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RD Sharma Solution for Class 9 Mathematics Chapter 4 - Algebraic Identities

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 9 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 4 - Algebraic Identities.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 9 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 9 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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RD Sharma Solution for Class 9 Mathematics Chapter 4 - Algebraic Identities Page/Excercise 4.1

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

Solution 2(i)

Solution 2(ii)

Solution 2(iii)

Solution 2(iv)

Solution 3(i)

Solution 3(ii)

Solution 3(iii)

Solution 3(iv)

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10(i)

Solution 10(ii)

Solution 11

Solution 12

Solution 13(i)

Solution 13(ii)

Solution 13(iii)

Solution 13(iv)

Solution 14

RD Sharma Solution for Class 9 Mathematics Chapter 4 - Algebraic Identities Page/Excercise 4.2

Solution 1(i)

Solution 1(ii)

Solution 1(iii)

Solution 1(iv)

Solution 1(v)

Solution 1(vi)

Solution 1(vii)

Solution 1(viii)

Solution 1(ix)

Solution 1 (x)

Solution 1 (xi)


Solution 1 (xii)


Solution 2

Solution 3

Solution 4

Solution 5

Solution 6(i)

Solution 6(ii)

Solution 6(iii)

Solution 6(iv)

Solution 6(v)

Solution 7(i)

Solution 7(ii)

Solution 7(iii)


RD Sharma Solution for Class 9 Mathematics Chapter 4 - Algebraic Identities Page/Excercise 4.3

Solution 1(i)

Solution 1(ii)


Solution 1(iii)

Solution 1(iv)

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11(i)

Solution 11(ii)

Solution 11(iii)

Solution 11(iv)

Solution 11(v)

Solution 11(vi)

Solution 12(i)

Solution 12(ii)

Solution 12(iii)

Solution 12(iv)

Solution 13

Solution 14(i)

Solution 14(ii)

Solution 15

Solution 16

Solution 17(ii)

Solution 17(iii)

Solution 17(iv)

Solution 17(i)

Solution 18

Solution 19

RD Sharma Solution for Class 9 Mathematics Chapter 4 - Algebraic Identities Page/Excercise 4.4

Solution 1 (i)

Solution 1 (ii)

Solution 1 (iii)

Solution 1 (iv)

Solution 1 (v)

Solution 1 (vi)


Solution 1 (vii)

Solution 1 (viii)

Solution 1 (ix)

Solution 1 (x)

Solution 1 (xi)

Solution 1 (xii)

Solution 2 (i)


Solution 2 (ii)


Solution 2 (iii)

Solution 2 (iv)

Solution 2 (v)

Solution 3


Solution 4

Solution 5

Solution 6 (iii)

Solution 6 (i)

Solution 6 (ii)


RD Sharma Solution for Class 9 Mathematics Chapter 4 - Algebraic Identities Page/Excercise 4.5

Solution 1 (i)

Solution 1 (ii)


Solution 1 (iii)


Solution 1 (iv)

Solution 2(i)

Solution 2(ii)

Solution 2(iii)

Solution 2(iv)

Solution 3

Solution 4

Solution 5

RD Sharma Solution for Class 9 Mathematics Chapter 4 - Algebraic Identities Page/Excercise 4.30

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

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

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

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

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

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

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

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).

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

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

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).

RD Sharma Solution for Class 9 Mathematics Chapter 4 - Algebraic Identities Page/Excercise 4.31

Solution 12

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).

Solution 13

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).

Solution 14

(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).

Solution 15

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

Solution 16

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

Solution 17

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

Solution 18

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

Solution 19

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).

Solution 20

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

Solution 21

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

Solution 22

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).

Solution 23

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

RD Sharma Solution for Class 9 Mathematics Chapter 4 - Algebraic Identities Page/Excercise 4.32

Solution 24

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

Solution 25

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).

Solution 26

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

Solution 27

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

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 4 - Algebraic Identities for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 4 - Algebraic Identities.

Text Book Solutions

CBSE IX - Mathematics

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