Class 9
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Mathematics
Exploring Algebraic Identities
NCERT Solutions
NCERT Solutions
Exploring Algebraic Identities
25 Solutions
Exercise:
All Exercises
End-of-Chapter Exercises
Exercise Set 4.1
Exercise Set 4.2
Exercise Set 4.3
Exercise Set 4.4
Exercise Set 4.5
Q1
End-of-Chapter Exercises
Use suitable identities to find the following products:
(i)
(
−
3
x
+
4
)
2
(-3 x+4)^{2}
(
−
3
x
+
4
)
2
(ii)
(
2
s
+
7
)
(
2
s
−
7
)
(2 s+7)(2 s-7)
(
2
s
+
7
)
(
2
s
−
7
)
(iii)
(
p
2
+
1
2
)
(
p
2
−
1
2
)
\left(p^{2}+\frac{1}{2}\right)\left(p^{2}-\frac{1}{2}\right)
(
p
2
+
2
1
)
(
p
2
−
2
1
)
(iv)
(
2
n
+
7
)
(
2
n
−
7
)
(2 n+7)(2 n-7)
(
2
n
+
7
)
(
2
n
−
7
)
(v)
(
s
−
2
t
)
(
s
2
+
2
s
t
+
4
t
2
)
(s-2 t)\left(s^{2}+2 s t+4 t^{2}\right)
(
s
−
2
t
)
(
s
2
+
2
s
t
+
4
t
2
)
(vi)
(
1
2
r
−
4
r
)
2
\left(\frac{1}{2 r}-4 r\right)^{2}
(
2
r
1
−
4
r
)
2
(vii)
(
−
3
m
+
4
k
−
l
)
2
(-3 m+4 k-l)^{2}
(
−
3
m
+
4
k
−
l
)
2
(viii)
(
x
−
1
3
y
)
3
\left(x-\frac{1}{3} y\right)^{3}
(
x
−
3
1
y
)
3
(ix)
(
7
2
k
−
2
3
m
)
3
\left(\frac{7}{2} k-\frac{2}{3} m\right)^{3}
(
2
7
k
−
3
2
m
)
3
Q2
End-of-Chapter Exercises
Find the values using suitable identities:
(i)
17 × 21
(ii)
104 × 96
(iii)
24 × 16
(iv)
147
3
147^{3}
14
7
3
(v)
199
3
199^{3}
19
9
3
(vi)
127
3
127^{3}
12
7
3
(vii)
(
−
107
)
3
(-107)^{3}
(
−
107
)
3
(viii)
(
−
299
)
3
(-299)^{3}
(
−
299
)
3
Q3
End-of-Chapter Exercises
Factor the following algebraic expressions:
(i)
4
y
2
+
1
+
1
16
y
2
4 y^{2}+1+\frac{1}{16 y^{2}}
4
y
2
+
1
+
16
y
2
1
(ii)
9
m
2
−
1
25
n
2
9 m^{2}-\frac{1}{25 n^{2}}
9
m
2
−
25
n
2
1
(iii)
27
b
3
−
1
64
b
3
\quad 27 b^{3}-\frac{1}{64 b^{3}}
27
b
3
−
64
b
3
1
(iv)
x
2
+
5
x
6
+
1
6
x^{2}+\frac{5 x}{6}+\frac{1}{6}
x
2
+
6
5
x
+
6
1
(v)
27
u
3
−
1
125
−
27
u
2
5
+
9
u
25
27 u^{3}-\frac{1}{125}-\frac{27 u^{2}}{5}+\frac{9 u}{25}
27
u
3
−
125
1
−
5
27
u
2
+
25
9
u
(vi)
64
y
3
+
1
125
z
3
64 y^{3}+\frac{1}{125} z^{3}
64
y
3
+
125
1
z
3
(vii)
p
3
+
27
q
3
+
r
3
−
9
p
q
r
p^{3}+27 q^{3}+r^{3}-9 p q r
p
3
+
27
q
3
+
r
3
−
9
pq
r
(viii)
9
m
2
−
12
m
+
4
9 m^{2}-12 m+4
9
m
2
−
12
m
+
4
(ix)
9
x
3
−
8
3
y
3
+
z
3
3
+
6
x
y
z
9 x^{3}-\frac{8}{3} y^{3}+\frac{z^{3}}{3}+6 x y z
9
x
3
−
3
8
y
3
+
3
z
3
+
6
x
yz
(x)
4
x
2
+
9
y
2
+
36
z
2
+
12
x
z
+
36
y
z
+
24
x
y
4 x^{2}+9 y^{2}+36 z^{2}+12 x z+36 y z+24 x y
4
x
2
+
9
y
2
+
36
z
2
+
12
x
z
+
36
yz
+
24
x
y
(xi)
27
u
3
−
1
216
−
9
u
2
2
+
u
4
27 u^{3}-\frac{1}{216}-\frac{9 u^{2}}{2}+\frac{u}{4}
27
u
3
−
216
1
−
2
9
u
2
+
4
u
Q4
End-of-Chapter Exercises
Simplify the following:
(i)
4
x
2
+
4
x
+
1
4
x
2
−
1
\frac{4 x^{2}+4 x+1}{4 x^{2}-1}
4
x
2
−
1
4
x
2
+
4
x
+
1
(ii)
9
(
3
a
3
−
24
b
3
)
9
a
2
−
36
b
2
\frac{9\left(3 a^{3}-24 b^{3}\right)}{9 a^{2}-36 b^{2}}
9
a
2
−
36
b
2
9
(
3
a
3
−
24
b
3
)
(iii)
s
3
+
125
t
3
s
2
−
2
s
t
−
35
t
2
\frac{s^{3}+125 t^{3}}{s^{2}-2 s t-35 t^{2}}
s
2
−
2
s
t
−
35
t
2
s
3
+
125
t
3
Note: Assume that the denominators are not equal to 0.
Q5
End-of-Chapter Exercises
Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.
(i)
25
a
2
−
30
a
b
+
9
b
2
25 a^{2}-30 a b+9 b^{2}
25
a
2
−
30
ab
+
9
b
2
(ii)
36
s
2
−
49
t
2
36 s^{2}-49 t^{2}
36
s
2
−
49
t
2
Q6
End-of-Chapter Exercises
Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units.
(i)
6
a
2
−
24
b
2
6 a^{2}-24 b^{2}
6
a
2
−
24
b
2
(ii)
3
p
s
2
−
15
p
s
+
12
p
3 p s^{2}-15 p s+12 p
3
p
s
2
−
15
p
s
+
12
p
Q7
End-of-Chapter Exercises
The village playground is shaped as a square of side 40 metres. A path of width
s
s
s
metres is created around the playground for people to walk. Find an expression for the area of the path in terms of
s
s
s
.
Q8
End-of-Chapter Exercises
If a number plus its reciprocal equals
10
3
\frac{10}{3}
3
10
, find the number.
Q9
End-of-Chapter Exercises
A rectangular pool has area
2
x
2
+
7
x
+
3
2 x^{2}+7 x+3
2
x
2
+
7
x
+
3
square hastas. If its width is
2
x
+
1
2 x+1
2
x
+
1
hastas, find its length. Hasta was a unit used to measure length.
Q10
End-of-Chapter Exercises
*10. If both
x
−
2
x-2
x
−
2
and
x
−
1
2
x-\frac{1}{2}
x
−
2
1
are factors of
p
x
2
+
5
x
+
r
p x^{2}+5 x+r
p
x
2
+
5
x
+
r
, show that
p
=
r
p=r
p
=
r
.
Q11
End-of-Chapter Exercises
*11. If
a
+
b
+
c
=
5
a+b+c=5
a
+
b
+
c
=
5
and
a
b
+
b
c
+
c
a
=
10
a b+b c+c a=10
ab
+
b
c
+
c
a
=
10
, then prove that
a
3
+
b
3
+
c
3
−
3
a
b
c
=
−
25
a^{3}+b^{3}+c^{3}-3 a b c=-25
a
3
+
b
3
+
c
3
−
3
ab
c
=
−
25
.
Q12
End-of-Chapter Exercises
*12. By factoring the expression, check that
n
3
−
n
n^{3}-n
n
3
−
n
is always divisible by 6 for all natural numbers
n
n
n
. Give reasons.
Q13
End-of-Chapter Exercises
*13. Find the value of
(i)
x
3
+
y
3
−
12
x
y
+
64
x^{3}+y^{3}-12 x y+64
x
3
+
y
3
−
12
x
y
+
64
, when
x
+
y
=
−
4
x+y=-4
x
+
y
=
−
4
(ii)
x
3
−
8
y
3
−
36
x
y
−
216
x^{3}-8 y^{3}-36 x y-216
x
3
−
8
y
3
−
36
x
y
−
216
, when
x
=
2
y
+
6
x=2 y+6
x
=
2
y
+
6
Q1
Exercise Set 4.1
Using the identity
(
a
+
b
)
2
=
a
2
+
2
a
b
+
b
2
(a+b)^{2}=a^{2}+2 a b+b^{2}
(
a
+
b
)
2
=
a
2
+
2
ab
+
b
2
, expand the following:
(i)
(
7
x
+
4
y
)
2
(7 x+4 y)^{2}
(
7
x
+
4
y
)
2
(ii)
(
7
5
x
+
3
2
y
)
2
\left(\frac{7}{5} x+\frac{3}{2} y\right)^{2}
(
5
7
x
+
2
3
y
)
2
(iii)
(
2.5
p
+
1.5
q
)
2
(2.5 p+1.5 q)^{2}
(
2.5
p
+
1.5
q
)
2
(iv)
(
3
4
s
+
8
t
)
2
\left(\frac{3}{4} s+8 t\right)^{2}
(
4
3
s
+
8
t
)
2
(v)
(
x
+
1
2
y
)
2
\left(x+\frac{1}{2 y}\right)^{2}
(
x
+
2
y
1
)
2
(vi)
(
1
x
+
1
y
)
2
\left(\frac{1}{x}+\frac{1}{y}\right)^{2}
(
x
1
+
y
1
)
2
Q2
Exercise Set 4.1
Using the same identity, find the values of the following:
(i)
(
64
)
2
(64)^{2}
(
64
)
2
(ii)
(
105
)
2
(105)^{2}
(
105
)
2
(iii)
(
205
)
2
(205)^{2}
(
205
)
2
Q1
Exercise Set 4.2
Factor completely:
(i)
9
x
2
+
24
x
y
+
16
y
2
9 x^{2}+24 x y+16 y^{2}
9
x
2
+
24
x
y
+
16
y
2
(ii)
4
s
2
+
20
s
t
+
25
t
2
4 s^{2}+20 s t+25 t^{2}
4
s
2
+
20
s
t
+
25
t
2
(iii)
49
x
2
+
28
x
y
+
4
y
2
49 x^{2}+28 x y+4 y^{2}
49
x
2
+
28
x
y
+
4
y
2
(iv)
64
p
2
+
32
3
p
q
+
4
9
q
2
64 p^{2}+\frac{32}{3} p q+\frac{4}{9} q^{2}
64
p
2
+
3
32
pq
+
9
4
q
2
*(v)
3
a
2
+
4
a
b
+
4
3
b
2
\quad 3 a^{2}+4 a b+\frac{4}{3} b^{2}
3
a
2
+
4
ab
+
3
4
b
2
*(vi)
9
5
s
2
+
6
s
v
+
5
v
2
\frac{9}{5} s^{2}+6 s v+5 v^{2}
5
9
s
2
+
6
s
v
+
5
v
2
(Hint: 2 was taken out as a common factor in Example 7. Is it possible to do something similar in Exercises (v) and (vi) above?)
Q2
Exercise Set 4.2
Find the values of the following using the identity
(
a
−
b
)
2
=
a
2
−
2
a
b
+
b
2
(a-b)^{2}=a^{2}-2 a b+b^{2}
(
a
−
b
)
2
=
a
2
−
2
ab
+
b
2
.
(i)
(
79
)
2
(79)^{2}
(
79
)
2
(ii)
(
193
)
2
(193)^{2}
(
193
)
2
(iii)
(
299
)
2
(299)^{2}
(
299
)
2
Q1
Exercise Set 4.3
Find the following squares using one of the above identities. Determine which of these identities will make these calculations easier.
(i)
117
2
117^{2}
11
7
2
(ii)
78
2
78^{2}
7
8
2
(iii)
198
2
198^{2}
19
8
2
(iv)
214
2
214^{2}
21
4
2
(v)
1104
2
\quad 1104^{2}
110
4
2
(vi)
1120
2
1120{ }^{2}
1120
2
Q2
Exercise Set 4.3
Factor using suitable identities:
(i)
16
y
2
−
24
y
+
9
16 y^{2}-24 y+9
16
y
2
−
24
y
+
9
(ii)
9
4
s
2
+
6
s
t
+
4
t
2
\frac{9}{4} s^{2}+6 s t+4 t^{2}
4
9
s
2
+
6
s
t
+
4
t
2
(iii)
m
2
9
+
m
k
3
+
k
2
4
+
3
n
k
+
2
m
n
+
9
n
2
\frac{m^{2}}{9}+\frac{m k}{3}+\frac{k^{2}}{4}+3 n k+2 m n+9 n^{2}
9
m
2
+
3
mk
+
4
k
2
+
3
nk
+
2
mn
+
9
n
2
(iv)
p
2
16
−
2
+
16
p
2
\frac{p^{2}}{16}-2+\frac{16}{p^{2}}
16
p
2
−
2
+
p
2
16
(v)
9
a
2
+
4
b
2
+
c
2
−
12
a
b
+
6
a
c
−
4
b
c
9 a^{2}+4 b^{2}+c^{2}-12 a b+6 a c-4 b c
9
a
2
+
4
b
2
+
c
2
−
12
ab
+
6
a
c
−
4
b
c
Q3
Exercise Set 4.3
Expand the following using the identity
(
a
+
b
+
c
)
2
=
a
2
+
b
2
+
c
2
+
2
a
b
+
2
b
c
+
2
c
a
:
(a+b+c)^{2}=a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a:
(
a
+
b
+
c
)
2
=
a
2
+
b
2
+
c
2
+
2
ab
+
2
b
c
+
2
c
a
:
(i)
(
p
+
3
q
+
7
r
)
2
(p+3 q+7 r)^{2}
(
p
+
3
q
+
7
r
)
2
(ii)
(
3
x
−
2
y
+
4
z
)
2
(3 x-2 y+4 z)^{2}
(
3
x
−
2
y
+
4
z
)
2
Q4
Exercise Set 4.3
Is this an identity?
(
a
+
b
−
c
)
2
+
(
a
−
b
+
c
)
2
+
(
a
−
b
−
c
)
2
=
2
a
2
+
2
b
2
+
2
c
2
.
(a+b-c)^{2}+(a-b+c)^{2}+(a-b-c)^{2}=2 a^{2}+2 b^{2}+2 c^{2}.
(
a
+
b
−
c
)
2
+
(
a
−
b
+
c
)
2
+
(
a
−
b
−
c
)
2
=
2
a
2
+
2
b
2
+
2
c
2
.
Q1
Exercise Set 4.4
Fill in the blanks to complete the following identities:
(i)
s
2
−
11
s
+
24
=
(
_
_
_
_
)
(
_
_
_
_
)
s^{2}-11 s+24=( \_\_\_\_ ) ( \_\_\_\_ )
s
2
−
11
s
+
24
=
(
____
)
(
____
)
(ii)
(
)
(
x
+
1
)
=
(
3
x
2
−
4
x
−
7
)
)(x+1)=\left(3 x^{2}-4 x-7\right)
)
(
x
+
1
)
=
(
3
x
2
−
4
x
−
7
)
(iii)
10
x
2
−
11
x
−
6
=
(
2
x
−
_
_
_
_
(
_
_
_
_
+
2
)
10 x^{2}-11 x-6=(2 x- \_\_\_\_ ( \_\_\_\_ + 2)
10
x
2
−
11
x
−
6
=
(
2
x
−
____
(
____
+
2
)
(iv)
6
x
2
+
7
x
+
2
=
(
_
_
_
_
(
)
_
_
_
_
6 x^{2}+7 x+2=( \_\_\_\_ () \_\_\_\_
6
x
2
+
7
x
+
2
=
(
____
(
)
____
Q2
Exercise Set 4.4
Select and use the identity that will help you to find the following products without multiplying directly:
(i)
(
41
)
2
(41)^{2}
(
41
)
2
(ii)
(
27
)
2
(27)^{2}
(
27
)
2
(iii)
(
23
×
17
)
(23 \times 17)
(
23
×
17
)
(iv)
(
135
)
2
(135)^{2}
(
135
)
2
(v)
(
97
)
2
(97)^{2}
(
97
)
2
(vi)
(
18
×
29
)
(18 \times 29)
(
18
×
29
)
(vii)
(
34
×
43
)
(34 \times 43)
(
34
×
43
)
(viii)
(
205
)
2
(205)^{2}
(
205
)
2
Q3
Exercise Set 4.4
Factor the following:
(i)
9
a
2
+
b
2
+
4
c
2
−
6
a
b
+
12
a
c
−
4
b
c
9 a^{2}+b^{2}+4 c^{2}-6 a b+12 a c-4 b c
9
a
2
+
b
2
+
4
c
2
−
6
ab
+
12
a
c
−
4
b
c
(ii)
16
s
2
+
25
t
2
−
40
s
t
16 s^{2}+25 t^{2}-40 s t
16
s
2
+
25
t
2
−
40
s
t
(iii)
r
2
−
r
−
42
r^{2}-r-42
r
2
−
r
−
42
(iv)
49
g
2
+
14
g
h
+
h
2
49 g^{2}+14 g h+h^{2}
49
g
2
+
14
g
h
+
h
2
(v)
64
u
2
+
121
v
2
+
4
w
2
−
176
u
v
−
32
u
w
+
44
v
w
64 u^{2}+121 v^{2}+4 w^{2}-176 u v-32 u w+44 v w
64
u
2
+
121
v
2
+
4
w
2
−
176
uv
−
32
u
w
+
44
v
w
Q1
Exercise Set 4.5
Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero:
(i)
3
p
2
−
3
p
q
−
18
q
2
p
2
+
3
p
q
−
10
q
2
\frac{3 p^{2}-3 p q-18 q^{2}}{p^{2}+3 p q-10 q^{2}}
p
2
+
3
pq
−
10
q
2
3
p
2
−
3
pq
−
18
q
2
(ii)
n
3
−
3
n
2
m
+
3
n
m
2
−
m
3
5
m
2
−
10
m
n
+
5
n
2
\frac{n^{3}-3 n^{2} m+3 n m^{2}-m^{3}}{5 m^{2}-10 m n+5 n^{2}}
5
m
2
−
10
mn
+
5
n
2
n
3
−
3
n
2
m
+
3
n
m
2
−
m
3
(iii)
w
3
−
v
3
+
x
3
+
3
w
v
x
w
2
+
v
2
+
x
2
−
2
w
v
−
2
v
x
+
2
w
x
\frac{w^{3}-v^{3}+x^{3}+3 w v x}{w^{2}+v^{2}+x^{2}-2 w v-2 v x+2 w x}
w
2
+
v
2
+
x
2
−
2
w
v
−
2
vx
+
2
w
x
w
3
−
v
3
+
x
3
+
3
w
vx
(iv)
4
y
2
−
20
y
z
+
25
z
2
(
25
z
2
−
4
y
2
)
\frac{4 y^{2}-20 y z+25 z^{2}}{\left(25 z^{2}-4 y^{2}\right)}
(
25
z
2
−
4
y
2
)
4
y
2
−
20
yz
+
25
z
2
(v)
(
x
2
+
x
−
6
)
(
x
2
−
7
x
+
12
)
(
x
2
−
6
x
+
8
)
(
x
2
−
9
)
\frac{\left(x^{2}+x-6\right)\left(x^{2}-7 x+12\right)}{\left(x^{2}-6 x+8\right)\left(x^{2}-9\right)}
(
x
2
−
6
x
+
8
)
(
x
2
−
9
)
(
x
2
+
x
−
6
)
(
x
2
−
7
x
+
12
)
(vi)
p
4
−
16
p
2
−
4
p
+
4
\frac{p^{4}-16}{p^{2}-4 p+4}
p
2
−
4
p
+
4
p
4
−
16
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