfy this equation.
(5)
 
(d)	Given that f (q ) = c is satisfied by only three values of q, find the value of c.
(2)
(Total 11 marks)

46.	Let f (x) = a (x − 4)2 + 8.
(a)	Write down the coordinates of the vertex of the curve of f.
(b)	Given that f (7) = −10, find the value of a.
(c)	Hence find the y-intercept of the curve of f.
(Total 6 marks)

47.	Solve the following equations.
(a)	ln (x + 2) = 3.
(b)	102x = 500.
(Total 6 marks)

48.	(a)	Express y = 2x2 – 12x + 23 in the form y = 2(x – c)2 + d.
The graph of y = x2 is transformed into the graph of y = 2x2 – 12x + 23 by the transformations
		a vertical stretch with scale factor k followed by
		a horizontal translation of p units followed by
		a vertical translation of q units.
(b)	Write down the value of
(i)	k;
(ii)	p;
(iii)	q.
(Total 6 marks)

49.	Consider the functions f and g where f (x) = 3x – 5 and g (x) = x – 2.
(a)	Find the inverse function, f −1.
(3)

(b)	Given that g–1 (x) = x + 2, find (g–1 ◦ f) (x).
(2)
 
(c)	Given also that (f −1 ◦ g) (x) , solve (f −1 ◦ g) (x) = (g–1 ◦ f) (x).
(2)
 
Let h (x) = , x  2.
(d)	(i)	Sketch the graph of h for −3 £ x £ 7 and −2 £ y £ 8, including any asymptotes.
(ii)	Write down the equations of the asymptotes.
(5)
 
(e)	The expression  may also be written as 3 + . Use this to answer the following.
(i)	Find dx.
(ii)	Hence, calculate the exact value of  h (x)dx.
(5)
(f)	On your sketch, shade the region whose area is represented by  h (x)dx.
(1)
(Total 18 marks)

50.	(a)	Let y = –16x2 + 160x –256. Given that y has a maximum value, find
(i)	the value of x giving the maximum value of y;
(ii)	this maximum value of y.
	The triangle XYZ has XZ = 6, YZ = x, XY = z as shown below. The perimeter of triangle XYZ is 16.
(4)

(b)	(i)	Express z in terms of x.
(ii)	Using the cosine rule, express z2 in terms of x and cos Z.
(iii)	Hence, show that cos Z = .
(7)
Let the area of triangle XYZ be A.
 
(c)	Show that A2 = 9x2 sin2 Z.
(2)
 
(d)	Hence, show that A2 = –16x2 + 160x – 256.
(4)

(e)	(i)	Hence, write down the maximum area for triangle XYZ.
(ii)	What type of triangle is the triangle with maximum area?
(3)
(Total 20 marks)

51.	Find the exact value of x in each of the following equations.
(a)	5x+1 = 625
 
(b)	loga (3x + 5) = 2
(Total 6 marks)

52.	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.
(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.
(Total 6 marks)

53.	The diagram below shows the graphs of f (x) = 1 + e2x, g (x) = 10x + 2, 0 £ x £ 1.5.

(a)	(i)	Write down an expression for the vertical distance p between the graphs of f and g.
(ii)	Given that p has a maximum value for 0 £ x £ 1.5, find the value of x at which this occurs.
(6)
 
	The graph of y = f (x) only is shown in the diagram below. When x = a, y = 5.

(b)	(i)	Find f –1(x).
(ii)	Hence show that a = ln 2.
(5)
 
(c)	The region shaded in the diagram is rotated through 360° about the x-axis. Write down an expression for the volume obtained.
(3)
(Total 14 marks)

54.	The equation of a curve may be written in the form y = a(x – p)(x – q). The curve intersects the x-axis at A(–2, 0) and B(4, 0). The curve of y = f (x) is shown in the diagram below.

(a)	(i)	Write down the value of p and of q.
(ii)	Given that the point (6, 8) is on the curve, find the value of a.
(iii)	Write the equation of the curve in the form y = ax2 + bx + c.
(5)
 
(b)	(i)	Find .
(ii)	A tangent is drawn to the curve at a point P. The gradient of this tangent is 7.
Find the coordinates of P.
(4)
 
(c)	The line L passes through B(4, 0), and is perpendicular to the tangent to the curve at point B.
(i)	Find the equation of L.
(ii)	Find the x-coordinate of the point where L intersects the curve again.
(6)
(Total 15 marks)

55.	A machine was purchased for $10000. Its value V after t years is given by V =100000e−0.3t. The machine must be replaced at the end of the year in which its value drops below $1500. Determine in how many years the machine will need to be replaced.

(Total 6 marks)

56.	Two weeks after its birth, an animal weighed 13 kg. At 10 weeks this animal weighed 53 kg. The increase in weight each week is constant.
(a)	Show that the relation between y, the weight in kg, and x, the time in weeks, can be written as y = 5x + 3
(2)
(b)	Write down the weight of the animal at birth.
(1)
(c)	Write down the weekly increase in weight of the animal.
(1)
(d)	Calculate how many weeks it will take for the animal to reach 98 kg.
(2)
(Total 6 marks)

57.	Consider the function f (x) =  + 8, x  10.
(a)	Write down the equation of
(i)	the vertical asymptote;
(ii)	the horizontal asymptote.
(2)
 
(b)	Find the
(i)	y-intercept;
(ii)	x-intercept.
(2)
 
(c)	Sketch the graph of f , clearly showing the above information.
(4)
 
(d)	Let g (x) = , x  0.
The graph of g is transformed into the graph of f using two transformations.
	The first is a translation with vector  Give a full geometric description of the second transformation.
(2)
(Total 10 marks)

58.	Let f (x) = 2x + 1.
(a)	On the grid below draw the graph of f (x) for 0 £ x £ 2.
(b)	Let g (x) = f (x +3) –2. On the grid below draw the graph of g (x) for –3 £ x £ –1.

Working:		
(Total 6 marks)

59.	There were 1420 doctors working in a city on 1 January 1994. After n years the number of doctors, D, working in the city is given by
D = 1420 + 100n.
(a)	(i)	How many doctors were there working in the city at the start of 2004?
(ii)	In what year were there first more than 2000 doctors working in the city?
(3)
 
	At the beginning of 1994 the city had a population of 1.2 million. After n years, the population, P, of the city is given by
P = 1 200 000 (1.025)n.
(b)	(i)	Find the population P at the beginning of 2004.
(ii)	Calculate the percentage growth in population between 1 January 1994 and 1 January 2004.
(iii)	In what year will the population first become greater than 2 million?
(7)
 
(c)	(i)	What was the average number of people per doctor at the beginning of 1994?
(ii)	After how many complete years will the number of people per doctor first fall below 600?
(5)
(Total 15 marks)

 
60.	The population p of bacteria at time t is given by p = 100e0.05t.
	Calculate
(a)	the value of p when t = 0;
(b)	the rate of increase of the population when t = 10.
Working:		
	Answers:
(a)	…………………………………………..
(b)	…………………………………………..	
(Total 6 marks)

61.	Part of the graph of the periodic function f  is shown below. The domain of f  is 0 £ x £ 15 and the period is 3.

(a)	Find
(i)	f (2);
(ii)	f ¢ (6.5);
(iii)	f ¢ (14).
(b)	How many solutions are there to the equation f (x) = 1 over the given domain?
Working:		
	Answers:
(a)	(i)   ………………………………………
	(ii)  ………………………………………
	(iii) ………………………………………
(b)  ……………………………………………	
(Total 6 marks)

62.	Let f (x) = 1 + 3 cos (2x) for 0 £ x £ π, and x is in radians.
(a)	(i)	Find f¢ (x).
(ii)	Find the values for x for which f¢ (x) = 0, giving your answers in terms of p.
(6)
 
	The function g (x) is defined as g (x) = f (2x) – 1, 0 £ x £ .
(b)	(i)	The graph of f may be transformed to the graph of g by a stretch in the x-direction with scale factor  followed by another transformation. Describe fully this other transformation.
(ii)	Find the solution to the equation g (x) = f (x)
(4)
(Total 10 marks)

63.	A family of functions is given by
f (x) = x2 + 3x + k, where k Î {1, 2, 3, 4, 5, 6, 7}.
	One of these functions is chosen at random. Calculate the probability that the curve of this function crosses the x-axis.
Working:		
	Answer:
..................................................................	
(Total 6 marks)

64.	The diagram shows part of the graph of the curve y = a (x – h)2 + k, where a, h, k Î .
.
(a)	The vertex is at the point (3, 1). Write down the value of h and of k.
(2)
(b)	The point P (5, 9) is on the graph. Show that a = 2.
(3)
(c)	Hence show that the equation of the curve can be written as
y = 2x2 – 12x + 19.
(1)
(d)	(i)	Find 
	A tangent is drawn to the curve at P (5, 9).
(ii)	Calculate the gradient of this tangent.
(iii)	Find the equation of this tangent.
(4)
(Total 10 marks)

65.	(a)	The diagram shows part of the graph of the function f (x) =  The curve passes through the point A (3, 10). The line (CD) is an asymptote.

	Find the value of
(i)	p;
(ii)	q.

(b)	The graph of f (x) is transformed as shown in the following diagram. The point A is transformed to A¢ (3, –10).

	Give a full geometric description of the transformation.
Working:		
	Answers:
(a)	(i)	...........................................................
	(ii)	...........................................................
(b)	..................................................................
	..................................................................	
(Total 6 marks)

66.	The mass m kg of a radio-active substance at time t hours is given by
	m = 4e–0.2t.
(a)	Write down the initial mass.
(b)	The mass is reduced to 1.5 kg. How long does this take?
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

67.	Let f (x) = 2x, and g (x) = , (x ¹ 2).
	Find
(a)	(g  f ) (3);
(b)	g–1 (5).
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

68.	$1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum number of months required for the value of the investment to exceed $3000.
Working:		
	Answer:
......................................................................	
(Total 6 marks)

69.	Each year for the past ﬁve years the population of a certain country has increased at a steady rate of 2.7% per annum. The present population is 15.2 million.
(a)	What was the population one year ago?
(b)	What was the population ﬁve years ago?
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

70.	Michele invested 1500 francs at an annual rate of interest of 5.25 percent,
compounded annually. 
(a)	Find the value of Michele’s investment after 3 years. Give your answer to the nearest franc.
(3)
(b)	How many complete years will it take for Michele’s initial investment to double in value?
(3)
(c)	What should the interest rate be if Michele’s initial investment were to double in value in 10 years?
(4)
(Total 10 marks)

71.	In the diagram below, the points O(0, 0) and A(8, 6) are ﬁxed. The angle 
varies as the point P(x, 10) moves along the horizontal line y = 10.

Diagram to scale
 
(a)	(i)	Show that 
(ii)	Write down a similar expression for OP in terms of x.
(2)

(b)	Hence, show that

(3)

(c)	Find, in degrees, the angle  when x = 8.
(2)
 
(d)	Find the positive value of x such that .
(4)
	Let the function f be deﬁned by

(e)	Consider the equation f (x) = 1.
(i)	Explain, in terms of the position of the points O, A, and P, why this
equation has a solution.
(ii)	Find the exact solution to the equation.
(5)
(Total 16 marks)

72.	The diagram below shows part of the graph of the function

	The graph intercepts the x-axis at A(–3, 0), B(5, 0) and the origin, O. There is a minimum point at P and a maximum point at Q.
 
(a)	The function may also be written in the form 
where a < b. Write down the value of
(i)	a;
(ii)	b.
(2)
 
(b)	Find
(i)	f ¢(x);
(ii)	the exact values of x at which f '(x) = 0;
(iii)	the value of the function at Q.
(7)

(c)	(i)	Find the equation of the tangent to the graph of f at O.
(ii)	This tangent cuts the graph of f at another point. Give the x-coordinate of this point.
(4)
(d)	Determine the area of the shaded region.
(2)
(Total 15 marks)

73.	(a)	Express f (x) = x2 – 6x + 14 in the form f (x) = (x – h)2 + k, where h and k are to be determined.
(b)	Hence, or otherwise, write down the coordinates of the vertex of the parabola with equation y – x2 – 6x + 14.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

74.	A group of ten leopards is introduced into a game park. After t years the number of leopards, N, is modelled by N = 10 e0.4t.
(a)	How many leopards are there after 2 years?
(b)	How long will it take for the number of leopards to reach 100? Give your answers to an appropriate degree of accuracy.
Give your answers to an appropriate degree of accuracy.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

75.	Consider the function 
(a)	Determine the inverse function f –1.
(b)	What is the domain of f –1?
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 4 marks)

76.	A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t seconds is given by
h = 2 + 20t – 5t2, t ³ 0
(a)	Find the initial height above the ground of the ball (that is, its height at the instant when it is released).
(2)
 
(b)	Show that the height of the ball after one second is 17 metres.
(2)

(c)	At a later time the ball is again at a height of 17 metres.
(i)	Write down an equation that t must satisfy when the ball is at a height of 17 metres.
(ii)	Solve the equation algebraically.
(4)
 
(d)	(i)	Find .
(ii)	Find the initial velocity of the ball (that is, its velocity at the instant when it is released).
(iii)	Find when the ball reaches its maximum height.
(iv)	Find the maximum height of the ball.
(7)
(Total 15 marks)

77.	f (x) = 4 sin.
	For what values of k will the equation f (x) = k have no solutions?
Working:		
	Answer:
......................................................................	
(Total 4 marks)

78.	Initially a tank contains 10 000 litres of liquid. At the time t = 0 minutes a tap is opened, and liquid then flows out of the tank. The volume of liquid, V litres, which remains in the tank after t minutes is given by
V = 10 000 (0.933t).
 
(a)	Find the value of V after 5 minutes.
(1)
 
(b)	Find how long, to the nearest second, it takes for half of the initial amount of liquid to flow out of the tank.
(3)
 
(c)	The tank is regarded as effectively empty when 95% of the liquid has flowed out. 
Show that it takes almost three-quarters of an hour for this to happen.
(3)
 
(d)	(i)	Find the value of 10 000 – V when t = 0.001 minutes.
(ii)	Hence or otherwise, estimate the initial flow rate of the liquid. 
Give your answer in litres per minute, correct to two significant figures.
(3)
(Total 10 marks)