Chapter 5 : Factorisation of Algebraic Expressions - Rd Sharma Solutions for Class 9 Maths CBSE

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Chapter 5 - Factorisation of Algebraic Expressions Excercise Ex. 5.1

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

Give possible expressions for the length and breadth of the rectangle having

35y2 + 13y - 12 as its area.

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

Question 35

Solution 35

Chapter 5 - Factorisation of Algebraic Expressions Excercise Ex. 5.2

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
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

Chapter 5 - Factorisation of Algebraic Expressions Excercise Ex. 5.3

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

Chapter 5 - Factorisation of Algebraic Expressions Excercise Ex. 5.4

Question 1
Solution 1
Question 2
Solution 2
Question 3
Factorise : 27x3 - y3 - z3 - 9xyz
Solution 3

Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7

Solution 7

Question 8

begin mathsize 12px style M u l t i p l y colon
open parentheses i close parentheses space x squared space plus y squared plus z squared space minus x y plus x z plus space y z space b y space x plus y minus z
open parentheses i i close parentheses space x squared space plus space 4 y to the power of 2 space end exponent plus space z squared space plus space 2 x y space plus x z minus 2 y z space b y space x minus 2 y minus z
open parentheses i i i close parentheses space space x squared space plus space 4 y to the power of 2 space end exponent plus space 2 x y space minus 3 x plus 6 y space plus 9 space b y space x space minus space 2 y space plus 3
open parentheses i v close parentheses space space 9 x squared space plus space 25 y to the power of 2 space end exponent space plus 15 x y space plus 12 x minus 20 y space plus 16 space b y space 3 x minus 5 y space plus 4 end style

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

Chapter 5 - Factorisation of Algebraic Expressions Excercise 5.25

Question 1

The factors of x3 - x2y - xy2 + y3 are

(a) (x + y)(x2 - xy + y2)

(b) (x + y)(x2 + xy + y2)

(c) (x + y)2(x - y)

(d) (x - y)2(x + y)

Solution 1

begin mathsize 12px style straight x cubed space minus space straight x squared straight y space minus space xy squared space plus space straight y cubed space equals space straight x cubed space plus space straight y cubed space minus space xy space left parenthesis straight x space plus space straight y right parenthesis
Now space by space identity space straight x cubed space plus space straight y cubed space equals space left parenthesis straight x space plus space straight y right parenthesis space left parenthesis straight x squared space plus space straight y squared space minus space xy right parenthesis comma space we space have
straight x cubed space minus space straight x squared straight y space minus space xy squared space plus space straight y cubed space equals space left parenthesis straight x space plus space straight y right parenthesis left parenthesis straight x squared space plus space straight y squared space minus space xy right parenthesis space minus space xy space left parenthesis straight x space plus space straight y right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x space plus space straight y right parenthesis left parenthesis straight x squared space plus space straight y squared space minus space xy space minus space xy right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x space plus space straight y right parenthesis left parenthesis straight x squared space plus straight y squared space minus space 2 xy right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals left parenthesis straight x space plus space straight y right parenthesis left parenthesis straight x space minus space straight y right parenthesis squared
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 2

The factors of x3 - 1 + y3 + 3xy are

(a) (x - 1 + y) (x2 + 1 + y2 + x + y - xy)

(b) (x + y + 1) ( x2 + y2 + 1 - xy - x - y)

(c) (x - 1 + y) (x2 - 1 - y2 + x + y + xy)

(d) 3(x + y - 1) (x2 + y2 - 1)

Solution 2

begin mathsize 12px style By space using space identity
straight a cubed space plus space straight b cubed space plus space straight c cubed space minus space 3 abc space equals space left parenthesis straight a space plus space straight b space plus space straight c right parenthesis space left parenthesis straight a squared space plus space straight b squared space plus space straight c squared space minus space ab space minus space bc space minus space ca right parenthesis
We space can space write comma
straight x cubed space minus space 1 space plus space straight y cubed space plus space 3 xy space
equals space left parenthesis straight x right parenthesis cubed space plus space left parenthesis negative 1 right parenthesis cubed space plus space left parenthesis straight y right parenthesis cubed space minus space 3 left parenthesis negative 1 right parenthesis space left parenthesis straight x right parenthesis space left parenthesis straight y right parenthesis
equals space left square bracket straight x space plus space left parenthesis negative 1 right parenthesis space plus space straight y right square bracket left square bracket straight x squared space plus space left parenthesis negative 1 right parenthesis squared space plus space straight y squared space minus space straight x left parenthesis negative 1 right parenthesis space minus space straight y left parenthesis negative 1 right parenthesis space minus space xy right square bracket
equals space left parenthesis straight x space minus space 1 space plus space straight y right parenthesis left parenthesis straight x squared space plus space 1 space plus space straight y squared space plus space straight x space plus straight y space minus space xy right parenthesis
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 3

The factors of 8a3 + b3 - 6ab + 1 are

(a) (2a + b - 1)(4a2 + b2 + 1 - 3ab - 2a)

(b) (2a - b + 1)(4a2 + b2 - 4ab + 1 - 2a + b)

(c) (2a + b + 1)(4a2 + b2 + 1 - 2ab - b - 2a)

(d) (2a - 1 + b)(4a2 + 1 - 4a - b - 2ab)

Solution 3

begin mathsize 12px style We space Know space the space identity
straight a cubed space plus space straight b cubed space plus space straight c cubed space minus space 3 abc space equals space left parenthesis straight a space plus space straight b space plus space straight c right parenthesis space left parenthesis straight a squared space plus space straight b squared space plus space straight c squared space minus space ab space minus space bc space minus space ca right parenthesis
So space by space using space this space identity comma space we space can space write space given space expression space as
left parenthesis 2 straight a right parenthesis cubed space plus space left parenthesis straight b right parenthesis cubed space plus space left parenthesis 1 right parenthesis cubed space minus space 3 left parenthesis 2 straight a right parenthesis space left parenthesis straight b right parenthesis space left parenthesis 1 right parenthesis
equals left parenthesis 2 straight a space plus space straight b space plus space 1 right parenthesis space left square bracket left parenthesis 2 straight a right parenthesis squared space plus space straight b squared space plus space 1 squared space minus space 2 straight a cross times straight b space minus space straight b cross times 1 space minus space 2 straight a cross times 1 right square bracket
equals left parenthesis 2 straight a space plus space straight b space plus space 1 right parenthesis space left parenthesis 4 straight a squared space plus space straight b squared space plus space 1 space minus space 2 ab space minus space straight b space minus space 2 straight a right square bracket
space Hence comma space correct space option space is space left parenthesis straight c right parenthesis. space end style

Question 4

(x + y)3 - (x - y)3 can be factorized as

(a) 2y(3x2 + y2)

(b) 2x(3x2 + y2)

(c) 2y(3y2 + x2)

(d) 2x(x2 + 3y2)

Solution 4

begin mathsize 12px style We space know space the space identity space straight a cubed space minus space straight b cubed space equals space left parenthesis straight a space minus space straight b right parenthesis space left parenthesis straight a squared space plus space straight b squared space plus space ab right parenthesis
Let space straight x space plus space straight y space equals space straight a space and space straight x space minus space straight y space equals space straight b
Then comma
straight a cubed space minus space straight b cubed space
equals space left parenthesis straight x space plus space straight y right parenthesis cubed space minus space left parenthesis straight x space minus space straight y right parenthesis cubed space space
equals space left square bracket left parenthesis up diagonal strike straight x space plus space straight y right parenthesis space minus space left parenthesis up diagonal strike straight x space minus space straight y right parenthesis right square bracket space left square bracket left parenthesis straight x space plus space straight y right parenthesis squared space plus space left parenthesis straight x space minus space straight y right parenthesis squared space plus space left parenthesis straight x space plus space straight y right parenthesis space left parenthesis straight x space minus space straight y right parenthesis right square bracket
equals 2 straight y left square bracket straight x squared space plus space straight y squared space plus up diagonal strike space 2 end strike xy space plus space straight x squared space plus space up diagonal strike straight y squared end strike space minus space up diagonal strike 2 xy space plus space straight x squared space minus space up diagonal strike straight y squared end strike right curly bracket
equals 2 straight y left parenthesis 3 straight x squared space plus space straight y squared right parenthesis
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 5

The expression (a - b)3 + (b - c)3 + (c - a)3 can be factorized as

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

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

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

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

Solution 5

begin mathsize 12px style By space know space that space straight a cubed space plus space straight b cubed space plus space straight c cubed space minus space 3 abc space equals space left parenthesis straight a space plus space straight b space plus space straight c right parenthesis space left parenthesis straight a squared space plus space straight b squared space plus space straight c squared space minus space ab space minus space bc space minus space ca right parenthesis
If 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 equals 3 abc
In space given space expression comma
let space space straight a space minus space straight b space equals straight A comma space space straight b space minus space straight c space equals space straight B comma space straight c space minus space straight a space equals space straight C
Now comma space straight a space minus space straight b space plus space straight b space minus space straight c space plus space straight c space minus space straight a space equals space 0
straight i. straight e. space straight A space plus space straight B space plus space straight C space equals space 0
rightwards double arrow straight A cubed space plus straight B cubed space plus space straight C cubed space space equals space 3 ABC
rightwards double arrow left parenthesis straight a space minus space straight b right parenthesis cubed space plus space left parenthesis straight b space minus space straight c right parenthesis cubed space plus space left parenthesis straight c space minus space straight a right parenthesis cubed space equals space 3 left parenthesis straight a space minus space straight b right parenthesis left parenthesis straight b space minus space straight c right parenthesis left parenthesis straight c space minus space straight a right parenthesis
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Question 6

begin mathsize 11px style The space value space of space fraction numerator open parentheses 2.3 close parentheses cubed space minus space 0.027 over denominator open parentheses 2.3 close parentheses squared space plus space 0.69 space plus space 0.09 end fraction space is
left parenthesis straight a right parenthesis space 2
left parenthesis straight b right parenthesis space 3
left parenthesis straight c right parenthesis space 2.327
left parenthesis straight d right parenthesis space 2.273 end style

Solution 6

begin mathsize 12px style fraction numerator open parentheses 2.3 close parentheses cubed minus 0.027 over denominator open parentheses 2.3 close parentheses squared plus 0.69 plus 0.09 end fraction
fraction numerator open parentheses 2.3 close parentheses cubed minus 0.027 over denominator open parentheses 2.3 close parentheses squared plus 0.09 plus 0.69 end fraction
equals fraction numerator open parentheses 2.3 close parentheses cubed minus open parentheses 0.3 close parentheses cubed over denominator open parentheses 2.3 close parentheses squared plus open parentheses 0.3 close parentheses squared plus open parentheses 2.3 close parentheses open parentheses 0.3 close parentheses end fraction
equals fraction numerator open parentheses 2.3 space minus space 0.3 close parentheses space open curly brackets open parentheses 2.3 close parentheses squared plus open parentheses 0.3 close parentheses squared plus open parentheses 2.3 close parentheses open parentheses 0.3 close parentheses close curly brackets over denominator open parentheses open parentheses 2.3 close parentheses squared plus open parentheses 0.3 close parentheses squared plus open parentheses 2.3 close parentheses open parentheses 0.3 close parentheses close parentheses end fraction
equals 2.3 space minus space 0.3
equals 2
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 7

begin mathsize 11px style The space value space of space fraction numerator open parentheses 0.013 close parentheses cubed plus open parentheses 0.007 close parentheses cubed over denominator open parentheses 0.013 close parentheses squared minus 0.013 cross times 0.007 plus open parentheses 0.007 close parentheses squared end fraction space is
left parenthesis straight a right parenthesis space 0.006
left parenthesis straight b right parenthesis space 0.02
left parenthesis straight c right parenthesis space 0.0091
left parenthesis straight d right parenthesis space 0.00185 end style

Solution 7

begin mathsize 12px style By space using space identity space straight a cubed space plus space straight b cubed space equals space left parenthesis straight a space plus space straight b right parenthesis left parenthesis straight a squared space minus space ab space plus space straight b squared right parenthesis comma space we space have
fraction numerator open parentheses 0.013 close parentheses cubed space plus space open parentheses 0.007 close parentheses cubed over denominator open parentheses 0.013 close parentheses squared space space minus space left parenthesis 0.013 right parenthesis left parenthesis 0.007 right parenthesis plus space open parentheses 0.007 close parentheses squared end fraction
equals fraction numerator open curly brackets open parentheses 0.013 close parentheses space plus space open parentheses 0.007 close parentheses close curly brackets open parentheses 0.013 close parentheses squared space minus space left parenthesis 0.013 right parenthesis left parenthesis 0.007 right parenthesis plus space open parentheses 0.007 close parentheses squared over denominator open parentheses 0.013 close parentheses squared space minus space left parenthesis 0.013 right parenthesis left parenthesis 0.007 right parenthesis plus space open parentheses 0.007 close parentheses squared end fraction
equals space 0.013 space plus space 0.007
equals space 0.020
equals space 0.02
Hence comma space correct space option space is space left parenthesis straight b right parenthesis. end style

Chapter 5 - Factorisation of Algebraic Expressions Excercise 5.26

Question 1

The factors of a2 - 1 - 2x - x2 are

(a) (a - x + 1)(a - x - 1)

(b) (a + x - 1)(a - x + 1)

(c) (a + x + 1)(a - x - 1)

(d) none of these

 

Solution 1

a2 - 1 - 2x - x2

= a2 - (1 + 2x + x2)

= a2 - (1 + x)2

= [a - (1 + x)][a + (1 + x)]

(a - x - 1)(a + x + 1)

Hence, correct option is (c).

Question 2

The factors of x4 + x2 + 25 are

(a) x2 + 3x + 5)(x2 - 3x + 5)

(b) (x2 + 3x + 5)(x2 + 3x - 5)

(c) (x2 + x + 5)(x2 - x + 5)

(d) none of these

Solution 2

begin mathsize 12px style For space making space perfect space square space to space straight x to the power of 4 plus straight x squared plus 25 comma
we space add space space plus 10 straight x to the power of 2 space end exponent and space minus 10 straight x squared space space to space space it. space
rightwards double arrow space straight x to the power of 4 plus straight x squared plus 25
equals straight x to the power of 4 space plus space straight x squared space plus space 25 space plus space 10 straight x squared space minus space 10 straight x squared
equals space left square bracket straight x to the power of 4 space plus space 10 straight x squared space plus space 25 right square bracket space minus space 9 straight x squared
equals space left parenthesis straight x squared space plus space 5 right parenthesis squared space minus space left parenthesis 3 straight x right parenthesis squared
equals left square bracket left parenthesis straight x squared space plus space 5 right parenthesis space plus space 3 straight x right square bracket space left square bracket left parenthesis straight x squared space plus space 5 right parenthesis space minus space 3 straight x right square bracket
equals left parenthesis straight x squared space plus space 3 straight x space plus space 5 right parenthesis space left parenthesis straight x squared space minus space 3 straight x space plus space 5 right parenthesis
Hence comma space correct space option space is space left parenthesis straight a right parenthesis.
end style

Question 3

The factors of x2 + 4y2 + 4y - 4xy - 2x - 8 are

(a) (x - 2y - 4)(x - 2y + 2)

(b) (x - y + 2)(x - 4y - 4)

(c) (x + 2y - 4)(x + 2y + 2)

(d) none of these

Solution 3

begin mathsize 12px style straight x squared space plus space 4 straight y squared space plus space 4 straight y space minus space 4 xy space minus space 2 straight x space minus space 8
equals space straight x squared space plus space left parenthesis 2 straight y right parenthesis squared space space minus space 2 cross times space straight x space left parenthesis 2 straight y right parenthesis space plus space 4 straight y space minus space 2 straight x space minus space 8
equals space left parenthesis straight x space minus space 2 straight y right parenthesis squared space plus space 4 straight y space minus space 2 straight x space minus space 8 space space.... open parentheses 1 close parentheses
Now space making space eq space open parentheses 1 close parentheses space straight a space perfect space sequare space by space adding space 1 space and space minus 1
left parenthesis straight x space minus space 2 straight y right parenthesis squared space plus space 4 straight y space minus space 2 straight x space minus space 8 space equals left parenthesis straight x space minus space 2 straight y right parenthesis squared space plus space 4 straight y space minus space 2 straight x space minus space 8 space plus space 1 space minus space 1
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals left parenthesis straight x space minus space 2 straight y right parenthesis squared space plus space left parenthesis 1 right parenthesis squared space minus space 2 space cross times space left parenthesis 1 right parenthesis space cross times space left parenthesis straight x space minus space 2 straight y right parenthesis space minus space 9
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals left parenthesis straight x space minus space 2 straight y space minus space 1 right parenthesis squared space minus space left parenthesis 3 right parenthesis squared
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left square bracket left parenthesis straight x space minus space 2 straight y space minus space 1 right parenthesis space minus space 3 right square bracket space left square bracket straight x space minus space 2 straight y space minus space 1 space plus space 3 right square bracket
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x space minus space 2 straight y space minus space 4 right parenthesis space left parenthesis straight x space minus space 2 straight y space plus space 2 right parenthesis
Hence comma space c orrect space Option space is space left parenthesis straight a right parenthesis.
end style

Question 4

The factors of x3 - 7x + 6 are

(a) x(x - 6)(x - 1)

(b) (x2 - 6)(x - 1)

(c) (x + 1)(x + 2)(x - 3)

(d) (x - 1)(x + 3)(x - 2)

Solution 4

begin mathsize 12px style straight x cubed space minus space 7 straight x space plus space 6 space space space space equals space straight x cubed space minus space 7 straight x space plus space 6 space plus space 1 space minus space 1 space left parenthesis by space adding space plus 1 space & space minus 1 space to space straight R. straight H. straight S right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals straight x cubed space minus space 7 straight x space plus space 7 space minus space 1
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals left parenthesis straight x cubed space minus space 1 right parenthesis space minus space 7 left parenthesis straight x space minus space 1 right parenthesis
Now space by space space identity space straight a cubed space minus space straight b cubed space equals space left parenthesis straight a space minus space straight b right parenthesis space left parenthesis straight a squared space plus space straight b squared space plus space ab right parenthesis comma space w e space g e t
straight x cubed space minus space 7 straight x space plus space 6 space equals space left parenthesis straight x cubed space minus space 1 right parenthesis space minus space 7 left parenthesis straight x space minus space 1 right parenthesis space
space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x space minus space 1 right parenthesis space left parenthesis straight x squared space plus space straight x space plus space 1 right parenthesis space minus space 7 left parenthesis straight x space minus space 1 right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x space minus space 1 right parenthesis space left parenthesis straight x squared space plus space straight x space plus space 1 space minus 7 right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x space minus space 1 right parenthesis space left parenthesis straight x squared space plus space straight x space minus space 6 right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x space minus space 1 right parenthesis left parenthesis straight x space plus space 3 right parenthesis left parenthesis straight x space minus space 2 right parenthesis
Hence comma space c orrect space option space is space left parenthesis straight d right parenthesis. end style

Question 5

The expression x4 + 4 can be factorized as

(a) (x2 + 2x + 2)(x2 - 2x + 2)

(b) (x2 + 2x + 2)(x2 + 2x - 2)

(c) (x2 - 2x - 2)(x2 - 2x + 2)

(d) (x2 + 2)(x2 - 2)

Solution 5

begin mathsize 12px style straight x to the power of 4 space plus space 4
equals space straight x to the power of 4 space plus space 4 space plus space 4 straight x squared space minus space 4 straight x squared
equals space left parenthesis straight x to the power of 4 space plus space 4 straight x squared space plus space 4 right parenthesis space minus space 4 straight x squared
equals space left parenthesis straight x squared space plus space 2 right parenthesis squared space minus space left parenthesis 2 straight x right parenthesis squared
equals space left parenthesis straight x squared space plus space 2 space minus space 2 straight x right parenthesis left parenthesis straight x squared space plus space 2 space plus space 2 straight x right parenthesis
equals space left parenthesis straight x squared space plus space 2 space minus space 2 straight x right parenthesis space left parenthesis straight x squared space plus space 2 straight x space plus space 2 right parenthesis
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

Question 6

If 3x = a + b + c, then the value of (x - a)3 + (x - b)3 + (x - c)3 - 3(x - a)(x - b)(x - c) is

(a) a + b + c

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

(c) 0

(d) none of these

Solution 6

begin mathsize 12px style 3 straight x space equals space straight a space plus space straight b space plus space straight c
rightwards double arrow straight a space plus space straight b space plus space straight c space minus space 3 straight x space equals space 0
rightwards double arrow 3 straight x space minus space left parenthesis straight a space plus space straight b space plus space straight c right parenthesis space equals space 0
rightwards double arrow left parenthesis straight x space minus space straight a right parenthesis space plus space left parenthesis straight x space minus space straight b right parenthesis space plus space left parenthesis straight x space minus space straight c right parenthesis space equals space 0 space space space.... open parentheses 1 close parentheses
Using space identity space if space straight a space plus space straight b space plus space straight c space equals space 0 space then comma 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
If space we space take space straight x space minus space straight a space equals space straight A comma space straight x space minus space straight b space equals space straight B comma space straight x space minus space straight c space equals space straight C space in space equation space left parenthesis 1 right parenthesis comma space we space get
straight A space plus space straight B space plus space straight C space equals space 0
rightwards double arrow straight A cubed space plus space straight B cubed space plus space straight C cubed space minus space 3 ABC space equals 0
rightwards double arrow left parenthesis straight x space minus space straight a right parenthesis cubed space plus space left parenthesis straight x space minus space straight b right parenthesis cubed space plus space left parenthesis straight x space minus space straight c right parenthesis cubed space minus space 3 left parenthesis straight x space minus space straight a right parenthesis left parenthesis straight x space minus space straight b right parenthesis left parenthesis straight x space minus space straight c right parenthesis space equals space 0 space space space space space space space space
Hence comma space correct space option space is space left parenthesis straight c right parenthesis. end style

Question 7

If (x + y)3 - (x - y)3 - 6y(x2 - y2) = ky2, then k =

(a) 1

(b) 2

(c) 4

(d) 8

Solution 7

begin mathsize 12px style Let space straight x space plus space straight y space equals space straight A space and space straight x space minus space straight y space equals space straight B
Now comma space space left parenthesis straight A space minus space straight B right parenthesis cubed space space equals space straight A cubed space minus space straight B cubed space minus space 3 AB left parenthesis straight A space minus space straight B right parenthesis
rightwards double arrow left square bracket left parenthesis straight x space plus space straight y right parenthesis space minus space left parenthesis straight x space minus space straight y right parenthesis right square bracket cubed space equals space left parenthesis straight x space plus space straight y right parenthesis cubed space minus space left parenthesis straight x space minus space straight y right parenthesis cubed space minus space 3 left parenthesis straight x space plus space straight y right parenthesis left parenthesis straight x space minus space straight y right parenthesis space left square bracket left parenthesis straight x space plus space straight y right parenthesis space minus space left parenthesis straight x space minus space straight y right parenthesis right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x space plus space straight y right parenthesis cubed space minus space left parenthesis straight x space minus space straight y right parenthesis cubed space minus space 3 left parenthesis straight x squared space minus space straight y squared right parenthesis left parenthesis 2 straight y right parenthesis
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis straight x space plus space straight y right parenthesis cubed space minus space left parenthesis straight x space minus space straight y right parenthesis cubed space minus space 6 straight y left parenthesis straight x squared space minus space straight y squared right parenthesis
But comma space left parenthesis straight x space plus space straight y right parenthesis cubed space minus space left parenthesis straight x space minus space straight y right parenthesis cubed space minus space 6 straight y space left parenthesis straight x squared space minus space straight y squared right parenthesis space equals space ky cubed
rightwards double arrow left square bracket left parenthesis straight x space plus space straight y right parenthesis space minus space left parenthesis straight x space minus space straight y right parenthesis right square bracket cubed space equals space left parenthesis 2 straight y right parenthesis cubed space space equals space straight k 8 straight y cubed
rightwards double arrow left parenthesis 2 straight y right parenthesis cubed space space equals space ky cubed
rightwards double arrow 8 straight y cubed space equals space ky cubed
rightwards double arrow straight k space equals space 8
Hence comma space correct space option space is space left parenthesis straight d right parenthesis. end style

Question 8

If x3 - 3x2 + 3x + 7 = (x + 1) (ax2 + bx + c), then a + b + c =

(a) 4

(b) 12

(c) -10

(d) 3

Solution 8

begin mathsize 12px style straight x cubed space minus space 3 straight x squared space plus space 3 straight x space plus space 7
equals space straight x cubed space minus space 3 straight x squared space plus space 3 straight x space minus space 1 space plus space 1 space plus space 7 space space space left parenthesis by space adding space plus 1 space & space minus 1 right parenthesis
equals space straight x cubed space minus space 3 straight x squared space plus space 3 straight x space plus space 1 space plus space 6
equals left parenthesis straight x right parenthesis cubed space plus space left parenthesis 1 right parenthesis cubed space minus space 3 straight x squared space plus space 3 straight x space plus space 6 space space space space
equals space left parenthesis straight x space plus space 1 right parenthesis left parenthesis straight x squared space plus space 1 space minus space straight x right parenthesis space minus space 3 straight x squared space plus space 3 straight x space plus space 6 space space space space space space space space space left curly bracket by space identity space straight a cubed space plus space straight b cubed space equals space left parenthesis straight a space plus space straight b right parenthesis space left parenthesis straight a squared space plus straight b squared space minus space ab right parenthesis right curly bracket
equals space left parenthesis straight x space plus space 1 right parenthesis left parenthesis straight x squared space minus space straight x space plus space 1 right parenthesis space minus space 3 left parenthesis straight x squared space minus space straight x space minus space 2 right parenthesis
equals space left parenthesis straight x space plus space 1 right parenthesis left parenthesis straight x squared space minus space straight x space plus space 1 right parenthesis space minus space 3 left parenthesis straight x space plus space 1 right parenthesis left parenthesis straight x space minus space 2 right parenthesis
equals space left parenthesis straight x space plus space 1 right parenthesis left square bracket straight x squared space minus space straight x space plus space 1 space minus space 3 left parenthesis straight x space minus space 2 right parenthesis right square bracket
equals space left parenthesis straight x space plus space 1 right parenthesis space left parenthesis straight x squared space minus space 4 straight x space plus space 7 right parenthesis
rightwards double arrow ax squared space plus space bx space plus space 6 space equals space straight x squared space minus space 4 straight x space plus space 7
rightwards double arrow straight a space equals 1 comma space space space straight b space equals space minus 4 comma space space space space space straight c space equals space 7
rightwards double arrow straight a space plus space straight b space plus space straight c space equals space 1 space minus space 4 space plus space 7 space equals space 4
Hence comma space correct space option space is space left parenthesis straight a right parenthesis. end style

CBSE Class 9 Maths Homework Help

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