NCERT Solutions for Class 8 Maths Chapter 9 Exercise 9.5
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Transcript of NCERT Solutions for Class 8 Maths Chapter 9 Exercise 9.5
Q1. Use a suitable identity to get each of the following products.
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
2 2 2 2
i 3 3
ii 2 5 2 5
iii 2 7 2 7
iv
v 1.1m 0.4 1.1m 0.4
vi
vii 6 7 6 7
viii
ix
1 13 3
2 2
3 3
2 4 2 4
x 7 9 7 9
x x
y y
a a
a
a a
x y x
b a b
x x
a c a c
y
a b a b
− −
+
+ +
+ +
− −
− +
+ − +
− +
− + −
+
+
− −
Difficulty level: Medium
Known:
Expressions
Unknown:
Simplification
Reasoning:
i) By using the distributive law, we can carry out the multiplication term by term.
ii) In multiplication of polynomials with polynomials, we should always look for like
terms, if any, and combine them.
Solution:
The products will be as follows.
(i)
( ) ( ) ( )
( ) ( )( ) ( ) ( )
2
2 2 2 2 2
2
3 3 3
2 3 3 2
6 9
x x x
x x a b a ab b
x x
+ + = +
= + + + = + +
= + +
NCERT Solutions Class 8 Maths Chapter 9 Exercise 9.5
(ii)
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
2
2 2 2 2 2
2
2 5 2 5 2 5
2 2 2 5 5 2
4 20 25
y y y
y y a b a ab b
y y
+ + = +
= + + + = + +
=
+ +
(iii)
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
2
2 2 2 2 2
2
2 7 2 7 2 7
2 2 2 7 7 2
4 28 49
a a a
a a a b a ab b
a a
− − = −
= − + − = − +
=
− +
(iv)
( ) ( ) ( )
2
22 2 2 2
2
1 1 13 3 3
2 2 2
1 13 2 3 2
2 2
19 3
4
a a a
a a a b a ab b
a a
− − = −
= − + − = − +
= − +
(v)
( ) ( ) ( ) ( ) ( ) ( )2 2 2 2
2
1.1m 0.4 1.1m 0.4 1.1m 0.4
1.21m 0.16
a b a b a b − + = − +
= −
− = −
(vi)
( ) ( ) ( ) ( )
( ) ( ) ( )( )
2 2 2 2 2 2 2 2
2 22 2 2 2
4 4
a b a b b a b a
b a a b a b a b
b a
+ − + = + −
= − + − = −
= −
(vii)
( ) ( ) ( ) ( ) ( ) ( )2 2 2 2
2
6 7 6 7 6 7
36 49
x x x a b a b a b
x
− + = − + −
=
= −
−
(viii)
( ) ( ) ( )
( ) ( )( ) ( ) ( )
2
2 2 2 2 2
2 2
2 2
2
a c a c a c
a a c c a b a ab b
a ac c
− + − + = − +
= − + − + + =
−
=
+ +
+
(ix)
( )
2
2 22 2 2
2 2
3 3 3
2 4 2 4 2 4
3 32 2
2 2 4 4
3 9
4 4 16
x y x y x y
x x y ya b a ab b
x xy y
+ + = +
= + + + = + +
= + +
(x)
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
2
2 2 2 2 2
2 2
7 9 7 9 7 9
7 2 7 9 9 2
49 126 81
a b a b a b
a a b b a b a ab b
a ab b
− − = −
= − + − = − +
= − +
Q2. Use the identity ( ) ( ) ( )2x a x b x a b x ab+ + = + + + to find the following
products.
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )( ) ( ) ( )
2 2
i 3 7
ii 4 5 4 1
iii 4 5 4 1
iv 4 5 4 1
v 2 5 2 3
vi 2 9 2 5
vii 4 2
x x
x x
x x
x x
x y x y
a a
xyz xyz
+ +
+ +
− −
+ −
+ +
+ +
− −
Difficulty level: Easy
Known:
( )( ) ( )2 x a x b x a b x ab+ + = + + +
Unknown:
Simplification
Solution:
The products will be as follows.
(i)
( ) ( ) ( ) ( ) ( )2
2
3 7 3 7 3 7
10 21
x x x x
x x
+ + = + + +
= + +
(ii)
( ) ( ) ( ) ( ) ( ) ( ) ( )2
2
4 5 4 1 4 5 1 4 5 1
16 24 5
x x x x
x x
+ + = + + +
= + +
(iii)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2
2
4 5 4 1 4 5 1 4 5 1
16 24 5
x x x x
x x
− − = + − + − + − −
= − +
(iv)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2
2
4 5 4 1 4 5 1 4 5 1
16 16 5
x x x x
x x
+ − = + + + − + + −
= + −
(v)
( ) ( ) ( ) ( ) ( ) ( ) ( )2
2 2
2 5 2 3 2 5 3 2 5 3
4 16 15
x y x y x y y x y y
x xy y
+ + = + + +
= + +
(vi)
( ) ( ) ( ) ( ) ( ) ( ) ( )2
2 2 2 2
4 2
2 9 2 5 2 9 5 2 9 5
4 28 45
a a a a
a a
+ + = + + +
= + +
(vii)
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2
2 2 2
4 2 4 2 4 2
6 8
xyz xyz xyz xyz
x y z xyz
− − = + − + − + − −
= − +
Q3. Find the following squares by using the identities.
( ) ( )
( ) ( )
( ) ( )
( )
( ) ( )
( ) ( )
2
2
2
2
2
2
2
2 3
3 2
i 7
ii 3
iii 6 5
iv
v 0.4 0.5
vi 2 5
b
xy z
x y
p
n
x
m
q
y y
−
+
−
−
+
+
Difficulty level: Medium
Known:
Expressions
Unknown:
Simplification
Reasoning: 2 2 2
2 2 2
2 2
( ) 2
( ) 2
( )( )
a b a ab b
a b a ab b
a b a b a b
+ = + +
− = − +
+ − = −
Solution:
(i)
( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2
2
7 2 7 7 2
14 49
b b b a b a ab b
b b
− = − + − = − +
= − +
(ii)
( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2
2 2 2
3 2 3 3 2
6 9
xy z xy xy z z a b a ab b
x y xyz z
+ = + + + = + +
=
+ +
(iii)
( ) ( ) ( ) ( ) ( ) ( )2 2 2 22 2 2 2 2
4 2 2
6 5 6 2 6 5 5 2
36 60 25
x y x x y y a b a ab b
x x y y
− = − + − = − +
− +
=
(iv)
( )2 2 2
2 2
2 2
22 3 2 2 3 32
3 2 3 3 2 2
4 92
9 4
2m n m m n n
m m
a a a b
n n
b b + = + +
+
+
+
+
=
= +
(v)
( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2
2 2
0.4 0.5 0.4 2 0.4 0.5 0.5 2
0.16 0.4 0.25
p q p p q q a b a ab b
p pq q
− = − + − = − +
− +
=
(vi)
( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 2
2 2 2 2
2 5 2 2 2 5 5 2
4 20 25
xy y xy xy y y a b a ab b
x y xy y
+ = + + + = + +
= +
+
Q4. Simplify.
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )
22 2
2 2
2 2
2 2
2 2
2 2
22 2 3 2
i
ii 2 5 2 5
iii 7 8 7 8
iv 4 5 5 4
v 2.5 1.5 1.5 2.5
vi 2
vii 2
a b
x x
m n m n
m n m n
p q p q
ab bc ab c
m n m m n
−
+ − −
− + +
+ + +
− − −
+ −
− +
Difficulty level: Medium
Known:
Expressions
Unknown:
Simplification
Reasoning: 2 2 2
2 2 2
2 2
( ) 2
( ) 2
( )( )
a b a ab b
a b a ab b
a b a b a b
+ = + +
− = − +
+ − = −
Solution:
( ) ( )
( ) ( )( ) ( ) ( )
22 2
2 2 22 2 2 2 2 2
4 2 2 4
2 2
2
i a b
a a b b a b a ab b
a a b b
−
= − + − = − +
= − +
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( )
( )
2 2
2 2 2 2
2 2 2
2 2 2
2 2
2 2
2 5 2 5
2 2 2 5 5 2 2 2 5 5
2
2
4 20 25 4 20 25
4 20 25 4 20 25
40
ii x x
x x x x
a b a ab b
a b a ab b
x x x x
x x x x
x
+
+ − −
= + + − − +
− = − +
+ = +
= + + −
=
− +
+ + −
+ −
=
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2 2
2 2 2 2
2 22 2 2 2
2
7 8 7 8
7 2 7 8 8 7 2 7 8 8
2 2
49 112
iii m n m n
m m n n m m n n
a b a ab b and a b a ab b
m
−
− + +
= + + + +
− = − + + = + +
= − 2 2 64 49 112mn n m+ + + 2
2 2
64
98 128
mn n
m n
+
= +
( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )
2 2
2 2 2 2 2 2 2
2 2 2 2
2 2
4 5 5 4
4 2 4 5 5 5 2 5 4 4 2
16 40 25 25 40 16
41 80 41
iv m n m n
m m n n m m n n a b a ab b
m mn n m mn n
m mn n
+ + +
= + + + + + + = + +
= + + + + +
= + +
( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( )( ) ( )
( )
2 2
2 2 2 2
2 2 2
2 2 2 2
2
2.5 1.5 1.5 2.5
2.5 2 2.5 1.5 1.5 1.5 2 1.5 2.5 2.5
2
6.25 7.5 2.25 2.25 7.5 6.25
6.25 7.5
v p q p q
p p q q p p q q
a b a ab b
p pq q p pq q
p
− − −
= − + − − +
− = − +
= − + − − +
=
−
2.25pq + 2 2.25q − 2 7.5p + 2
2 24
] 6.25
4
pq q
p q
−
= −
( ) ( )
( ) ( )( ) ( ) ( )
2 2
2 2 22 2 2
2 2 2
2
2 2 2
2
vi ab bc ab c
ab ab bc bc ab c a b a ab b
a b ab c
+ −
= + + − + =
= +
+ +
2 2 2 2b c ab c+ −2 2 2 2 a b b c= +
( ) ( )
( ) ( ) ( ) ( ) ( )
22 2 3 2
2 2 22 2 2 2 3 2 2 2
4 3 2
2
2 2 2
2
vii m n m m n
m m n m n m m n a b a ab b
m m n
− +
= − + + − = − +
= − 4 2 3 22n m m n+ +4 4 2 m n m= +
Q5. Show that
( ) ( ) ( )
( ) ( ) ( )
( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2
2 2
2 2 2
2
2 2
i 3 7 84 3 7
ii 9 5 180 9 5
ii4 3 16 9
23 4 9 1
i
iv 4 3 4 3 48
0
6
v
m n mn
x x x
p q pq p q
pq q pq q pq
a b a b b c b
m
c c a c a
n
+ − = −
− + = +
+ − − =
− + + − + + −
− + = +
+
=
Difficulty level: Hard
Known:
LHS and RHS expression
Unknown:
Verification of LHS = RHS
Solution:
( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
2
2 2
2
2
2
2 2
2
i . . 3 7 84
3 2 3 7 7 84
9 42 49 84
9 42 49
. . 3 7
3 2 3 7 7
9 42 49
. . . .
L H S x x
x x x
x x x
x x
R H S x
x x
x x
L H S R H S
= + −
= + + −
= + + −
= − +
= −
= − +
= − +
=
( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
2
2 2
2 2
2 2
2
2 2
2 2
ii . . 9 5 180
9 2 9 5 5 180
81 90 25 180
81 90 25
. . 9 5
9 2 9 5 5
81 90 25
. . . .
L H S p q pq
p p q q pq
p pq q pq
p pq q
R H S p q
p p q q
p pq q
L H S R H S
= − +
= − + +
= − + +
= + +
= +
= + +
= + +
=
( )2
2 2
2
4 3iii . . 2
3 4
4 4 3 32 2
3 3 4 4
162
9
L H S m n mn
m m n n mn
m mn
= − +
= − + +
= − 292
16n mn+ +
2 216 9
9 16
. . . .
m n
L H S R H S
= +
=
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
2 2
2 2 2 2
2 2 2 2 2 2 2 2
2 2
iv . . 4 3 4 3
4 2 4 3 3 4 2 4 3 3
16 24 9 16 24 9
16
L H S pq q pq q
pq pq q q pq pq q q
p q pq q p q pq q
p q
= + − −
= + + − − +
= + + − − +
= 2 224 9pq q+ + 2 216 p q− 2 224 9pq q+ −
248
. . . .
pq
L H S R H S
=
=
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )2 2 2 2 2 2
2
v . . L H S a b a b b c b c c a c a
a b b c c a
a
= − + + − + + − +
= − + − + −
= 2b− 2b+ 2c− 2c+ 2a−
0
. . . . .L H S R H S
=
=
Q6.Using identities, evaluate.
( )
( )
( )
( )
( ) ( )
( )
( )
( )
( )
2
2
2
2
2
2
i 71
ii 99
iii 102
iv 998
v 5.2
vi 297 303
vii 78 82
viii 8.9
ix 1.05 9.5
Difficulty level: Hard
Known:
Expressions
Unknown:
Values of the expressions
Reasoning: 2 2 2
2 2 2
2 2
( ) 2
( ) 2
( )( )
a b a ab b
a b a ab b
a b a b a b
+ = + +
− = − +
+ − = −
Solution:
( ) ( )
( ) ( ) ( ) ( ) ( )
22
2 2 2 2 2
i 71 70 1
70 2 70 1 1 2
4900 140 1
5041
a b a ab b
= +
= + + + = + +
= + +
=
( )
( ) ( ) ( ) ( ) ( )
2 2
2 2 2 2 2
ii 99 1 00 1
100 2 100 1 1 2
10000 200 1
980
(
1
)
a b a ab b
= −
= − + − = − +
= − +
=
( ) ( )
( ) ( ) ( ) ( ) ( )
22
2 2 2 2 2
iii 102 100 2
100 2 100 2 2 2
10000 400 4
10404
a b a ab b
= +
= + + + =
=
+
=
+ +
+
( ) ( )
( ) ( ) ( ) ( ) ( )
22
2 2 2 2 2
iv 998 1000 2
1000 2 1000 2 2 2
1000000 4000 4
996004
a b a ab b
= −
= − + − = − +
= − +
=
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
2 2
2 2 2 2 2
v 5.2 5.0 0.2
5.0 2 5.0 0.2 0.2 2
25 2 0.04
27.04
a b a ab b
= +
= + + + = + +
= + +
=
( ) ( ) ( )
( ) ( ) ( ) ( )2 2 2 2
vi 297 303 300 3 300 3
300 3
90000 9
89991
a b a b a b
= − +
= − + − = −
= −
=
( ) ( ) ( )
( ) ( ) ( ) ( )2 2 2 2
vii 78 82 80 2 80 2
80 2
6400 4
6396
a b a b a b
= − +
= − + − = −
= −
=
( ) ( )
( ) ( ) ( ) ( ) ( )
22
2 2 2 2 2
viii 8.9 9.0 0.1
9.0 2 9.0 0.1 0.1 2
81 1.8 0.01
79.21
a b a ab b
= −
= − + − = − +
= −
=
+
( )
( ) ( )
( ) ( )
( ) ( )
2 2
2 2
ix 1.05 9.5 1.05 0.95 10
1 0.05 1 0.05 10
1 0.05 10
1 0.0025 10
0.9975 10
9.975
a b a b a b
=
= + −
= −
= − + − =
=
=
−
Q 7. Using ( ) ( )2 2 ,a b a b a b− = + − find
( )
( ) ( ) ( )
( )
( )
2 2
2 2
2 2
2 2
i 51 – 49
ii 1.02 0.98
iii 153 147
iv 12.1 7.9
−
−
−
Difficulty level: Medium
Known:
( ) ( )2 2 ,a b a b a b− = + −
Unknown:
Results of the given expression with their corresponding values
Solution:
( ) ( ) ( )
( ) ( )
2 2i 51 49 51 49 51 49
100 2 200
− = + −
= =
( ) ( ) ( ) ( ) ( )
( ) ( )
2 2ii 1.02 0.98 1.02 0.98 1.02 0.98
2 0.04
0.08
− = + −
=
=
( ) ( ) ( )
( ) ( )
2 2iii 153 147 153 147 153 147
300 6
1800
− = + −
=
=
( ) ( ) ( )
( ) ( )
2 2iv 12.1 7.9 12.1 7.9 12.1 7.9
20.0 4.2
84
− = + −
=
=
Q8. Using ( ) ( ) ( )2x a x b x a b x ab+ + = + + + , find
( )
( )
( )
( )
i 103 104
ii 5.1 5.2
iii 103 98
iv 9.7 9.8
Difficulty level: Medium
Known:
( ) ( ) ( )2x a x b x a b x ab+ + = + + +
Unknown:
Results of the given expression with their corresponding values
Solution:
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )2
i 103 104 100 3 100 4
100 3 4 100 3 4
10000 700 12
10712
= + +
= + + +
= + +
=
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )2
ii 5.1 5.2 5 0.1 5 0.2
5 0.1 0.2 5 0.1 0.2
25 1.5 0.02
26.52
= + +
= + + +
= + +
=
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )2
iii 103 98 100 3 100 2
100 3 2 100 3 2
10000 100 6
10094
= + −
= + + − + −
= + −
=
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( )
2
iv 9.7 9.8 10 0.3 10 0.2
10 0.3 0.2 10 0.3 0.2
100 0.5 10 0.06
100 5 0.06
95 0.06
95.06
= − −
= + − + − + − −
= + − +
= − +
= +
=
