Algebraic Identities- 1 RD Sharma Solutions | Mathematics (Maths) Class 9 PDF Download

 Page 1


Q u e s t i o n : 1
Evaluate each of the following using identities:
i
2x -
1
x
2
ii
(2x + y) (2x - y)
iii
(a
2
b - b
2
a)
2
iv
(a - 0.1) (a + 0.1)
v
(1.5x
2 
- 0.3y
2
) (1.5x
2 
+ 0.3y
2
)
S o l u t i o n :
In the given problem, we have to evaluate expressions by using identities.
i Given 
We shall use the identity
By applying identity we get 
Hence the value of is
ii We have been given
We shall use the identity 
Here, , 
By applying identity we get 
Hence the value of is 
( )
Page 2


Q u e s t i o n : 1
Evaluate each of the following using identities:
i
2x -
1
x
2
ii
(2x + y) (2x - y)
iii
(a
2
b - b
2
a)
2
iv
(a - 0.1) (a + 0.1)
v
(1.5x
2 
- 0.3y
2
) (1.5x
2 
+ 0.3y
2
)
S o l u t i o n :
In the given problem, we have to evaluate expressions by using identities.
i Given 
We shall use the identity
By applying identity we get 
Hence the value of is
ii We have been given
We shall use the identity 
Here, , 
By applying identity we get 
Hence the value of is 
( )
iii The given expression is
We shall use the identity 
Here
By applying identity we get 
Hence the value of is
iv The given expression is
(a +0. 1)(a -0. 1)
We shall use the identity 
Here 
By applying identity we get 
(a +0. 1)(a -0. 1) = (a)
2
-(0. 1)
2
                           = (a ×a)-(0. 1 ×0. 1)                           = a
2
-0. 01
Hence the value of(a +0. 1)(a -0. 1)
is 
v The given expression is 
We shall use the identity 
Here  
By applying identity we get 
Hence the value of is
Q u e s t i o n : 2
Evaluate each of the following using identities:
i
399
2
ii
0.98
2
iii
991 ? 1009
Page 3


Q u e s t i o n : 1
Evaluate each of the following using identities:
i
2x -
1
x
2
ii
(2x + y) (2x - y)
iii
(a
2
b - b
2
a)
2
iv
(a - 0.1) (a + 0.1)
v
(1.5x
2 
- 0.3y
2
) (1.5x
2 
+ 0.3y
2
)
S o l u t i o n :
In the given problem, we have to evaluate expressions by using identities.
i Given 
We shall use the identity
By applying identity we get 
Hence the value of is
ii We have been given
We shall use the identity 
Here, , 
By applying identity we get 
Hence the value of is 
( )
iii The given expression is
We shall use the identity 
Here
By applying identity we get 
Hence the value of is
iv The given expression is
(a +0. 1)(a -0. 1)
We shall use the identity 
Here 
By applying identity we get 
(a +0. 1)(a -0. 1) = (a)
2
-(0. 1)
2
                           = (a ×a)-(0. 1 ×0. 1)                           = a
2
-0. 01
Hence the value of(a +0. 1)(a -0. 1)
is 
v The given expression is 
We shall use the identity 
Here  
By applying identity we get 
Hence the value of is
Q u e s t i o n : 2
Evaluate each of the following using identities:
i
399
2
ii
0.98
2
iii
991 ? 1009
iv
117 ? 83
S o l u t i o n :
In the given problem, we have to evaluate expressions by using identities.
i Given 
We can write as 
We shall use the Identity  
Where , 
By applying in identity we get 
(400 -1)
2
= (400)
2
-2 ×400 ×1 +(1)
2
= 400 ×400 -800 +1 = 160000 -800 +1 = 159201
Hence the value of is 
ii We have been given 
We can write  as 
We shall use the identity 
Where ,
By applying in identity we get 
Hence the value of is
iii The given expression is 
We have 
So we can express and in the terms of as 
We shall use the identity 
(x -y)(x +y) = x
2
-y
2
Here 
By applying in identity we get 
Hence the value of is 
iv The given expression is 
Page 4


Q u e s t i o n : 1
Evaluate each of the following using identities:
i
2x -
1
x
2
ii
(2x + y) (2x - y)
iii
(a
2
b - b
2
a)
2
iv
(a - 0.1) (a + 0.1)
v
(1.5x
2 
- 0.3y
2
) (1.5x
2 
+ 0.3y
2
)
S o l u t i o n :
In the given problem, we have to evaluate expressions by using identities.
i Given 
We shall use the identity
By applying identity we get 
Hence the value of is
ii We have been given
We shall use the identity 
Here, , 
By applying identity we get 
Hence the value of is 
( )
iii The given expression is
We shall use the identity 
Here
By applying identity we get 
Hence the value of is
iv The given expression is
(a +0. 1)(a -0. 1)
We shall use the identity 
Here 
By applying identity we get 
(a +0. 1)(a -0. 1) = (a)
2
-(0. 1)
2
                           = (a ×a)-(0. 1 ×0. 1)                           = a
2
-0. 01
Hence the value of(a +0. 1)(a -0. 1)
is 
v The given expression is 
We shall use the identity 
Here  
By applying identity we get 
Hence the value of is
Q u e s t i o n : 2
Evaluate each of the following using identities:
i
399
2
ii
0.98
2
iii
991 ? 1009
iv
117 ? 83
S o l u t i o n :
In the given problem, we have to evaluate expressions by using identities.
i Given 
We can write as 
We shall use the Identity  
Where , 
By applying in identity we get 
(400 -1)
2
= (400)
2
-2 ×400 ×1 +(1)
2
= 400 ×400 -800 +1 = 160000 -800 +1 = 159201
Hence the value of is 
ii We have been given 
We can write  as 
We shall use the identity 
Where ,
By applying in identity we get 
Hence the value of is
iii The given expression is 
We have 
So we can express and in the terms of as 
We shall use the identity 
(x -y)(x +y) = x
2
-y
2
Here 
By applying in identity we get 
Hence the value of is 
iv The given expression is 
We have
So we can express and in the terms of 100 as
We shall use the identity (x -y)(x +y) = x
2
-y
2
Here
By applying in identity we get 
Hence the value of  is 
Q u e s t i o n : 3
Simplify each of the following:
i
175 × 175 + 2 × 175 × 25 + 25 × 25
ii
322 × 322 - 2 × 322 × 22 + 22 × 22
iii
0.76 × 0.76 + 2 × 0.76 × 0.24 + 0.24 × × 0.24
iv
7.83 ×7.83-1.17×1.17
6.66
S o l u t i o n :
In the given problem, we have to simplify expressions
i Given 
Put 
Hence the equation becomes ,
That is 
Hence the value of is 
Page 5


Q u e s t i o n : 1
Evaluate each of the following using identities:
i
2x -
1
x
2
ii
(2x + y) (2x - y)
iii
(a
2
b - b
2
a)
2
iv
(a - 0.1) (a + 0.1)
v
(1.5x
2 
- 0.3y
2
) (1.5x
2 
+ 0.3y
2
)
S o l u t i o n :
In the given problem, we have to evaluate expressions by using identities.
i Given 
We shall use the identity
By applying identity we get 
Hence the value of is
ii We have been given
We shall use the identity 
Here, , 
By applying identity we get 
Hence the value of is 
( )
iii The given expression is
We shall use the identity 
Here
By applying identity we get 
Hence the value of is
iv The given expression is
(a +0. 1)(a -0. 1)
We shall use the identity 
Here 
By applying identity we get 
(a +0. 1)(a -0. 1) = (a)
2
-(0. 1)
2
                           = (a ×a)-(0. 1 ×0. 1)                           = a
2
-0. 01
Hence the value of(a +0. 1)(a -0. 1)
is 
v The given expression is 
We shall use the identity 
Here  
By applying identity we get 
Hence the value of is
Q u e s t i o n : 2
Evaluate each of the following using identities:
i
399
2
ii
0.98
2
iii
991 ? 1009
iv
117 ? 83
S o l u t i o n :
In the given problem, we have to evaluate expressions by using identities.
i Given 
We can write as 
We shall use the Identity  
Where , 
By applying in identity we get 
(400 -1)
2
= (400)
2
-2 ×400 ×1 +(1)
2
= 400 ×400 -800 +1 = 160000 -800 +1 = 159201
Hence the value of is 
ii We have been given 
We can write  as 
We shall use the identity 
Where ,
By applying in identity we get 
Hence the value of is
iii The given expression is 
We have 
So we can express and in the terms of as 
We shall use the identity 
(x -y)(x +y) = x
2
-y
2
Here 
By applying in identity we get 
Hence the value of is 
iv The given expression is 
We have
So we can express and in the terms of 100 as
We shall use the identity (x -y)(x +y) = x
2
-y
2
Here
By applying in identity we get 
Hence the value of  is 
Q u e s t i o n : 3
Simplify each of the following:
i
175 × 175 + 2 × 175 × 25 + 25 × 25
ii
322 × 322 - 2 × 322 × 22 + 22 × 22
iii
0.76 × 0.76 + 2 × 0.76 × 0.24 + 0.24 × × 0.24
iv
7.83 ×7.83-1.17×1.17
6.66
S o l u t i o n :
In the given problem, we have to simplify expressions
i Given 
Put 
Hence the equation becomes ,
That is 
Hence the value of is 
ii We have been given 
Put 
Hence the equation becomes 
That is
Hence the value of is 
iii Given 
Put 
Hence the equation becomes 
That is 
Hence the value of is 
iv We have been given 
Put 
Hence the equation becomes 
Hence the value of  is 
Q u e s t i o n : 4
If x +
1
x
= 11,
find the value of x
2
+
1
x
2
.
S o l u t i o n :