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Learn Exercise 2.5 with Free Lessons & Tips

Factorise: (i)

(ii) 

It is known that,

(i) 

(ii) 

Comments

Write the following cubes in expanded form: (i)  (ii)  (iii)  (iv) 

It is known that,

 

(i) 

 

(ii) 

 

(iii) 

 

(vi) 

Comments

Evaluate the following using suitable identities: (i)  (ii)  (iii) 

(i) (99)= (100 − 1)3

= (100)3 − (1)3 − 3(100) (1) (100 − 1)

 

= 970299

(ii) (102)3 = (100 + 2)3

= (100)3 + (2)3 + 3(100) (2) (100 + 2)

 

= 1061208.00

(iii) (998)3= (1000 − 2)3

= (1000)3 − (2)3 − 3(1000) (2) (1000 − 2)

= 994011992.00

Comments

 Factorise each of the following: (i)  (ii)  (iii)  (iv)  (v)  

It is known that,

(i) 




 

(ii) 




 

(iii) 

(iv) 

(v) 

Comments

 Verify: (i)  (ii)  

(i) It is known that,

(ii) It is known that,

Comments

10. Factorise each of the following: (i)  (ii)  [Hint : See Question 9.]

(i) 

(ii) 

 

Comments

Factorise : (i) 

It is known that,

Comments

Verify that : 

It is known that,

Comments

If x + y + z = 0, show that

It is known that,

Put x + y + z = 0,

Comments

Without actually calculating the cubes, find the value of each of the following:
(i)
(ii) 

(i) 

Let x = −12, y = 7, and z = 5

It can be observed that,

x + y + z = − 12 + 7 + 5 = 0

It is known that if x + y + z = 0, then

∴ 

= −1260

(ii) 

Let x = 28, y = −15, and z = −13

It can be observed that,

x + y + z = 28 + (−15) + (−13) = 28 − 28 = 0

It is known that if x + y + z = 0, then

Comments

Give possible expressions for the length and breadth of each of the following
rectangles, in which their areas are given: (i) Area:  (ii) Area: 

Area = Length × Breadth

The expression given for the area of the rectangle has to be factorised. One of its factors will be its length and the other will be its breadth.

(i) 

Therefore, possible length = 5a − 3

And, possible breadth = 5a − 4

(ii) 

Therefore, possible length = 5y + 4

And, possible breadth = 7y − 3

Comments

What are the possible expressions for the dimensions of the cuboids whose volumes
are given below? (i) Volume : (ii) Volume : 

Volume of cuboid = Length × Breadth × Height

The expression given for the volume of the cuboid has to be factorised. One of its factors will be its length, one will be its breadth, and one will be its height.

(i)

One of the possible solutions is as follows.

Length = 3, Breadth = x, Height = x − 4

(ii)

One of the possible solutions is as follows.

Length = 4k, Breadth = 3y + 5, Height = y − 1

Comments

Use suitable identities to find the following products: (i) (x + 4) (x + 10)
(ii) (x + 8) (x – 10)
(iii) (3x + 4) (3x – 5) (iv)  (v) (3 – 2x) (3 + 2x)

(i) By using the identity ,

 

(ii) By using the identity ,

 

(iii) 

By using the identity ,

 

(iv) By using the identity ,

(v) By using the identity ,

Comments

Evaluate the following products without multiplying directly: (i) 103x107 (ii) 95x96 (iii) 104x96

(i) 103 × 107 = (100 + 3) (100 + 7)

= (100)2 + (3 + 7) 100 + (3) (7)

[By using the identity, where

x = 100, a = 3, and b = 7]

= 10000 + 1000 + 21

= 11021

(ii) 95 × 96 = (100 − 5) (100 − 4)

= (100)2 + (− 5 − 4) 100 + (− 5) (− 4)

[By using the identity, where

x = 100, a = −5, and b = −4]

= 10000 − 900 + 20

= 9120

(iii) 104 × 96 = (100 + 4) (100 − 4)

= (100)2 − (4)2 

= 10000 − 16

= 9984

Comments

Factorise the following using appropriate identities: (i)  (ii)  (iii)  

(i) 

 

(ii) 

 

(iii) 

Comments

Expand each of the following, using suitable identities:(i)   (ii)  (iii)  (iv)  (v) (vi) 

It is known that,

 

(i) 

 

(ii) 

 

(iii) 

 

(iv) 

 

(v) 

 

(vi) 

Comments

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