Cambridge IGCSE and O Level Additional Mathematics Workbook by Val Hanrahan, Jeanette Powell

8 Straight line graphs

8 The results of an experiment are shown in the table:

p2 4 6 8 10 12

q 7.1 10.0 12.2 14.1 15.8 17.3

The relationship between the two variables, p and q, is of the form q = Apb
where A and b are constants.

a) Show that the relationship may be written as lg q = b lg p + lg A.

b) What graph must be plotted to test this model?

c) Plot the graph on the axes provided and use it to estimate the values of b and A.

lg q

1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 lg p

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9 Circular measure

1 Express each angle in radians. Leave your answer in terms of π. 0.3c is another way of
writing 0.3 radians.
a) 60° c) 27°

b) 270° d) 108°

2 Express each angle in degrees. Answer to 1 d.p. where necessary.

a) π c) 0.3c
3
3π
b) 2π d) 5
9

3 Complete the table, which gives information about some sectors of circles.

Radius, r (cm) Angle at centre, Arc length, s (cm) Area, A (cm2)
θ (degrees)
12 150 20
8 12
75
15 100
30 60

4 Complete the table, which gives information about some sectors of circles.
Leave your answers as a multiple of π where possible.

Radius, r (cm) Angle at centre θ Arc length, s (cm) Area A (cm2)
8 (radians)
15 15 20π
2π 12 50
6 3

π
4

2π
5

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9 Circular measure a) Calculate the area of the sector OAB.
c) Find the area of the shaded segment.
5 Look at this diagram:

A

120°

O 5 cm B

b) Calculate the area of the triangle OAB.

6 The diagram shows two circles, each of radius 5 cm, with each one passing through
the centre of the other.

a) Calculate the area of the shaded region.

b) Calculate the perimeter of the shaded region.

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Questions 5–7

7 The shaded region in the diagram is the top of a desk that is to be covered in leather.
AB and DC are arcs of circles with centre O and radii and angle as shown.

A 5π B
6 c 100 cm
D O 50 cm

a) Work out the area of the desk to be covered. Give the answer in square metres.

b) (i) The leather is sold in rectangular strips 140 cm wide, and is sold in units of 10 cm.
What length must be purchased?

(ii) How much is wasted?

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10[Page 57]
Trigonometry

1 Find the length of x in each triangle. Give your answer to 2 d.p.
a)

x

27º
12 cm

b)

18 cm
53º

x

c)

5.6 cm

x 72º

2 Write the following in terms of a single trigonometric function.

a) sinθ b) sinθ c) cosθ × tanθ
cosθ tanθ

3 Simplify: b) tan2 θ (1 - sin2 θ)
a) cos2 θ (1 + tan2 θ)

54 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

4 In the diagram, OA = 1 cm, angle Questions 1–5
AOB = angle BOC = angle COD = 60° and
angle OAB = angle OBC = angle OCD = 90°. C
a) Find the length of OD.
B
D 60º 60º60º

O 1 cm A

b) Show that the perimeter of OABCD is (9 + 7 3) cm.

5 Work out the values of the following quantities without using a calculator.
Show your working carefully.
a) sin2 30° - cos2 30° tan2 30°

b) sin2 π4 - cos2 π4 tan2 π4

c) sin2 60° - cos2 60° tan2 60°

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10 Trigonometry

6 a) By plotting suitable points, draw the curve of y = cos x for 0°  x  360° on the grid below.

y

1

0.5

0 30º 60º 90º 120º 150º 180º 210º 240º 270º 300º 330º 360º 390º 420º 450º x
–0.5

–1

b) Solve the equation cos x = 0.4 for 0°  x  360° and illustrate the roots on your sketch.

c) Write down, without using your calculator, the solution to the equation cos x = -0.4
for 0°  x  360°.

7 Without using your calculator, write the following as fractions or using surds.

a) sin 60° b) cos 120° c) tan 150°

8 Solve the following equations for 0  x  2π without using your calculator.

a) sin θ = 3 b) cos θ = 3 c) tan θ = -1
2 2

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Questions 6–11

9 Without using a calculator show that: b) 3 cos 2 π = sin 2 π
a) sin2 30° + sin2 45° = sin2 60° 3 3

10 Solve the following equations for -360°  x  360°.

a) sin (x - 30°) = 0.6 b) cos (x + 60°) = 0.4 c) tan (x - 45°) = 1

11 Starting with the graph of y = sin x, state the transformations that can be used to sketch
each of the following curves.

a) y = sin 3x b) y = 2 sin 3x c) y = 2 sin 3x - 1

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10 Trigonometry

12 Apply these transformations to the graph of y = sin x. State the equation, amplitude
and period of each transformed graph.

a) A stretch of scale factor 1 parallel to the x-axis.
2

y

1.5

y = sinx

1

0.5

–90º 0 90º 180º 270º 360º 450º x

–1
–1.5

b) A translation of 90° in the negative x direction.

y

1.5

y = sinx

1

0.5

–90º 0 90º 180º 270º 360º 450º x

–1
–1.5

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Question 12

c) A stretch of scale factor 2 parallel to the y-axis followed by a translation of 1 unit
vertically downwards.

y y = sinx

1

0.5

–90º 0 90º 180º 270º 360º 450º x
–0.5

–1

–1.5

–2

–2.5

–3

d) A translation of 1 unit vertically downwards followed by a stretch of scale factor 2
parallel to the y-axis.

y y = sinx

1

0.5

–90º 0 90º 180º 270º 360º 450º x
0.5
–1
–1.5
–2
–2.5
–3
–3.5
–4
–4.5

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10 Trigonometry

13 State the transformations required, in the correct order, to obtain the graph below
from the graph of y = sin x.

y

3
2
1

–90° –45° 0 45° 90° 135° 180° 225° 270° 315° 360° x
–1

14 State the transformations required, in the correct order, to obtain the graph below
from the graph of y = tan x.

y

3 45° 90° 135° 180° 225° 270° 315° 360° x

1
–90° –45° 0

15 State the transformations required, in the correct order, to obtain the graph below
from the graph of y = cos x.

y

3

2
1.5

1

–90° –45° 0 45° 90° 135° 180° 225° 270° 315° 360° x

–1

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Questions 13–18

16 Simplify:

a) sinθ + cosθ c) cosθ 1
cosθ sinθ (1 + tan2θ )

b) (1 + tan2 θ ) d) 1 − sec 2 θ
(1 − sin2 θ ) 1 − cosec 2 θ

17 Solve cot x = sin x for 0°  x  360°.

18 Solve tan x + cot x = 2 sec x for 0  x  2π.

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11 Permutations and combinations

1 Without using a calculator, evaluate the following.

a) 6! b) 8! c) 5! × 8!
6! 6! × 3!

2 Simplify the following.

a) (n + 2)! b) (n − 2)! c) (2n + 1)!
n! n! (2n − 1)!

3 Factorise: b) (n + 1)! - (n - 1)!
a) 7! - 5!

4 How many different five letter arrangements can be formed from the letters
A, B, C, D and E if letters cannot be repeated?

5 Eight friends are going to the theatre together and they all have tickets for
adjacent seats in the same row. In how many ways can they be seated?

6 How many different arrangements are there of the letters in each word?

a) CHINA b) ISLAND c) DAUGHTER d) UNIVERSAL

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Questions 1–12

7 A security keypad has the numbers 1, 2, 3, 4 and the letters A, B, C, D on it.
In order to unlock it, a passcode with five of these numbers and letters is needed.
How many possible passcodes are there if:

a) there are no restrictions b) there must be at least two letters and at least
two numbers?

8 Without using a calculator, evaluate the following.

a) 10P2 b) 8P2 c) 6P2 d) 4P2

9 Without using a calculator, evaluate the following.

a) 10C8 b) 8C6 c) 6C4 d) 4C2

10 There are ten cleaners at a supermarket, one of whom is supervisor. If only 6 cleaners are
needed for each shift, including the supervisor, how many possible teams are there?

11 A quiz team of 4 people is chosen at random from 5 girls and 7 boys.
In how many ways can the team be chosen if:

a) there are no restrictions b) there must be equal numbers of boys and girls

c) there must be more boys than girls?

12 Four letters are picked from the word MAJESTIC. In how many of these choices
is there at least one of the letters A, E or I among the letters?

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12 Series The word ‘expand’
means ‘write out
In the questions on this page, simplify the terms in your expansions as far term by term’. So
as possible. expanding (x + 1)2
1 Expand the following binomial expressions: gives x2 + 2x + 1.

a) (1 + x)5 b) (1 - x)5 c) (1 + 2x)5

2 Expand the following binomial expressions:

a) (2x + y)3 b) (2x - y)3 c) (2x + 3y)3

3 Find the first three terms, in ascending powers of x, in the expansions of:

a) (3 - x)5 ( )b) 3x 5
2
−

4 Find the first three terms, in descending powers of x, in the expansion of the following:

2( )a) 1 4 3( )b) 2 4
x x
− −

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Questions 1–7

5 Find:
a) the coefficient of x2 in the expansion of (1 + 2x)6

b) the coefficient of x3 in the expansion of (1 + 2x)7.

6 a) Expand (1 - 2x)4.

b) Hence expand (1 + x)(1 - 2x)4.

7 Identify which of the following sequences are arithmetic, stating the common difference
where appropriate.

Sequence Arithmetic? Yes / No Common difference
a) 1, 5, 9, 13, …
b) 2, 4, 8, 16 ,…

c) 5, 3, 1, -1, …
d) 1, 1, 2, 2, 3, 3, …

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12 Series

8 The first term of an arithmetic sequence is 5 and the fourth term is 14. Find:
a) the common difference
b) the tenth term
c) the sum of the first ten terms.

9 An arithmetic progression of 15 terms has first term 7 and last term -49.
a) What is the common difference?

b) Find the sum of the arithmetic progression.

10 The 8th term of an arithmetic progression is 9 times the 2nd term.
The sum of the 2nd and 3rd terms is 10.
a) Write down a pair of simultaneous equations for the first term a and the common difference d.

b) Solve the equations to find the values of a and d.

c) Find the sum of the first 20 terms of the progression.

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Questions 8–13

11 A ball rolls down a slope. The distances it travels in successive seconds are
4 cm, 12 cm, 20 cm, 28 cm, etc., and are in an arithmetic progression.
How many seconds elapse before it has travelled 9 metres?

12 a) How many terms of the arithmetic progression 15, 13, 11, … make a total of 55?

b) Explain why there are two possible answers to this question.

13 Are the following sequences geometric? If so, state the common ratio and calculate the eighth term.

Sequence Geometric? Yes / No 8th term
a) 2, 6, 18, 54, …
b) 2, 6, 10, 14, …

c) 1, -1, 1, -1, …
d) 4, -12, 36, -108,

…
e) 8, 4, 2, 0, …
f) 1, 0, 0, 0, …

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12 Series

14 A geometric sequence has first term -2 and common ratio 2. The sequence has 10 terms.
a) Find the last term.

b) Find the sum of the terms in the sequence.

15 a) How many terms are there in the sequence 27, 9, 3, … 217?

b) Find the sum of the terms in this sequence.

16 The 1st term of a geometric progression is positive, the 5th term is 128 and the 11th term is 524 288.
a) Find two possible values for the common ratio.

b) Find the first term.

c) Find two possible values for the sum of the first seven terms.

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Questions 14–18

17 The first three terms of an infinite geometric sequence are 100, -60 and 36.
a) Write down the common ratio of the progression.

b) Find the sum of the first 10 terms.

c) Find the sum to infinity of its terms.

18 In each month, the growth of a bush is three-quarters of the growth the previous month.
The bush is initially 1.2 m tall and grows 12 cm in the first month.
a) What is the tallest the bush will grow?

b) After how many months is it within 5% of its maximum height?

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13 Vectors in two dimensions

1 Express the following vectors (i) in component form and (ii) in column vector form.

a) y 1 2 3 4x c) y

4 4
3 3
2 2
1 1

–1 0 −1−10 1 2 3 4 5 x

b) y 1 2 3 4 5 6x d) y

3 4 1 2x
2 3
1 2
1
−1−10
–5 –4 –3 –2 –1 0

a) b) c) d)

(i) Components
(ii) Column vectors

2 (i) Draw the following vectors on the grid below:

a) -3i b) 2i + 5j c) 4i + 2j d) -4i + 2j

(ii) Find the modulus of each vector. c) d)
a) b)

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Questions 1–5

3 For each of the following vectors (i) draw a diagram and (ii) find its magnitude.

a)  −2 b)  0 c)  5 d)  −1
 0  3  −2  1

4 a) A, B, C and D have coordinates (-1, 3), b) Write down the position vectors of the
(1, 5), (4, 3) and (1, 1). Draw quadrilateral points A, B, C and D.
ABCD on the grid.
A
y B
C
5 D
4
3 1 2 3 4 5 6 7x
2
1

–1–10

c) Write down the vectors:

(i) AB (ii) BC (iii) A D (iv) D C

5 ABC is a triangle with AB = 2i and AC = i + 2j.

a) Write down the vector BC. c) Describe the triangle ABC.

b) Sketch the triangle ABC. d) Find AD if ABDC is a parallelogram.

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13 Vectors in two dimensions

6 Simplify the following. b) 3(i + j) - 2(i - j) c) 2(2i - 3j) - 3(-2i + 3j)
a) (3i + j) – (3i - j)

7 p = 2i – j, q = i + 2j and r = -i + 3j

Find the following vectors and work out their lengths:

a) p + q + r b) 2p - 3q + 4r c) 3(p + 2q) - 2(2p - 3q)

8 The diagram shows an isosceles trapezium OPQR where OP =  4 and OQ =  9 .
 3  3

y

4 Q
3P

2

1 R
O
–1–10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 x

a) Write down the vectors PQ, QR and OR as column vectors.

PQ QR OR

b) Write down the vector RP as a column vector.

c) When produced, OP and RQ meet at the point B. Add B to the diagram,
and use one word to describe the triangle OBR.

9 Find unit vectors parallel to each of the following:

a) 5i + 12j b) i + j c)  3 d)  −2
 −5  −5

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Questions 6–10

10 A(1, 2), B(4, 6), C(8, 9) and D(5, 5) form the vertices of a quadrilateral.

a) Draw the quadrilateral.

y 1 2 3 4 5x

5
4
3
2
1

–5 –4 –3 –2 –1–10
–2
–3
–4
–5

b) Write the following sides of the quadrilateral as column vectors.

(i) AB (iii) AD

(ii) BC (iv) DC

c) Find the lengths of the sides of the quadrilateral.

d) Describe the quadrilateral as fully as possible.

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13 Vectors in two dimensions

11 C     O naka wants to measure the area and perimeter of a field that he is
buying, so he asks a surveyor to measure it for him. The surveyor then
–4i – 2j

presents him with this sketch.
D

          i + 5 j    Onaka doesn’t understand how vectors work, so he starts by adding up
i – 3j the four pieces of information on the diagram.

A 2i B

a) What answer does this give him? Explain the result.

b) What answer should he get for the perimeter if each unit represents 100 m?

c) Taking A to be at the origin, write down the position vectors of the other three
corners of the field.

d) Find the area of the field.

12 Jenny starts at O and travels north west for 2 hours at 6 km h-1, and then east until she arrives
at Q which is 20.1 km from O. Find the bearing of Q from O, giving your answer to the nearest
degree.

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14 Differentiation

In questions 1 to 5 differentiate the given functions with respect to x.

1 a) y = x6 b) y = 3x2 c) y = -2x d) y = -4

1 2 c) y = 2 4 x d) y = 2x
3
2 a) y = x 3 b) y = -3x 3

3 a) y = 4x3 - 3x4 b) y = 7x2 + x - 5 c) y = 2x3 - 3x2 + 4

4 a) f(x) = 2 b) f(x) = 2 x − x x c) f(x) = 3 x 1 - 1 x − 1
x3 2 2 2 2

5 a) y = (x - 1)(2x + 1) b) y = 3x2 - 5 c) y = x2(2x - 3)
x

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14 Differentiation

6 a) Draw the curve y = (x + 1)(x - 2) b) Find the gradient of the curve at the points of
intersection with the x and y axes.

y 1 2 3 4x

10
9
8
7
6
5
4
3
2
1

–3 –2 –1–10
–2

7 a) Draw the curve y = x2 - 4 and the line b) Use algebra to find the coordinates of the
y = 3x on the same axes. points where the two graphs intersect.

y c) Find the gradient of the curve at the
points of intersection.
13
12 1 2 3 4 5x
11
10

9
8
7
6
5
4
3
2
1

–2 –1–10
–2
–3
–4
–5

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Questions 6–8

8 Complete the following for equations a and b.

a) y = x3 − 3x2 − 9x + 15 b) y = x4 − 8x2 + 16
(i) Find dy and the values of x for
dx (i) Find dy and the values of x for which
which dy = 0 dx
dx dy = 0
dx

(ii) Classify the points on the curve with (ii) Classify the points on the curve with
these x-values these x-values

(iii)  F ind the corresponding y-values (iii)  F ind the corresponding y-values

(iv)  S ketch the curve. (iv)  S ketch the curve.

y y

xx

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14 Differentiation

9 The graph of y = x2 + ax + b passes through the point (-1, 10) and its gradient at that point is -7.
a) Find the values of a and b.

b) Find the coordinates of the stationary point of the curve.

10 a) Find the stationary points of the function y = (x + 1)2(x - 1) and Classifying a stationary
classify them. point means determining
whether it is a maximum,
a minimum or a point of
inflection.

b) Sketch the curve.

y

x

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Questions 9–11

11 For each of the following curves

a) y = 3x3 - 4x - 4 b) y = x4 - 6x2 + 8x - 5

(i) find dy and d2y (i) find dy and d2y
dx dx2 dx dx2

(ii) find any stationary points (ii) find any stationary points

(iii) use the second derivative test to determine (iii) u se the second derivative test to
  their nature determine their nature

(iv) sketch the curve. (iv) sketch the curve.

y y

xx

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14 Differentiation

12 An open tank with a capacity of 32 m3 is to be constructed with a square base and vertical sides.
a) Find an expression for the height h m of the tank in terms of the length x m
of a side of the base.

b) To reduce costs, it will be constructed using the smallest possible area of sheet metal. Find its
dimensions and use the second derivative test to show that your answer is a minimum.

13 A curve has equation y = x2 - 7x + 10.

a) Find the gradient function dy .
dx

b) Find the gradient of the curve at the point P(4, -2).

c) Find the equation of the tangent at P.
d) Find the equation of the normal at P.

80 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 12–14

14 The diagram shows a sketch the curve y = x4 - 4x2.

y

5
4
3
2
1

–4 –3 –2 –1–1 1 2 3 4 x

–2
–3
–4

–5

a) Differentiate y = x4 - 4x2.

b) Find the equations of the tangent and normal to the curve at the point (2, 0).

c) Find the equations of the tangent and normal to the curve at the point (-2, 0).

d) State the equations of the tangent and normal to the curve at the point (0, 0).

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14 Differentiation

15 a) The equation of a curve is y = x4 + x3. Find dy and d2y .
dx dx2

b) Find the coordinates of the stationary points on the curve.

c) Classify the stationary points, using the second derivative test where possible.

d) Sketch the curve y = x4 + x3 labelling the stationary points.

y

x

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Questions 15–18

16 Differentiate each of the following functions with respect to x.

a) y = 2 sin x + 3 cos x d) y = ln 3x

b) y = 2 tan x - 3 cos x e) y = 4ex

c) y = 3 ln x f) y = 3e-x

17 Use the product rule to differentiate each of the following functions with respect to x.

a) y = xex d) y = x tan x

b) y = xe-x e) y = x2 sin x

c) y = x ln x f) y = x3 cos x

18 Use the quotient rule to differentiate each of the following functions with respect to x.

a) y = ex d) y = ex
sin x
x

b) y = x e) y = ln x
ex x

c) y = sin x f) y= x
ex ln x

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14 Differentiation

19 Use the chain rule to differentiate each of the following functions with respect to x.

a) y = sin4 x c) y = 1 + 2x

b) y = tan2 x d) y = 3 1 + 3x

20 Use an appropriate method to differentiate each of the following functions with respect to x.

a) y = ex cos x c) y = 1 tan θ
- cos θ

b) y = ln x d) y = (1 − cos θ)2
ex

21 You are given that y = u and that u = 2x2 + 1.
a) Show that the point (2, 3) lies on the graph of y against x.

b) Find the values of du , dy and dy at the point (2, 3).
dx du dx

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15 Integration

1 Find y for each of the following gradient functions:

a) dy = 4x - 5 b) dy = 2x2 - 5x - 4 c) dy = (x + 2)(2x - 3)
dx dx dx

2 Find f(x) for each of the following gradient functions.

a) f´(x) = x3 - 3 b) f´(x) = 4 + 3x - x2 c) f´(x) = (2x + 3)2

3 Find the following indefinite integrals.

∫a) (4x + 3)dx b) ∫(2x4 - 1)dx ∫c) (x3 - 2x)dx

4 Find the following indefinite integrals.

∫a) (2x - 3)2dx ∫b) (x + 3)(x - 2)dx ∫c) (1 - 2x)2dx

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15 Integration

5 Find the equation of the curve y = f(x) that passes through the specified point
for each of the following gradient functions.

a) dy = 4x + 1; (1, 3)
dx

b) dy = 1 - 2x3; (4, 0)
dx

c) f´(x) = (3x - 2)2; (0, -4)

d) f´(x) = (x - 2)(x + 3); (-1, -2)

6 Curve C passes through the point (4, 10); its gradient at any point is given by dy = 3x2 - 6x + 1.
dx

a) Find the equation of the curve C.

b) Show that the point (2, -12) lies on the curve.

86 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 5–7

7 Evaluate the following definite integrals. Do not use a calculator.

∫a) 34xdx ∫  f) 0 (5 - 4x)dx
1  -1 

∫  b) 5 6x2dx ∫g) 3(2x + 1)2dx
-1  0 

c) ∫ - 12(x - 3)dx ∫  h) 2 (2x - 3)2dx
-2 

∫  d) 2 (x2 - 3x)dx ∫  i) 1 (x + 1)(2x - 1)dx
-1  -1 

∫e) -2 (x3 + x)dx ∫j) 3x(x + 1)(x + 2)dx
-4  1 

Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook 87

15 Integration

8 Find the area of each of the shaded regions. Do not use a calculator.

a) y c) y

y = x3

y = 4 – x2

–2 –1 0 1 2x
–1

0 1 2x

–1

b) y d) y

20 y = x2 + 4x + 3 4 y = x3 – 2x2 – 2x + 2
10
–1 0 3
2
1

1 2x –1 0 0.5 x
–1

–2

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Questions 8–10

9 The graph shows the curve y = x2 - 4x + 3. Calculate the area of the shaded region.

y 1 2 3 4x

8
7
6
5
4
3
2
1

–1 –10

10 The graph shows the curve y = x3 - 5x2 + 6x.

y

3

2
1A

–1 0 1 2 B3 4 x
–1

a) Find the area of each shaded region, A and B. Do not use a calculator.

b) State the total area enclosed between the curve and the x-axis for 0  x  3.
Do not use a calculator.



Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook 89

15 Integration

11 a) Sketch the curve y = (x + 1)(x - 1)(x - 3) and shade the areas enclosed between
the curve and the x-axis.

y

x

b) Find the total area enclosed between the curve and the x-axis. Do not use a calculator.

12 a) Sketch the curve y = (x + 1)2(x - 2) and shade the areas enclosed between
the curve and the x-axis.

y

x

b) Find the area you have shaded. Do not use a calculator.

90 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 11–14

13 Find the following indefinite integrals.

∫a) 1 3  dx ∫b) e2x - 3dx
2x - ∫d) sin(2x - 3)dx

∫c) (2x - 3)3dx

∫e) cos(2x - 3)dx ∫f) sec2(2x - 3)dx

14 Evaluate the following definite integrals. Do not use a calculator.

∫ a) 42 1  dx ∫b) 14 e2x + 1dx
1 2x +

∫ c) 4(2x + 1)3dx d) ∫π2 sin a2x + π4 b dx
1 
0

e) ∫π2 cos a2x + π4 b dx ∫f) π cos ax2 - π4 b dx
2
0
- π
2

Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook 91

16 Kinematics

1 In each of the following cases:
(i) find expressions for the velocity and acceleration at time t
(ii) find the initial position, velocity and acceleration
(iii) find the time and position when the velocity is zero.

a) s = 3t2 - t - 4
(i)

(ii)

(iii)

b) s = 4t - t3
(i)

(ii)

(iii)

c) s = 5t4 - 2t2 + 3
(i)

(ii)

(iii)

92 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 1–3

2 A particle is projected in a straight line from a point O. After t seconds its displacement,
s metres, from O is given by s = 3t - t3.
a) Find expressions for the velocity and acceleration at time t.

b) Find the time when the particle is instantaneously at rest.

c) Find the velocity when t = 2 and interpret your result.

d) Find the initial acceleration.

3 A ball is thrown vertically upwards and its height, h metres, above the ground after
t seconds is given by h = 2 + 10t - 5t2.
a) From what height is the ball projected?
b) When is the ball instantaneously at rest?
c) What is the greatest height reached by the ball?
d) After what length of time does the ball reach the ground?
e) At what speed is the ball travelling when it reaches the ground?

Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook 93

16 Kinematics

4 A hot air balloon is rising at a rate of 0.6 m s-1 and is at a height of 28 m when it starts
to experience a downward acceleration of 0.2 m s-1.
a) Find the height reached by the balloon before it starts to descend.

b) How long does the balloon take to return to the ground?

c) At what speed is the balloon travelling when it reaches the ground?

5 The height of a ball thrown up in the air is given by h = 10t - 5t2 + 2, where h is the height
above ground level.
a) Find an expression for the velocity of the ball.

b) Find the maximum height reached by the ball and the time when this occurs.

c) Find the acceleration of the ball.

d) Find the time taken for the ball to reach the ground.

94 Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook

Questions 4–7

6 Find expressions for the velocity, v, and displacement, s, at time t in each of the following:
a) a = 2 + 4t; when t = 0, v = 3 and s = 0.

b) a = 6t2 - 2t; when t = 0, v = 6 and s = 2.

c) a = 4; when t = 0, v = 2 and s = 3.

7 The acceleration of a particle a m s-2, at time t seconds is given by a = 5 - 4t.
When t = 0 the particle is moving at 3 m s-1 in the positive direction,
and is 2 m from the point O.
a) Find expressions for the velocity and displacement in terms of t.

b) Find when the particle is instantaneously at rest and its displacement from O at that time.

Photocopying prohibited Cambridge IGCSE™ and O Level Additional Mathematics Workbook 95

Reinforce learning and deepen understanding
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