1.	Let f(x) = cos(x2) and g(x) = ex, for –1.5 ≤ x ≤ 0.5.
	Find the area of the region enclosed by the graphs of f and g.
(Total 6 marks)

2.	The following diagram shows a waterwheel with a bucket. The wheel rotates at a constant rate in an anticlockwise (counterclockwise) direction.
		
diagram not to scale
	The diameter of the wheel is 8 metres. The centre of the wheel, A, is 2 metres above the water level. After t seconds, the height of the bucket above the water level is given by h = a sin bt + 2.
 
(a)	Show that a = 4.
(2)
 
	The wheel turns at a rate of one rotation every 30 seconds.
(b)	Show that b = .
(2)
 
	In the first rotation, there are two values of t when the bucket is descending at a rate of
0.5 m s–1.
(c)	Find these values of t.
(6)

(d)	Determine whether the bucket is underwater at the second value of t.
(4)
(Total 14 marks)

3.	A gradient function is given by . When x = 0, y = 8. Find the value of y when
x = 1.
(Total 8 marks)

4.	Let f(x) = x cos x, for 0 ≤ x ≤ 6.
(a)	Find f′(x).
(3)

(b)	On the grid below, sketch the graph of y = f′(x).
		
(4)
(Total 7 marks)

5.	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)

6.	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)

7.	The velocity v m s–1 of an object after t seconds is given by v(t) = , for 0 ≤ t ≤ 25.
(a)	On the grid below, sketch the graph of v, clearly indicating the maximum point.
		
(3)
 
	Let d be the distance travelled in the first nine seconds.
(b)	(i)	Write down an expression for d.
(ii)	Hence, write down the value of d.
(4)
(Total 7 marks)

8.	Let f′(x) = –24x3 + 9x2 + 3x + 1.
(a)	There are two points of inflexion on the graph of f. Write down the x-coordinates of these points.
(3)
 
(b)	Let g(x) = f″(x). Explain why the graph of g has no points of inflexion.
(2)
(Total 5 marks)

9.	Let f(x) = x ln(4 – x2), 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. Find the 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 and x = c.
	Find the area of R.
(4)
(Total 12 marks)

10.	In this question s represents displacement in metres and t represents time in seconds.
	The velocity v m s–1 of a moving body is given by v = 40 – at where a is a non-zero constant.
(a)	(i)	If s = 100 when t = 0, find an expression for s in terms of a and t.
(ii)	If s = 0 when t = 0, write down an expression for s in terms of a and t.
(6)
 
	Trains approaching a station start to slow down when they pass a point P. As a train slows down, its velocity is given by v = 40 – at, where t = 0 at P. The station is 500 m from P.
(b)	A train M slows down so that it comes to a stop at the station.
(i)	Find the time it takes train M to come to a stop, giving your answer in terms of a.
(ii)	Hence show that a = .
(6)
 
(c)	For a different train N, the value of a is 4.
Show that this train will stop before it reaches the station.
(5)
(Total 17 marks)

11.	Let f(x) = x(x – 5)2, for 0 ≤ x ≤ 6. The following diagram shows the graph of f.
		
	Let R be the region enclosed by the x-axis and the curve of f.

(a)	Find the area of R.
(3)
 
(b)	Find the volume of the solid formed when R is rotated through 360° about the x-axis.
(4)
 
(c)	The diagram below shows a part of the graph of a quadratic function g(x) = x(a – x). The graph of g crosses the x-axis when x = a.
		
	The area of the shaded region is equal to the area of R. Find the value of a.
(7)
(Total 14 marks)

12.	Let f(x) = 3sinx + 4 cos x, for –2π ≤ x ≤ 2π.
(a)	Sketch the graph of f.
(3)

(b)	Write down
(i)	the amplitude;
(ii)	the period;
(iii)	the x-intercept that lies between  and 0.
(3)
 
(c)	Hence write f(x) in the form p sin (qx + r).
(3)
 
(d)	Write down one value of x such that f′(x) = 0.
(2)
 
(e)	Write down the two values of k for which the equation f(x) = k has exactly two solutions.
(2)
 
(f)	Let g(x) = ln(x + 1), for 0 ≤ x ≤ π. There is a value of x, between 0 and 1, for which the gradient of f is equal to the gradient of g. Find this value of x.
(5)
(Total 18 marks)

13.	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)

14.	Let f (x) = ex (1 – x2).
(a)	Show that f ′ (x) = ex (1 – 2x – x2).
(3)
 
	Part of the graph of y = f (x), for – 6 £ x £ 2, is shown below. The x-coordinates of the local minimum and maximum points are r and s respectively.

(b)	Write down the equation of the horizontal asymptote.
(1)
 
(c)	Write down the value of r and of s.
(4)

(d)	Let L be the normal to the curve of f at P(0, 1). Show that L has equation x + y = 1.
(4)
 
(e)	Let R be the region enclosed by the curve y = f (x) and the line L.
(i)	Find an expression for the area of R.
(ii)	Calculate the area of R.
(5)
(Total 17 marks)

15.	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)

16.	Let f(x) = e2x cos x, –1 ≤ x ≤ 2.
(a)	Show that f′(x) = e2x (2 cos x – sin x).
(3)
 
	Let the line L be the normal to the curve of f at x = 0.
(b)	Find the equation of L.
(5)
 
	The graph of f and the line L intersect at the point (0, 1) and at a second point P.
(c)	(i)	Find the x-coordinate of P.
(ii)	Find the area of the region enclosed by the graph of f and the line L.
(6)
(Total 14 marks)

17.	The function f is defined as f (x) = ex sin x, where x is in radians. Part of the curve of f is shown below.

	There is a point of inflexion at A, and a local maximum point at B. The curve of f intersects the x-axis at the point C.
(a)	Write down the x-coordinate of the point C.
(1)

(b)	(i)	Find f ¢ (x).
(ii)	Write down the value of f ¢ (x) at the point B.
(4)
 
(c)	Show that f ²(x) = 2ex cos x.
(2)
 
(d)	(i)	Write down the value of f ²(x) at A, the point of inflexion.
(ii)	Hence, calculate the coordinates of A.
(4)
 
(e)	Let R be the region enclosed by the curve and the x-axis, between the origin and C.
(i)	Write down an expression for the area of R.
(ii)	Find the area of R.
(4)
(Total 15 marks)

18.	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)

19.	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)

20.	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)

21.	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)

22.	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)

23.	Consider the function f (x) = cos x + sin x.
(a)	(i)	Show that f (–) = 0.
(ii)	Find in terms of p, the smallest positive value of x which satisfies f (x) = 0.
(3)

	The diagram shows the graph of y = ex (cos x + sin x), – 2 £ x £ 3. The graph has a maximum turning point at C(a, b) and a point of inflexion at D.

(b)	Find .
(3)
 
(c)	Find the exact value of a and of b.
(4)
 
(d)	Show that at D, y = .
(5)
 
(e)	Find the area of the shaded region.
(2)
(Total 17 marks)

24.	Let the function f  be defined by f (x) = , x ¹ –1.
(a)	(i)	Write down the equation of the vertical asymptote of the graph of f .
(ii)	Write down the equation of the horizontal asymptote of the graph of f .
(iii)	Sketch the graph of f in the domain –3 £ x £ 3.
(4)
 
(b)	(i)	Using the fact that f ¢(x) = , show that the second derivative
f² (x) = .
 
(ii)	Find the x-coordinates of the points of inflexion of the graph of f .
(6)
 
(c)	The table below gives some values of f (x) and 2f (x).
x	f (x)	2f (x)	
1	1	2	
1.4	0.534188	1.068376	
1.8	0.292740	0.585480	
2.2	0.171703	0.343407	
2.6	0.107666	0.215332	
3	0.071429	0.142857	
(i)	Use the trapezium rule with five sub-intervals to approximate the integral 

(ii)	Given that = 0.637599, use a diagram to explain why your answer is greater than this.
(5)
(Total 15 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.	Note: Radians are used throughout this question.
(a)	Draw the graph of y = p + x cos x, 0 £ x £ 5, on millimetre square graph paper, using a scale of 2 cm per unit. Make clear
(i)	the integer values of x and y on each axis;
(ii)	the approximate positions of the x-intercepts and the turning points.
(5)
 
(b)	Without the use of a calculator, show that p is a solution of the equation
p + x cos x = 0.
(3)
 
(c)	Find another solution of the equation p + x cos x = 0 for 0 £ x £ 5, giving your answer to six significant figures.
(2)
 
(d)	Let R be the region enclosed by the graph and the axes for 0 £ x £ p. Shade R on your diagram, and write down an integral which represents the area of R .
(2)

(e)	Evaluate the integral in part (d) to an accuracy of six significant figures. (If you consider it necessary, you can make use of the result 
(3)
(Total 15 marks)

27.	The function f is given by

(a)	(i)	To display the graph of y = f (x) for –10 £ x £ 10, a suitable interval for y, a £ y £ b must be chosen. Suggest appropriate values for a and b .
(ii)	Give the equation of the asymptote of the graph.
(3)
 
(b)	Show that 
(4)
 
(c)	Use your answer to part (b) to find the coordinates of the maximum point of the graph.
(3)
 
(d)	(i)	Either by inspection or by using an appropriate substitution, find

(ii)	Hence find the exact area of the region enclosed by the graph of f, the x-axis and the y-axis.
(8)
(Total 18 marks)

28.	In this question you should note that radians are used throughout.
	(a)	(i)	Sketch the graph of y = x2 cos x, for 0 £ x £ 2 making clear the approximate positions of the positive x-intercept, the maximum point and the end-points.
(ii)	Write down the approximate coordinates of the positive x-intercept, the maximum point and the end-points.
(7)
(b)	Find the exact value of the positive x-intercept for 0 £ x £ 2.
(2)
	Let R be the region in the first quadrant enclosed by the graph and the x-axis.
(c)	(i)	Shade R on your diagram.
(ii)	Write down an integral which represents the area of R.
(3)
(d)	Evaluate the integral in part (c)(ii), either by using a graphic display calculator, or by using the following information.
(x2 sin x + 2x cos x – 2 sin x) = x2 cos x.
(3)
(Total 15 marks)

29.	In this part of the question, radians are used throughout.
	The function f is given by
f (x) = (sin x)2 cos x.
	The following diagram shows part of the graph of y = f (x).

	The point A is a maximum point, the point B lies on the x-axis, and the point C is a point of inflexion.
 
(a)	Give the period of f.
(1)
 
(b)	From consideration of the graph of y = f (x), find to an accuracy of one significant figure the range of  f.
(1)

(c)	(i)	Find f¢ (x).
(ii)	Hence show that at the point A, cos x = .
(iii)	Find the exact maximum value.
(9)
 
(d)	Find the exact value of the x-coordinate at the point B.
(1)
 
(e)	(i)	Find dx.
(ii)	Find the area of the shaded region in the diagram.
(4)
 
(f)	Given that f² (x) = 9(cos x)3 – 7 cos x, find the x-coordinate at the point C.
(4)
(Total 20 marks)

30.	The following diagram shows the graph of f(x) = .
		
	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)	Find f′(x).
(ii)	Show that f″(x) = (4x2 – 2).
(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)

31.	The diagram below shows a plan for a window in the shape of a trapezium.
		
	Three sides of the window are 2 m long. The angle between the sloping sides of the window and the base is θ, where 0 < θ < .
(a)	Show that the area of the window is given by y = 4 sin θ + 2 sin 2θ.
(5)
 
(b)	Zoe wants a window to have an area of 5 m2. Find the two possible values of θ.
(4)
 
(c)	John wants two windows which have the same area A but different values of θ.
	Find all possible values for A.
(7)
(Total 16 marks)

32.	The acceleration, a m s–2, of a particle at time t seconds is given by
		a =  + 3sin 2t, for t ≥ 1.
	The particle is at rest when t = 1.
	Find the velocity of the particle when t = 5.
(Total 7 marks)

33.	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)

34.	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)

35.	Let f(x) = x3 – 4x + 1.
(a)	Expand (x + h)3.
(2)
 
(b)	Use the formula f′(x) =  to show that the derivative of f(x) is 3x2 – 4.
(4)
 
(c)	The tangent to the curve of f at the point P(1, –2) is parallel to the tangent at a point Q. Find the coordinates of Q.
(4)
 
(d)	The graph of f is decreasing for p < x < q. Find the value of p and of q.
(3)

(e)	Write down the range of values for the gradient of f.
(2)
(Total 15 marks)

36.	Consider the curve y = ln(3x – 1). Let P be the point on the curve where x = 2.
(a)	Write down the gradient of the curve at P.
(2)
 
(b)	The normal to the curve at P cuts the x-axis at R. Find the coordinates of R.
(5)
(Total 7 marks)

37.	Consider the curve with equation f(x) = px2 + qx, where p and q are constants.
The point A(1, 3) lies on the curve. The tangent to the curve at A has gradient 8.
Find the value of p and of q.
(Total 7 marks)

38.	A farmer wishes to create a rectangular enclosure, ABCD, of area 525 m2, as shown below.
		
	The fencing used for side AB costs $11 per metre. The fencing for the other three sides costs $3 per metre. The farmer creates an enclosure so that the cost is a minimum.
Find this minimum cost.
(Total 7 marks)

39.	The following diagram shows the graphs of f (x) = ln (3x – 2) + 1 and g (x) = – 4 cos (0.5x) + 2, for 1 £ x £ 10.

(a)	Let A be the area of the region enclosed by the curves of f and g.
(i)	Find an expression for A.
(ii)	Calculate the value of A.
(6)
 
(b)	(i)	Find f ′ (x).
(ii)	Find g′ (x).
(4)
 
(c)	There are two values of x for which the gradient of f is equal to the gradient of g. Find both these values of x.
(4)
(Total 14 marks)

40.	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)

41.	The function f is defined by f : x  –0.5x2 + 2x + 2.5.
	Let N be the normal to the curve at the point where the graph intercepts the y-axis.
(a)	Show that the equation of N may be written as y = –0.5x + 2.5.
(4)
 
(b)	Find the coordinates of the other point of intersection of the normal and the curve.
(5)

(c)	Let R be the region enclosed between the curve and N. Find the area of R.
(4)
(Total 13 marks)

42.	Let f (x) = x3 − 3x2 − 24x +1.
	The tangents to the curve of f at the points P and Q are parallel to the x-axis, where P is to the left of Q.
(a)	Calculate the coordinates of P and of Q.
Let N1 and N2 be the normals to the curve at P and Q respectively.
 
(b)	Write down the coordinates of the points where
(i)	the tangent at P intersects N2;
(ii)	the tangent at Q intersects N1.
(Total 6 marks)

43.	The velocity, v, in m s−1 of a particle moving in a straight line is given by v = e3t−2, where t is the time in seconds.
(a)	Find the acceleration of the particle at t =1.
(b)	At what value of t does the particle have a velocity of 22.3 m s−1?
(c)	Find the distance travelled in the first second.
(Total 6 marks)

44.	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)

45.	The following diagram shows part of the graph of y = cos x for 0 £ x £ 2p. Regions A and B are shaded.

(a)	Write down an expression for the area of A.
(1)
 
(b)	Calculate the area of A.
(1)
 
(c)	Find the total area of the shaded regions.
(4)
(Total 6 marks)

46.	Consider the function f (x) = 4x3 + 2x. Find the equation of the normal to the curve of f at the point where x =1.
(Total 6 marks)

47.	A Ferris wheel with centre O and a radius of 15 metres is represented in the diagram below. Initially seat A is at ground level. The next seat is B, where  = .

(a)	Find the length of the arc AB.
(2)
 
(b)	Find the area of the sector AOB.
(2)
 
(c)	The wheel turns clockwise through an angle of . Find the height of A above the ground.
(3)
 
	The height, h metres, of seat C above the ground after t minutes, can be modelled by the function
		h (t) = 15 − 15 cos .
(d)	(i)	Find the height of seat C when t = .
(ii)	Find the initial height of seat C.
(iii)	Find the time at which seat C first reaches its highest point.
(8)
 
(e)	Find h′ (t).
(2)
 
(f)	For 0 £ t £ p,
(i)	sketch the graph of h′;
(ii)	find the time at which the height is changing most rapidly.
(5)
(Total 22 marks)

48.	A part of the graph of y = 2x – x2 is given in the diagram below.

The shaded region is revolved through 360° about the x-axis.
(a)	Write down an expression for this volume of revolution.
(b)	Calculate this volume.
(Total 6 marks)

49.	The diagram below shows the graph of ƒ (x) = x2 e–x for 0 £ x £ 6. There are points of inflexion at A and C and there is a maximum at B.

(a)	Using the product rule for differentiation, find ƒ′ (x).
(b)	Find the exact value of the y-coordinate of B.
(c)	The second derivative of ƒ is ƒ′′ (x) = (x2 – 4x + 2) e–x. Use this result to find the exact value of the x-coordinate of C.
(Total 6 marks)

50.	The displacement s metres at time t seconds is given by
		s = 5 cos 3t + t2 + 10, for t ³ 0.
(a)	Write down the minimum value of s.
(b)	Find the acceleration, a, at time t.
(c)	Find the value of t when the maximum value of a first occurs.
(Total 6 marks)

51.	Let f (x) = – x2 + x + 4.
(a)	(i)	Write down f ′ (x).
(ii)	Find the equation of the normal to the curve of f at (2, 3).
(iii)	This normal intersects the curve of f at (2, 3) and at one other point P.
Find the x-coordinate of P.
(9)
Part of the graph of f is given below.

(b)	Let R be the region under the curve of f from x = −1 to x = 2.
(i)	Write down an expression for the area of R.
(ii)	Calculate this area.
(iii)	The region R is revolved through 360° about the x-axis. Write down an expression for the volume of the solid formed.
(6)
(c)	Find  giving your answer in terms of k.
(6)
(Total 21 marks)

52.	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)

53.	The shaded region in the diagram below is bounded by f (x) = , x = a, and the x-axis. The shaded region is revolved around the x-axis through 360°. The volume of the solid formed is 0.845p.

Find the value of a.
(Total 6 marks)

54.	The function f is defined by 
(a)	Write down
(i)	f ¢(x);
(ii)	f ¢(0).
(2)
 
(b)	Let N be the normal to the curve at the point where the graph intercepts the y-axis. Show that the equation of N may be written as y = –0.5x + 2.5.
(3)
 
	Let 
(c)	(i)	Find the solutions of f (x) = g (x).
(ii)	Hence find the coordinates of the other point of intersection of the normal and the curve.
(6)

(d)	Let R be the region enclosed between the curve and N.
(i)	Write down an expression for the area of R.
(ii)	Hence write down the area of R.
(5)
(Total 16 marks)

55.	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)

56.	Consider the function 
	A sketch of part of the graph of h is given below.

	The line (AB) is a vertical asymptote. The point P is a point of inflexion.
 
(a)	Write down the equation of the vertical asymptote.
(1)
 
(b)	Find h′(x), writing your answer in the form

	where a and n are constants to be determined.
(4)

(c)	Given that , calculate the coordinates of P.
(3)
(Total 8 marks)

57.	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)

58.	The function g (x) is defined for –3 £ x £ 3. The behaviour of g¢(x) and g²(x) is given in the tables below.
x	–3 < x < –2	–2	–2 < x < 1	1	1 < x < 3	
g¢(x)	negative	0	positive	0	negative	

x	–3 < x < –	–	–< x < 3	
g²(x)	positive	0	negative	
	Use the information above to answer the following. In each case, justify your answer.
 
(a)	Write down the value of x for which g has a maximum.
(2)
(b)	On which intervals is the value of g decreasing?
(2)
(c)	Write down the value of x for which the graph of g has a point of inflexion.
(2)
(d)	Given that g (–3) = 1, sketch the graph of g. On the sketch, clearly indicate the position of the maximum point, the minimum point, and the point of inflexion.
(3)
(Total 9 marks)

59.	A particle moves with a velocity v m s−1 given by v = 25 − 4t2 where t ³ 0.
(a)	The displacement, s metres, is 10 when t is 3. Find an expression for s in terms of t.
(6)
(b)	Find t when s reaches its maximum value.
(3)
(c)	The particle has a positive displacement for m £ t £ n. Find the value of m and the value of n.
(3)
(Total 12 marks)

60.	The graph of y = sin 2x from 0£ x £ p is shown below.

The area of the shaded region is 0.85. Find the value of k.
(Total 6 marks)

61.	Let f (x) = x3 – 2x2 – 1.
(a)	Find f¢ (x).
(b)	Find the gradient of the curve of f (x) at the point (2, –1).
Working:		
	Answers:
(a)	…………………………………………
(b)	…………………………………………	
(Total 6 marks)

62.	A car starts by moving from a fixed point A. Its velocity, v m s–1 after t seconds is given by v = 4t + 5 – 5e–t. Let d be the displacement from A when t = 4.
(a)	Write down an integral which represents d.
(b)	Calculate the value of d.
Working:		
	Answers:
(a)	…………………………………………..
(b)	…………………………………………..	
(Total 6 marks)

63.	(a)	Consider the function f (x) = 2 + . The diagram below is a sketch of part of the graph of y = f (x).

	Copy and complete the sketch of f (x).
(2)
 
(b)	(i)	Write down the x-intercepts and y-intercepts of f (x).
(ii)	Write down the equations of the asymptotes of f (x).
(4)
 
(c)	(i)	Find f ¢ (x).
(ii)	There are no maximum or minimum points on the graph of f (x).
Use your expression for f ¢ (x) to explain why.
(3)
 
	The region enclosed by the graph of f (x), the x-axis and the lines x = 2 and x = 4, is labelled A, as shown below.

(d)	(i)	Find dx.
(ii)	Write down an expression that represents the area labelled A.
(iii)	Find the area of A.
(7)
(Total 16 marks)

64.	The displacement s metres of a car, t seconds after leaving a fixed point A, is given by
s = 10t – 0.5t2.
(a)	Calculate the velocity when t = 0.
(b)	Calculate the value of t when the velocity is zero.
(c)	Calculate the displacement of the car from A when the velocity is zero.
Working:		
	Answers:
(a)	…………………………………………..
(b)	…………………………………………..
(c)	…………………………………………..	
(Total 6 marks)

65.	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)

66.	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)

67.	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)

68.	An aircraft lands on a runway. Its velocity v m s–1 at time t seconds after landing is given by the equation v = 50 + 50e–0.5t, where 0 £ t £ 4.
(a)	Find the velocity of the aircraft
(i)	when it lands;
(ii)	when t = 4.
(4)
 
(b)	Write down an integral which represents the distance travelled in the first four seconds.
(3)
 
(c)	Calculate the distance travelled in the first four seconds.
(2)
 
	After four seconds, the aircraft slows down (decelerates) at a constant rate and comes to rest when t = 11.
(d)	Sketch a graph of velocity against time for 0 £ t £ 11. Clearly label the axes and mark on the graph the point where t = 4.
(5)
 
(e)	Find the constant rate at which the aircraft is slowing down (decelerating) between t = 4 and t = 11.
(2)
(f)	Calculate the distance travelled by the aircraft between t = 4 and t = 11.
(2)
(Total 18 marks)

69.	(a)	Find ò(1 + 3 sin (x + 2))dx.
(b)	The diagram shows part of the graph of the function f (x) = 1 + 3 sin (x + 2).
The area of the shaded region is given by .

	Find the value of a.
Working:		
	Answers:
(a)	..................................................................
(b)	..................................................................	
(Total 6 marks)

70.	Consider the function f (x) = 1 + e–2x.
(a)	(i)	Find f ¢(x).
(ii)	Explain briefly how this shows that f (x) is a decreasing function for all values of x (ie that f (x) always decreases in value as x increases).
(2)

	Let P be the point on the graph of f  where x = –.
(b)	Find an expression in terms of e for
(i)	the y-coordinate of P;
(ii)	the gradient of the tangent to the curve at P.
(2)

(c)	Find the equation of the tangent to the curve at P, giving your answer in the form y = ax + b.
(3)

(d)	(i)	Sketch the curve of f  for –1 £ x £ 2.
(ii)	Draw the tangent at x = –.
(iii)	Shade the area enclosed by the curve, the tangent and the y-axis.
(iv)	Find this area.
(7)
(Total 14 marks)

71.	Note: Radians are used throughout this question.
	A mass is suspended from the ceiling on a spring. It is pulled down to point P and then released. It oscillates up and down.

	Its distance, s cm, from the ceiling, is modelled by the function s = 48 + 10 cos 2πt where t is the time in seconds from release.
(a)	(i)	What is the distance of the point P from the ceiling?
(ii)	How long is it until the mass is next at P?
(5)

(b)	(i)	Find .
(ii)	Where is the mass when the velocity is zero?
(7)
	A second mass is suspended on another spring. Its distance r cm from the ceiling is modelled by the function r = 60 + 15 cos 4pt. The two masses are released at the same instant.
(c)	Find the value of t when they are first at the same distance below the ceiling.
(2)

(d)	In the first three seconds, how many times are the two masses at the same height?
(2)
(Total 16 marks)

72.	Consider the function f given by f (x) = ,	x ¹ 1.
A part of the graph of f is given below.

	The graph has a vertical asymptote and a horizontal asymptote, as shown.
(a)	Write down the equation of the vertical asymptote.
(1)

(b)	f (100) = 1.91	f (–100) = 2.09	f (1000) = 1.99
(i)	Evaluate f (–1000).
(ii)	Write down the equation of the horizontal asymptote.
(2)

(c)	Show that f ¢(x) =,	x ¹ 1.
(3)

	The second derivative is given by f ²(x) = ,	x ¹ 1.
(d)	Using values of f ¢(x) and f ²(x) explain why a minimum must occur at x = 3.
(2)

(e)	There is a point of inflexion on the graph of f. Write down the coordinates of this point.
(2)
(Total 10 marks)

73.	The graph of y = x3 – 10x2 +12x + 23 has a maximum point between x = –1 and x = 3. Find the coordinates of this maximum point.
Working:		
	Answer:
......................................................................	
(Total 6 marks)

74.	The point P () lies on the graph of the curve of y = sin (2x –1).
	Find the gradient of the tangent to the curve at P.
Working:		
	Answer:
.......................................................................	
(Total 4 marks)

75.	The function f is given by	
(a)	(i)	Show that	
	Hence
(ii)	prove that the graph of f can have only one local maximum or minimum point;

(iii)	ﬁnd the coordinates of the maximum point on the graph of f.
(6)

(b)	By showing that the second derivative	 or otherwise,
ﬁnd the coordinates of the point of inﬂexion on the graph of f.
(6)

(c)	The region S is enclosed by the graph of f , the x-axis, and the vertical line
through the maximum point of f , as shown in the diagram below.

(i)	Would the trapezium rule overestimate or underestimate the area of S? Justify your answer by drawing a diagram or otherwise.
(3)

(ii)	Find , by using the substitution u = ln 2x, or otherwise.
(4)

(iii)	Using , ﬁnd the area of S.
(4)

(d)	The Newton–Raphson method is to be used to solve the equation f (x) = 0. 
(i)	Show that it is not possible to ﬁnd a solution using a starting value of
x1 = 1.
(3)
(ii)	Starting with x1 = 0.4, calculate successive approximations x2, x3, ...
for the root of the equation until the absolute error is less than 0.01.
Give all answers correct to ﬁve decimal places.
(4)
(Total 30 marks)

76.	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)

77.	A ball is dropped vertically from a great height. Its velocity v is given by
	v = 50 – 50e–0.2t, t ³ 0
	where v is in metres per second and t is in seconds.
(a)	Find the value of v when
(i)	t = 0;
(ii)	t = 10.
(2)

(b)	(i)	Find an expression for the acceleration, a, as a function of t.
(ii)	What is the value of a when t = 0?
(3)

(c)	(i)	As t becomes large, what value does v approach?
(ii)	As t becomes large, what value does a approach?
(iii)	Explain the relationship between the answers to parts (i) and (ii).
(3)

(d)	Let y metres be the distance fallen after t seconds.
(i)	Show that y = 50t + 250e–0.2t + k, where k is a constant.
(ii)	Given that y = 0 when t = 0, find the value of k.
(iii)	Find the time required to fall 250 m, giving your answer correct to four significant figures.
(7)
(Total 15 marks)

78.	Radian measure is used, where appropriate, throughout the question.
	Consider the function 
	The graph of this function has a vertical and a horizontal asymptote.
(a)	Write down the equation of
(i)	the vertical asymptote;
(ii)	the horizontal asymptote.
(2)

(b)	Find, simplifying the answer as much as possible.
(3)

(c)	How many points of inflexion does the graph of this function have?
(1)
(Total 6 marks)

79.	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)

80.	Find the coordinates of the point on the graph of y = x2 – x at which the tangent is parallel to the line y = 5x.
Working:		
	Answer:
......................................................................	
(Total 4 marks)

81.	The main runway at Concordville airport is 2 km long. An airplane, landing at Concordville, touches down at point T, and immediately starts to slow down. The point A is at the southern end of the runway. A marker is located at point P on the runway.

Not to scale

	As the airplane slows down, its distance, s, from A, is given by
s = c + 100t – 4t2,
	where t is the time in seconds after touchdown, and c metres is the distance of T from A.

(a)	The airplane touches down 800 m from A, (ie c = 800).
(i)	Find the distance travelled by the airplane in the first 5 seconds after touchdown.
(2)
(ii)	Write down an expression for the velocity of the airplane at time t seconds after touchdown, and hence find the velocity after 5 seconds.
(3)

	The airplane passes the marker at P with a velocity of 36 m s–1. Find
(iii)	how many seconds after touchdown it passes the marker; 
(2)
(iv)	the distance from P to A.
(3)
(b)	Show that if the airplane touches down before reaching the point P, it can stop before reaching the northern end, B, of the runway.
(5)
(Total 15 marks)

82.	(a)	Sketch the graph of y = p sin x – x, –3 £ x £ 3, on millimetre square paper, using a scale of 2 cm per unit on each axis.
	Label and number both axes and indicate clearly the approximate positions of the 
x-intercepts and the local maximum and minimum points.
(5)

(b)	Find the solution of the equation
	p sin x – x = 0,          x > 0.
(1)

(c)	Find the indefinite integral

	and hence, or otherwise, calculate the area of the region enclosed by the graph, the 
x-axis and the line x = 1.
(4)
(Total 10 marks)

83.	The diagram shows the graph of the function y = 1 + , 0 < x £ 4. Find the exact value of the area of the shaded region.

Working:		
	Answer:
......................................................................	
(Total 4 marks)

84.	A rock-climber slips off a rock-face and falls vertically. At first he falls freely, but after 2 seconds a safety rope slows him down. The height h metres of the rock-climber after t seconds of the fall is given by:
h = 50 – 5t2, 	0 £ t £ 2
h = 90 – 40t + 5t2, 	2 £ t £ 5

(a)	Find the height of the rock-climber when t = 2.
(1)

(b)	Sketch a graph of h against t for 0 £ t £ 5.
(4)

(c)	Find  for:
(i)	0 £ t £ 2
(ii)	2 £ t £ 5
(2)

(d)	Find the velocity of the rock-climber when t = 2.
(2)

(e)	Find the times when the velocity of the rock-climber is zero.
(3)

(f)	Find the minimum height of the rock-climber for 0 £ t £ 5.
(3)
(Total 15 marks)

85.	The diagram shows the graph of the function f given by
f (x) = A sin + B,
	for 0 £ x £ 5, where A and B are constants, and x is measured in radians.

	The graph includes the points (1, 3) and (5, 3), which are maximum points of the graph.
(a)	Write down the values of f (1) and f (5).
(2)
(b)	Show that the period of f is 4.
(2)

	The point (3, –1) is a minimum point of the graph.
(c)	Show that A = 2, and find the value of B.
(5)
(d)	Show that f¢ (x) = p cos .
(4)

	The line y = k – px is a tangent line to the graph for 0 £ x £ 5.
(e)	Find
(i)	the point where this tangent meets the curve;
(ii)	the value of k.
(6)
(f)	Solve the equation f (x) = 2 for 0 £ x £ 5.
(5)
(Total 24 marks)

86.	(a)	Find the equation of the tangent line to the curve y = ln x at the point (e, 1), and verify that the origin is on this line.
(4)
(b)	Show that  (x ln x – x) = ln x.
(2)
(c)	The diagram shows the region enclosed by the curve y = ln x, the tangent line in part (a), and the line y = 0.

	Use the result of part (b) to show that the area of this region is e – 1.
(4)
(Total 10 marks)

87.	A curve has equation y = x(x – 4)2.
(a)	For this curve find
(i)	the x-intercepts;
(ii)	the coordinates of the maximum point;
(iii)	the coordinates of the point of inflexion.
(9)
(b)	Use your answers to part (a) to sketch a graph of the curve for 0 £ x £ 4, clearly indicating the features you have found in part (a). 
(3)

(c)	(i)	On your sketch indicate by shading the region whose area is given by the following integral:

(ii)	Explain, using your answer to part (a), why the value of this integral is greater than 0 but less than 40.
(3)
(Total 15 marks)