1.
The diagram represents the graph of the function
f : x a (x – p)(x – q).
y
– 12
2
x
C
(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)
1
2.
The function f is given by
f (x) = 2 x + 1 , x Î
x-3
(a)
(i)
, x ¹ 3.
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)
-7
and hence find the equation of the tangent at
( x - 3) 2
the point S where x = 4.
Show that f¢ (x) =
(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)
2
3.
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.
y
B
A
0
C
x
Find the coordinates of A, B and C.
Working:
Answer:
......................................................................
(Total 4 marks)
4.
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.
3
Identify which diagram represents which graph.
y
I
y
II
4
2
1
2
1
– –p
2
–p
1
– –p
2
–p
1
–p
2
p
x
3
–p
2
p
x
3
–p
2
–2
y
III
–1
1
–p
2
y
IV
3
5
2
4
1
3
x
–p
1
– –p
2
1
–p
2
p
2
3
–p
2
1
x
–3
–p
1
– –p
2
1
–p
2
p
3
–p
2
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 4 marks)
4
5.
The function f is given by f (x) =
1n ( x - 2) . Find the domain of the function.
Working:
Answer:
......................................................................
(Total 4 marks)
6.
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)
5
7.
Let f (x) =
x , and g (x) = 2x. Solve the equation
(f –1 o g)(x) = 0.25.
Working:
Answer:
......................................................................
(Total 4 marks)
6
8.
Two functions f, g are defined as follows:
f : x ® 3x + 5
g : x ® 2(1 – x)
Find
(a)
f –1(2);
(b)
(g o f )(–4).
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
7
9.
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)
8
10.
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.
y
y
(a)
1
0
y=x2
x
0
x
1
y
(b)
(c)
y
7
4
3
0
1
x
0
1
x
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(c) ..................................................................
(Total 4 marks)
9
11.
f (x) = 4 sin æç 3 x + p ö÷ .
2ø
è
For what values of k will the equation f (x) = k have no solutions?
Working:
Answer:
......................................................................
(Total 4 marks)
12.
The diagram shows the graph of the function y = ax2 + bx + c.
y
x
10
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)
13.
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)
11
(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)
14.
(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)
12
15.
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)
13
16.
Consider the function f : x a x + 1, x ³ – 1
(a)
Determine the inverse function f –1.
(b)
What is the domain of f –1?
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
17.
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
dh
.
dt
14
(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)
18.
The diagram shows part of the graph with equation y = x2 + px + q. The graph cuts the x-axis at
–2 and 3.
y
6
4
2
–3
–2
–1
0
1
2
3
4
x
–2
–4
–6
15
Find the value of
(a)
p;
(b)
q.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
19.
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)
16
20.
The function f is defined by
f :xa
3 – 2x ,
3
x£ .
2
Evaluate f –1(5).
Working:
Answer:
.......................................................................
(Total 4 marks)
17
21.
The following diagram shows the graph of y = f (x). It has minimum and maximum points at
1
(0, 0) and ( 1, ).
2
y
3.5
3
2.5
2
1.5
1
0.5
–2
–1
0
1
2
3
x
–0.5
–1
–1.5
–2
–2.5
3
.
2
(a)
On the same diagram, draw the graph of y = f ( x – 1) +
(b)
What are the coordinates of the minimum and maximum points of
3
y = f ( x – 1) + ?
2
Working:
Answer:
18
(b) ................................................................
(Total 4 marks)
22.
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)
23.
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)
19
(c)
Give reasons why f (x) ≥ 0 for all values of x.
(2)
(Total 16 marks)
20
24.
æ xö
The diagram below shows the graph of y = x sin ç ÷ , for 0 £ x < m, and 0 £ y < n, where x is
è3ø
in radians and m and n are integers.
y
n
n–1
0
m–1
m
x
Find the value of
(a)
m;
(b)
n.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
21
25.
Given that f (x) = 2e3x, find the inverse function f –1(x).
Working:
Answer:
.......................................................................
(Total 4 marks)
26.
The diagram below shows part of the graph of the function
f : x a – x 3 + 2 x 2 + 15 x .
y
40
35
30
25
20
15
10
5
A
–3
–2
–1 –5
–10
–15
P
–20
Q
B
1
2
3
4
5
x
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.
22
(a)
The function may also be written in the form f : x a – x( x – a ) ( x – b),
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)
23
27.
Let f (x) = 2x, and g (x) =
x
, (x ¹ 2).
x–2
Find
(a)
(g o f ) (3);
(b)
g–1 (5).
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
28.
Consider functions of the form y = e–kx
(a)
Show that
ò
1
0
e – kx dx =
1
(1 – e–k).
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)
24
(c)
(i)
Find
dy
in terms of k, where y = e–kx.
dx
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)
29.
Consider the functions f : x a 4(x – 1) and g : x a
(a)
Find g–1.
(b)
Solve the equation ( f ° g–1) (x) = 4.
6– x
.
2
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
25
30.
$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)
26
31.
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).
8
7
6
E
5
4
A
3
B
2
D
1
C
–4
–3
–2
–1
0
1
2
3
4
5
–1
–2
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)
27
32.
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.
y
B
A
–1
(a)
0
1
C
D 2 x
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
dy
.
dx
π
.
2
(6)
(d)
(i)
Write down the integral which represents the shaded area.
(ii)
Evaluate this integral.
(5)
28
(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)
33.
Let g (x) = x4 – 2x3 + x2 – 2.
(a)
Solve g (x) = 0.
(2)
Let f (x) =
2x 3
+ 1 . A part of the graph of f (x) is shown below.
g ( x)
y
C
A
(b)
0
B
x
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)
29
(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)
34.
Let f (x) = e–x, and g (x) =
(a)
f –1 (x);
(b)
(g ° f ) (x).
x
, x ¹ –1. Find
1+ x
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
30
35.
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)
31
36.
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.
(i)
y
(ii)
(iii)
y
x
y
x
x
Complete the table below, by writing the letter for the two relations that each sketch could
represent.
sketch
relation letters
(i)
(ii)
(iii)
32
37.
The diagram shows part of the graph of the curve y = a (x – h)2 + k, where a, h, k Î
.
y
20
15
10
P(5, 9)
5
0
(a)
1
2
3
4
5
6
x
.
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
dy
.
dx
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)
33
38.
The equation kx2 + 3x + 1 = 0 has exactly one solution. Find the value of k.
Working:
Answer:
..................................................................
(Total 6 marks)
34
39.
(a)
The diagram shows part of the graph of the function f (x) =
q
. The curve passes
x– p
through the point A (3, 10). The line (CD) is an asymptote.
y
15
C
10
A
5
–15
–10
–5
0
5
10
15 x
-5
-10
-15
D
Find the value of
(i)
p;
(ii)
q.
35
(b)
The graph of f (x) is transformed as shown in the following diagram. The point A is
transformed to A¢ (3, –10).
y
15
C
10
5
–15
–10
–5
5
0
10
15 x
–5
–10
A
–15
D
36
Give a full geometric description of the transformation.
Working:
Answers:
(a) (i) ...........................................................
(ii) ...........................................................
(b) ..................................................................
..................................................................
(Total 6 marks)
40.
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)
37
41.
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)
42.
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)
38
43.
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)
39
44.
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)
40
45.
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.
f( x)
4
3
2
1
0
0
(a)
(b)
1
2
3
4
5
6
7
8
9
10
x
Find
(i)
f (2);
(ii)
f ¢ (6.5);
(iii)
f ¢ (14).
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)
41
46.
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).
y
4
P(3, 2)
2
–1
1
2
3
4
5
6
x
–2
–4
–6
–8
–10
–12
(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)
42
47.
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).
y
4
3
2
–5
–4
–3
–2
S
1
(a, 0)
–1
(b, 0)
1
–1
R
2
3
4
5
x
–2
–3
–4
–5
–6
–7
(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)
43
48.
The functions f and g are defined by f : a 3 x, g : x a x + 2 .
(a)
Find an expression for (f ° g) (x).
(b)
Show that f –l (18) + g–l (18) = 22.
.....................................................................................................................................
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(Total 6 marks)
44
49.
The function f is defined by f ( x) =
(a)
3
9 – x2
, for –3 < x < 3.
On the grid below, sketch the graph of f.
45
(b)
Write down the equation of each vertical asymptote.
(c)
Write down the range of the function f.
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
.....................................................................................................................................
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(Total 6 marks)
50.
The diagram below shows the graphs of f (x) = 1 + e2x, g (x) = 10x + 2, 0 £ x £ 1.5.
y
f
g
16
12
p
8
4
0.5
1
1.5
x
46
(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.
y
16
12
8
5
4
0.5 a
(b)
(i)
Find f –1(x).
(ii)
Hence show that a = ln 2.
1
1.5
x
(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)
47
51.
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)
52.
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.
y
4
2
–4
A
–2
0
2
B
4
6 x
–2
–4
–6
48
(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)
dy
.
dx
(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)
49
53.
Consider the functions f (x) = 2x and g (x) =
(a)
Calculate (f ◦ g) (4).
(b)
Find g−1(x).
(c)
Write down the domain of g−1.
1
, x ¹ 3.
x -3
Working:
Answers:
(a) .....................................................
(b) .....................................................
(c) .....................................................
(Total 6 marks)
50
54.
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.
Working:
Answers:
........................................................
(Total 6 marks)
51
55.
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).
y
B
0
1
(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.
2
x
Working:
Answers:
(a) .................................................
(b) .................................................
(Total 6 marks)
52
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 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)
53
(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
(e)
1
is given below.
f
Write down the equations of the two vertical asymptotes.
(2)
(f)
The tangent to the graph of
1
is horizontal at P. Write down the x-coordinate of P.
f
(2)
(Total 15 marks)
54
58.
Let f (x) = x3 − 4 and g (x) = 2x.
(a)
Find (g ◦ f ) (−2).
(b)
Find f −1 (x).
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(Total 6 marks)
59.
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)
x+3
, solve (f −1 ◦ g) (x) = (g–1 ◦ f) (x).
3
(2)
55
Let h (x) =
f ( x)
, x ¹ 2.
g ( x)
(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
3x - 5
1
may also be written as 3 +
. Use this to answer the
x -3
x-2
following.
ò h (x) dx.
(i)
Find
(ii)
Hence, calculate the exact value of
ò
5
3
h (x)dx.
(5)
(f)
On your sketch, shade the region whose area is represented by
ò
5
3
h (x)dx.
(1)
(Total 18 marks)
60.
Find the exact value of x in each of the following equations.
(a)
5x+1 = 625
56
(b)
loga (3x + 5) = 2
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(Total 6 marks)
61.
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)
57
(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)
62.
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
of k.
, has two solutions. Write down the range of values
(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)
58
63.
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).
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(Total 6 marks)
59
64.
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).
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
60
(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)
61
65.
(a)
Given that (2x)2 + (2x) −12 can be written as (2x + a)(2x + b), where a, b Î
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.
, find the
..............................................................................................................................................
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(Total 6 marks)
62
66.
Let f (x) =
x + 4 , 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.
.
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(Total 6 marks)
63
67.
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.
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(Total 6 marks)
68.
The function f (x) is defined as f (x) = 3 +
(a)
1
5
,x ¹
.
2
2x - 5
Sketch the curve of f for −5 £ x £ 5, showing the asymptotes.
(3)
64
(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.
æ
6
1
ò ççè 9 + 2 x - 5 + (2 x - 5)
ö
÷ dx.
÷
ø
(i)
Find
(ii)
æ 28
ö
Hence, given that V = πç + 3 ln 3 ÷ , find the value of a.
è 3
ø
2
(10)
(Total 17 marks)
65
69.
Let f (x) = p -
3x
, where p, qÎ
x - q2
+
2
.
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)
(ii)
Show that f ′ (x) =
(
).
3 x 2 +1
(x
2
)
-1
2
Hence, show that there are no maximum or minimum points on the graph of f.
(8)
66
(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)
70.
Let f (x) = loga x, x > 0.
(a)
Write down the value of
(i)
f (a);
(ii)
f (1);
(iii)
f (a4 ).
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(3)
67
(b)
The diagram below shows part of the graph of f.
y
2
1
–2
–1
f
0
1
2
x
–1
–2
On the same diagram, sketch the graph of f−1.
(3)
(Total 6 marks)
71.
æ 5 ö
1
÷÷ , x ¹
Consider the function f (x) e(2x–1) + çç
.
2
è (2 x -1) ø
(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.
ò
2
1
ò
2
0
(iii)
f (x)dx
f (x)dx
Justify your answer.
(3)
68
(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)
72.
(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)
73.
Let f (x) = ln (x + 5) + ln 2, for x > –5.
69
(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
.
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(3)
(Total 7 marks)
70
74.
Find the equation of the normal to the curve with equation
y = x3 + 1
at the point (1, 2).
Working:
Answer:
.........................................................................
(Total 4 marks)
75.
Differentiate with respect to x
(a)
(b)
3 - 4x
esin x
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
71
76.
The function f
is such that f ² (x) = 2x – 2.
When the graph of f is drawn, it has a minimum point at (3, –7).
(a)
Show that f¢ (x) = x2 – 2x – 3 and hence find f (x).
(6)
(b)
Find f (0), f (–1) and f¢ (–1).
(3)
(c)
Hence sketch the graph of f, labelling it with the information obtained in part (b).
(4)
(Note: It is not necessary to find the coordinates of the points where the graph
cuts the x-axis.)
(Total 13 marks)
x
77.
The diagram shows part of the graph of y = e 2 .
y
x
y = e2
P
ln2
(a)
x
Find the coordinates of the point P, where the graph meets the y-axis.
(2)
The shaded region between the graph and the x-axis, bounded by x = 0
and x = ln 2, is rotated through 360° about the x-axis.
(b)
Write down an integral which represents the volume of the solid obtained.
(4)
(c)
Show that this volume is p.
(5)
(Total 11 marks)
72
78.
The parabola shown has equation y2 = 9x.
y 2 = 9x
y
P
M
Q
(a)
x
Verify that the point P (4, 6) is on the parabola.
(2)
The line (PQ) is the normal to the parabola at the point P, and cuts the x-axis at Q.
(b)
(i)
Find the equation of (PQ) in the form ax + by + c = 0.
(5)
(ii)
Find the coordinates of Q.
(2)
S is the point æç 9 , 0 ö÷.
è4 ø
(c)
Verify that SP = SQ.
(4)
(d)
The line (PM) is parallel to the x-axis. From part (c), explain why
(QP) bisects the angle SP̂M.
(3)
(Total 16 marks)
73
79.
The diagram shows part of the graph of y = 12x2(1 – x).
y
x
0
(a)
Write down an integral which represents the area of the shaded region.
(b)
Find the area of the shaded region.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
74
80.
Differentiate with respect to x:
(a)
(x2 + l)2.
(b)
1n(3x – 1).
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
81.
The diagram shows part of the graph of y = 1 . The area of the shaded region is 2 units.
x
y
0
1
a
x
75
Find the exact value of a.
Working:
Answer:
......................................................................
(Total 4 marks)
82.
(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 d (x ln x – x) = ln x.
dx
(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.
y
1
0
(e, 1)
1
2
x
3
Use the result of part (b) to show that the area of this region is
1
2
e – 1.
(4)
(Total 10 marks)
76
83.
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:
ò
(ii)
4
0
x( x - 4) 2 dx.
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)
84.
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)
77
85.
If f ¢(x) = cos x, and f æç p ö÷ = – 2, find f (x).
è2ø
Working:
Answer:
......................................................................
(Total 4 marks)
86.
Let f (x) = x3.
f (5 + h) - f (5)
for h = 0.1.
h
(a)
Evaluate
(b)
What number does
f (5 + h) - f (5)
approach as h approaches zero?
h
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
87.
The diagram shows part of the graph of the curve with equation
y = e2x cos x.
78
y
P(a, b)
0
(a)
Show that
x
dy
= e2x (2 cos x – sin x).
dx
(2)
(b)
Find
d2 y
.
dx 2
(4)
There is an inflexion point at P (a, b).
(c)
Use the results from parts (a) and (b) to prove that:
(i)
tan a = 3 ;
4
(3)
(ii)
the gradient of the curve at P is e2a.
(5)
(Total 14 marks)
79
88.
A curve with equation y =f (x) passes through the point (1, 1). Its gradient function is
f¢ (x) = –2x + 3.
Find the equation of the curve.
Working:
Answer:
......................................................................
(Total 4 marks)
89.
Given that f (x) = (2x + 5)3 find
(a)
f¢ (x);
(b)
ò f ( x)dx.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
80
90.
The diagram shows the graph of the function y = 1 + 1 , 0 < x £ 4. Find the exact value of the
x
area of the shaded region.
4
3
1
y = 1+ –
x
2
1 13
1
0
1
2
3
4
Working:
Answer:
......................................................................
(Total 4 marks)
81
91.
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.
d (x2 sin x + 2x cos x – 2 sin x) = x2 cos x.
dx
(3)
(Total 15 marks)
82
92.
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).
y
A
C
B
x
O
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)
83
(c)
(i)
Find f¢ (x).
(ii)
Hence show that at the point A, cos x =
(iii)
Find the exact maximum value.
1.
3
(9)
(d)
Find the exact value of the x-coordinate at the point B.
(1)
(e)
ò f (x) dx.
(i)
Find
(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)
93.
Given the function f (x) = x2 – 3bx + (c + 2), determine the values of b and c such that f (1) = 0
and f¢ (3) = 0.
Working:
Answer:
....................................................................
(Total 4 marks)
84
94.
The function f is given by
f ( x) =1 –
(a)
2x
1+ x 2
(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 f ¢ ( x) =
2x 2 – 2
(1 + x 2 ) 2
.
(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
ò f ( x) dx
(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)
85
95.
Find
(a)
ò sin (3x + 7)dx;
(b)
òe
–4 x
dx .
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 4 marks)
96.
The function f is given by
(a)
(i)
Show that
f ( x) =
f ¢ ( x) =
1n 2 x
,
x
1 – 1n 2 x
x2
x > 0.
.
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
f ¢¢ ( x) =
2 1n 2 x – 3
x3
find the coordinates of the point of inflexion on the graph of f.
or otherwise,
(6)
86
(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.
y
y = f (x )
h
0
(i)
x
Would the trapezium rule overestimate or underestimate the area of S? Justify your
answer by drawing a diagram or otherwise.
(3)
(ii)
Find
ò f ( x) dx , by using the substitution u = ln 2x, or otherwise.
(4)
(iii)
Using
ò f ( x) dx , 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)
87
97.
The derivative of the function f is given by f ¢(x) =
1
– 0.5 sin x, for x ¹ –1.
1+ x
The graph of f passes through the point (0, 2). Find an expression for f (x).
Working:
Answer:
......................................................................
(Total 6 marks)
88
98.
Figure 1 shows the graphs of the functions f1, f2,
f3,
f4.
Figure 2 includes the graphs of the derivatives of the functions shown in Figure 1, eg the
derivative of f1 is shown in diagram (d).
Figure 1
Figure 2
y
y
f1
(a)
O
O
x
x
y
y
f2
(b)
O
O
x
y
x
y
f3
(c)
O
x
O
y
x
y
f4
(d)
O
x
O
x
y
(e)
O
x
89
Complete the table below by matching each function with its derivative.
Function
Derivative diagram
f1
(d)
f2
f3
f4
Working:
(Total 6 marks)
99.
Let the function f
(a)
be defined by f (x) =
2
, x ¹ –1.
1 + x3
(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) =
f² (x) =
(
– 6x 2
, show that the second derivative
(1 + x 3 ) 2
)
12 x 2 x 3 – 1
.
(1 + x 3 ) 3
90
(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
(i)
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
Use the trapezium rule with five sub-intervals to approximate the integral
3
ò f (x )dx.
1
(ii)
Given that
3
ò f (x )dx = 0.637599, use a diagram to explain why your answer is
1
greater than this.
(5)
(Total 15 marks)
91
100. Let f (x) =
x 3 . Find
(a)
f¢ (x);
(b)
ò f ( x)dx.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
92
101. The diagram shows part of the curve y = sin x. The shaded region is bounded by the curve and
3π
.
the lines y = 0 and x =
4
y
3p
4
Given that sin
3π
=
4
3π
2
and cos
=–
2
4
p
x
2
, calculate the exact area of the shaded
2
region.
Working:
Answer:
......................................................................
(Total 6 marks)
93
102. The diagram below shows the shaded region R enclosed by the graph of y = 2x 1 + x 2 , the
x-axis, and the vertical line x = k.
y
y = 2x 1+x 2
R
k
(a)
Find
x
dy
.
dx
(3)
(b)
Using the substitution u = 1 + x2 or otherwise, show that
3
ò
2 x 1 + x 2 dx =
2
(1 + x2) 2 + c.
3
(3)
(c)
Given that the area of R equals 1, find the value of k.
(3)
(Total 9 marks)
94
x
103. Let f (x) = e 3 + 5 cos 2x. Find f¢ (x).
Working:
Answer:
..................................................................
(Total 6 marks)
104. Given that
(a)
ò
(b)
ò
3
1
3
1
ò
3
1
g ( x)dx = 10, deduce the value of
1
g ( x)dx;
2
( g ( x) + 4)dx.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
95
105. Consider the function f (x) = cos x + sin x.
(a)
π
) = 0.
4
(i)
Show that f (–
(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.
y
6
C(a, b)
4
D
2
–2
(b)
Find
–1
1
2
3
x
dy
.
dx
(3)
(c)
Find the exact value of a and of b.
(4)
96
π
(d)
Show that at D, y =
2e 4 .
(5)
(e)
Find the area of the shaded region.
(2)
(Total 17 marks)
106. It is given that
dy
= x3+2x – 1 and that y = 13 when x = 2.
dx
Find y in terms of x.
Working:
Answer:
..................................................................
(Total 6 marks)
97
107. (a)
(b)
Find ò(1 + 3 sin (x + 2))dx.
The diagram shows part of the graph of the function f (x) = 1 + 3 sin (x + 2).
ò
The area of the shaded region is given by
a
0
f ( x)dx .
y
4
2
–4
–2
0
2
4
x
–2
Find the value of a.
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
98
108. 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
(b)
where x = –
1
.
2
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.
1
.
2
(7)
(Total 14 marks)
99
109. 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.
diagram not to
scale
P
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)
ds
.
dt
(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)
100
110. Consider the function f given by f (x) =
2 x 2 – 13 x + 20
,
( x – 1) 2
x ¹ 1.
A part of the graph of f is given below.
y
x
0
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) =
9 x – 27
(x – 1)3
,
x ¹ 1.
(3)
The second derivative is given by f ²(x) =
(d)
72 – 18 x
, x ¹ 1.
( x – 1) 4
Using values of f ¢(x) and f ²(x) explain why a minimum must occur at x = 3.
(2)
101
(e)
There is a point of inflexion on the graph of f. Write down the coordinates of this point.
(2)
(Total 10 marks)
111. The derivative of the function f is given by f ¢ (x) = e–2x +
1
, x < 1.
1- x
The graph of y = f (x) passes through the point (0, 4). Find an expression for f (x).
Working:
Answer:
…………………………………………........
(Total 6 marks)
102
112. Let f be a function such that
(a)
(b)
ò
3
0
f ( x ) dx = 8 .
Deduce the value of
3
(i)
ò
(ii)
ò ( f ( x) + 2) dx.
If
0
2 f ( x) dx;
3
0
ò
d
c
f ( x - 2)dx = 8 , write down the value of c and of d.
Working:
Answers:
(a) (i) ........................................................
(ii) .......................................................
(b) c = ......................., d = .......................
(Total 6 marks)
113. Let f (x) =
(a)
1
.
1+ x2
Write down the equation of the horizontal asymptote of the graph of f.
(1)
(b)
Find f ¢ (x).
(3)
103
(c)
The second derivative is given by f ²(x) =
6x 2 - 2
.
(1 + x 2 ) 3
Let A be the point on the curve of f where the gradient of the tangent is a maximum. Find
the x-coordinate of A.
(4)
(d)
Let R be the region under the graph of f, between x = –
1
1
and x =
,
2
2
as shaded in the diagram below
y
2
1
R
–1
– 12
1
1
2
x
–1
Write down the definite integral which represents the area of R.
(2)
(Total 10 marks)
114. The table below shows some values of two functions, f and g, and of their derivatives f ¢ and g ¢.
x
1
2
3
4
f (x)
5
4
–1
3
g (x)
1
–2
2
–5
f ¢(x)
5
6
0
7
g¢ (x)
–6
–4
–3
4
104
Calculate the following.
(a)
d (f (x) + g (x)), when x = 4;
dx
(b)
ò (g' ( x) + 6)dx .
3
1
Working:
Answers:
(a) ..................................................................
(b) ..................................................................
(Total 6 marks)
115. 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 < – 1
2
–1
2
–1<x<3
2
g²(x)
positive
0
negative
Use the information above to answer the following. In each case, justify your answer.
105
(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)
116. Let f (x) = (2x + 7)3 and g (x) cos2 (4x). Find
(a)
f ′ (x);
(b)
g′ (x).
Working:
Answers:
(a) .................................................
(b) .................................................
(Total 6 marks)
106
117. The following diagram shows a rectangular area ABCD enclosed on three sides by 60 m of
fencing, and on the fourth by a wall AB.
Find the width of the rectangle that gives its maximum area.
Working:
Answers:
........................................................
(Total 6 marks)
107
118. 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)
119. The graph of a function g is given in the diagram below.
The gradient of the curve has its maximum value at point B and its minimum value at point D.
The tangent is horizontal at points C and E.
(a)
Complete the table below, by stating whether the first derivative g′ is positive or negative,
and whether the second derivative g′′ is positive or negative.
Interval
g′
g′′
a<x<b
e<x<ƒ
108
(b)
Complete the table below by noting the points on the graph described by the following
conditions.
Conditions
Point
g′ (x) = 0, g′′ (x) < 0
g′ (x) < 0, g′′ (x) = 0
(Total 6 marks)
109
120. 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)
110
121. Consider the function ƒ : x a 3x2 – 5x + k.
(a)
Write down ƒ′ (x).
The equation of the tangent to the graph of ƒ at x = p is y = 7x – 9. Find the value of
(b)
p;
(c)
k.
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
(Total 6 marks)
111
122. 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)
112
123. 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)
113
124. 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)
125. Consider the function f (x) = 4x3 + 2x. Find the equation of the normal to the curve of f at the
point where x =1.
114
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
(Total 6 marks)
115
126. On the axes below, sketch a curve y = f (x) which satisfies the following conditions.
x
f (x)
−2 £ x < 0
0
–1
0 < x <1
1
1<x£2
2
f ′ (x)
f ′′ (x)
negative
positive
0
positive
positive
positive
positive
0
positive
negative
(Total 6 marks)
116
127. A Ferris wheel with centre O and a radius of 15 metres is represented in the diagram below.
π
Initially seat A is at ground level. The next seat is B, where AÔB =
.
6
(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
2π
. Find the height of A above the
3
ground.
(3)
The height, h metres, of seat C above the ground after t minutes, can be modelled by the
function
πö
æ
h (t) = 15 − 15 cos ç 2t + ÷ .
4ø
è
(d)
π
.
4
(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)
117
(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)
118
128. The diagram below shows part of the graph of the gradient function, y = f ′ (x).
y
p
(a)
q
x
r
On the grid below, sketch a graph of y = f ² (x), clearly indicating the x-intercept.
y
p
q
r
x
(2)
(b)
Complete the table, for the graph of y = f (x).
x-coordinate
(i)
Maximum point on f
(ii)
Inflexion point on f
(2)
(c)
Justify your answer to part (b) (ii).
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
..............................................................................................................................................
(2)
(Total 6 marks)
119
129. 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)
120