Vedanta Excel in Opt. Maths Book 6 Final (2078)

Coordinate Geometry

4Coordinate Geometry

4.0 Introduction

A coordinate geometry is a branch of geometry where the position of the points on
the plane is defined with the help of an ordered pair of numbers. The ordered pair
of numbers is called coordinates.

For example: An ordered pair of numbers (3, 5) represents a point on the plane. The
point is 3 units right from the origin and 5 units from X-axis. On the graph the point
(3, 5) is denoted by P.

Y

6 P(3, 5)
5
4
3
2
1

X' O 1 2 3 4 5 6 X

Y'

Let us study about number lines, X-axis, Y-axis, four different quadrants, locating a
point on a graph, distance between given two points.

4.1 Number lines and coordinate axes

Let us draw a horizontal number line XOX'. In a horizontal number line, the positive
numbers are written along the right side of the zero (0) and the negative numbers
are written along the left side of the 0.

negative numbers positive numbers

X' X
–8 –7 –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6 +7 +8

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Coordinate Geometry

Again let us draw a vertical number line YOY'. Y

In a vertical number line, the positive numbers are written above the zero +8

(0) and the negative numbers are written below the zero (0). +7

Now, let's combine these two number lines in such a way that the zero +6

mark of both number lines meet at the same point and the number lines +5
intersect each other perpendicularly.

Y +4
+8 +3

+7 +2

+6 +1

+5 0

+4 –1

+3 –2

+2 –3

+1 –4

X' – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 O + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 X –5
–1 –6

–2 –7

–3 –8 Y'

–4

–5

–6

–7

–8 Y
Y'

In this case, the horizontal number line XOX'

is called X-axis. The vertical number line YOY'

is called Y-axis. The zero denoted by O is called

the origin. Now, the number lines are called the O X
coordinate axes (axes is the plural form of axis). X'

The region covered by the coordinate axes is called

the coordinate plane.

Furthermore, the intersected X-axis and Y-axis Y'
divide the coordinate plane into four regions. The

52 vedanta Excel in Additional Mathematics - Book 6

Coordinate Geometry
regions are called the quadrants.
The region XOY is called the first quadrant. The region X'OY is called the second
quadrant. The region X'OY' is called the third quadrant.
The region XOY' is called the fourth quadrant.

4.2 Coordinates of a point in different quadrants

In the adjoining graph, the point P lies in the Y
first quadrant. We can find its position in terms P(4, 3)

of ordered pair in which the first number is Rise

the number along OX (X-axis) and the second X' Run X
number is the number of square rooms along OY O

(Y-axis). So, the position of the point P is (4, 3).

Here, 4 is called the x-coordinate and 3 is called

the y-coordinate. Y'

Note :

The position of any point in the coordinate plane is obtained as an ordered pair.

The ordered pair is (x-coordinate, y-coordinate) or simply (x, y) which is like 'run
and rise'.

(i) The coordinates of any point in the first Y
quadrant is (x, y).
(X, Y)
(ii) The coordinates of any point in the second (–X, Y) O

quadrant is (–x, y). X' (X, –Y) X

(iii) The coordinates of any point in the third (–X, –Y)
quadrant is (–x, –y).

(iv) The coordinates of any point in the fourth Y'
quadrant is (x, –y).

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Coordinate Geometry

In the adjoining graph, Y A(3, 4)
B(–3, 4)
(i) The coordinates of the point A are (3, 4) and
written as A(3, 4).

(ii) The coordinates of the point B are (–3, 4) and

written as B(–3, 4). X' O(0, 0) X

(iii) The coordinates of the point C are (–3, –4)
and written as C(–3, –4).

(iv) The coordinates of the point D are (3, –4) and C(–3, –4) C(3, –4)
written as D(3, –4).
Y'

Note :

(i) The coordinates of the origin O is (0,0).

(ii) The coordinates of a point in X-axis is (x, 0) or (–x, 0) where x is the number
of square units counting from origin along X-axis.

(iii) The coordinates of a point on Y-axis is (0, y) or (0, –y), where y is the number
of square units counting from origin along Y-axis.

4.3 Plotting points in different quadrants

We use graph paper to plot the given points in different quadrants. To plot a point,
we should first draw the coordinate axes XOX' and YOY' on the graph paper.

Examples:

Let's learn to plot the given points in different quadrants from the following
illustration:

Plotting A(3, 2) Plotting B(–3, 2)
Y Y
1st quadrant
B (–3, 2)
A (3,2)
2 2nd quadrant
32 –3
X' X X' O (0,0) X
O (0,0)

Y' Y'
A(3, 2) lies in the 1st quadrant B(–3, 2) lies in the 2ndquadrant

54 vedanta Excel in Additional Mathematics - Book 6

Plotting C(–3, –2) Coordinate Geometry
Y
Plotting D(3, –2)
Y

–3 X 3
X' –2 O (0,0) X' O (0,0) –2 X

3rd C(–3,–2) D(3,–2)
quadrant 4th quadrant

Y' Y'
C(–3, –2) lies in the 3rd quadrant D(3, –2) lies in the 4th quadrant

Thus, to plot x - coordinate, run along X–axis and then rise along Y-axis to plot y -
coordinate.

Y

B(–3, 2) A(3, 2) Then
Then rise
rise First run
O(0,0) X
First run
Then
X' down

Then D(3, –2)
down

C(–3, –2)

Y'

Exercise 4.1

Short Questions :
1. Fill in the blanks, the position of a point P is 5 units from the origin along X-axis

and 5 units rise from the Y-axis.
(a) its coordinates in the first quadrant is .......................
(b) its coordinates in the second quadrant is .......................
(c) its coordinates in the third quadrant is .......................
(d) its coordinates in the fourth quadrant is .......................

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Coordinate Geometry

2. (a) The coordinates of the origin are .......................

(b) The coordinates of a point on X-axis are .......................

(c) The coordinates of a pint on Y-axis are .......................

3. Write the coordinate of the given points from the graph a quickly as possible.

(a) The coordinates of the point A are .............. Y
(b) The coordinates of the point B are .............. BA

(c) The coordinates of the point C are .............. X' O (0,0) X

(d) The coordinates of the point D are .............. CD
Y'

4. Plot these points in the graph.

A(4, 5) B(–4, 5) Y

C(–4, –5) D(4, –5)

E(4, 0) F(0, 5)

G(–5, 0) H(0, –5) X' O (0,0) X

Y'

Long Questions :

5. From the adjoining graph, find the coordinates of the points P, Q, R, S, and U.

Y

P

QU

X' O (0,0) X

S
R

Y'

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Coordinate Geometry

6. Find the coordinates of the vertices of the following figures:

(a) Y (b) Y
A R

B P

X' O (0,0) X X' O (0,0) X

C Q
Y' Y'

7. On the graph paper, draw XOX' and YOY' axes which are mutually perpendicular
at O (origin). Then, plot these points.

(a) A(4, 5) (b) B(–5, 4)

(c) C(–3, –5) (d) D(–5, –6)

8. Plot the following points in the coordinate plane on the graph. Join them in
order by using a ruler to make triangles.

(a) A(2, 4), B(6, 8), C(5, 2) (b) P(3, 6), Q(–6, –2), R(2, –2)

(c) D(4, 6), E(5, 6), F(3, 7) (d) U(4, 4), V(–4, 4), W(–4, –4)

Show to your teacher.

4.4 Distance between two points

In the graph given below, A and B are any two points. The coordinates of A are (2,
1) and that of B are (4, 4).

Y Y
B(4,4) B(4,4)

3

X' A(2,1) X X' A(2,1) 2 C
O(0,0) X
O(0,0)

Y' Y'
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57

Coordinate Geometry

AB is the distance between the points A and B
In the right angled triangle ABC, ‘C is the right angle.
So AB is the hypotenuse (h),

BC is the perpendicular (p).
And, AC is the base (b).
Here, AC = 2 units
And, BC = 3 units
Now, by using Phythagoras Theorem,

h2 = p2 + b2
or, AB2 = BC2 + AC2
or, AB2 = 32 + 22
or, AB2 = 9 + 4
or, AB2 = 13
or, AB = 13 units
? The distance between the points A and B = AB = 13 units.
Example:
The length of each square unit of the adjoining graph is 1 cm use Phythagoras
theorem to find the distance between the points P and Q.

Y
Q(4,6)

P (1,2) R(4,2)

X' X
O (0,0)
Y'

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Coordinate Geometry

Now, measure the length of PQ by using a ruler. What do you find? Discuss with
your friends.

PR QR Length of PR2 + QR2 Length of PQ PQ2 Remarks
PR2 QR2 by using ruler

PQ2 = PR2 + QR2

4.5 Distance formula

Let A(x1, y1) and B(x2, y2) be two points on the plane. Draw perpendicular AM and
BN on OX. Also draw perpendicular AP on BN.

Now, AM = y1 Y B(x2, y2)

BN = y2

OM = x1

ON = x2 A(x1, y1) P
AMNP is a rectangle.

AP = MN X' O M NX

AM = PN Y'
Then, AP = MN

= ON – OM

= x2 – x1
BP = BN – PN

= y2 – y1
ABP is a right angled triangle. By Pytahgoras theorem.

we have,

AB2 = AP2 + BP2

= (x2 – x1)2 + (y2 – y1)2

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Coordinate Geometry

If AB = d, distance between A and B,
d2 = (x2 – x1)2 + (y2 – y1)2

? d = (x2 – x1)2 + (y2 – y1)2
If A(0, 0) and B(x1, y1), we have,

d = x12 + y12

Worked Out Examples

Example 1. Plot the following points on graph A(3, 3), B(9, 3) and C(9, 11). Then
join.

(a) AB (b) BC

(c) AC

Also find the lengths of AB, BC and AC.

Solution: Here, the given three points are A(3, 3), B(9, 3) and C(9, 11).

Plotting the points A, B and C on a graph,
Y

C(9, 11)

A(3, 3) B(9, 3)

X' O X

Y'

We get right angled 'ABC by joining them.

Then, we find the lengths of AB, BC, and AC

(a) For length AB,

A(3, 3) = (x1, y1)
B(9, 3) = (x2, y2)

60 vedanta Excel in Additional Mathematics - Book 6

using distance formula, Coordinate Geometry
AB = (x2 – x1)2 + (y2 – y1)2 61

= (9 – 3)2 + (3 – 3)2
= 62 + 0
= 36
= 6 units
(b) For length BC,
B(9, 3) = (x1, y1)
C(9, 11) = (x2, y2)
using distance formula, we get,
BC = (x2 – x1)2 + (y2 – y1)2
= (9 – 9)2 + (11 – 3)2
= 0 + 82
= 64
= 8 units
(c) For length AC,
A(3, 3) = (x1, y1)
C(9, 11) = (x2, y2)
using distance formula, we get,
AC = (x2 – x1)2 + (y2 – y1)2
= (9 – 3)2 + (11 – 3)2
= 62 + 82
= 36 + 64
= 100
= 10 units

vedanta Excel in Additional Mathematics - Book 6

Coordinate Geometry

Exercise 4.2

1. Find the distance between A and B using Pythagoras theorem.

(a) (b)
Y Y

R(8, 13)

A(1, 4) B (9, 6) P(2, 5) Q(8, 5)
C(9, 4) B(12, 15)
X' O (0, 0) X X' O (0, 0)
Y' Y' X

(c) Y (d) Y

B(13, 10)

A(1, 4) C(13, 4)

X' O (0, 0) X X' A(2, 3) C(12, 3)
Y' X
O(0, 0)
Y'

2. Find the distance between the following pair of points.

(a) A(4, 5) and B(7, 9)

(b) M(4, 5) and N(1, 8)

(c) E(2, 2) and F(9, 4)

(d) M(6, 7) and N(0, 6)

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Coordinate Geometry

(e) P(5, 5) and Q(5, 0)
(f) T(–2, –2) and U(–6, –7)
(g) R(–1, –4) and S(6, 2)
Long Questions :
3. From the following set of points prove that AB = MN.
(a) A(2, 4), B(4, 6), M(5, 3) and N(3, 5)
(b) A(6, 7), B(9, 11), M(7, 2) and (4, 6)
(c) A(3, 4), B(11, 10), M(2, 3) and N(10, 9)
(d) A(1, 1), B(10, 13), M(6, 7) and N(–3, –5)
4. From the given points, find the length of line segments AB and CD, which one
is longer?
(a) A(2, 3) and B(5, 3), C(6, 7) and D(7, 8)
(b) A(4, 4) and B(8, 8), C(4, 7) and D(7, 8)
(c) A(8, 2) and B(6, 7), C(8, 9) and D(12, 14)

1. (a) 2 17 units (b) 10 units (c) 6 5 units (d) 2 61 units

2. (a) 5 units (b) 3 2 units (c) 53 units (d) 37 units

(e) 5 units (f) 41 units (g) 85 units

4. (a) AB > CD AB = 3, CD = 2, AB > CD (b) AB = 4 2, CD = 10, AB > CD

(c) AB = 29, CD = 41, AB < CD

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Matrices

5Matrices

5.1 Matrix - Introduction

In a class test, Hary secured 48 marks in Mathematics, 45 marks in Science and 44
marks in English out 50 full marks.

Now, let’s present these marks in a table:

Subjects Mathematics Science English
Marks obtained 48 45 44

The marks obtained by Hary presented on the above table can be arranged as:
[48 45 44]

Here, the marks are arranged in a row enclosing by the brackets [ ].
Again, let’s present these marks in table by another ways.

Subjects Marks
Mathematics 48
Science 45
English 44

The marks obtained by Hary presented on the above table can also be arranged as :

48
45
44
Here, the marks are arranged in a column enclosed by the brackets [ ].

The arrangement of numbers in rows and column enclosing by brackets [ ] is called
matrix. Matrices is the plural form of matrix.

Usually, matrices are represented by capital letters like A, B, C, D, ....., X, Y, Z.

For example:

A = [48 45 44], B = 2 4 8 and so on.
3 ,C= 7 5

4

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Matrices

Furthermore, the table given below shows the marks obtained by Jomes, Jerry, and
Kristina in a test examination of 50 full marks of each subject.

Subjects James Name of Students Kristina
44 Jerry 36
Mathematics 43 38 42
Science 40 42 45
English 40

Now, the marks can be arranged in the matrix form marks:

44 38 36
A = 43 42 42

40 40 45

Thus, the marks are arranged in 3 rows and in 3 column in the given matrix.

Again, let’s present the above marks obtained by three students in the table by
another way.

Name of Mathematics Subjects English
Students 44 Science 40
38 40
James 36 43 45
Jerry 42
Kristina 42

In this case, the marks can be arranged in the matrix form.

44 43 40
A = 38 42 40

36 42 45

The data written in the form of a matrix are called the members or elements of the
matrix.

Definition: A rectangular arrangements of elements in rows and columns enclosed
by a pair of round or square brackets is called a matrix.

5.2 Rows and Columns of a Matrix

In a matrix, numbers are arranged horizontally and vertically. The horizontal
arrangements are called rows and the vertical arrangements are called columns.

For example :

A = [4 5 6] It has only one row.

vedanta Excel in Additional Mathematics - Book 6 65

Matrices

15 It has only one column.
B = 12 It has two rows and two columns.

13

C= 6 7
8 9

P= 8 9 12 It has two rows and three columns.
11 14 12

In a matrix, pq r First row (R1)
A= s tu Second row (R2)
vwx Third row (R3)

First Second Third
column column column
(C1) (C2) (C3)

The members of the first row are p, q, r; the second row has s, t, u; and the third row
has v, w, x.

Also, the members of the first column are p, s, v; the second column has q, t, w, and
the third column has r, u, x.

5.3 Order of a Matrix

Let us consider a matrix. A = 1 2 3
4 5 6
Here, number of rows of matrix A = 2

number of columns of matrix A = 3.

It means that the matrix A has 2 rows and 3 columns.

This matrix is called order of 2 × 3 (read as 2 by 3).

Definition: The number of "rows × columns" of a matrix is known as the order of
the matrix.

Example: Let P = 1 2
3 4
It has 2 rows and 2 columns.

Thus, the order of matrix P is 2 × 2. It is also written as P2 × 2 to say that the matrix
P is of order 2 × 2.

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5.4 Equal Matrices

Two matrices are said to be equal if their orders are equal and the corresponding
elements are the same.

For example:

If A = 4 7 and B= 4 7 , then matrix A = matrix B
6 8 6 8

Also, if x 8 = 5 w , then x = 5, w = 8, z = 9 and y = 4.
9 y z 4

5.5 Operations on Matrices

Let us study the following two matrices of sales of newspapers in two shops in two
days.

Shop M

Day Annapurna Newspapers Naya Patrika
40 Kantipur 60
Sunday 45 50 65
Monday 60

Shop N

Day Annapurna Newspapers Naya Patrika
44 Kantipur 62
Sunday 48 60 68
Monday 65

Then, we can find the total number of newspapers sold in two days. To find the total
number of newspapers sold in two days, we can add the corresponding elements:

i.e., M + N = 40 + 44 50 + 60 60 + 62
45 + 48 60 + 65 65 + 68

= 84 110 122
93 125 133

Addition, subtraction, and multiplication are operations on matrices. Here, we
study only about addition and subtraction of matrices.

(a) Addition of Matrices

Let P = 1 2 and Q = 6 2 be two matrices. They are of the same order.
4 5 1 3
Then, the sum or addition of matrices P and Q is denoted by P + Q. Then, their

corresponding elements are added to find P + Q.

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Matrices

i.e., P + Q = 1 2 + 6 2
4 5 1 3

= 1+6 2+2
4+1 5+3

= 7 4
5 8

Two or more matrices of the same orders can be added.

(b) Subtraction of Matrices

Let M = 2 4 and N = 1 2 be two matrices.
8 2 6 0
Then, the difference of two matrices M and N is denoted by M – N and it is

calculated by subtracting their corresponding elements.

i.e., M – N = 2 4 – 1 2
8 2 6 0

= 2–1 4–2
8–6 2–0

= 1 2
2 2

Two matrices of same orders can be subtracted.

Worked out Examples

Example 1. State the orders of the following matrices:
Solution:
(a) M= 1 2 (b) N= 2 46
4 5 3 21

12 2 4]

(c) P = 3 4 (d) Q = [1

56

a

(e) R = b

c

(a) Here, M = 1 2
4 5

In matrix M,

number of rows = 2

number of columns = 2

Thus, M is a matrix of order 2 × 2.

68 vedanta Excel in Additional Mathematics - Book 6

Matrices

(b) Here, N = 2 4 6
3 2 1

In matrix N,

number of rows = 2

number of columns = 3

Thus, N is a matrix of order 2 × 3.

12
(c) Here, P = 3 4

56

In matrix P,

number of rows = 3

number of columns = 2

Thus, order of matrix P is 3 × 2.

(d) Here, Q = [1 2 4]

In matrix Q,

number of rows = 1

number of columns = 3

Thus, order of matrix Q is 1 × 3.

a
(e) Here, R = b

c

In matrix R,

number of rows = 3

number of columns = 1

Thus, order of matrix R is 3 × 1.

Example 2. Find the values of x, y, z, and w of x y = 4 5 .
Solution: z w 6 7

Here, x y = 4 5
z w 6 6

Equating the corresponding elements of the equal matrices

we get, x = 4, y = 5, z = 6 and w = 7.

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Matrices

Example 3. Add two matrices P and Q if P = 2 3 and Q = 6 7 .
8 4 3 2

Solution: Here, P= 2 3 and Q = 6 7
8 4 3 2

The given matrices are of same order we can find P + Q.

Now, P + Q = 2 3 + 6 7
8 4 3 2

= 2+6 3+7
8+3 4+2

= 8 10
11 6

Example 4. If A = 1 2 and B = 3 4 , show that A + B = B + A.
5 3 7 4

Solution: Here, A = 1 2 and B = 3 4
5 3 7 4

Now, A + B = 1 2 + 3 4
5 3 7 4

= 1+3 2+4
5+7 3+4

= 4 6
12 7

Again, B + A = 3 4 + 1 2
7 4 5 3

= 3+1 4+2
7+5 4+3

= 4 6
12 7

? A + B = B + A Proved.

Note :

In matrix addition, A + B = B + A.

It is called commutative property of matrix addition.

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Example 5. If A = 10 3 and B = 4 5 , find A – B and B – A.
7 4 2 3

Solution: Here, A = 10 3 and B = 4 5
7 4 2 3

Now, A – B = 10 3 – 4 5
7 4 2 3

= 10 – 4 3–5
7–2 4–3

= 6 –2
5 1

Again, B – A = 4 5 – 10 3
2 3 7 4

= 4 – 10 5–3
2–7 3–4

= –6 2
–5 –1

Here, A – B and B – A are not equal.

Example 6. Mr. Hanuman secured marks of 45 in English, 40 in Mathematics
and 38 in Science in a class test of 50 marks. Write the marks in a
table as well as in matrix form. (use 1 × 3 matrix form)

Solution: The marks secured by Mr. Hanuman can be written in a table as below:

English Mathematics Science
45 40 38

In matrix form [45 40 38].

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Matrices

Exercise 5

Short Questions :

1. State the orders of the following matrices:

(a) A = [1 2 3] (b) B= 4 8
2 3

(c) C= 2 (d) D= 1 2 3
7 4 5 6

24 123

(e) E = 5 6 (f) P = 4 5 6

79 789

2. Find the values of a , b, c and d in the following equal matrices.

(a) a 8 = 5 c (b) 8 4a = 4c 2
7 b d 10 3b 15 9 d

3. (a) Find the values of x and y if x–5 8 = 2 8 .
9 y+6 9 8

(b) Find the values of p and q if p–4 12 = 2–p 12 .
20 2q + 1 20 q+8

4. Add the following matrices:

(a) A = [1 2 3] and B = [2 7 5]

(b) P = [6 2 4] and Q = [3 4 5]

(c) E= 2 3 and F = 6 2
4 5 7 4

(d) T= –2 4 and U = 6 2
5 6 4 8

(e) M= 0 2 and N = 2 3
4 6 4 5

(f) G= 2 4 and H = –2 –4
6 7 –6 –7

(g) T= 1 2 3 and P = 6 2 7
4 5 6 8 4 8

5. Find P – Q in the following cases.

(a) P= 2 4 and Q = 1 2
6 7 3 5

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(b) P= 6 7 and Q = 2 3
8 2 3 4

(c) P= 2 4 and Q = 2 8
6 14 3 4

(d) P = [4 5 6] and Q = [2 4 6]

47

(e) P = 5 and Q = 8

69

(f) P= 2 4 6 and Q = 3 2 7
3 2 4 9 2 4

Long Questions :

6. (a) The marks obtained by Romeo a class 6 student in a class test are given as:
Mathematics 47, Science 44, English 46. Present the given marks in a table.
Then, arrange the marks in a matrix form. (use 1 × 3 matrix form)

(b) Shushma obtained 43 marks in mathematics and 42 marks in science in an
exam. In the same exam, her friend Kajal obtained 48 marks in mathematics
and 42 marks in science. Present their marks in a table. Then, arrange the
marks in a matrix form.

(c) In an examination Sambhu got 45 marks in computer, 42 marks in Nepali,
and 43 marks in social studies. In the same exam, his friend Parvati got 46
marks, 40 marks, and 45 marks respectively in the three above subjects.
Tabulate the marks obtained by these two students. Then, arrange their
marks in a matrix form. (use 2 × 3 matrix form)

(d) A shopkeeper sold 22 kg of Sugar on Sunday, 25 kg on Monday, and 30 kg
on Tuesday. Similarly, she sold 28 kg of rice on Sunday, 30 kg on Monday,
and 34 kg on Tuesday. She also sold 21 kg of Wheat flour on Sunday, 28 kg
on Monday, and 33 kg on Tuesday. Present the sales of these different items
on different days in a table. Then, arrange the data in a matrix form.

7. (a) If A = 2 4 and B = 7 4 , show that A + B = B + A.
6 10 3 2

(b) If P = 4 5 and Q = 2 8 , show that P + Q = Q + P.
6 7 7 4

(c) If G = –4 –3 and H = 14 6 , show that G + H = H + G.
2 –1 4 8

8. (a) If A = 2 4 and B = 7 5 , find A – B and B – A.
6 7 8 10

(b) If M = 2 4 and N = 10 6 , find M – N and N – M.
5 6 8 2

vedanta Excel in Additional Mathematics - Book 6 73

Matrices

(c) If T = 2 4 and U = –2 8 , then T – U and U – T.
–6 7 4 9

1. (a) 1 × 3 (b) 2 × 2 (c) 2 × 1 (d) 2 × 3

(e) 3 × 2 (f) 3 × 3

2. (a) a = 5, b = 10, c = 8, d = 7 (b) a = 1 , b = 3, c = 2, d = 15
3. (a) x = 7, y = 2 2
(b) p = 3, q = 7

4. (a) [3 9 8] (b) [9 6 9] (c) 8 5 (d) 4 6
11 9 9 14
2 5 0 0 7 4 10
(e) 8 11 (f) 0 0 (g) 12 9 14

5. (a) 1 2 (b) 4 4 (c) 0 –4 (d) [2 1 0]
3 2 5 –2 3 10

–3 (f) –1 2 –1
(e) – 3 –6 0 0

–3

6. (a) [47 44 46] (b) 43 42 (c) 45 42 43
48 42 46 40 45
Sugar Rice Wheat

Sunday 22 28 21

(d) Monday 25 30 28

Tuesday 30 34 33

8. (a) –5 – 1 , 5 1 (b) –8 – 2 , 8 2
–2 – 3 2 3 –3 – 4 3 4
4 – 4 4 4
(c) – 10 – 2 , 10 2

74 vedanta Excel in Additional Mathematics - Book 6

Class : 6 Model Questions Matrices
Full marks : 50
Annual Examination
Time : 1.5 hours

Attempt all the questions.

Group A [9×2 = 18]

1. Find the values of 'x' and 'y' if (4x + 3, y + 2) = (11, 4).

2. If P = 1 2 and Q = 6 3 , find P + Q.
3 4 1 4

3. If A = 2 4 and B = 4 6 , find A – B.
6 7 2 7

4. Convert 81° into grade and radian measure.

5. In a right angled triangle, if one of acute angle is 27°, find the other angle in
grade.

6. From the given right angled triangle, find the unknown side.
C

5 cm

AB
3 cm

7. Simplify : 4sinA – 3cosA + sinA + 8cosA
8. Prove that : sinA + cosA . tanA = 2sinA
9. Find the distance between A(6, 7) and B(3, 3).

vedanta Excel in Additional Mathematics - Book 6 75

Matrices

Group B [8×4 = 32]

10. If 3x°, 5x° and 2x° are angles of a triangle, find each angle in degree.

11. Convert the following degree in radian and grade.

(a) 45° (b) 72°

12. From the given right angled triangle, find six trigonometric ratios of angle T.

C

2
3

AT B
1

13. Prove the following :

(a) sin2A + cos2A = 1 (b) (1 – cos2T) . sec2T = tan2T

14. Plot the given points on a graph paper A(3, 3), B(9, 3), and C(9, 11).

Join AB, BC, and CA show that it is a right angled triangle.
15. If A(6, 7), B(9, 11), C(7, 2) and D(4, 6) are four points, then show that AB = CD.

16. If P = 2 4 and Q = –6 2 , prove that P + Q = Q + P.
6 7 5 –4

17. If M = 3 6 and N = 2 7 , find M – N and N – M, are they equal?
7 8 –3 –4

76 vedanta Excel in Additional Mathematics - Book 6