**Question** 1:

Use suitable identities to find the following products:

(i) (ii)

(iii) (iv)

(v)

**Answer**:

(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 ,

**Question** 2:

Evaluate the following products without multiplying directly:

(i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96

**Answer**:

(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

**Question** 3:

Factorise the following using appropriate identities:

(i) 9*x*^{2} + 6*xy* + *y*^{2}

(ii)

(iii)

**Answer**:

(i)

(ii)

(iii)

**Question** 4:

Expand each of the following, using suitable identities:

(i) (ii)

(iii) (iv)

(v) (vi)

**Answer**:

It is known that,

(i)

(ii)

(iii)

(iv)

(v)

(vi)

**Question** 5:

Factorise:

(i)

(ii)

**Answer**:

It is known that,

(i)

(ii)

**Question** 6:

Write the following cubes in expanded form:

(i) (ii)

(iii) (iv)

**Answer**:

It is known that,

(i)

(ii)

(iii)

(vi)

**Question** 7:

Evaluate the following using suitable identities:

(i) (99)^{3} (ii) (102)^{3} (iii) (998)^{3}

**Answer**:

It is known that,

(i) (99)^{3 }= (100 − 1)^{3}

= (100)^{3} − (1)^{3} − 3(100) (1) (100 − 1)

= 1000000 − 1 − 300(99)

= 1000000 − 1 − 29700

= 970299

(ii) (102)^{3} = (100 + 2)^{3}

= (100)^{3} + (2)^{3} + 3(100) (2) (100 + 2)

= 1000000 + 8 + 600 (102)

= 1000000 + 8 + 61200

= 1061208

(iii) (998)^{3}= (1000 − 2)^{3}

= (1000)^{3} − (2)^{3} − 3(1000) (2) (1000 − 2)

= 1000000000 − 8 − 6000(998)

= 1000000000 − 8 − 5988000

= 1000000000 − 5988008

= 994011992

**Question** 8:

Factorise each of the following:

(i) (ii)

(iii) (iv)

(v)

**Answer**:

It is known that,

(i)

(ii)

(iii)

(iv)

(v)

**Question** 9:

Verify:

(i)

(ii)

**Answer**:

(i) It is known that,

(ii) It is known that,

**Question** 10:

Factorise each of the following:

(i)

(ii)

[**Hint: **See **Question** 9.]

**Answer**:

(i)

(ii)

**Question** 11:

Factorise:

**Answer**:

It is known that,

**Question** 12:

Verify that

**Answer**:

It is known that,

**Question** 13:

If *x + y + z* = 0, show that

x3 + y3 + z3 = 3xyz.

**Answer**:

It is known that,

Put *x + y + z* = 0,

**Question** 14:

Without actually calculating the cubes, find the value of each of the following:

(i)

(ii)

**Answer**:

(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

**Question** 15:

Give possible expressions for the length and breadth of each of thefollowing rectangles, in which their areas are given:

**Answer**:

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 = 5*a* − 3

And, possible breadth = 5*a* − 4

(ii)

Therefore, possible length = 5*y* + 4

And, possible breadth = 7*y* − 3

**Question** 16:

What are the possible expressions for the dimensions of the cuboids whose volumes are given below?

**Answer**:

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

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