SELINA Solutions for Class 9 Maths Chapter 4 - Expansion

Access TopperLearning’s free Selina Solutions for ICSE Class 9 Mathematics Chapter 4 Expansion to learn about finding the square of a given algebraic expression using expansion. Also, go through the solutions to understand expansion of cubes. Highly-experienced Maths experts have created the answers for textbook exercises in a step-wise format to help it easy for students to understand and learn.

With thorough revision using Selina textbook solutions, you will learn to evaluate identities using the right methods. If you need explanation on the basics of expansion, check our ICSE Class 9 Maths concept videos and other chapter resources.

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Chapter 4 - Expansion Exercise Ex. 4(A)

Question 1

Find the square of:

(i) 2a + b

(ii) 3a + 7b

(iii) 3a - 4b

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 1

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 2

Use identities to evaluate:

(i) (101)2

 

(ii) (502)2

 

(iii) (97)2

 

(iv) (998)2

Solution 2

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 3

Evalute:

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii) 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 3

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion


(ii) 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 4

Evaluate:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

(ii) (4a +3b)2 - (4a - 3b)2 + 48ab.

Solution 4

(i)Consider the given expression:

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii)Consider the given expression:

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 5

If a + b = 7 and ab = 10; find a - b.

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 6

If a -b = 7 and ab = 18; find a + b.

Solution 6

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 7

If x + y = Selina Solutions Icse Class 9 Mathematics Chapter - Expansionand xy = Selina Solutions Icse Class 9 Mathematics Chapter - Expansion; find:

(i) x - y 

(ii) x2- y2

Solution 7

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 8

If a - b = 0.9 and ab = 0.36; find:

(i) a + b

(ii) a2 - b2.

Solution 8

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii) 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 9

If a - b = 4 and a + b = 6; find

(i) a2 + b2

(ii) ab

Solution 9

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 10

If a + Selina Solutions Icse Class 9 Mathematics Chapter - Expansion= 6 and  a ≠ 0 find :

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Solution 10

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 11

If a - Selina Solutions Icse Class 9 Mathematics Chapter - Expansion= 8 and a ≠0, find :

 

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Solution 11

(i)

 Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion 

Question 12

If a2 - 3a + 1 = 0, and a≠ 0; find:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 12

(i)

 Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 13

If a2 - 5a - 1 = 0 and a ≠ 0; find:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 13

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 14

If 3x + 4y = 16 and xy = 4; find the value of 9x2 + 16y2.

Solution 14

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 15

The number x is 2 more than the number y. If the sum of the squares of x and y is 34, then find the product of x and y.

 

Solution 15

Given x is 2 more than y, so x = y + 2



Sum of squares of x and y is 34, so x2 + y2 = 34.


Replace x = y + 2 in the above equation and solve for y.


We get (y + 2)2 + y2 = 34


2y2 + 4y - 30 = 0


y2 + 2y - 15 = 0


(y + 5)(y - 3) = 0


So y = -5 or 3


For y = -5, x =-3


For y = 3, x = 5




Product of x and y is 15 in both cases.

 

Question 16

The difference between two positive numbers is 5 and the sum of their squares is 73. Find the product of these numbers.

 

Solution 16

Let the two positive numbers be a and b.

 

Given difference between them is 5 and sum of squares is 73.


So a - b = 5, a2 + b2 = 73

 

Squaring on both sides gives

 

(a - b)2 = 52

 

a2 + b2 - 2ab = 25

 

But a2 + b2 = 73

 

So 2ab = 73 - 25 = 48

 

ab = 24

 

So, the product of numbers is 24.

Chapter 4 - Expansion Exercise Ex. 4(B)

Question 1

Find the cube of :

(i) 3a- 2b

 

(ii) 5a + 3b

 

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(iv) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion 

Solution 1

(i) 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion


(ii) 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion


(iii) 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion


(iv) 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 2

If a2 + Selina Solutions Icse Class 9 Mathematics Chapter - Expansion= 47 and ≠ 0 find:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 2

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 3

If a2 + Selina Solutions Icse Class 9 Mathematics Chapter - Expansion= 18; a ≠ 0 find:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 3

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 4

If a + Selina Solutions Icse Class 9 Mathematics Chapter - Expansion= p and a ≠ 0 ; then show that:

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 5

If a + 2b = 5; then show that:

a3 + 8b3 + 30ab = 125.

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 6

If Selina Solutions Icse Class 9 Mathematics Chapter - Expansion and a ≠ 0 ; then show: a3 + Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Solution 6

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

Question 7

If a + 2b + c = 0; then show that:

a3 + 8b3 + c3 = 6abc.

Solution 7

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

Question 8

Use property to evaluate:

 

 

(i) 133 + (-8)3 + (-5)3

 

 

(ii)73 + 33 + (-10)3

 

 

(iii) 93 - 53 - 43

 

 

(iv) 383 + (-26)3 + (-12)3

 

Solution 8

Property is if a + b + c = 0 then a3 + b3 + c3 = 3abc

 

 

 

 

 

(i) a = 13, b = -8 and c = -5

 

 

133 + (-8)3 + (-5)3 = 3(13)(-8)(-5) = 1560

 

 

 

 

 

(ii) a = 7, b = 3, c = -10

 

 

73 + 33 + (-10)3 = 3(7)(3)(-10) = -630

 

 

 

 

 

(iii)a = 9, b = -5, c = -4

 

 

93 - 53 - 43 = 93 + (-5)3 + (-4)3 = 3(9)(-5)(-4) = 540

 

 

 

 

 

(iv) a = 38, b = -26, c = -12

 

 

383 + (-26)3 + (-12)3 = 3(38)(-26)(-12) = 35568

 

 

 

 

Question 9

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 9

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 10

If Selina Solutions Icse Class 9 Mathematics Chapter - Expansion and a - Selina Solutions Icse Class 9 Mathematics Chapter - Expansion= 4; find:

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(iii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Solution 10

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion


(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion


(iii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 11

IfSelina Solutions Icse Class 9 Mathematics Chapter - Expansion and  x + Selina Solutions Icse Class 9 Mathematics Chapter - Expansion= 2; then show that:

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 11

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Thus from equations (1), (2) and (3), we have

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Question 12

If 2x - 3y = 10 and xy = 16; find the value of 8x3 - 27y3.

Solution 12

Given that 2x - 3y = 10, xy = 16

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 13

Expand :

(i)  (3x + 5y + 2z) (3x - 5y + 2z)

(ii)  (3x - 5y - 2z) (3x - 5y + 2z)

Solution 13

(i)

(3x + 5y + 2z) (3x - 5y + 2z)

 

= {(3x + 2z) + (5y)} {(3x + 2z) - (5y)}

= (3x + 2z)2 - (5y)2

{since (a + b) (a - b) = a2 - b2}

= 9x2 + 4z2 + 2 × 3x × 2z - 25y2

= 9x2 + 4z2 + 12xz - 25y2

= 9x2 + 4z- 25y2 + 12xz

 

(ii)

(3x - 5y - 2z) (3x - 5y + 2z)

 

= {(3x - 5y) - (2z)} {(3x - 5y) + (2z)}

 

= (3x - 5y)2 - (2z)2{since(a + b) (a - b) = a2 - b2}

= 9x2 + 25y2 - 2 × 3x × 5y - 4z2

 

= 9x2 + 25y2- 30xy - 4z2

= 9x2 +25y2 - 4z2 - 30xy

 

Question 14

The sum of two numbers is 9 and their product is 20. Find the sum of their

(i) Squares (ii) Cubes

 

Solution 14

Given sum of two numbers is 9 and their product is 20.


Let the numbers be a and b.


a + b = 9


ab = 20





Squaring on both sides gives


(a+b)2 = 92


a2 + b2 + 2ab = 81


a2 + b2 + 40 = 81




So sum of squares is 81 - 40 = 41




Cubing on both sides gives


(a + b)3 = 93


a3 + b3 + 3ab(a + b) = 729


a3 + b3 + 60(9) = 729


a3 + b3 = 729 - 540 = 189




So the sum of cubes is 189.

 

 

 

 

Question 15

Two positive numbers x and y are such that x > y. If the difference of these numbers is 5 and their product is 24, find:

 

 

(i) Sum of these numbers

 

 

(ii) Difference of their cubes

 

 

(iii) Sum of their cubes.

 

Solution 15

Given x - y = 5 and xy = 24 (x>y)

 

 

(x + y)2 = (x - y)2 + 4xy = 25 + 96 = 121

 

 

So, x + y = 11; sum of these numbers is 11.

 

 

 

 

 

Cubing on both sides gives

 

 

(x - y)3 = 53

 

 

x3 - y3 - 3xy(x - y) = 125

 

 

x3 - y3 - 72(5) = 125

 

 

x3 - y3= 125 + 360 = 485

 

 

So, difference of their cubes is 485.

 

 

 

 

 

Cubing both sides, we get

 

 

(x + y)3 = 113

 

 

x3 + y3 + 3xy(x + y) = 1331

 

 

x3 + y3 = 1331 - 72(11) = 1331 - 792 = 539

 

 

So, sum of their cubes is 539.

 

 

 

 

Question 16

If 4x2 + y2 = a and xy = b, find the value of 2x + y.

Solution 16

xy = b ….(i)

4x2 + y2 = a ….(ii)

Now, (2x + y)2 = (2x)2 + 4xy + y2

= 4x2 + y2 + 4xy

= a + 4b ….[From (i) and (ii)]

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Chapter 4 - Expansion Exercise Ex. 4(C)

Question 1

Expand:

(i) (x + 8) (x + 10)

(ii) (x + 8) (x - 10)

(iii) (x - 8) (x + 10)

(iv) (x - 8) (x - 10) 

Solution 1

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 2

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 2

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 3

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 3

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion 

Question 4

If a + b + c = 12 and a2 + b2 + c2 = 50; find ab + bc + ca.

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 5

If a2 + b2 + c2 = 35 and ab + bc + ca = 23; find a + b + c.

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 6

If a + b + c = p and ab + bc + ca = q; find a2 + b2 + c2.

Solution 6

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 7

If a2 + b2 + c2 = 50 and ab + bc + ca = 47, find a + b + c.

Solution 7

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 8

If x+ y - z = 4 and x2 + y2 + z2 = 30, then find the value of xy - yz - zx.

Solution 8

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Chapter 4 - Expansion Exercise Ex. 4(D)

Question 1

If x + 2y + 3z = 0 and x3 + 4y3 + 9z3 = 18xyz; evaluate:

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Solution 1

Given that x3 + 4y3 + 9z3 = 18xyz and x + 2y + 3z = 0

Therefore, x + 2y = - 3z, 2y + 3z = -x and 3z + x = -2y

Now

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 2

If a + Selina Solutions Icse Class 9 Mathematics Chapter - Expansion= m and ≠ 0 ; find in terms of 'm'; the value of :

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Solution 2

(i)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Question 3

In the expansion of (2x2 - 8) (x - 4)2; find the value of

(i) coefficient of x3

(ii) coefficient of x2

(iii) constant term.

Solution 3

 Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 4

If x > 0 and Selina Solutions Icse Class 9 Mathematics Chapter - Expansionfind: Selina Solutions Icse Class 9 Mathematics Chapter - Expansion.

 

 

Solution 4

Given that

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Question 5

If 2(x2 + 1) = 5x, find :

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion(ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Solution 5

(i)

 

2(x2 + 1} = 5x

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

Dividing by x, we have

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

 

 

 

(ii)

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

 

 

 

 

 

 

 

 

 

Question 6

If a2 + b2 = 34 and ab = 12; find:

(i) 3(a + b)2 + 5(a - b)2

(ii) 7(a - b)2 - 2(a + b)2

 

 

 

Solution 6

a2 + b2 = 34, ab= 12

(a + b)2 = a2 + b2 + 2ab

 

 

= 34 + 2 x 12 = 34 + 24 = 58  

(a - b)2 = a2 + b2 - 2ab

 

 

= 34 - 2 x 12 = 34- 24 = 10

(i) 3(a + b)2 + 5(a - b)2

 

 

= 3 x 58 + 5 x 10 = 174 + 50

= 224

 

 

 

 

(ii) 7(a - b)2 - 2(a + b)2

= 7 x 10 - 2 x 58 = 70 - 116 = -46

 

 

 

Question 7

If 3x - Selina Solutions Icse Class 9 Mathematics Chapter - Expansion and x ≠ 0 find : Selina Solutions Icse Class 9 Mathematics Chapter - Expansion.

 

 

Solution 7

Given 3x - Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

We need to findSelina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

 

 

 

Question 8

If x2 + Selina Solutions Icse Class 9 Mathematics Chapter - Expansion= 7 and  x ≠ 0; find the value of:

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion.

 

 

Solution 8

Given that Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

We need to find the value of Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

Consider the given equation:

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Question 9

If x =Selina Solutions Icse Class 9 Mathematics Chapter - Expansion and x ≠ 5 find Selina Solutions Icse Class 9 Mathematics Chapter - Expansion.

 

Solution 9

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

By cross multiplication,

=> x (x - 5) = 1 => x2 - 5x = 1 => x2 - 1 = 5x

Dividing both sides by x,

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

 

 

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

 

 

 

 

Question 10

If x = Selina Solutions Icse Class 9 Mathematics Chapter - Expansion and x ≠ 5 find Selina Solutions Icse Class 9 Mathematics Chapter - Expansion.

 

 

 

Solution 10

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

By cross multiplication,

=> x (5 - x) = 1 => x2 - 5x =-1 => x2 + 1 = 5x

Dividing both sides by x,

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

 

 

 

Question 11

If 3a + 5b + 4c = 0, show that:

27a3 + 125b3 + 64c3 = 180 abc

Solution 11

Given that 3a + 5b + 4c = 0

 

 

3a + 5b = -4c

 

Cubing both sides,

(3a + 5b)3 = (-4c)3

=>(3a)3 + (5b)3 + 3 x 3a x 5b (3a + 5b) = -64c3

=>27a3 + 125b3 + 45ab x (-4c) = -64c3

=>27a3 + 125b3 - 180abc = -64c3

=>27a3 + 125b3 + 64c3 = 180abc

 

 

Hence proved.

Question 12

The sum of two numbers is 7 and the sum of their cubes is 133, find the sum of their square.

Solution 12

Let a, b be the two numbers

.'. a + b = 7 and a3 + b3 = 133

 

(a + b)3 = a3 + b3 + 3ab (a + b)

=> (7)3 = 133 + 3ab (7)

 

=> 343 = 133 + 21ab => 21ab = 343 - 133 = 210

 

=> 21ab = 210 => ab= 2I

Now a2 + b2 = (a + b)2 - 2ab

=72 - 2 x 10 = 49 - 20 = 29

 

 

 

Question 13

In each of the following, find the value of 'a':

(i) 4x2 + ax + 9 = (2x + 3)2

(ii) 4x2 + ax + 9 = (2x - 3)2

(iii) 9x2 + (7a - 5)x + 25 = (3x + 5)2

Solution 13

(i) 4x2 + ax + 9 = (2x + 3)2

Comparing coefficients of x terms, we get

ax = 12x

so, a = 12

 

(ii) 4x2 + ax + 9 = (2x - 3)2

Comparing coefficients of x terms, we get

ax = -12x

so, a = -12

(iii) 9x2 + (7a - 5)x + 25 = (3x + 5)2

Comparing coefficients of x terms, we get

(7a - 5)x = 30x

7a - 5 = 30

7a = 35

a = 5

 

 

Question 14

If Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

(i) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion (ii) Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 14

Given

 

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

 

 

 

 

 

 

 

 

 

 

 

 

Question 15

The difference between two positive numbers is 4 and the difference between their cubes is 316.

Find:

(i) Their product

(ii) The sum of their squares

Solution 15

Given difference between two positive numbers is 4 and difference between their cubes is 316.


Let the positive numbers be a and b

a - b = 4

a3 - b3 = 316

Cubing both sides,

(a - b)3 = 64

a3 - b3 - 3ab(a - b) = 64


Given a3 - b3 = 316

So 316 - 64 = 3ab(4)

252 = 12ab

So ab = 21; product of numbers is 21


Squaring both sides, we get

(a - b)2 = 16

a2 + b2 - 2ab = 16

a2 + b2 = 16 + 42 = 58

Sum of their squares is 58.

 

Chapter 4 - Expansion Exercise Ex. 4(E)

Question 1

Simplify:

(i) (x + 6)(x + 4)(x - 2)

(ii) (x - 6)(x - 4)(x + 2)

(iii) (x - 6)(x - 4)(x - 2)

(iv) (x + 6)(x - 4)(x - 2) 

Solution 1

Using identity:

(x + a)(x + b)(x + c) = x3 + (a + b + c)x2 + (ab + bc + ca)x + abc

(i) (x + 6)(x + 4)(x - 2)

= x3 + (6 + 4 - 2)x2 + [6 × 4 + 4 × (-2) + (-2) × 6]x + 6 × 4 × (-2)

= x3 + 8x2 + (24 - 8 - 12)x - 48

= x3 + 8x2 + 4x - 48

 

(ii) (x - 6)(x - 4)(x + 2)

= x3 + (-6 - 4 + 2)x2 + [-6 × (-4) + (-4) × 2 + 2 × (-6)]x + (-6) × (-4) × 2

= x3 - 8x2 + (24 - 8 - 12)x + 48

= x3 - 8x2 + 4x + 48

(iii) (x - 6)(x - 4)(x - 2)

= x3 + (-6 - 4 - 2)x2 + [-6 × (-4) + (-4) × (-2) + (-2) × (-6)]x + (-6) × (-4) × (-2)

= x3 - 12x2 + (24 + 8 + 12)x - 48

= x3 - 12x2 + 44x - 48

 

(iv) (x + 6)(x - 4)(x - 2)

= x3 + (6 - 4 - 2)x2 + [6 × (-4) + (-4) × (-2) + (-2) × 6]x + 6 × (-4) × (-2)

= x3 - 0x2 + (-24 + 8 - 12)x + 48

= x3 - 28x + 48 

Question 2

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 2

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion 

Question 3

Using suitable identity, evaluate

(i) (104)3

(ii) (97)3 

Solution 3

Using identity: (a ± b)3 = a3 ± b3 ± 3ab(a ± b)

(i) (104)3 = (100 + 4)3

= (100)3 + (4)3 + 3 × 100 × 4(100 + 4)

= 1000000 + 64 + 1200 × 104

= 1000000 + 64 + 124800

= 1124864

 

(ii) (97)3 = (100 - 3)3

= (100)3 - (3)3 - 3 × 100 × 3(100 - 3)

= 1000000 - 27 - 900 × 97

= 1000000 - 27 - 87300

= 912673

Question 4

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 4

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion 

Question 5

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 5

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion 

Question 6

If a - 2b + 3c = 0; state the value of a3 - 8b3 + 27c3.

Solution 6

a3 - 8b3 + 27c3 = a3 + (-2b)3 + (3c)3

Since a - 2b + 3c = 0, we have

a3 - 8b3 + 27c3 = a3 + (-2b)3 + (3c)3

= 3(a)( -2b)(3c)

= -18abc 

Question 7

If x + 5y = 10; find the value of x3 + 125y3 + 150xy - 1000.

Solution 7

x + 5y = 10

(x + 5y)3 = 103

x3 + (5y)3 + 3(x)(5y)(x + 5y) = 1000

x3 + (5y)3 + 3(x)(5y)(10) = 1000

= x3 + (5y)3 + 150xy = 1000

= x3 + (5y)3 + 150xy - 1000 = 0 

Question 8

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion

Solution 8

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion 

Question 9

If a + b = 11 and a2 + b2 = 65; find a3 + b3.

Solution 9

Selina Solutions Icse Class 9 Mathematics Chapter - Expansion 

Question 10

Prove that:

x2+ y2 + z2 - xy - yz - zx  is always positive.

Solution 10

x2 + y2 + z2 - xy - yz - zx

= 2(x2 + y2 + z2 - xy - yz - zx)

= 2x2 + 2y2 + 2z2 - 2xy - 2yz - 2zx

= x2 + x2 + y2 + y2 + z2 + z2 - 2xy - 2yz - 2zx

= (x2 + y2 - 2xy) + (z2 + x2 - 2zx) + (y2 + z2 - 2yz)

= (x - y)2 + (z - x)2 + (y - z)2

Since square of any number is positive, the given equation is always positive.

Question 11

Find:

(i) (a + b)(a + b)

(ii) (a + b)(a + b)(a + b)

(iii) (a - b)(a - b)(a - b) by using the result of part (ii)

Solution 11

(i) (a + b)(a + b) = (a + b)2

= a × a + a × b + b × a + b × b

= a2 + ab + ab + b2

= a2 + b2 + 2ab

 

(ii) (a + b)(a + b)(a + b)

= (a × a + a × b + b × a + b × b)(a + b)

= (a2 + ab + ab + b2)(a + b)

= (a2 + b2 + 2ab)(a + b)

= a2 × a + a2 × b + b2 × a + b2 × b + 2ab × a + 2ab × b

= a3 + a2 b + ab2 + b3 + 2a2b + 2ab2

= a3 + b3 + 3a2b + 3ab2

 

(iii) (a - b)(a - b)(a - b)

In result (ii), replacing b by -b, we get

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

= a3 + (-b)3 + 3a2(-b) + 3a(-b)2

= a3 - b3 - 3a2b + 3ab2