Chapter 4 : Algebraic Identities - Rd Sharma Solutions for Class 9 Maths CBSE

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Chapter 4 - Algebraic Identities Excercise Ex. 4.1

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Chapter 4 - Algebraic Identities Excercise Ex. 4.2

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Write the following in the expanded form:

(x + 2y + 4z)2
Solution 10
Question 11
Write the following in the expanded form:

(2x - y + z)2
Solution 11

Question 12
Write the following in the expanded form:

(-2x + 3y + 2z)2
Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22
Solution 22
Question 23
Solution 23
Question 24
Solution 24

Chapter 4 - Algebraic Identities Excercise Ex. 4.3

Question 1
Solution 1
Question 2
Solution 2

Question 3
Solution 3
Question 4
Solution 4
Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Chapter 4 - Algebraic Identities Excercise Ex. 4.4

Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6

Solution 6

Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13

Question 14
Solution 14

Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18

Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
Question 22
Solution 22
Question 23
Solution 23

Chapter 4 - Algebraic Identities Excercise Ex. 4.5

Question 1
Solution 1
Question 2
Solution 2

Question 3

Solution 3

Question 4
Solution 4
Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Chapter 4 - Algebraic Identities Excercise 4.30

Question 1

begin mathsize 11px style If space straight x plus 1 over straight x equals 5 comma space then space straight x squared plus 1 over straight x squared space equals
left parenthesis straight a right parenthesis space 25
left parenthesis straight b right parenthesis space 10
left parenthesis straight c right parenthesis space 23
left parenthesis straight d right parenthesis space 27 end style

Solution 1

begin mathsize 11px style By space using space Identity space open parentheses straight a plus straight b close parentheses squared space equals space straight a squared plus straight b squared plus 2 ab comma space we space have
open parentheses space straight x plus 1 over straight x close parentheses squared space equals straight x squared plus open parentheses 1 over straight x close parentheses squared space plus space 2 cross times up diagonal strike straight x cross times fraction numerator 1 over denominator up diagonal strike straight x end fraction
rightwards double arrow open parentheses straight x plus 1 over straight x close parentheses squared space equals space straight x squared plus 1 over straight x squared plus 2 space space space space space
rightwards double arrow open parentheses 5 close parentheses squared space equals space straight x squared plus 1 over straight x squared plus 2 space space space space space space space open curly brackets straight x plus 1 over straight x equals 5 space given close curly brackets
rightwards double arrow straight x squared plus 1 over straight x squared equals 25 minus 2
rightwards double arrow straight x squared plus 1 over straight x squared space equals space 23
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 2

begin mathsize 11px style If space straight x plus 1 over straight x equals 2 comma space then space straight x cubed plus 1 over straight x cubed equals
left parenthesis straight a right parenthesis space 64
left parenthesis straight b right parenthesis space 14
left parenthesis straight c right parenthesis space 8
left parenthesis straight d right parenthesis space 2
end style

Solution 2

begin mathsize 11px style By space using space identity comma space
open parentheses straight a plus straight b close parentheses cubed space equals space straight a cubed plus straight b cubed plus 3 ab open parentheses straight a space plus space straight b close parentheses
open parentheses space straight x plus 1 over straight x close parentheses cubed space equals space straight x cubed plus 1 over straight x cubed space plus space 3. up diagonal strike straight x. fraction numerator 1 over denominator up diagonal strike straight x end fraction open parentheses space straight x plus 1 over straight x close parentheses space equals space straight x cubed plus 1 over straight x cubed plus 3 open parentheses straight x plus 1 over straight x close parentheses
Now space space straight x plus 1 over straight x equals 2
rightwards double arrow open parentheses 2 close parentheses cubed space equals space straight x cubed plus 1 over straight x cubed plus 3 open parentheses 2 close parentheses
rightwards double arrow straight x cubed plus 1 over straight x cubed equals open parentheses 2 close parentheses cubed space minus space 3 cross times 2 equals 8 minus 6 equals 2
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 3

begin mathsize 11px style If space straight x plus 1 over straight x equals 4 comma space then space straight x to the power of 4 plus 1 over straight x to the power of 4 space equals
left parenthesis straight a right parenthesis space 196
left parenthesis straight b right parenthesis space 194
left parenthesis straight c right parenthesis space 192
left parenthesis straight d right parenthesis space 190 end style

Solution 3

begin mathsize 11px style open parentheses straight x plus 1 over straight x close parentheses squared space equals space straight x squared plus 1 over straight x squared plus 2
open parentheses straight x plus 1 over straight x close parentheses equals 4 space space space left parenthesis given right parenthesis
rightwards double arrow straight x squared plus 1 over straight x squared space equals open parentheses 4 close parentheses squared space minus space 2 equals 16 minus 2 equals space 14 space space space space space.... left parenthesis 1 right parenthesis
Squaring space equation space left parenthesis 1 right parenthesis comma
open parentheses straight x squared plus 1 over straight x squared close parentheses squared space equals space open parentheses 14 close parentheses squared
rightwards double arrow open parentheses straight x squared close parentheses squared plus open parentheses 1 over straight x squared close parentheses squared plus 2. straight x squared.1 over straight x squared space equals space 196
rightwards double arrow straight x to the power of 4 plus 1 over straight x to the power of 4 equals 196 minus 2
rightwards double arrow straight x to the power of 4 plus 1 over straight x to the power of 4 space equals space 194
Hence comma space co rrect space option space is space left parenthesis straight b right parenthesis. end style

Question 4

begin mathsize 11px style If space straight x plus 1 over straight x equals 3 comma space then space straight x to the power of 6 plus 1 over straight x to the power of 6 space equals
left parenthesis straight a right parenthesis space 927
left parenthesis straight b right parenthesis space 414
left parenthesis straight c right parenthesis space 364
left parenthesis straight d right parenthesis space 322 end style

Solution 4

begin mathsize 11px style open parentheses space straight x plus 1 over straight x close parentheses squared space equals space space straight x squared plus 1 over straight x squared space plus space 2
space straight x plus 1 over straight x space equals space 3 space left parenthesis g i v e n right parenthesis
rightwards double arrow x squared plus 1 over straight x squared space equals space open parentheses 3 close parentheses squared space minus space 2
rightwards double arrow straight x squared plus 1 over straight x squared space equals space 7 space space space space space space.... left parenthesis 1 right parenthesis
Cubing space both space side space of space equation space left parenthesis 1 right parenthesis comma space w e space h a v e
open parentheses straight x squared plus 1 over straight x squared close parentheses cubed space equals open parentheses 7 close parentheses cubed
rightwards double arrow open parentheses straight x squared close parentheses cubed plus open parentheses 1 over straight x squared close parentheses cubed space plus space 3. straight x squared.1 over straight x squared open parentheses straight x squared plus 1 over straight x squared close parentheses space equals space 7 cubed
rightwards double arrow straight x to the power of 6 plus 1 over straight x to the power of 6 space plus space 3 open parentheses 7 close parentheses equals 7 cubed space space space space space
rightwards double arrow straight x to the power of 6 plus 1 over straight x to the power of 6 equals 343 minus 21
rightwards double arrow straight x to the power of 6 plus 1 over straight x to the power of 6 equals 322
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 5

begin mathsize 11px style If space straight x squared plus 1 over straight x squared equals 102 comma space then space straight x minus 1 over straight x space equals
left parenthesis straight a right parenthesis space 8
left parenthesis straight b right parenthesis space 10
left parenthesis straight c right parenthesis space 12
left parenthesis straight d right parenthesis space 13 end style

Solution 5

begin mathsize 11px style Correct space option space left parenthesis straight b right parenthesis
open parentheses space straight x minus 1 over straight x close parentheses squared space equals straight x squared plus 1 over straight x squared minus 2. straight x.1 over straight x
open parentheses space straight x minus 1 over straight x close parentheses squared space space equals straight x squared plus 1 over straight x squared minus 2 space space
space space space space space space space space space space space space space space space space space space space space equals 102 minus 2 space space space space space space space open curly brackets straight x squared plus 1 over straight x squared equals 102 close curly brackets
space space space space space space space space space space space space space space space space space space space space equals 100
rightwards double arrow straight x minus 1 over straight x equals space square root of 100
rightwards double arrow straight x minus 1 over straight x space equals space 10
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

begin mathsize 11px style If space straight x cubed plus 1 over straight x cubed equals 110 comma space then space straight x plus 1 over straight x space equals
left parenthesis straight a right parenthesis space 5
left parenthesis straight b right parenthesis space 10
left parenthesis straight c right parenthesis space 15
left parenthesis straight d right parenthesis space none space of space these end style

Solution 6

begin mathsize 11px style open parentheses space straight x plus 1 over straight x close parentheses cubed space equals space straight x cubed plus 1 over straight x cubed plus 3 open parentheses straight x plus 1 over straight x close parentheses
rightwards double arrow open parentheses space straight x plus 1 over straight x close parentheses cubed minus space 3 open parentheses straight x plus 1 over straight x close parentheses equals straight x cubed plus 1 over straight x cubed
rightwards double arrow open parentheses space straight x plus 1 over straight x close parentheses cubed space minus space 3 open parentheses straight x plus 1 over straight x close parentheses space equals space 110
Let space straight x plus 1 over straight x equals straight t
rightwards double arrow straight t cubed space minus space 3 straight t space minus space 110 space equals space 0
straight t space equals space 5 space is space one space of space it apostrophe straight s space solution space which space is space real comma
other space two space solutions space are space imaginary
rightwards double arrow straight x plus 1 over straight x equals 5 space space space space space space space space space space space space space space space space space space space space
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 7

begin mathsize 11px style If space straight x cubed minus 1 over straight x cubed equals 14 comma space then space straight x minus 1 over straight x space equals
left parenthesis straight a right parenthesis space 5
left parenthesis straight b right parenthesis space 4
left parenthesis straight c right parenthesis space 3
left parenthesis straight d right parenthesis space 2 end style

Solution 7

begin mathsize 11px style space open parentheses straight x minus 1 over straight x close parentheses cubed space equals space straight x cubed minus 1 over straight x cubed space minus space 3 up diagonal strike straight x fraction numerator 1 over denominator up diagonal strike straight x end fraction space open parentheses straight x minus 1 over straight x close parentheses
straight x cubed minus 1 over straight x cubed equals open parentheses straight x minus 1 over straight x close parentheses cubed space plus space 3 space open parentheses straight x minus 1 over straight x close parentheses
rightwards double arrow open parentheses straight x minus 1 over straight x close parentheses cubed space plus space 3 space open parentheses straight x minus 1 over straight x close parentheses minus straight x cubed minus 1 over straight x cubed equals 0
rightwards double arrow open parentheses straight x minus 1 over straight x close parentheses cubed space plus space 3 space open parentheses straight x minus 1 over straight x close parentheses minus 14 equals 0
Let space straight x minus 1 over straight x equals space straight t
rightwards double arrow straight t cubed space plus space 3 straight t space minus space 14 space equals space 0
rightwards double arrow straight t cubed minus 2 straight t squared plus 2 straight t squared minus 4 straight t plus 7 straight t minus 14 equals 0
rightwards double arrow straight t left parenthesis straight t minus 2 right parenthesis plus 2 straight t left parenthesis straight t minus 2 right parenthesis plus 7 left parenthesis straight t minus 2 right parenthesis equals 0
rightwards double arrow left parenthesis straight t minus 2 right parenthesis left parenthesis straight t plus 2 straight t plus 7 right parenthesis equals 0
straight t squared space plus space 2 straight t space plus space 7 space equals space 0 space has space no space real space roots
so comma space straight t space equals space 2 space is space straight a space solution
rightwards double arrow straight x minus 1 over straight x equals 2
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 8

If a + b + c = 9 and ab + bc + ca = 23, then a+ b+ c2 =

(a) 35

(b) 58

(c) 127

(d) none of these

Solution 8

We know that (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

Here, a + b + c = 9, ab + bc + ca = 23

Thus, we have

(9)2 = a2 + b2 + c2 + 2(23)

81 = a2 + b2 + c2 + 46

a2 + b2 + c2 = 81 - 46

a2 + b2 + c2 = 35

Hence, correct option is (a).

Question 9

(a - b)3 + (b - c)3 + (c - a)3 =

(a) (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

(b) (a - b)(b - c)( c - a)

(c) 3(a - b)(b - c)(c - a)

(d) none of these

Solution 9

begin mathsize 11px style Let
straight a space minus space straight b space equals space straight A
straight b space minus space straight c space equals space straight B
straight c space minus space straight a space equals space straight C
Now space left parenthesis straight A space plus space straight B space plus space straight C right parenthesis cubed space equals space straight A cubed space plus space straight B cubed space plus space straight C cubed space plus space 3 open parentheses straight A space plus space straight B close parentheses open parentheses straight B plus space straight C close parentheses open parentheses straight C space plus space straight A close parentheses
rightwards double arrow straight A cubed space plus space straight B cubed space plus space straight C cubed space equals space open parentheses straight A space plus space straight B space plus space straight C close parentheses cubed space minus space 3 open parentheses straight A space plus space straight B close parentheses open parentheses space straight B space plus space straight C close parentheses open parentheses straight C space plus space straight A close parentheses
Now space putting space values space of space straight A comma space straight B space and space straight C comma space we space get
open parentheses straight a minus straight b close parentheses cubed space plus space open parentheses straight b minus straight c close parentheses cubed space plus space open parentheses straight c minus straight a close parentheses cubed space equals space open parentheses up diagonal strike straight a minus up diagonal strike straight b plus up diagonal strike straight b minus up diagonal strike straight c plus up diagonal strike straight c minus up diagonal strike straight a close parentheses cubed space minus space 3 space open parentheses straight a minus up diagonal strike straight b plus up diagonal strike straight b minus straight c close parentheses space open parentheses straight b minus up diagonal strike straight c space plus space up diagonal strike straight c minus straight a close parentheses space open parentheses straight c minus up diagonal strike straight a plus up diagonal strike straight a minus straight b close parentheses
rightwards double arrow open parentheses straight a minus straight b close parentheses cubed space plus space open parentheses straight b minus straight c close parentheses cubed space plus space open parentheses straight c minus straight a close parentheses cubed space equals space 0 space minus space 3 open parentheses straight a minus straight c close parentheses open parentheses straight b minus straight a close parentheses open parentheses straight c minus straight b close parentheses
rightwards double arrow open parentheses straight a minus straight b close parentheses cubed space plus space open parentheses straight b minus straight c close parentheses cubed space plus space open parentheses straight c minus straight a close parentheses cubed space equals space 3 open parentheses straight a minus straight b close parentheses open parentheses straight b minus straight c close parentheses open parentheses straight c minus straight a close parentheses
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 10

begin mathsize 11px style If space straight a over straight b plus straight b over straight a equals negative 1 comma space then space straight a cubed space minus space straight b cubed space equals
left parenthesis straight a right parenthesis space 1
left parenthesis straight b right parenthesis space minus 1
left parenthesis straight c right parenthesis space 1 half
left parenthesis straight d right parenthesis space 0 end style

Solution 10

begin mathsize 11px style straight a over straight b plus straight b over straight a equals negative 1 space
rightwards double arrow fraction numerator straight a squared plus straight b squared over denominator ab end fraction equals negative 1
rightwards double arrow straight a squared plus straight b squared plus ab equals 0
Now space using space identity comma
straight a cubed minus straight b cubed space
equals space open parentheses straight a minus straight b close parentheses space open parentheses straight a squared plus straight b squared plus ab close parentheses
equals space left parenthesis straight a minus straight b right parenthesis left parenthesis 0 right parenthesis space space space space space space space space space space space left parenthesis because space straight a squared plus straight b squared plus ab equals 0 right parenthesis
equals 0
Hence comma space straight c orrect space option space is space left parenthesis straight d right parenthesis.
end style

Question 11

If a - b = -8 and ab = -12, then a3 - b3 =

(a) -244

(b) -240

(c) -224

(d) -260

Solution 11

a - b = -8

(a - b)2 = 64

a2 + b2 - 2ab = 64

a2 + b2 - 2ab + 3ab = 64 + 3ab

a2 + b2 + ab = 64 + 3(-12)

a2 + b2 + ab = 64 - 36

a2 + b2 + ab = 28

Now a3 - b3 = (a - b)(a2 + b2 + ab)

                 = (-8)(28)

                 = -224

Hence, correct option is (c).

Chapter 4 - Algebraic Identities Excercise 4.31

Question 1

If the volume of a cuboid is 3x2 - 27, then its possible dimensions are

(a) 3, x2, -27x

(b) 3, x - 3, x + 3

(c) 3, x2, 27x

(d) 3, 3, 3

Solution 1

Volume of a cuboid of side a, b and c = abc

Now, Volume = 3x2 - 27    (given)

abc = 3(x2 - 9)

abc = 3(x - 3)(x + 3)

So, possible dimensions are 3, x - 3 and x + 3

Hence, correct option is (b).

Question 2

75 × 75 + 2 × 75 × 25 + 25 × 25 is equal to

(a) 10000

(b) 6250

(c) 7500

(d) 3750

Solution 2

Given expression is 75 × 75 + 2 × 75 × 25 + 25 × 25

Let 75 = a and 25 = b

Then, we have

× a + 2 × a × b + b × b

a2 + 2ab + b2

= (a + b)2

= (75 + 25)2

= (100)2

= 10000

Hence, correct option is (a).

Question 3

(x - y)( x + y)(x2 + y2)(x4 + y4) is equal to

(a) x16 - y16

(b) x8 - y8

(c) x8 + y8

(d) x16 + y16

Solution 3

(x - y)(x + y) = x2 - y2  [by identity (a + b)(a - b) = a2 - b2]

(x2 - y2)(x2 + y2) = x4 - y4

(x4 - y4)(x4 + y4) = x8 - y8

Now,

(x - y)(x + y)(x2 + y2)(x4 + y4)

= (x2 - y2)(x2 + y2)(x4 + y4)

= (x4 - y4)(x4 + y4)

= x8 - y8

Hence, correct option is (b).

Question 4

begin mathsize 11px style If space straight x to the power of 4 plus 1 over straight x to the power of 4 equals 623 comma space then space straight x plus 1 over straight x space equals
left parenthesis straight a right parenthesis space 27
left parenthesis straight b right parenthesis space 25
left parenthesis straight c right parenthesis space 3 square root of 3
left parenthesis straight d right parenthesis space minus 3 square root of 3 end style

Solution 4

begin mathsize 11px style open parentheses straight x plus 1 over straight x close parentheses squared space equals space straight x squared plus 1 over straight x squared plus 2. straight x.1 over straight x equals straight x squared plus 1 over straight x squared plus 2
rightwards double arrow straight x squared plus 1 over straight x squared space equals space open curly brackets open parentheses straight x plus 1 over straight x close parentheses squared minus 2 close curly brackets
Squaring space both space sides comma
space open parentheses straight x squared plus 1 over straight x squared close parentheses squared equals open curly brackets open parentheses space straight x plus 1 over straight x close parentheses squared minus 2 close curly brackets squared
rightwards double arrow straight x to the power of 4 plus 1 over straight x to the power of 4 plus 2. straight x squared.1 over straight x squared equals open curly brackets open parentheses space straight x plus 1 over straight x close parentheses squared space minus space 2 close curly brackets squared
rightwards double arrow straight x to the power of 4 plus 1 over straight x to the power of 4 plus 2 equals open curly brackets open parentheses space straight x plus 1 over straight x close parentheses squared space minus space 2 close curly brackets squared space equals open parentheses 623 close parentheses space plus space 2
rightwards double arrow 623 plus 2 equals open curly brackets open parentheses space straight x plus 1 over straight x close parentheses squared space minus space 2 close curly brackets squared space space space space space space space open curly brackets straight x to the power of 4 plus 1 over straight x to the power of 4 equals 623 close curly brackets
rightwards double arrow 625 equals open curly brackets open parentheses space straight x plus 1 over straight x close parentheses squared space minus space 2 close curly brackets squared
rightwards double arrow open parentheses space straight x plus 1 over straight x close parentheses squared space minus space 2 space equals square root of 625 space equals space 25
rightwards double arrow open parentheses space straight x plus 1 over straight x close parentheses squared space equals space 25 plus 2 space equals space 27
rightwards double arrow open parentheses space straight x plus 1 over straight x close parentheses space equals space square root of 27
straight x plus 1 over straight x equals 3 square root of 3
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 5

begin mathsize 11px style If space straight x to the power of 4 plus 1 over straight x to the power of 4 equals 194 comma space then space straight x cubed plus 1 over straight x cubed space equals
left parenthesis straight a right parenthesis space 76
left parenthesis straight b right parenthesis space 52
left parenthesis straight c right parenthesis space 64
left parenthesis straight d right parenthesis space none space of space these end style

Solution 5

begin mathsize 11px style straight x to the power of 4 plus 1 over straight x to the power of 4 equals 194 space
Now space open parentheses straight x squared plus 1 over straight x squared close parentheses squared space equals space straight x to the power of 4 plus 1 over straight x to the power of 4 plus 2
rightwards double arrow open parentheses straight x squared plus 1 over straight x squared close parentheses squared equals space 194 space plus space 2 space equals space 196
rightwards double arrow straight x squared plus 1 over straight x squared space equals space 14 space space space space space space.... left parenthesis 1 right parenthesis
Now space space open parentheses straight x plus 1 over straight x close parentheses squared space equals space straight x squared plus 1 over straight x squared plus 2 space space space space space space space space space open curly brackets straight x squared plus 1 over straight x squared space equals space 14 close curly brackets
rightwards double arrow open parentheses straight x plus 1 over straight x close parentheses squared equals 14 space plus space 2 space equals space 16 space space space space space space left square bracket From space left parenthesis 1 right parenthesis right square bracket
rightwards double arrow straight x plus 1 over straight x space equals space square root of 16
rightwards double arrow straight x plus 1 over straight x space equals space 4 space space space space space space space space.................. circle enclose 3
By space identity space straight a cubed plus straight b cubed equals open parentheses straight a plus straight b close parentheses space open parentheses straight a squared plus straight b squared minus ab close parentheses
space straight x cubed plus 1 over straight x cubed space equals open parentheses space straight x plus 1 over straight x close parentheses space open parentheses space straight x squared plus 1 over straight x squared space minus 1 close parentheses
space space space space space space space space space space space space space space space space space equals open parentheses 4 close parentheses open parentheses 14 minus 1 close parentheses
space space space space space space space space space space space space space space space space space space equals 4 cross times 13
space space space space space space space space space space space space space space space space space space equals 52
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

begin mathsize 11px style If space straight x minus 1 over straight x equals 15 over 4 comma space then space straight x plus 1 over straight x space equals
left parenthesis straight a right parenthesis space 4
left parenthesis straight b right parenthesis space 17 over 4
left parenthesis straight c right parenthesis space 13 over 4
left parenthesis straight d right parenthesis space 1 fourth end style

Solution 6

begin mathsize 11px style space open parentheses straight x minus 1 over straight x close parentheses squared space equals space straight x squared plus 1 over straight x squared space minus space 2 space space space...... left parenthesis 1 right parenthesis
open parentheses straight x plus 1 over straight x close parentheses squared space equals space space straight x squared plus 1 over straight x squared space plus space 2 space space..... left parenthesis 2 right parenthesis
SUbtracting space eq. space left parenthesis 2 right parenthesis space from space eq. space left parenthesis 1 right parenthesis comma space we space get
space open parentheses straight x minus 1 over straight x close parentheses squared space minus space open parentheses straight x plus 1 over straight x close parentheses squared space equals space minus 4
rightwards double arrow space open parentheses 15 over 4 close parentheses squared space minus space space open parentheses straight x plus 1 over straight x close parentheses squared space equals space minus 4
rightwards double arrow space open parentheses straight x plus 1 over straight x close parentheses squared space equals space open parentheses 15 over 4 close parentheses squared space plus space 4 space equals space 225 over 16 plus 4
rightwards double arrow space open parentheses straight x plus 1 over straight x close parentheses squared space equals space fraction numerator 225 plus 64 over denominator 16 end fraction space equals space 189 over 16
rightwards double arrow space open parentheses straight x plus 1 over straight x close parentheses space equals space square root of 289 over 16 end root
rightwards double arrow space straight x plus 1 over straight x equals 17 over 4
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 7

begin mathsize 11px style If space 3 straight x plus 2 over straight x equals 7 comma space then space open parentheses 9 straight x squared minus 4 over straight x squared close parentheses space equals space
left parenthesis straight a right parenthesis space 25
left parenthesis straight b right parenthesis space 35
left parenthesis straight c right parenthesis space 49
left parenthesis straight d right parenthesis space 30 end style

Solution 7

begin mathsize 11px style open parentheses 3 straight x plus 2 over straight x close parentheses squared space equals space 9 straight x squared plus 4 over straight x squared plus 12 space space space space space..... left parenthesis 1 right parenthesis
open parentheses 3 straight x minus 2 over straight x close parentheses squared space equals space 9 straight x squared plus 4 over straight x squared minus 12 space space space space space..... left parenthesis 2 right parenthesis
Subtracting space equation space left parenthesis 1 right parenthesis space from space eq. space left parenthesis 2 right parenthesis comma space we space get
open parentheses 3 straight x minus 2 over straight x close parentheses squared space minus space open parentheses 3 straight x plus 2 over straight x close parentheses squared space equals space minus 24
rightwards double arrow open parentheses 3 straight x minus 2 over straight x close parentheses squared space equals space open parentheses 7 close parentheses squared space minus space 24 space equals space 25
rightwards double arrow 3 straight x minus 2 over straight x space equals space 5
Now comma space open parentheses 3 straight x plus 2 over straight x close parentheses space open parentheses 3 straight x minus 2 over straight x close parentheses space equals space 7 space cross times space 5
rightwards double arrow open parentheses 9 straight x squared minus 4 over straight x squared close parentheses equals 35
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 8

If a2 + b2 + c2 - ab - bc - ca = 0, then

(a) a + b = c

(b) b + c = a

(c) c + a = b

(d) a = b = c

Solution 8

a2 + b2 + c2 - ab - bc - ca = 0

Multiplying by 2 on both the sides, we have

2(a2 + b2 + c2 - ab - bc - ca) = 0

2a+ 2b2 + 2c2 - 2ab - 2bc - 2ca = 0

a2 + a2 + b2 + b2 + c+ c2 - 2ab - 2bc - 2ca = 0

(a2 + b2 - 2ab) + (b2 + c2 - 2bc) + (a2 + c2 - 2ac) = 0

(a - b)2 + (b - c)2 + (a - c)2 = 0

(a - b)2 = 0(b - c)2 = 0, (a - c)2 = 0

(a - b) = 0(b - c) = 0, (a - c) = 0

a = b, b = c, a = c

or we can say a = b = c

Hence, correct option is (d).

Question 9

begin mathsize 11px style If space straight a space plus space straight b space plus space straight c space equals space 0 comma space then space straight a squared over bc plus straight b squared over ca plus straight c squared over ab equals
left parenthesis straight a right parenthesis space 0
left parenthesis straight b right parenthesis space 1
left parenthesis straight c right parenthesis space minus 1
left parenthesis straight d right parenthesis space 3 end style

Solution 9

begin mathsize 11px style straight a cubed space plus space straight b cubed space plus space straight c cubed space minus space 3 abc space equals space open parentheses straight a space plus space straight b space plus space straight c close parentheses space open parentheses straight a squared space plus space straight b squared plus space straight c squared minus space ab space minus space bc space minus space ca close parentheses
If space straight a space plus space straight b space plus space straight c space equals space 0 comma space then space
table attributes columnalign right center left columnspacing 2px 2px 2px end attributes row cell straight a to the power of 3 space end exponent plus space straight b cubed space plus space straight c cubed space minus space 3 abc end cell equals 0 row cell rightwards double arrow straight a cubed space plus space straight b cubed space plus space straight c cubed end cell equals cell 3 abc space space space space space.... left parenthesis 1 right parenthesis end cell end table
Now comma space consider space space straight a squared over bc plus straight b squared over ca plus straight c squared over ab
Multiplying space and space dividng space by space straight a comma space straight b comma space and space straight c space in space straight a squared over bc comma space straight b squared over ca space and space straight c squared over ab space respectively comma space we space get
straight a cubed over abc plus straight b cubed over bca plus straight c cubed over cab space
equals space fraction numerator straight a cubed plus straight b cubed plus straight c cubed over denominator abc end fraction
equals fraction numerator 3 abc over denominator abc end fraction space space space space space space space space space.... left square bracket from space left parenthesis 1 right parenthesis right square bracket
equals 3
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 10

If a1/3 + b1/3 + c1/3 = 0, then

(a) a + b + c = 0

(b) (a + b + c)3 = 27abc

(c) a + b + c = 3abc

(d) a3 + b3 + c3 = 0

Solution 10

begin mathsize 11px style Let space straight a to the power of 1 divided by 3 end exponent space equals space straight A comma space space space straight b to the power of 1 divided by 3 end exponent space equals space straight B space space and space straight c to the power of 1 divided by 3 end exponent space equals space straight C
Now comma space straight A space plus space straight B space plus space straight C space equals space 0 space open parentheses given close parentheses
straight I straight f space straight A space plus space straight B space plus space straight C space equals space 0 comma space then space straight A cubed space plus space straight B cubed space plus space straight C cubed space minus space 3 ABC space equals space 0
rightwards double arrow space straight A cubed space plus straight B cubed space plus straight C cubed space minus space 3 ABC space equals space 0
rightwards double arrow space straight A cubed space plus space straight B cubed space plus space straight C cubed space equals space 3 ABC space space space space.... left parenthesis 1 right parenthesis
open curly brackets straight A space equals space straight a to the power of 1 divided by 3 end exponent space space space space straight B space equals space straight b to the power of 1 divided by 3 end exponent space space space space space space space space straight C space equals space straight c to the power of 1 divided by 3 end exponent
straight A cubed space equals space straight a space space space space space space space straight B cubed space equals space straight b space space space space space space space space space space space space straight C cubed space equals space straight c close curly brackets
Then comma space equation space left parenthesis 1 right parenthesis space becomes
straight a space plus space straight b space plus space straight c space equals space 3 open parentheses abc close parentheses to the power of 1 divided by 3 end exponent
Cubing space both space sides space of space above space equation comma space we space get
open parentheses straight a space plus space straight b space plus space straight c close parentheses cubed space equals space 27 abc
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 11

If a + b + c = 9 and ab + bc + ca = 23, then a+ b+ c3 - 3abc =

(a) 108

(b) 207

(c) 669

(d) 729

Solution 11

Given, a + b + c = 9

Hence, (a + b + c)2 = 81

So, a2 + b2 + c2 + 2ab + 2bc + 2ca = 81

i.e. a2 + b2 + c2 + 2(ab + bc + ca) = 81

i.e. a2 + b2 + c2 + 2(23) = 81

i.e. a2 + b2 + c2 = 81 - 46 = 35

Now, a3 + b3 + c3 - 3abc

= (a + b + c)(a2 + b2 + c2 - ab - bc - ca)

= (a + b + c)[(a2 + b2 + c2) - (ab + bc + ca)]

= (9)[35 - 23]

= 9 × 12

= 108

Hence, correct option is (a).

Question 12

begin mathsize 11px style fraction numerator open parentheses straight a squared minus straight b squared close parentheses cubed space plus space open parentheses straight b squared space minus space straight c squared close parentheses cubed space plus space open parentheses straight c squared minus space straight a squared close parentheses cubed over denominator open parentheses straight a space minus space straight b close parentheses cubed space plus space open parentheses straight b space minus space straight c close parentheses cubed space plus space open parentheses straight c space minus space straight a close parentheses cubed end fraction space equals
left parenthesis straight a right parenthesis space 3 open parentheses straight a space plus space straight b close parentheses open parentheses straight b space plus space straight c close parentheses open parentheses straight c space plus space straight a close parentheses
left parenthesis straight b right parenthesis space 3 open parentheses straight a space minus space straight b close parentheses open parentheses straight b space minus space straight c close parentheses open parentheses straight c space minus space straight a close parentheses
left parenthesis straight c right parenthesis space open parentheses straight a space minus space straight b close parentheses open parentheses straight b space minus space straight c close parentheses open parentheses straight c space minus space straight a close parentheses
left parenthesis straight d right parenthesis space none space of space these end style

Solution 12

begin mathsize 14px style If space straight a plus straight b plus straight c equals 0 space then comma space straight a cubed plus straight b cubed plus straight c cubed equals 3 abc
Now comma space open parentheses straight a squared minus straight b squared close parentheses plus open parentheses straight b squared minus straight c squared close parentheses plus open parentheses straight c squared minus straight a squared close parentheses equals straight a squared minus straight b squared plus straight b squared minus straight c squared plus straight c squared minus straight a squared equals 0
rightwards double arrow open parentheses straight a squared minus straight b squared close parentheses cubed plus open parentheses straight b squared minus straight c squared close parentheses cubed plus open parentheses straight c squared minus straight a squared close parentheses cubed equals 3 open parentheses straight a squared minus straight b squared close parentheses open parentheses straight b squared minus straight c squared close parentheses open parentheses straight c squared minus straight a squared close parentheses space space space space space
Again comma space open parentheses straight a minus straight b close parentheses plus open parentheses straight b minus straight c close parentheses plus open parentheses straight c minus straight a close parentheses equals straight a minus straight b plus straight b minus straight c plus straight c minus straight a equals 0
rightwards double arrow open parentheses straight a minus straight b close parentheses cubed plus open parentheses straight b minus straight c close parentheses cubed plus open parentheses straight c minus straight a close parentheses cubed equals 3 open parentheses straight a minus straight b close parentheses open parentheses straight b minus straight c close parentheses open parentheses straight c minus straight a close parentheses
Thus comma space we space have
fraction numerator open parentheses straight a squared minus straight b squared close parentheses cubed plus open parentheses straight b squared minus straight c squared close parentheses cubed plus open parentheses straight c squared minus straight a squared close parentheses cubed over denominator open parentheses straight a minus straight b close parentheses cubed plus open parentheses straight b minus straight c close parentheses cubed plus open parentheses straight c minus straight a close parentheses cubed end fraction
equals fraction numerator 3 open parentheses straight a squared minus straight b squared close parentheses open parentheses straight b squared minus straight c squared close parentheses open parentheses straight c squared minus straight a squared close parentheses space space over denominator 3 open parentheses straight a minus straight b close parentheses open parentheses straight b minus straight c close parentheses open parentheses straight c minus straight a close parentheses end fraction
equals fraction numerator open parentheses straight a minus straight b close parentheses left parenthesis straight a plus straight b right parenthesis open parentheses straight b minus straight c close parentheses left parenthesis straight b plus straight c right parenthesis open parentheses straight c minus straight a close parentheses left parenthesis straight c plus straight a right parenthesis over denominator open parentheses straight a minus straight b close parentheses open parentheses straight b minus straight c close parentheses open parentheses straight c minus straight a close parentheses end fraction
equals open parentheses straight a plus straight b close parentheses open parentheses straight b plus straight c close parentheses open parentheses straight c plus straight a close parentheses
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Chapter 4 - Algebraic Identities Excercise 4.32

Question 1

The product (a + b)(a - b)(a2 - ab + b2) (a2 + ab + b2) is equal to

(a) a6 + b6

(b) a6 - b6

(c) a3 - b3

(d) a3 + b3

Solution 1

begin mathsize 11px style open parentheses straight a plus straight b close parentheses space open parentheses straight a minus straight b close parentheses space open parentheses straight a squared space minus ab plus straight b squared close parentheses space open parentheses straight a squared plus ab plus straight b squared close parentheses
equals open parentheses straight a squared minus straight b squared close parentheses space open parentheses straight a squared plus straight b squared minus ab close parentheses space open parentheses straight a squared plus straight b squared plus ab close parentheses
equals open parentheses straight a squared minus straight b squared close parentheses space open curly brackets open parentheses straight a squared plus straight b squared close parentheses squared space minus space open parentheses ab close parentheses squared close curly brackets
equals open parentheses straight a squared minus straight b squared close parentheses space open curly brackets straight a to the power of 4 plus straight b to the power of 4 plus 2 straight a squared straight b squared minus straight a squared straight b squared close curly brackets
equals open parentheses straight a squared minus straight b squared close parentheses space open curly brackets straight a to the power of 4 plus straight b to the power of 4 plus straight a squared straight b squared close curly brackets
equals open curly brackets straight a to the power of 6 plus straight a squared up diagonal strike straight b to the power of 4 end strike plus straight a to the power of 4 up diagonal strike straight b squared end strike minus straight b squared up diagonal strike straight a to the power of 4 end strike minus straight b to the power of 6 minus straight b to the power of 4 up diagonal strike straight a squared end strike close curly brackets
equals straight a to the power of 6 minus straight b to the power of 6
Hence comma space straight c orrect space option space is space left parenthesis straight b right parenthesis. end style

Question 2

The product (x2 - 1)(x4 + x2 + 1) is equal to

(a) x8 - 1

(b) x8 + 1

(c) x6 - 1

(d) x6 + 1

Solution 2

Given expression is (x2 - 1)(x4 + x2 + 1)

Let x2 = A and 1 = B

Then, we have

(A - B)(A2 + AB + B2)

= A3 - B3

= (x2)3 - (1)3

= x6 - 1

Hence, correct option is (c).

Question 3

begin mathsize 11px style If space straight a over straight b plus straight b over straight a equals 1 comma space then space straight a cubed space plus space straight b cubed space equals
left parenthesis straight a right parenthesis space 1
left parenthesis straight b right parenthesis space minus 1
left parenthesis straight c right parenthesis space 1 half
left parenthesis straight d right parenthesis space 0 end style

Solution 3

begin mathsize 11px style straight a over straight b plus straight b over straight a equals 1 space rightwards double arrow straight a squared plus straight b squared minus ab equals 0
Now space by space identity space straight a cubed plus straight b cubed equals space space open parentheses straight a plus straight b close parentheses space open parentheses straight a squared plus straight b squared minus ab close parentheses comma
if space straight a squared plus straight b squared minus ab space equals space 0 comma
then space straight a cubed plus straight b cubed equals 0
Hence comma space straight c orrect space option space is space left parenthesis straight d right parenthesis. end style

Question 4

begin mathsize 11px style If space 49 straight a squared space minus space straight b space equals space open parentheses 7 straight a space plus space 1 half close parentheses space open parentheses 7 straight a space minus space 1 half space close parentheses comma space then space the space value space of space straight b space is
left parenthesis straight a right parenthesis space 0
left parenthesis straight b right parenthesis space 1 fourth
left parenthesis straight c right parenthesis space fraction numerator 1 over denominator square root of 2 end fraction
left parenthesis straight d right parenthesis space 1 half end style

Solution 4

begin mathsize 11px style open parentheses 7 straight a plus 1 half close parentheses space open parentheses 7 straight a space minus space 1 half close parentheses space equals space open parentheses 7 straight a close parentheses squared space minus space open parentheses 1 half close parentheses squared
left square bracket by space using space identity space open parentheses straight a plus straight b close parentheses space open parentheses straight a minus straight b close parentheses space equals space straight a squared minus straight b squared right square bracket
rightwards double arrow open parentheses 7 straight a plus 1 half close parentheses space open parentheses 7 straight a space minus space 1 half close parentheses space equals 49 straight a squared minus 1 fourth
rightwards double arrow 49 straight a squared minus straight b equals 49 straight a squared minus 1 fourth
rightwards double arrow straight b equals 1 fourth
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

CBSE Class 9 Maths Homework Help

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