Shu OPT matheamtics Book 8 2077 final com (1)

3. (a) In P (a, b) P' (–a, b), what is the name of reflection.

(b) What is the image of P(x, y) after reflection through x-axis?

Section B

4. Draw the image of the following figures after reflection through the
line l.

(a) (b) 
A P

B C Q R


(c) (d) Y
M 
N

P

O W X

(e)

MN

A 

5. Reflect MAN and OMN given in the graph along side, in the following
axes.

(i) In x-axis (ii) In y-axis

(a) (b)

Y Y
M M

A N N
X' X

Y' O X
(d) Y

(c)

Y

XM O M
X

X' O A X

A N Y' N


Infinity Optional Mathematics Book - 8 151

6. Find the coordinates of the image of the following points under the
reflection in X-axis.

a) A (4, 5) b) B (–4, 5)

c) C (–3, –5) d) D (7, –3)

7. Find the coordinates of the image of the following points under the
reflection in Y-axis.

a) M (–2, 3) b) N (4, 5)

c) P (–3, –4) d) (2, –3)

8. Obtained the coordinates of the images of the following points under
the reflection in the line y = x.

a) D (10, 8) b) E (8, – 7)

c) F (7, 2) d) D (–4, –3)

9. Find the coordinates of the image of the following points under the
reflection in the line y = – x.

a) G (–9, –6) b) H (6, 8)

c) I (–4, 5) d) J (5, –3)

Section C
10. A (2, 3), B (–3, 4) and C (1, –2) are the vertices of a ∆ABC. Find the image of

∆PQR under the reflection about.
(i) X-axis (ii) Y-axis. Present the ∆ABC and its image ∆A'B'C' on the same

graph?
11. Find the image of the ∆PUN where vertices are P (–1, 5), U (4, 9) and N

(10, 3) under the reflection about (i) X-axis, (ii) Y-axis. Present the ∆PUN and
its image ∆P'U'N' on the graph.
12. Obtain the image of the ∆SUN where vertices are S (4, 2), U (–3, 6) and
N (–1, –4) under the reflection about (i) the line y = x (ii) the line y = –x.
Present the ∆SUN and its image ∆S'U'N' on the same graph.
13. P (–2, 5), Q (3, 8) and R (5, 2) are the vertices of a ∆PQR. Find the image of
∆PQR under the reflection in the line (i) y = x (ii) y = –x. Present the triangle
PQR and its image ∆P'Q'R' on the graph.

152 Infinity Optional Mathematics Book - 8

9.2 Rotation A
Review:

Look at the following figures

A'

80º 180º

O M A M'
Fig (i) Fig (ii)

i) How far is the point A' from A in figure (i).
ii) What is the relation between AM and AM' in figure (ii).
iii) Are OA, OA', AM and AM' equal?

The figure (i) describes the rotation of a point A about the point O by 80º in
anticlockwise direction, so that the image of A is A'.
Similarly, M' is the image of M after 180º rotation.
In the above figures, AO = A'O and MA = AM', MA and AM' are related to each
other by
i) Centre A ii) Angle of rotation 180º (iii) Direction (Clock wise).
Thus, rotation is a type of transformation in which every point of object turns
through the same angle about a fixed point in a direction. The fixed point is known
as centre of rotation and the angle of turning is called the angle of rotation.
If the rotation is anticlock wise, then the angle is positive and the rotation is
clockwise then the angle is negative.
Rotation using Coordinates:
In this case, we shall discuss about the rotation of figures through some special
angles such as 90º and 180º in anticlockwise and clockwise directions about the
centre at origin.
The rotation through 90º is also called a quarter-turn and the rotation through 180º
is called a half turn.

Infinity Optional Mathematics Book - 8 153

Rotation through 90º in anticlockwise about the centre at origin.
Y Scale : 1 box = 1 units

A'(–4, 3) A (3, 4)

X' O X

Y'

In the above graph, the point A(3, 4) is in the 1st quadrant. It is rotated through
positive 90º about centre origin i.e. R AOA' = 90º and OA = OA'. The image A' is formed

in the second quadrant and the coordinates of the image A' is (–4, 3).

A(3, 4) A'(–4, 3)

Thus, in rotation through +90°

P(x, y) P'(– y, x)

Rotation through (–90º) about centre origin (in clockwise direction)
Y Scale : 1 box = 1 units
A (4, 5)

X' O X

A'(5, –4)
Y'

In the above graph, the point A(4, 5) is in the first quadrant. It is rotated through
negative 90º about centre origin. R AOA' = 90º in clockwise and OA = OA'. The

image A' is formed in the second quadrant and the coordinates of the image is

A'(5,– 4)

A(4, 5) A'(5, –4)

Thus, in rotation through –90°.

P(x, y) P'(y, –x)

Rotation through (± 180º) about centre origin (anticlockwise and clockwise
both)

When a point is rotated through 180º in anticlockwise or in clockwise direction
about origin as the centre of rotation, the coordinates of the image are the same.

154 Infinity Optional Mathematics Book - 8

Y Scale : 1 box = 1 units

A (6, 3)

180º

X' O X
A'(–6, –3)
– 180º

Y'

In the above graph, the point A(6, 3) is in the first quadrant. It is rotated trough
180º or – 180º about centre origin i.e. AOA' = ±180º and OA = OA'. The image A' is
formed in the third quadrant and the coordinates of the image is A'(–6, –3).

A(6, 3) A'(–6, –3)

Thus,in rotation through ±180°

P(x, y) P'(–x, – y)

WORKED OUT EXAMPLES

1. Find the image of ∆ABC under the rotation A C
through (+90º) about the given centre of rotation C
O. B
O
Solution: Here,
A
O is the centre of rotation OA, OB and OC are joined.
C'
Taking OA, OB and OC as radii with centre O, drawn B
arcs through A, B and C. At O, making  AOA'
= 90º,  BOB' = 90º and COC' = 90º meeting O
B'
respective arcs at A', B' and C'. Joined A', B'

and C'. Then ∆A'B'C' is formed which is the

image of ∆ABC. A'


Infinity Optional Mathematics Book - 8 155

2. Find the image of the point M(2, 3) under the rotation about the origin
through the angle (i) + 90º (ii) – 90º ( iii) 180º.

Solution: Here,

M(2, 3) is an object point.

(i) In rotation through +90°

P(x, y) P'(–y, x)

\ M(2, 3) M'(–3, 2)

(ii) In rotation through – 90°

P(x, y) P'(y, –x)

\ M(2, 3) M'(3, – 2)

(iii) In rotation through 180°,

P(x, y) P'(–x, –y)

\ M(2, 3) M'(–2, –3)

3. P (–4, 2), Q(3, 7) and R(–1, –6) are the vertices of ∆PQR. Find the
coordinates of its image under the rotation through 90º in clockwise
direction about centre origin. Present the ∆PQR and its image on the
same graph paper.

Solution: Here,

P (–4, 2), Q(3, 7) and R(–1, –6) are the vertices of ∆PQR.

In rotation through – 90°. Y Scale : 1 box = 1 units
Q
P(x, y) P'(y, –x)

Now, P'(2, 4) R' P P'
P(–4, 2) Q' (7, –3) X' O
Q (3, 7) X
R (–1, –6) R' (–6, 1) Q'

∴ P' (2, 4), Q' (7, –3) and R' (–6, 1) R
∆PQR and ∆P'Q'R' is shown on the Y'
same graph.

156 Infinity Optional Mathematics Book - 8

Exercise 9.2

Section 'A'

1. (a) What do you mean by rotation?

(b) Define centre of rotation and angle of rotation.

(c) What are the two direction of rotation? Write them.

2. (a) If P(a, b) P'(–b, a), then write the angle of rotation and centre
of rotation.

(b) If M(m, n) M' (–m, –n) then write the angle of rotation and

centre of rotation.

3. In the given figures, measure the RAOA' and RBOB' with the help of
protractor. Also write their values.

B
A' A

O O

B'

Section 'B'
4. Draw the images of the following figures rotating through the given

angles about the given centre of rotation.

(a) P b) A

B

QR C
O D

Rotation through (+60º) O
Rotation through (–50º)

Infinity Optional Mathematics Book - 8 157

c) M d) D H
C

NO J O

O

Rotation through (+90º) Rotation through (–80º)

5. Find the image of the following points under the rotation through
+90º about the centre origin.

(a) A (4, 5) (b) B(–2, 3)

(c) C (–3, –5) (d) D(4, –1)

6. Find the image of the following points under the rotation through
–90º about the origin.

(a) M(3, 4) (b) N(–2, 4)

(c) Q (–3, –2) (d) R (6, – 1)

7. Find the image of the following points under the rotation through
180º about centre origin.

(a) E(–5, –2) (b) F(–3, 2)

(c) G(7, – 1) (d) H(–8, – 2)

Section 'C'

8. ∆ABC has vertices A(3, 1), B(5, 0) and C(2, – 2). Rotate the vertices of
the ∆ABC about the origin through

(i) 90º in the anticlockwise direction

(ii) 90º in the clockwise direction

(iii) 180º

Find the image of the vertices of the ∆ABC. Draw the ∆ABC and its images on
the graph for each case.

9. A(2, –3), B(–1, 1), C(4, 1) and D(3, –3) are the vertices of a parallelogram
ABCD. Find the coordinates of the vertices of image parallelogram
A'B'C'D' under the rotation about the origin through

(i) quarter turn in the anticlockwise direction;

(ii) half turn

(iii) quarter turn in the clockwise direction

Draw the parallelogram ABCD and it images on the graph for each case.

158 Infinity Optional Mathematics Book - 8

9.3 Translation (Displacement)

Review:

A A'

B B'

C C'

In the given figure, object ∆ABC is displaced to image ∆A'B'C' in which AA' = BB' =
CC' and AA'//BB'//CC'.

(i) How do the ∆ABC and ∆A'B'C' relates?

These two triangles are congrument. Thus, it is the transformation of
geometrical figures in which each vertex of a figure is displaced by equal
distance to the same direction. It is also called displacement.

The displacement of a geometrical figure has magnitude as well as direction
So, it is a vector quantity.

Translation using coordinates
→

Let the coordinates of the point A be (a, b). So, OA =

(→

The directed line segment OA is used to translate
( a Scale : 1 box = 1 units
b
(
Y
(
the point P(x, y), it shifts the point P by a units P'(x', y')
b
in the horizontal direction and then b units in P' P(x, y)

vertical direction to reach the point P'(x', y'). Aa

∴ x' = x + a ... (i) and y' = y + b ... (ii) X' b

Oa B X
Y'
The relation (i) and (ii) are called equat→ion of

translation. The directed line segment OA is

called the translation vector. Translation vector is denoted by T.

→

∴ T = OA =
( a
b

Hence,

(a

P (x, y) T = b P' (x + a, y + b)

Infinity Optional Mathematics Book - 8 159

WORKED OUT EXAMPLES

1. Find the image of quadrilateral PQRS under the translation
vector →a .

Solution: Here, P

Quadrilateral PQRS is the object figure. Now, →a S Q

P P'

R

S Q S' Q'

R R'

P'Q'R'S' is the image of quadrilateral PQRS such that PP' = QQ' = RR' = SS' =
→a and PP'//QQ'//RR'//SS'//→a .

2. Find the image of the point A(3, 4) under the translation vector

( ( a (–1
(T) = b = 2.
(
Solution: Here, (

a –1((
((
( ( Translation vector (T) = b = 2
A(3, 4) is an object point.

(a

We have, P (x, y) T = b P' (x + a, y + b)

( T= –1
2
A (3, 4) A' (3 – 1, 4 + 2) = A' (2, 6)

Hence, A'(2, 6) is the image of the point A(3, 4)

3. P(– 2, 2), Q(2, 2) and R(2, 6) are the vertices of a ∆PQR. Find the image

(3

of ∆PQR, under the translation vector T = –2 . Present the ∆PQR and
its image on the same graph paper.
Solution:
Here, P (–2, 2) , Q(2, 2) and R(2, 6) are the vertices of a ∆PQR

160 Infinity Optional Mathematics Book - 8

3a(

( ( Translation vector (T) = –2 = b (
We have, P (x, y)
3 ((P' (x + a, y + b) Y Scale : 1 box = 1 units
P' (–2 + 3, 2 – 2) = P' (1, 0) R
( Now, P (–2, 2) T = –2 R'
Q
Q (2, 2) Q' (2 + 3, 2 – 2) = Q' (5, 0)

R (2, 6) R' (2 + 3, 6 – 2) = R' (5, 4) P

∴ P' (1, 0), Q(5, 0) and R'(5, 4) are the O P' Q' X
coordinates of the vertices of image X' Y'
∆P'Q'R'.

∆ PQR and image ∆P'Q'R' are shown on a graph paper.

Exercise 9.3

Section 'A'

1. (a) Define translation?

(a

(b) In formula, P (x, y) T = b P' (x + a, y + b) , what do a and b represent?
(c) Does the translation is a vector quantity?

2. (a) Under the translation 'T' a point A(4, 2) A'(7, 4). Write down
the components of T.

(b) If a point B(3, 1) is translated by 5 units right and 4 units up then write
the translation vector.

Section 'B'
3. Translate the following objects in the given direction and magnitude.

(a) A (b) A 5 cm

3 cm B
BC C

Infinity Optional Mathematics Book - 8 161

(c) D C (d) T
A 6 cm
V
B SU

4 cm

(4. ( –1
(a) Find the image of the following points under the translation( 2 .

i) A(3, 5) ii) M(5, – 4) iii) N(–2, 5) iv) K(–3, – 4) (

(b) Find the images of A (2, 3), B(–2, 4), C(5, – 7) and D(–2, –3) after(
(
( –3

translation vector (T) = 4 .

5. (a) Find the images of the points under the translation with equation
x' = x + 2, y' = y + 3.

i) D(–3, 1) ii) E(2, 5) iii) F(–4, –1) iv) G(7, – 5)

(b) Find the points which is mapped to A' (3, 4) under the translation with
equation x' = x + 1, y' = y + 3.

(c) Find the point which is mapped to B'(5, – 3) under the translation
x' = x – 3 and y' = y + 1.
Section 'C'

6. The vertices of ∆ABC are A(1, 1), B(6, 3) and C(4, 7). Find the coordinates of
4

(the vertices of the image ∆A'B'C' under the translation 3 . Present ∆ ABC

and its image on the same graph paper.

7. On a squared paper, draw a triangle having vertices P(6, 2), Q(8, 6) and
R(2, 8). Translate it by 2 units right and 3 units up and write down the
coordinates of image. Represent this translation in a graph.

8. A quadrilateral ABCD has vertices, A(1, 3), B(2, 5), C(1, 5) and D(–1, 5). Find
3

(the image of quadrilateral ABCD under the translation vector (T) = 2 .

Present the quadrilateral ABCD and its image on the same graph paper.

9. A rectangle GITA has vertices, G(3, 1), I(5, 1), T(5, 4) and A(3, 4). Find the
–3

(image of rectangle GITA under the translation vector (T) = 2 . Present the

quadrilateral GITA and its image on the same graph paper.

162 Infinity Optional Mathematics Book - 8

9.4 Enlargement

In the given figure, ∆A'B'C' is the enlargement A'
of ∆ABC and the measure of distance from O
to the points on A'B'C' is twice the measure A B'
of distance from O to a points on ABC.
B
Now, OA' = 2OA O C C'
OB' = 2OB

OC' = 2OC

Where O is the centre of enlargement and 2 is the scale factor. The above relation
can be written as.

OA' = OB' = OC' = 2
OA OB OC

This enlargement is denoted by E[O, 2]

Enlargement using coordinates

The geometrical figures can be enlarged with the

i) Centre of enlargement at O(0, 0) and the scale factor 'k' and

ii) Centre of enlargement at G(a, b) and the scale factor 'k'.

Here, we will discuss, if the centre of enlargement at O(0, 0) and having scale factor
'K'.

Let, a point B(2, –1) is transformed with the centre of enlargement O(0, 0) and scale
factor (K) = 2 then the image of the point B will be as

B(2, – 1) B'(2 × 2, – 1 × 2) = B' (4, – 2)

Hence, the coordinates of the point P(x, y) after enlargement using scale factor k
will be (kx, ky)

∴ P(x, y) E[(0, 0), k] P'(kx, ky)


Infinity Optional Mathematics Book - 8 163

WORKED OUT EXAMPLES
1. Find the image of ∆PQR under the enlargement with centre as O and

scale factor (k) = 3
P

Q

OR
Solution: Here,
Centre of enlargement (O) and scale factor (k) = 3

P'

Q'

P
Q R'

R
O
∴ P'Q'R' is the image of ∆PQR under the enlargement O and scale factor k = 3.

164 Infinity Optional Mathematics Book - 8

2. Find the image of the point G(3, 2) under the enlargement E[(0, 0), 2].
Solution: Here,
G(3, 2) is an object point.
We have,
Centre of enlargement (0, 0) and scale factor (k) = 2 then

P(x, y) E[(0, 0), k] P'(kx, ky)

Now,

G(3, 2) G'(2 × 3, 2 × 2) = G'(6, 4)

∴ G (3, 2) G'(6, 4)

3. A (2, 1), B(0, 3) and C(–2, 1) are the vertices of a triangle. Find the
coordinates of the image ∆A'B'C' under the enlargement with scale
factor 2 and centre at origin. Present the ∆ ABC and its image on the
same graph paper

Solution: Here,

A (2, 1), B(0, 3) and C(–2, 1) are the vertices of a triangle ABC. Centre of
enlargement (0, 0) and scale factor (k) = 2

We have,
P(x, y) E[(0, 0), k] P'(kx, ky)

Now,

A(2, 1) A'(2 × 2, 2 × 1) = A'(4, 2)

B (0, 3) B' (2 × 0, 2 × 3) = B'(0, 6)

C (–2, 1) C' (2 × –2, 2 × 1) = C' (–4, 2)

∴ A' (4, 2), B'(0, 6) and C'(– 4, 2) are the coordinates of the vertices of image ∆
A'B'C'. ∆ABC and image ∆A'B'C' are shown on the same graph paper.

Y Scale : 1 box = 1 units

B'

B

C' A'
AX
X' C
O

Y'

Infinity Optional Mathematics Book - 8 165

Exercise 9.4

Section 'A'
1. (a) Define enlargement?
(b) Write the difference between an enlargement and reduction.
2. (a) In formula, P(x, y) E[(0, 0), k] P'(kx, ky) what do (0, 0) and 'k' represent?
(b) Are the object and image congruent in enlargement.

Section 'B'
3. Find the image of the following figures after enlargement taking

centre at O and the following scale factor (k).

(a) A (b) P

Q

BC S

O
OR

[k = 2] [k = 3]

(c) D E (d) W X

GF ZY
O O

[k = 2] [k = 3]

4. If E[(0, 0), 2] is the enlargement with centre origin and 2 as the scale factor,

then find the image of the following points under the enlargement.

(a) E(4, 2) (b) F(–3, 3)

(c) G(5, – 3) (d) H(–3, –6)

5. Find the image of the points A(2, 0), B(3, 1), C(5, 7) and D(–4, –2) under the

enlargement with centre (0, 0) and scale factor –2.

Section 'C'

6. A(3, 1), B(5, 0) and C(2, –2) are the vertices of ∆ABC. Find the image of the

∆ABC under the enlargement E[(0, 0), 2]. Present the ∆ABC and its image

∆A'B'C' on the same graph paper.

7. P(2, –3), Q(–1, 1) and R(1, 3) are the vertices of ∆PQR. Find the image of ∆PQR

under the enlargement E[(0, 0), 3]. Present the ∆PQR and its image ∆P'Q'R'

on the same graph paper.

8. D(0, – 1), I(1, 3), N(2, 2) and A(1, –2) are the vertices of a parallelogram DINA.

Find the image of parallelogram DINA under the enlargement E[(0, 0), 2].

Present the parallelogram DINA and its image on the same graph paper.

166 Infinity Optional Mathematics Book - 8

UNIT

10 STATISTICS

10.1 Measure of central tendency

Review:

30 students obtained following marks in a class test out of full marks 20 are as
follow.
7, 9, 12, 15, 18, 7, 15, 16
8, 13, 11, 12, 14, 15, 16, 17
18, 17, 7, 9, 8, 17, 18 15
9, 12, 13, 7, 14, 16
Study on the above data and discuss the following questions.
(i) What are the minimum and maximum marks in a class test?
(ii) Can you draw the frequency distribution table using tally bars?
(iii) What is the difference between raw data and arrayed data?
(iv) Can you draw the cumulative frequency (c.f) table with the help of frequency

distribution table.
After discussing on the above questions may be you have a question.
What is statistics?
Statistics is the branch of mathematics in which facts and information are collected,
sorted, displayed and analysed. Statistics are used to make decision and prediction
about the future plans and policies.
Measures of Central tendency:
The measure of Central tendency gives a single central value that represents the
characteristics of entire data. A single central value is the best representative of
the given data towards which the values of all other data are approaching.
Average of the given data in the measure of central tendency. There are three
types of averages which are commonly used as the measure of central tendency.
They are
(i) mean (ii) median and (iii) mode.

Infinity Optional Mathematics Book - 8 167

Arithmetic mean

Arithmetic mean is the most common type of average. It is the number obtained
by dividing the sum of all the items by the number of items. Arithmetic mean is
denoted by (X).

∴ Arithmetic mean (X) = Sum of all the items
Number of items

(a) The mean for individual data (non-repeated data)

5, 7, 13, 15, 10, 18, 9 are the given data.

Now, mean (X)= 5 + 7 + 13 + 15 + 10 + 18 + 9 = 77 = 11
7 7

Let x1, x2, x3, ............ xn the N values of a variable

ΣX = x1 + x2 + x3 + ...... xn

\ Arithmetic mean (X) = ΣX
N

(b) The mean of discrete data (mean of individual repeated data)

x 10 15 20 25 30
f 34642

Mean (X) = Σfx
N
Where N = Σf

fx = f × x (The product of each item and its corresponding frequency

WORKED OUT EXAMPLES

1. Find the arithmetic mean of 40, 100, 60, 80, 120, 160.
Solution: Here,
The given data are 40, 100, 60, 80, 120, 160
The sum of terms (ΣX) = 40 + 100 + 60 + 80 + 120 + 160 = 560
The number of terms (N) = 6

168 Infinity Optional Mathematics Book - 8

By formula,

Arithmetic mean (X) = ΣX = 560 = 93.33
N 6
2. Find the mean of the following data

Marls obtained 10 15 20 25 30
No. of students 25364

Solution:

Tabulating the given data in ascending order for the calculation of mean

Marks obtained (x) No. of students (f) fx
20
10 2 75
60
15 5 150
120
20 3 Σfx = 425

25 6

30 4

Σf = N = 20

By formula Mean (X) = Σfx = 425
N 20

= 21.25

Hence, Mean (X) = 21.25
3. If the mean of the given data is 46, find the value of m.
20, 46, 34, 38, 2m, 42, 58, 66

Solution: Here

Mean (X) = 46

The sum of the terms (ΣX) = 20 + 46 + 34 + 38 + 2m + 42 + 58 + 66

= 2m + 304

Total numbers of terms (N) = 8

By formula

Mean (X) = Σfx
N
2m + 304
or, 46 = 8

or, 368 = 2m + 304

Infinity Optional Mathematics Book - 8 169

or, 368 – 304 = 2 m

or, 64 =m
2

\ m = 34

Hence, the value of m = 34
4. If the mean of the data given below is 17, find the value of 'a'.

Marks obtained 5 10 15 20 25 30
No. of students 2 5 10 a 4 2

Solution: Here

Mean (X) = 17
To find: The values of a

Tabulating the given data for the calculation of missing frequecny a.

Marks obtained (x) No. of students (f) fx
5 2 10
10 5 50
15 10 150
20 a 20a
25 4 100
30 2 60
Σfx = 370 + 20a
Σf = N = 23 + a

By formula,

Mean (x) = Σfx
N
370 + 20a
or, 17 = 23 + a

or, 391 + 17a = 370 + 20a

or, 391 – 370 = 20a – 17a

or, 21 = 3a

or, 21 =a
3

\ a = 7

Hence, the value of a is 7
170 Infinity Optional Mathematics Book - 8

Exercise 10.1

Section A

1. (a) Define arithmetic mean

(b) What is central tendency?

2. (a) Define frequency

(b) What is the meaning of fx in discrete series?

3. (a) In a formula (X) = Σx , what does ΣX represent?
N

(b) Write the formula, to find the mean in discrete series.

Section B

4. Find the arithmetic mean of the following set of observations.

(a) 30, 36, 25, 44, 40

(b) 2, 3, 4, 5, 8, 10, 15, 20

(c) 20ºC, 16ºC, 22ºC, 18ºC, 27ºC

(d) 10 kg, 25 kg, 35 kg, 40 kg, 50 kg, 60 kg

5. (a) Find the arithmetic mean of all even numbers between 20 to 30.

(b) Find the arithmetic mean of the prime numbers between 10 and 30.

6. (a) The age of 7 workers in years are 22, 26, 18, 24, 20, 16, 14. Find the mean
age.

(b) The expenditure (in rupees) of 15 days of a man is as follows. Find his
average expenditure.

20, 22, 25, 30, 28, 35, 40, 32, 34, 35, 21, 27, 29, 36, 40.

7 (a) Find the value of a from the given data whose mean (X) is 34.

20, 24, 32, 40, a, 52

(b) Find the value of p if the mean of 2, 3, 4, 6, p and 8 is 5.

8. If the mean of b, b + 4, b + 8, b + 12, b + 16 is 22, find the value of b.

9. (a) If (X) = 50 and Σfx = 750, find the number of terms

Infinity Optional Mathematics Book - 8 171

(b) If (X) = 40 and Σfx = 840, find Σf.

10. (a) If (X) = 12 and Σfx = 70 + 10a, and the number of frequency (N) = 5 + a,
and N = 18 + 2a, find the value of a.

(b) If mean (X)=20 and Σfx = 40a + 200 and N = 18 + 2a, find the values of a.

Section C

11. Find the arithmetic mean of the following data.

(a) Marks obtained 5 15 25 35 45
No. of students 8 10 9 12 11

(b) Age (in years) 11 12 13 14 15

No. of students 8 27 33 20 12

(c) Marks obtained 65 75 85 95
10
No. of students 10 20 10

(d) Age (in years) 25 30 35 45 55

No. of people 7 2 9 10 3

12. (a) Find the value of unknown item g, if the mean of given data is 15.

Marks 5 10 15 20 25
No. of students 2
5 10 g 4

(b) The mean of the following data is 11, what is the value of m

Age (in years) 5 8 11 14 17
No. of students 2 4 8 m2

(c) If the mean of the following data is 12, find the value of p.

x 4 8 p 16 20
f 5 3 12 5 4

172 Infinity Optional Mathematics Book - 8

10.2 Median and Quartiles:-

Look at the following data
4, 7, 10, 13 16, 19, 22

3 items 3 items

Middle term

In the above data, the numbers are arranged in ascending order. The fourth

item 13 has three items before it and three items after it. So, 13 is the middle item

in the series.

Therefore, 13 is called the median of the given data.

Pictorically Median

0% 100%
50% 50%

Definition: Median is the middle value of the observation when the items are

arranged in ascending order or descending order. If there is no one middle

value, the average of the two middle values is the median. Median is denoted

by Md or Md. It is also called 2nd quartile.

(i) Median for individual series:

To find the median of an ungroupped data, we should arrange them in

ascending or descending order. Let, the total numbers of observation be N.

The value of median (Md) = Value of N+1 th
2 Item.

(ii) Median for discrete series:

To find the median of a discrete series of frequency distribution, we should

display the data in ascending or descending order in a cumulative frequency

Mtaebdlei.aTnh(Me md)e d=iavnaliuseoboftaiNne2+d1byth item
Where N = Sum of the frequencies (Σf)

Quartiles

Quartiles are those values that divide the data arranged in ascending or
descending order into four equal parts. A distribution is divided into four equal
parts by there quartiles. So there are three quartiles, they are first quartile or lower

Infinity Optional Mathematics Book - 8 173

quartile, second quartile or median and 3rd quartile or upper quartile.

Pictorically

A Q2 B
Q1 75%
Q3
25%

AB
Q1 Q2 Q3
50% 50%

AB
Q1 Q2
75% Q3
25%

– The first or lower quartile (Q1) is the point below which 25% of the items lie
and above which 75% of the items lie.

– The second quartile (Q2) is the point below which 50% of the items lie and
above which 50% of the items lie. So second quartile is the median.

– The third or upper quartile (Q3) is the point below which 75% of the items lie
and above which 25% of the items lie.

To find quartiles, arrange the given data in ascending order,

(i) For individual Series:-

– The first quartile (Q1) = value of N+1 th
4 item

– The 2nd quartile (Q2) = value of 2 N+1 th
4 item

= value of N+1 th
2 item

– The third quartile (Q3) = value of 3 N+1 th
4 item

Where N = total number of observations.

(ii) For descrete Series:-

– The first quartile (Q1) = value of N+1 th
4 item

– The second quartile (Q2) = value of 2 N+1 th
4 item

= value of N+1 th
2 item

174 Infinity Optional Mathematics Book - 8

– The third quartile (Q3) = value of 3 N+1 th
4 item

Where, N = Sum of the frequencies (Σf)

Note:- To find quartiles in individual and discrete series, we use same
formula but the meaning of N is different.

WORKED OUT EXAMPLES

1. Find the median of the given data.

5, 12, 19, 26, 33, 40, 47

Solution:-

Here, the given data in ascending order are 5, 12, 19, 26, 33, 40, 47

Number of observation (N) = 7

We know that,

Median (Md) = value of N+1 th
item
2
= value of 7+1 th
item
2

= value of 4th item.

= 26

∴ Median (Md) = 26

2. If the given data is in ascending order and the median is 70, find the
value of a. 50, 60, 3a + 5 , 80, 90.
2
Solution:- Here
3a + 5
The given data in ascending order is 50, 60, , 80, 90.
2
Median (Md) = 70

No. of observation (N) = 5

By formula

Median (Md) = value N+1 th item
2
th
70 = value of 5+1 item
2

or, 70 = value of 3rd item

Infinity Optional Mathematics Book - 8 175

or, 70 = 3a + 5 [∴from given data]
2

or, 140 = 3a + 5

or, a = 135 = 45
3

Hence, the required value of a = 45

3. Compute the median from the data given in the table below.

x 10 15 20 25 30 35 40
f 8 6 10 12 16 7 14

Solution:-

Here, tabulating the given data in ascending order for the calculation of

median. f c.f.
x

10 8 8

15 6 14

20 10 24

25 12 36

30 16 52

35 7 59

40 14 73

Σf = N = 73

The position of median = N+1 th
2 item

= N+1 th
2 item

73 + 1 th
2
= item

= 37th item

In c.f. column, 52 is just greater than 37 so its corresponding value is 30

∴ Median (Md) = 30

4. Find the first quartile (Q1) and the third quartile (Q3) from the data
given below. 4, 5, 6, 7, 8, 9, 10.

Solution:- Here,

Arranging the given data in ascending order for the calculation of Q1 and Q3,
we get

176 Infinity Optional Mathematics Book - 8

4, 5, 6, 7. 8. 9, 10

No. of observation (N) = 7 th
item
The first quartile (Q1) = N+1
4 th
= item
7+1
4

= 2th item

∴ First quartile (Q1) = 5 N+1 th
Again, 4 item

The 3rd quartile (Q1) = 3

= 3 × 2th item

= 6th item

∴The third quartile (Q3) = 9

5. The marks obtained by 10 students of class 8 in mathematics are
given below. Find the first quartile (Q1) and third quartile (Q3).

15, 20, 18, 10, 12, 5, 8, 13, 14, 16

Solution:- Here,

Arranging the given data in ascending order, we get

5, 8, 10, 12, 13, 14, 15, 16, 18, 20.

Number of observation (N) = 10 th
item
The first quartile (Q1) = the value of N+1
4 th
= the value of item
10 + 1
4

= the value of 2.75th item

Now,

First quartile (Q1) = 2nd item + 0.75 (3rd item – 2nd item)

= 8 + 0.75 (10 – 8)

= 8 + 0.75 × 2

= 8 + 1.5

= 9.5

Similarly, = the value of 3 N+1 th
4 item
Third quartile (Q3)
= the value of 3 (2.75)th item

Infinity Optional Mathematics Book - 8 177

= the value of (8.25)th item

∴Third quartile (Q3) = 8th item + 0.25 (9th item – 8th item)

= 16 + 0.25 (18 – 16)

= 16 + 0.25 × 2

= 16 + 0.50

= 16.50

6. Find the first and third quartiles from the table given below.

Marks obtained 10 20 30 40 50 60

No. of students 5 7 10 8 2 3

Solution:- Here,

Tabulating the given data in ascending order for the calculation of Q1 and Q3.

Marks obtained No. of students c.f.

10 5 5

20 7 12

30 10 22

40 8 30
32
50 2 35

60 3

Σf = N = 35
th
The position of first quartile (Q1) = N+ 1 item
4 1
= th
35 + item
4

= 9th item

In c.f. column, 12 is just greater than 9. So its corresponding value of 20

\ First quartile (Q1) = 20

Again, th
item
The position of 3rd quartile = 3 N+1
4
= 3 × 9th item


= 27th item

In c.f. column, 30 is just greater than 27. So its corresponding value is 40.

∴ Third quartile (Q3) = 40

178 Infinity Optional Mathematics Book - 8

Exercise 10.2

Section A
1. (a) What is median? Define it.
(b) What is quartiles?
(c) Define first quartile.
(d) Define second quartile.
(e) Define third quartile.
2. (a) Write the name of quartile which divides the data below 25%.
(b) Write the name of quartile which divides the data below 50% and above

50%.
(c) Write the name of quartile which divides the given data below 75%.
(d) Which quartile divides the given data above 75% write it.

3. (a) What is the another name of median? Write it.

(b) What does 'N' represent in the formula third quartile (Q3) = the value of
th
3 N+1 item in discrete series.
4

(c) Find the position of third quartile in a data having 11 items.

Section B

4. Find the median of the following set of observation

(i) 12, 14, 16, 18, 20

(ii) 20, 22, 24, 26, 28, 30, 32

(iii) 5, 8, 11, 14, 17, 20

(iv) 64, 60, 70, 72, 65, 80, 85

(v) 60, 62, 67, 54, 59, 61, 62, 56, 60, 58

5. (a) If the given data is in ascending order and the median is 80, find the

value of x. : 60, 70, 5x + 10 , 90, 100.
4

(b) P + 1, 2P – 1, P + 7 and 3P + 4 are in ascending order. If the median is

12, find the value of P.

(c) If the given data is in descending order and the median is 11, find the

value of x : 28, 24, 3x – 5, 2x – 3, 12, 10.

Infinity Optional Mathematics Book - 8 179

6. Find the first quartile of the following set of data.

(a) 4, 7, 10, 13, 16, 18, 11

(b) 40, 20, 10, 30, 8, 12, 16

(c) 5, 6, 7, 11, 8, 10, 9

(d) 40, 30, 50, 20, 10, 35, 30, 25.

7. Find the third quaritl (Q3) from the given set of data.
(a) 4, 12, 8, 14, 16, 9, 7

(b) 15, 17, 18, 20, 21, 22, 24

(c) 8, 6, 5, 12, 10, 3, 8, 9, 10, 13, 7, 9, 11

(d) 10, 12, 14, 20, 16, 18, 19, 20

Section C

8. Find the median of the following data.

(a) Wages (in Rs.) 35 45 55 65 75
7
No. of workers 5 10 12 6

(b) Marks obtained 15 25 35 45 55

No. of students 4 6 10 7 3

(c) Marks obtained 20 30 40 50 60 70 80
No. of workers 8 12 15 10 8 7 1

(d) x 10 20 30 40 50 60
f 4 8 10 7 5 1

9. Find the lower quartile (Q1) and the upper quartile (Q3) from the
date given below.

(a) Marks obtained 32 36 40 44 48 52

No. of students 2 5 9 63 2

(b) Marks obtained 20 30 50 60 70 80 95
No. of students 4 5 8 12 11 6 5

(c) Wages (in Rs.) 50 60 70 80 90 100
No. of workers 6 10 15 13 8 3

(d) x 10 15 20 25 30 35
f
6 5 4 78 9

180 Infinity Optional Mathematics Book - 8

10.3 Mode

Mode is a value which is repeated maximum number of times. In case of

ungrouped data it can be easily located by its maximum repetition in the data.

A distribution that has two modes is called bimodel. In case of continuous series,

mode is computed by the following formula.
Mode (Mo) =  + f1 – f0 × i

2f1 – f0 – f2
Where,  = the Lower Limit of the model class
f0 = the frequency of class preceding the model class.
f1 = the frequency of the model class
f2 = the frequency of the class succeeding the model class
i= the width of the class interval.

WORKED OUT EXAMPLES

1. Find the mode of the following set of observation.

5, 10, 12, 7, 15, 7, 25, 19, 7, 25, 12, 7, 10.
Solution: Here,
Arranging the given data in ascending order.
5, 7, 7, 7, 7, 10, 10, 12, 12, 15, 19, 25, 25
Now, 7 has the highest frequency. ( Frequency of 7 is 4.)
Mode (Mo) = 7

2. Find the mode of the following distribution

Marks obtained 40 50 60 70 80 90 100
2
No. of students 10 15 14 20 11 8

Solution:- Here

From the given table

20 is the highest frequency and its corresponding value is 70.

∴ Mode (M0) = 70

Infinity Optional Mathematics Book - 8 181

Exercise 10.3

Section A

1. (a) Define mode.

(b) Write the mode value of the data.2, 3, 4, 4, 5

2. (a) What does (f1) represent in the formula Mode (Mo) =+ f1 – f0 ×i
2f1 – f0 – f2

(b) What do you mean by bimodel.

Section B

3. From the data given below, find the mode

(a) 5, 7, 2, 3, 5, 8, 10, 12, 5, 5, 2, 3.

(b) 10, 11, 12, 13, 10, 12, 13, 15, 13, 17, 13.

(c) 2, 3, 4, 2, 3, 3, 3, 4, 4, 2, 5, 5, 3, 4, 2.

(d) 20, 30, 35, 27, 27, 25, 27, 27, 20, 35.

4. Find the model size of the following distribution

(a) Age (in years) 14 15 18 20 22 24 25

No. of people 7 12 9 13 15 14 11

(b) Marks 20 30 40 50 60 70
No. of students 5 15 14 21 13 5

(c) Wages 100 200 300 400 500
No. of workers 15 17 18 20 10

(d) x 5 10 15 20 25 30
f 2 4 6 15 5 2

182 Infinity Optional Mathematics Book - 8

ANSWER SHEET

Exercise 1.1

4. (a) x = 2, y = 5 Section 'B' x = –3, y = 4
(c) x = 4, y = – 1 (b) x = 1, y = 6
(e) x = 1, y = –7 (d) x = –6, y = 2
(f)

Exercise 1.3

5. (a) {(2, 1), (4, 2)} (b) {(2, 1), (4, 1), (4, 2)}

6. (a) {(3,3), (2, 2)}, Domain = {2, 3}, Range = {2, 3}

(b) {(1, 2), (1, 3), (2, 2)}, Domain = {1, 2}, Range = {2, 3}

(c) {(1, 2), (1, 3), (2, 3)}, Domain = {1, 2}, Range = {2, 3}

Exercise 2.2

1. (a) – 3x – 2 (b) 4x2 – 10x + 6
(c) –5x2 + 5x – 15 (d) 8x4 + 3x3 + 3x2 + 5x – 20
(e) 8x7 + 4x6 + x3 – 5.
2. (a) –4x + 6 (b) 7x2 – 5x + 11
(c) –11x2 + 9x + 7 (d) x3 – 2x2 + 1
(e) x5 – 8x4 + 9x3 – 4x – 7
3. (a) 15x3 – 24x2 (b) – 20x2 – 15x
(c) 8x3 – 27 (d) 14x3 – 31x2 – 17x – 2
(e) 4x4 – 6x3 – 17x2 – 9x – 14
4. (a) – 9x + 4 (b) –2x2 + 10x – 5
(c) –14x + 5 (d) –2x2 – 3x + 2
(e) 4x3 + 2x2 – x

Exercise 3.1

6. (a) 5, 8, 11, 14, 17 (b) 3, 7, 11, 15, 19

(c) 12, 32, 43, 45, 65 (d) 14 , 81, 116, 312, 614

Infinity Optional Mathematics Book - 8 183

(e) –6, 7, –8, 9, –10 (f) 17, –1 , 111 , –1 , 1
9 13 15

7. (a) tn = n + 1, t10 = 11 (b) tn = 2n + 3, t10 = 23

(c) tn = 25 – 5n, t10 = – 25 (d) tn = 7n + 4, t10 = 74

(e) tn = 21n, t10 = 1 (f) tn = n2, t10 = 100
20

Exercise 3.2

1. (a) 40 (b) 40 (c) 85 (d) 61 (e) 99 (f) – 4

Σ5 Σ Σ6 5

2. (a) n (b) 2n (c) 2n + 1
n=1 n=1 n=1

4 (5 – 5n) (e) 4 n 7 1
(f)
Σ Σ Σ (d)
n=1 n+1 n = 1 n2
n=1

Exercise 4.1

12 10 12 1 –1
7. (a) 3 4 (b) 3 2 (c) 2 4 (d) 4 2

8. (a) p = 4, q = 2 (b) p = 2, q = – 3
(c) p = – 4, q = 2
9. (a) a = –2, b = –3, c = –4, d = 1 (d) p = –5, q = – 3
(c) a = 0, b = –1, c = –3, d = 6 (b) a = 1, b = –1, c = 4 d = –1
(d) a = –1, b = 5, c = –39, d = 7
Exercise 4.2

3. (a) (2, 4) 6 65 42
(b) 3 (c) 9 3 (d) 7 1
1

3 5 8 (f) 3 –4
(e) –4 5 3 –1 5
–3 –5

4. (a) (4, 3) (b) –2 0 4 52
–2 (c) –10 8 (d) 0 0
–3

3 0 4 –1 –1
(e) 6 1 10 (f) –3 –1

22

184 Infinity Optional Mathematics Book - 8

5. (a) (7 1 9) –16 6 0 7 28
(b) –6 (c) 0 5 (d) 21 1

6. (a) x = 9, y = 1, z = –2 3
(c) x = – 10 , y = 4 (b) x = , y = 14

3 2
(d) x = 2, y = 9
Exercise 5.2

1. (a) (x2 – x1)2 + (y2 – y1)2 (b) (c – a)2 + (d – b)2

(c) a2 + b2

2. (a) (Longest side)2 = the sum of the square of other two side

(b) In parallelogram diagonal are not equal but in rectangle length of
diagonals are equal

(c) Square = diagonals equal, Rhombus = diagonals not equal

3. (a) 5 units (b) 5 units

4. (a) 2 10 units (b) 13 units (c) 3 3 units

(d) c2 + a2 units

6. (a) 8, 2 (b) – 3, 7 (c) –1, 5

Exercise 5.3

4. (i) (3, 3) (ii) (0, 0) 5. (i) (5, 19) (ii) (12, –14)
(iv) (–2, –5)
6. (i) (4, –3) (ii) (0, –3) ((iii) 72, 9 (
7. (a) C(1, 1) (b) H(11, 2) 2
8. (a) 2:3 (b) 1:2
(c) 1, 6

Exercise 5.4

2. (a) (1, 4) (b) (i) yes (ii) no (iii) no (iv) yes (c) yes,

3. (a) –23 (b) p = 4, q = –5 c) k = 6,

4. (i) x – 4 = 0 (ii) y + 2 = 0 (iii) y – 3 = 0 iv) x – y = 0

5. (i) x2 + y2 = 25 (ii) x2 + y2 – 4x –6y + 4 = 0 (iii) 4x + 6y – 23 = 0

6. (i) x2 + y2 = 25 (ii) x = 0

7. i) x2 + y2 = 25 ii) y = 0

Infinity Optional Mathematics Book - 8 185

Exercise 6.1

2. (a) 98146" (b) 68718" (c) 504540" (d) 428450"

(e) 1458" (f) 1256085" (g) 775836" (h) 155600"

(i) 820070" (j) 7589"

3. (a) 50.258° (b) 125.944° (c) 35.3° (d) 140.012°

4. (a) 60.526g (b) 8.0706g (c) 181.56g (d) 20.005g

5. (a) 30g (b) 70g (c) 90g (d) 120g (e) 100g

(f) 200g (g) 300g (h) 400g

6. (a) 90º (b) 45º (c) 90º (d) 108º (e) 180º

(f) 270º (g) 360º (h) 540º

7. 20º 9. 36º 10. 80g 11. 45º and 90º 12. 50g and 30g

Exercise 6.2

2. (a)  c (b) 23 c (c) 76 c (d) 4 c
6

(e) 32 c (f) 1230 c

3. (a) 108° (b) 40° (c) 50° (d) 120°

4. (a) 80g (b) 140g (c) 10g (d) 32g

5. (a) 5 : 36 (b) 2 : 7

6. (a) 31 c (b) 117°
180


7. (a) 305° (b) 55 c
36

9. (a) 1000 g (b) 130 c (c) 20g (d) 20g,40g,140g
9

(e) 4 c 7 c 2 c 30g, 80g (g) 100g (h) 270 c
20 5
, , (f)

10. (a) (i) 90°, c (ii) 120°, 2 c (iii) 144°, 4 C
2 3 5


(ii) 40°, 400 g
9
(b) (i) 72°, 80g, (ii) 45°, 50g

(c) 8 (d) 54°, 81°, 108°, 135°, 162° (e) 12° (f) 10

11. (a) (i) 270° (ii) 150° (iii) 240°

 c 5 c (iii) 356 c
2 6
(b) (i) (ii)

186 Infinity Optional Mathematics Book - 8

(c) (i) 255 c (ii) 90° (iii) 180°
2

Exercise 7.1

2. (a) 5cm (b) 8cm (c) 12cm

3. (a) 3 cm (b) 10 cm (c) 12 cm

Exercise 7.2

1. DDEF, DEFF, DEFE 2. GI , GGHI , GH 3. AACB, ABCC, ABCB, AC
IH IH AB

4. (a) 53 , 5 , 4 (b) 54 , 5 , 3 (c) 5 , 5 , 3
4 3 3 4 4 3 4

5. (a) 3cm, 4 , 3 , 4 , 5 , 5 , 3 (b) 13cm, 153, 12 , 152, 153, 1123, 12
5 5 3 4 3 4 13 5

(c) 20cm, 4 , 3 , 4 , 5 , 5 , 3
5 5 3 4 3 4

6. (a) 153, 1132, 5 , 153, 1132, 152 (b) 12cm (c) 8cm, 6cm
12

Exercise 7.3

2. 32 3. 13 4. b , a2 – b2 5. –259
5 7. 11 a a

6. x2 + y2 10
x2 – y2

Exercise 7.4

2. (a) 83 (b) 24 (c) 13 (d) 18

(e) 147 (f) 53 (g) 8 (h) 94

(i) –21 (j) 3

5. (a) 169 (b) 23 (c) 54
6. (a) 30º (b) 60º (c) 30º (d) 0º

(e) 45º (f) 90º

Exercise 7.5

1. (a) sin2­θ – cos2θ (b) sin3­θ + cos3θ

Infinity Optional Mathematics Book - 8 187

(c) 35tan2­θ – 31cosθ . tanθ + 6cos2θ (d) 1 – tan4­θ

(e) 6sin4­θ – sin2θ . cos2θ – 2cos4θ

2. (a) 6sinA – 11cosA (b) sin2θ­ + 6sinθ

(c) 18sin2A – 4tan2A – 2 (d) (sin2A – cos2A) (sinA – cosA)2

(e) – 4sinA . secA (f) 1 2–csoisnA2A
(g) sec22Ata–ntAan2A (h) 1
3. (a) (sinA – cosA) (sinA + cosA) (b) cos2θ (cosθ – tanθ) (cosθ + tanθ)

(c) (sec2θ + cos2θ) (secθ + cosθ) (secθ – cosθ) (d) (sinθ – 2) (4sinθ + 3)

(e) (tanq – 2) (3 tan q + 4) + (4 sin q – 3) (sinq – 2)

Exercise 7.6

1. (a) 60º (b) 47º (c) 18º (d) 2º (e) 10º
5. (a) 10º (b) 10º (c) 10º (d) 6º (e) 6º
(f) 18º

Exercise 7.7

1. (a) 30º, 20cm, 10 3cm (b) 60º, 5 cm, 53 cm
(c) 45º, 3 2cm, 3cm 2 2
2. (a) 60º, 30º, 1cm
(c) 90º, 45º, 45º, 3 2cm, 3 2cm (d) 60º, 3cm, 2cm

(b) 30º, 60º, 6 3cm

(d) 30º, 60º, 2 cm

Exercise 7.8

1. (a) 28.87m (b) 103.92m (c) 48m (d) 54.64m
(d) 60º
2. (a) 100 3m (b) 80m (c) 10 3m (h) 129.9m
(l) 45º
(e) 15ft (f) 134.22m (g) 32.47m (d) 30.46m.

(i) 50 3cm (j) 30º (k) 23.09m

3. (a) 43.3m (b) 23.32m (c) 60º

Exercise 8.1

(→ →

CD = EF = (–5, –3)
→ (( 2 ,
( –4
4. a) AB = (4, 4)

(→ 6 (→ –6 and (→ 5 (
0 –2 –4
HG = IJ = PQ =

188 Infinity Optional Mathematics Book - 8

→ → ( (→ →

AB = BC = CD = DA =
( (5. –5 ( 7 –7 ( 5
–5 ( 0 4 (( 1
→
(( 2 units, 45º b) |→a | = 6 units, 330º
5. (a) |AB| = 3

(c) 8 units, 210º (d) 6 2 , 135º

(7. (a) –2 , 5 units, 206.56º ((b) 23 ,4 3 units, 79.10º
–1 6

( (d) –33 , 3 2 units, 135º 8. 120º, 4 units

Exercise 8.2

2. unit vectors: (ii), (iii), (vi) Null vectors : (i)

3. (a) 9, 13 (b) 8, 3

5. (a) x = 4, y = 3 (b) p = 3, q = 3 (c) a = 2, b = 1

(→ → →→

6. (a) AB = CD = AB = MN =
((
((
4 ((b) 1
3 3
→

(b) AB
→→

( (7. (a) MN = – PQ =
1 → = 1
3 1
= – RN

Exercise 8.3

→ ((
(a) 3 p = ((b)
(2. 6 –2
9 –6

((
( (4. (a) 4 –3
–6 (b) –6

(( ( (
(5. (i) 4 ( ( ((ii) –3 –6 2
6 –9 (iii) –9 iv) 3
2

(( ( (
2 10 –1
5 (iii) 4 iv) –3
6

( ( ( (6. (a) i) 8 (ii)

(( ((
( (6 2 ( ((b) 3 , 1
2 4
7. (a) –3 , 7
5 5 –15 9
4 (ii) –3 (iii) 10 (iv) –6
(( (( ( 15 (
( ((
(v) –6
((
( ( ( (8. (i)
( –1

(vi) 20
15 37
5 , –3 , 34 units, 2 17 units
( ( ( (9. (a) 2 , –6 , 5 units, 61 units (b)

10. (a) 218 units, 118.30º (b) 3 units, 90º

→ (( → ((
→
AC = TK =
UW =
(11. –7 ( ( (→ 10 –4 –8
5 0 –3 0
MO
=

Infinity Optional Mathematics Book - 8 189

Exercise 9.1

2. (a) Reflection, rotation, translation and enlargement

(b) Equal

3. (a) Y-axis (b) (x, –y)

6. (a) A' (4, –5) (b) B' (–4, –5)

(c) C' (–3, 5) (d) D'(7, 3)

7. (a) M'(2, 3) (b) N' (–4, 5) (c) P'(3, –4) (d) (–2,–3)

8. (a) D'(8, 10) (b) E' (–7, 8) (c) F' (2, 7) (d) D(–3, –4)

9. (a) G'(6, 9) (b) H' (– 8, –6) (c) I' (–5, 4) (d) J' (3, –5)

10. (i) A' (2, –3) B'(–3, –4) and C' (1, 2)

(ii) A' (–2, 3) B' (3, 4) and C' (–1, –2)

11. (i) P' (–1, –5) U' (4, –9) and N' (10, –3)

(ii) P'(1, 5) U' (–4, 9) and N' (–10, 3)

12. (i) S' (2, 4) U' (6, –3) and N' (–4, –1)

(ii) S' (–2, – 4) U' (–6, 3) and N'(4, 1)

13. (i) P' (5, –2), Q' (8, 3) and R' (2, 5)

(ii) P' (–5, 2) Q' (–8, –3) and R' (–2, –5)

Exercise 9.2

2. (a) + 90º, (0, 0) (b) 180º, (0, 0)

5. (a) A' (–5, 4) (b) B' (–3, –2) (c) C'(5, –3) (d) D'(1, 4)

6. (a) M' (4, –3) (b) N' (4, 2) (c) Q'(–2, 3) (d) R'(–1, –6)

7. (a) E' (5, 2) (b) F' (3, –2) (c) G' (–7, 1) (d) H'(8, 2)

8. (i) A' (–1, 3), B'(0, 5) C'(2, 2) (ii) A' (1, – 3), B'(0, –5), C'(–2, –2)

(iii) A' (–3, –1), B'(–5, 0) C' (–2, 2)

9. (i) A' (3, 2), B'(–1, –1), C' (–1, 4), D' (3, –3)

(ii) A' (–2, 3), B'(1, –1), C'(–4, –1) and D'(–3, 3)

(iii) A' (–3, –2), B' (1, 1), C'(1, –4) and D'(–3, –3)

Exercise 9.3

1. b) a = x - component of translation vector.

b = y-component of translation vector.

(3 ((
(5
2. a) T = 2 iii) N' (–3, 7) iv) K' (–4, –2)
4. a) i) A' (2, 7) b) T = 4
ii) M' (4, –2)

b) A' (–1, 7), B'(–5, 8), C'(2, –3) and D' (–5, 1)

5. a) i) D' (–1, 4) ii) E' (4, 8) iii) F' (–2, 2) iv) G' (9, –2)

b) A(2, 1) c) B(8, –4)

6. A' (5, 4), B' (10, 6) and C' (8, 10) 7. P' (8, 5), Q' (10, 9) and R' (4, 11)

8. A' (4, 5), B'(5, 7) and C' (4, 7) 9. G' (0, 3), I' (2, 3), T' (2, 6) and A' (0, 6)

190 Infinity Optional Mathematics Book - 8

Exercise 9.4

2. (a) (0, 0) represent centre and k represent scale factor (b) No.
H'(–6, –12)
4. (a) E' (8, 4) (b) F' (–6, 6) (c) G' (10, –6) (d)

5. A' (–4, 0), B'(–6, –2), C' (–10, –14) and D' (8, 4)

6. A' (6, 2), B' (10, 0) and C' (4, –4)

7. P' (6, –9), Q' (–3, 3) and R' (3, 9)

8. D' (0, –2), I' (2, 6), N' (4, 4) and A' (2, –4)

Exercise 10.1

2. (b) Product of f and x.

3. (a) Sx represent (sum) total of x. (b) X = Sfx
N

4. (i) 35 (ii) 8.375 (iii) 20.6 (iv) 36.66

5. (a) 25 (b) 18.66 (b) 21 (d) 37.58
6. (a) 20 (b) 30.26 (c) 79
7. (a) 36 (b) 7
8. 14 9. a) 15
10. (a) 5 (b) 2
11. (a) 26.6 (b) 13.01

12. (a) 1 b) 4 (c) 12

Exercise 10.2

2. (a) First quartile (Q1) (b) Second quartile (Q2)

(c) Third quartile (Q3) (d) First quartile (Q1) (c) 9th position
3. (a) Second quartile (Q2) (b) Sum of the frequency
4. (i) 16 (ii) 26 (iii) 12. 12.5

(iv) 70 (v) 60

5. (a) 62 (b) 6 (c) 6

6. (a) 7 (b) 10 (c) 6 (d) 21.25

7. (a) 14 (b) 22 (c) 10.5 (d) 19.75

8. (a) 55 (b) 35 (c) 40 (d) 30

9. (a) 35, 44 (b) 50, 70 (c) 60, 80 (d) 15, 30

Exercise 10.3

1. (b) 4

2. (a) f1 represent the frequency of the model class
3. (a) 5 (b) 13 (c) 3 (d) 27

4. (a) 22 (b) 50 (c) 400 (d) 20

Infinity Optional Mathematics Book - 8 191

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192 Infinity Optional Mathematics Book - 8