#### Chapters

Chapter 2: Data Handling

Chapter 3: Square-Square Root and Cube-Cube Root

Chapter 4: Linear Equation In One Variable

Chapter 5: Understanding Quadrilaterals and Practical Geometry

Chapter 6: Visualising Solid Shapes

Chapter 7: Algebraic Expression, Identities and Factorisation

Chapter 8: Exponents and Powers

Chapter 9: Comparing Quantities

Chapter 10: Direct and Inverse Proportions

Chapter 11: Mensuration

Chapter 12: Introduct To Graphs

Chapter 13: Playing With Numbers

## Chapter 7: Algebraic Expression, Identities and Factorisation

### NCERT solutions for Mathematics Exemplar Class 8 Chapter 7 Algebraic Expression, Identities and Factorisation Exercise [Pages 224 - 240]

#### Choose the correct alternative:

The product of a monomial and a binomial is a ______.

Monomial

Binomial

Trinomial

None of these

In a polynomial, the exponents of the variables are always ______.

Integers

Positive integers

Non-negative integers

Non-positive integers

Which of the following is correct?

`(a - b)^2 = a^2 + 2ab - b^2`

`(a - b)^2 = a^2 - 2ab + b^2`

`(a - b)^2 = a^2 - b^2`

`(a + b)^2 = a^2 + 2ab - b^2`

The sum of –7pq and 2pq is ______.

–9pq

9pq

5pq

– 5pq

If we subtract –3x^{2}y^{2} from x^{2}y^{2}, then we get ______.

– 4x

^{2}y^{2}– 2x

^{2}y^{2}2x

^{2}y^{2}4x

^{2}y^{2}

Like term as 4m^{3}n^{2} is ______.

4m

^{2}n^{2}– 6m

^{3}n^{2}6pm

^{3}n^{2}4m

^{3}n

Which of the following is a binomial?

7 × a + a

6a

^{2}+ 7b + 2c4a × 3b × 2c

6(a

^{2}+ b)

Sum of a – b + ab, b + c – bc and c – a – ac is ______.

2c + ab – ac – bc

2c – ab – ac – bc

2c + ab + ac + bc

2c – ab + ac + bc

Product of the following monomials 4p, – 7q^{3}, –7pq is ______.

196 p

^{2}q^{4}196 pq

^{4}– 196 p

^{2}q^{4}196 p

^{2}q^{3}

Area of a rectangle with length 4ab and breadth 6b^{2} is ______.

24a

^{2}b^{2}24ab

^{3}24ab

^{2}24ab

Volume of a rectangular box (cuboid) with length = 2ab, breadth = 3ac and height = 2ac is ______.

12a

^{3}bc^{2}12a

^{3}bc12a

^{2}bc2ab + 3ac + 2ac

Product of 6a^{2} – 7b + 5ab and 2ab is ______.

12a

^{3}b – 14ab^{2}+ 10ab12a

^{3}b – 14ab^{2}+ 10a^{2}b^{2}6a

^{2}– 7b + 7ab12a

^{2}b – 7ab^{2}+ 10ab

Square of 3x – 4y is ______.

9x

^{2}– 16y^{2}6x

^{2}– 8y^{2}9x

^{2}+ 16y^{2}+ 24xy9x

^{2}+ 16y^{2}– 24xy

Which of the following are like terms?

5xyz2, – 3xy2z

– 5xyz

^{2}, 7xyz^{2}5xyz

^{2}, 5x^{2}yz5xyz

^{2}, x^{2}y^{2}z^{2}

Coefficient of y in the term `(-y)/3` is ______.

– 1

– 3

`(-1)/3`

`1/3`

a^{2} – b^{2} is equal to ______.

(a – b)

^{2}(a – b)(a – b)

(a + b)(a – b)

(a + b)(a + b)

Common factor of 17abc, 34ab^{2}, 51a^{2}b is ______.

17abc

17ab

17ac

17a

^{2}b^{2}c

Square of 9x – 7xy is ______.

81x

^{2}+ 49x^{2}y^{2}81x

^{2}– 49x^{2}y^{2}81x

^{2}+ 49x^{2}y^{2}–126x^{2}y81x

^{2}+ 49x^{2}y^{2}– 63x^{2}y

Factorised form of 23xy – 46x + 54y – 108 is ______.

(23x + 54)(y – 2)

(23x + 54y)(y – 2)

(23xy + 54y)(– 46x – 108)

(23x + 54)(y + 2)

Factorised form of r^{2} – 10r + 21 is ______.

(r – 1)(r – 4)

(r – 7)(r – 3)

(r – 7)(r + 3)

(r + 7)(r + 3)

Factorised form of p^{2} – 17p – 38 is ______.

(p – 19)(p + 2)

(p – 19)(p – 2)

p + 19)(p + 2)

(p + 19)(p – 2)

On dividing 57p^{2}qr by 114pq, we get ______.

`1/4 pr`

`3/4 pr`

`1/2 pr`

`2pr`

On dividing p(4p^{2} – 16) by 4p(p – 2), we get ______.

2p + 4

2p – 4

p + 2

p + 2

The common factor of 3ab and 2cd is ______.

1

– 1

a

c

An irreducible factor of 24x^{2}y^{2} is ______.

x

^{2}y

^{2}x

24x

Number of factors of (a + b)^{2} is ______.

4

3

2

1

The factorised form of 3x – 24 is ______.

3x × 24

3(x – 8)

24(x – 3)

3(x – 12)

The factors of x^{2} – 4 are ______.

(x – 2), (x – 2)

(x + 2), (x – 2)

(x + 2), (x + 2)

(x – 4), (x – 4)

The value of (– 27x^{2}y) ÷ (– 9xy) is ______.

3xy

– 3xy

– 3x

3x

The value of (2x^{2} + 4) ÷ 2 is ______.

2x

^{2}+ 2x

^{2}+ 2x

^{2}+ 42x

^{2}+ 4

The value of (3x^{3} +9x^{2} + 27x) ÷ 3x is ______.

x

^{2}+ 9 + 27x3x

^{3}+ 3x^{2}+ 27x3x

^{3}+ 9x^{2}+ 9x

^{2}+ 3x + 9

The value of (a + b)^{2} + (a – b)^{2} is ______.

2a + 2b

2a – 2b

2a

^{2}+ 2b^{2}2a

^{2}– 2b^{2}

The value of (a + b)^{2} – (a – b)^{2} is ______.

4ab

– 4ab

2a

^{2}+ 2b^{2}2a

^{2}– 2b^{2}

#### Fill in the blanks:

The product of two terms with like signs is a ______ term.

The product of two terms with unlike signs is a ______ term.

a(b + c) = ax ____ × ax _____.

(a – b) ______ = a^{2} – 2ab + b^{2}

a^{2} – b^{2} = (a + b) ______.

(a – b)^{2} + ______ = a^{2} – b^{2}

(a + b)^{2} – 2ab = ______ + ______.

(x + a)(x + b) = x^{2} + (a + b) x + ______.

The product of two polynomials is a ______.

Common factor of ax^{2} + bx is ______.

Factorised form of 18mn + 10mnp is ______.

Factorised form of 4y^{2} – 12y + 9 is ______.

38x^{3}y^{2}z ÷ 19xy^{2} is equal to ______.

Volume of a rectangular box with length 2x, breadth 3y and height 4z is ______.

67^{2} – 37^{2} = (67 – 37) × ______ = ______.

1032 – 1022 = ______ × (103 – 102) = ______.

Area of a rectangular plot with sides 4x^{2} and 3y^{2} is ______.

Volume of a rectangular box with l = b = h = 2x is ______.

The coefficient in – 37abc is ______.

Number of terms in the expression a^{2} + bc × d is ______.

The sum of areas of two squares with sides 4a and 4b is ______.

The common factor method of factorisation for a polynomial is based on ______ property.

The side of the square of area 9y^{2} is ______.

On simplification `(3x + 3)/3` = ______.

The factorisation of 2x + 4y is ______.

#### State whether the following statement is True or False:

(a + b)^{2} = a^{2} + b^{2}

True

False

(a – b)^{2} = a^{2} – b^{2}

True

False

(a + b)(a – b) = a^{2} – b^{2}

True

False

The product of two negative terms is a negative term.

True

False

The product of one negative and one positive term is a negative term

True

False

The coefficient of the term – 6x^{2}y^{2} is – 6.

True

False

p^{2}q + q^{2}r + r^{2}q is a binomial.

True

False

The factors of a^{2} – 2ab + b^{2} are (a + b) and (a + b).

True

False

h is a factor of 2π(h + r).

True

False

Some of the factors of `n^2/2 + n/2` are `1/2`, n and `(n + 1)`

True

False

An equation is true for all values of its variables.

True

False

`x^2 + (a + b)x + ab = (a + b)(x + ab)`

True

False

Common factor of 11pq^{2}, 121p^{2}q^{3}, 1331p^{2}q is 11p^{2}q^{2}.

True

False

Common factor of 12a^{2}b^{2} + 4ab^{2} – 32 is 4.

True

False

Factorisation of – 3a^{2} + 3ab + 3ac is 3a(– a – b – c).

True

False

Factorised form of p^{2} + 30p + 216 is (p + 18)(p – 12).

True

False

The difference of the squares of two consecutive numbers is their sum

True

False

abc + bca + cab is a monomial.

True

False

On dividing `p/3` by `3/p`, the quotient is 9.

True

False

The value of p for 51^{2} – 49^{2} = 100p is 2.

True

False

(9x – 51) ÷ 9 is x – 51

True

False

The value of (a + 1)(a – 1)(a^{2} + 1) is a^{4} – 1.

True

False

**Add:** 7a^{2}bc, – 3abc^{2}, 3a^{2}bc, 2abc^{2}

**Add:** 9ax, + 3by – cz, – 5by + ax + 3cz

**Add:** xy^{2}z^{2} + 3x^{2}y^{2}z – 4x^{2}yz^{2}, – 9x^{2}y^{2}z + 3xy^{2}z^{2} + x^{2}yz^{2}

**Add:** 5x^{2} – 3xy + 4y^{2} – 9, 7y^{2} + 5xy – 2x^{2} + 13

**Add:** 2p^{4} – 3p^{3} + p^{2} – 5p +7, –3p^{4} – 7p^{3} – 3p^{2} – p – 12

**Add:** 3a(a – b + c), 2b(a – b + c)

**Add: **3a(2b + 5c), 3c(2a + 2b)

**Subtract:** 5a^{2}b^{2}c^{2} from – 7a^{2}b^{2}c^{2}^{ }

**Subtract:** 6x^{2} – 4xy + 5y^{2} from 8y^{2} + 6xy – 3x^{2}

**Subtract: **2ab^{2}c^{2} + 4a^{2}b^{2}c – 5a^{2}bc^{2} from –10a^{2}b^{2}c + 4ab^{2}c^{2} + 2a^{2}bc^{2}

**Subtract:** 3t^{4} – 4t^{3} + 2t^{2} – 6t + 6 from – 4t^{4} + 8t^{3} – 4t^{2} – 2t + 11

**Subtract:** 2ab + 5bc – 7ac from 5ab – 2bc – 2ac + 10abc

**Subtract: **7p(3q + 7p) from 8p(2p – 7q)

**Subtract: **–3p^{2} + 3pq + 3px from 3p(– p – a – r)

**Multiply the following:**

– 7pq^{2}r^{3}, – 13p^{3}q^{2}r

**Multiply the following: **

3x^{2}y^{2}z^{2}, 17xyz

**Multiply the following:**

15xy^{2}, 17yz^{2}

**Multiply the following: **

–5a^{2}bc, 11ab, 13abc^{2}

**Multiply the following: **

–3x^{2}y, (5y – xy)

**Multiply the following: **

abc, (bc + ca)

**Multiply the following: **

7pqr, (p – q + r)

**Multiply the following: **

x^{2}y^{2}z^{2}, (xy – yz + zx)

**Multiply the following: **

(p + 6), (q – 7)

**Multiply the following: **

6mn, 0mn

**Multiply the following: **

a, a^{5}, a^{6}

**Multiply the following: **

–7st, –1, – 13st^{2}

**Multiply the following:**

b^{3}, 3b^{2}, 7ab^{5}

**Multiply the following: **

`- 100/9 rs; 3/4 r^3s^2`

**Multiply the following: **

(a^{2} – b^{2}), (a^{2} + b^{2})

**Multiply the following: **

(ab + c), (ab + c)

**Multiply the following: **

(pq – 2r), (pq – 2r)

**Multiply the following: **

`(3/4x - 4/3 y), (2/3x + 3/2y)`

**Multiply the following: **

`3/2 p^2 + 2/3 q^2, (2p^2 - 3q^2)`

**Multiply the following: **

(x^{2} – 5x + 6), (2x + 7)

**Multiply the following:**

(3x^{2} + 4x – 8), (2x^{2} – 4x + 3)

**Multiply the following:**

(2x – 2y – 3), (x + y + 5)

**Simplify:** `(3x + 2y)^2 + (3x - 2y)^2`

**Simplify:** `(3x + 2y)^2 - (3x - 2y)^2`

**Simplify: **`(7/9 a + 9/7 b)^2 - ab`

**Simplify: **`(3/4x - 4/3y)^2 + 2xy`

**Simplify:** `(1.5p + 1.2q)^2 - (1.5p - 1.2q)^2`

**Simplify: **`(2.5m + 1.5q)^2 + (2.5m - 1.5q)^2`

**Simplify: **`(x^2 - 4) + (x^2 + 4) + 16`

**Simplify: **`(ab - c)^2 + 2abc`

**Simplify: **`(a - b) (a^2 + b^2 + ab) - (a + b) (a^2 + b^2 - ab)`

**Simplify: **`(b^2 - 49) (b + 7) + 343`

**Simplify:** `(4.5a + 1.5b)^2 + (4.5b + 1.5a)^2`

**Simplify: **`(pq - qr)^2 + 4pq^2r`

**Simplify: **`(s^2t + tq^2)^2 - (2stq)^2`

**Expand the following, using suitable identities.**

`(xy + yz)^2`

**Expand the following, using suitable identities.**

`(x^2y - xy^2)^2`

**Expand the following, using suitable identities.**

`(4/5a + 5/4b)^2`

**Expand the following, using suitable identities.**

`(2/3x - 3/2y)^2`

**Expand the following, using suitable identities.**

`(4/5p + 5/3q)^2`

**Expand the following, using suitable identities.**

`(x + 3)(x + 7)`

**Expand the following, using suitable identities.**

(2x + 9)(2x – 7)

**Expand the following, using suitable identities.**

`((4x)/5 + y/4)((4x)/5 + (3y)/4)`

**Expand the following, using suitable identities.**

`((2x)/3 - 2/3)((2x)/3 + (2a)/3)`

**Expand the following, using suitable identities.**

`(2x - 5y) (2x - 5y)`

**Expand the following, using suitable identities.**

`((2a)/3 + b/3)((2a)/3 - b/3)`

**Expand the following, using suitable identities.**

`(x^2 + y^2)(x^2 - y^2)`

**Expand the following, using suitable identities.**

`(a^2 + b^2)^2`

**Expand the following, using suitable identities.**

`(7x + 5)^2`

**Expand the following, using suitable identities.**

`(0.9p - 0.5q)^2`

**Using suitable identities, evaluate the following.**

(52)^{2}

**Using suitable identities, evaluate the following.**

(49)^{2}

**Using suitable identities, evaluate the following.**

(103)^{2}

**Using suitable identities, evaluate the following.**

(98)^{2}

**Using suitable identities, evaluate the following.**

(1005)^{2}

**Using suitable identities, evaluate the following.**

(995)^{2}

**Using suitable identities, evaluate the following.**

47 × 53

**Using suitable identities, evaluate the following.**

52 × 53

**Using suitable identities, evaluate the following.**

105 × 95

**Using suitable identities, evaluate the following.**

104 × 97

**Using suitable identities, evaluate the following.**

101 × 103

**Using suitable identities, evaluate the following.**

98 × 103

**Using suitable identities, evaluate the following.**

(9.9)^{2}

**Using suitable identities, evaluate the following.**

9.8 × 10.2

**Using suitable identities, evaluate the following.**

10.1 × 10.2

**Using suitable identities, evaluate the following.**

(35.4)^{2} – (14.6)^{2}

**Using suitable identities, evaluate the following.**

(69.3)^{2} – (30.7)^{2}

**Using suitable identities, evaluate the following.**

(9.7)^{2} – (0.3)^{2}

**Using suitable identities, evaluate the following.**

(132)^{2} – (68)^{2}

**Using suitable identities, evaluate the following.**

(339)^{2} – (161)^{2}

**Using suitable identities, evaluate the following.**

(729)^{2} – (271)^{2}

**Write the greatest common factor in the following terms.**

– 18a^{2}, 108a

**Write the greatest common factor in the following terms.**

3x^{2}y, 18xy^{2}, – 6xy

**Write the greatest common factor in the following terms.**

2xy, –y^{2}, 2x^{2}y

**Write the greatest common factor in the following terms.**

l^{2}m^{2}n, lm^{2}n^{2}, l^{2}mn^{2}

**Write the greatest common factor in the following terms.**

21pqr, –7p^{2}q^{2}r^{2}, 49p^{2}qr

**Write the greatest common factor in the following terms.**

qrxy, pryz, rxyz

**Write the greatest common factor in the following terms. **

3x^{3}y^{2}z, – 6xy^{3}z^{2}, 12x^{2}yz^{3}

**Write the greatest common factor in the following terms. **

63p^{2}a^{2}r^{2}s, – 9pq^{2}r^{2}s^{2}, 15p^{2}qr^{2}s^{2}, – 60p^{2}a^{2}rs^{2}^{ }

**Write the greatest common factor in the following terms. **

13x^{2}y, 169xy

**Write the greatest common factor in the following terms. **

11x^{2}, 12y^{2}

**Factorise the following expression.**

6ab + 12bc

**Factorise the following expression.**

–xy – ay

**Factorise the following expression.**

ax^{3} – bx^{2} + cx

**Factorise the following expression.**

l^{2}m^{2}n – lm^{2}n^{2} – l^{2}mn^{2}

**Factorise the following expression.**

3pqr – 6p^{2}q^{2}r^{2} – 15r^{2}

**Factorise the following expression.**

x^{3}y^{2} + x^{2}y^{3} – xy^{4} + xy

**Factorise the following expression.**

4xy^{2} – 10x^{2}y + 16x^{2}y^{2} + 2xy

**Factorise the following expression.**

2a^{3 }– 3a^{2}b + 5ab^{2} – ab

**Factorise the following expression.**

63p^{2}q^{2}r^{2}s – 9pq^{2}r^{2}s^{2} + 15p^{2}qr^{2}s^{2} – 60p^{2}q^{2}rs^{2}

**Factorise the following expression.**

24x^{2}yz^{3} – 6xy^{3}z^{2} + 15x^{2}y^{2}z – 5xyz

**Factorise the following expression.**

a^{3} + a^{2} + a + 1

**Factorise the following expression.**

lx + my + mx + ly

**Factorise the following expression.**

a^{3}x – x^{4} + a^{2}x^{2} – ax^{3 }

**Factorise the following expression.**

2x^{2} – 2y + 4xy – x

**Factorise the following expression.**

y^{2} + 8zx – 2xy – 4yz

**Factorise the following expression.**

ax^{2}y – bxyz – ax^{2}z + bxy^{2}^{ }

**Factorise the following expression.**

a^{2}b + a^{2}c + ab + ac + b^{2}c + c^{2}b

**Factorise the following expression.**

2ax^{2} + 4axy + 3bx^{2} + 2ay^{2} + 6bxy + 3by^{2 }

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

x^{2} + 6x + 9

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

x^{2} + 12x + 36

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

x^{2} + 14x + 49

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

x^{2} + 2x + 1

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

4x^{2} + 4x + 1

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

a^{2}x^{2} + 2ax + 1

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

a^{2}x^{2} + 2abx + b^{2}

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

a^{2}x^{2} + 2abxy + b^{2}y^{2 }

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

4x^{2} + 12x + 9

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

16x^{2} + 40x + 25

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

9x^{2} + 24x + 16

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

9x^{2} + 30x + 25

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

2x^{3} + 24x^{2} + 72x

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

a^{2}x^{3} + 2abx^{2} + b^{2}x

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

4x^{4} + 12x^{3} + 9x^{2}

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

`x^2/4 + 2x + 4`

**Factorise the following, using the identity a ^{2} + 2ab + b^{2} = (a + b)^{2} **

`9x^4 + 12x^3 + y^2/9`

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

x^{2} – 8x + 16

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

x^{2} – 10x + 25

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

y^{2} – 14y + 49

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

p^{2} – 2p + 1

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

4a^{2} – 4ab + b^{2}

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

p^{2}y^{2} – 2py + 1

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

a^{2}y^{2} – 2aby + b^{2}

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

9x^{2} – 12x + 4

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

4y^{2} – 12y + 9

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

`x^2/4 - 2x + 4`

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

a^{2}y^{3} – 2aby^{2} + b^{2}y

**Factorise the following, using the identity a ^{2} – 2ab + b^{2} = (a – b)^{2}.**

`9y^2 - 4xy + (4x^2)/9`

**Factorise the following.**

x^{2} + 15x + 26

**Factorise the following.**

x^{2} + 9x + 20

**Factorise the following.**

y^{2} + 18x + 65

**Factorise the following.**

p^{2} + 14p + 13

**Factorise the following.**

y^{2} + 4y – 21

**Factorise the following.**

y^{2} – 2y – 15

**Factorise the following.**

18 + 11x + x^{2}

**Factorise the following.**

x^{2} – 10x + 21

**Factorise the following.**

x^{2} – 17x + 60

**Factorise the following.**

x^{2} + 4x – 77

**Factorise the following.**

y^{2} + 7y + 12

**Factorise the following.**

p^{2} – 13p – 30

**Factorise the following.**

a^{2} – 16p – 80

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

x^{2} – 9

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

4x^{2} – 25y^{2}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

4x^{2} – 49y^{2}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

3a^{2}b^{3} – 27a^{4}b

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

28ay^{2} – 175ax^{2}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

9x^{2} – 1

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

25ax^{2} – 25a

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

`x^2/9 - y^2/25`

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

`(2p^2)/25 - 32q^2`

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

49x^{2} – 36y^{2}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

`y^3 - y/9`

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

`x^2/25 - 625`

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

`x^2/8 - y^2/18`

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

`(4x^2)/9 - (9y^2)/16`

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

`(x^3y)/9 - (xy^3)/16`

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

1331x^{3}y – 11y^{3}x

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

`1/36a^2b^2 - 16/49b^2c^2`

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

a^{4} – (a – b)^{4}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

x^{4} – 1

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

y^{4} – 625

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

p^{5} – 16p

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

16x^{4} – 81

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

x^{4} – y^{4}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

y^{4} – 81

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

16x^{4} – 625y^{4}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

(a – b)^{2} – (b – c)^{2}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

(x + y)^{4} – (x – y)^{4}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

x^{4} – y^{4} + x^{2} – y^{2}

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

8a^{3} – 2a

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

`x^2 - y^2/100`

**Factorise the following using the identity a ^{2} – b^{2} = (a + b)(a – b).**

9x^{2} – (3y + z)^{2}

**The following expression are the area of rectangles. Find the possible length and breadth of these rectangles.**

x^{2} – 6x + 8

**The following expression are the area of rectangles. Find the possible length and breadth of these rectangles.**

x^{2} – 3x + 2

**The following expression are the area of rectangles. Find the possible length and breadth of these rectangles.**

x^{2} – 7x + 10

x^{2} + 19x – 20

x^{2} + 9x + 20

**Carry out the following divisions:**

51x^{3}y^{2}z ÷ 17xyz

**Carry out the following divisions:**

76x^{3}yz^{3} ÷ 19x^{2}y^{2}

**Carry out the following divisions:**

17ab^{2}c^{3} ÷ (– abc^{2})

**Carry out the following divisions:**

–121p^{3}q^{3}r^{3} ÷ (–11xy^{2}z^{3})

**Perform the following divisions:**

(3pqr – 6p^{2}q^{2}r^{2}) ÷ 3pq

**Perform the following divisions:**

(ax^{3} – bx^{2} + cx) ÷ (– dx)

**Perform the following divisions:**

(x^{3}y^{3} + x^{2}y^{3} – xy^{4} + xy) ÷ xy

**Perform the following divisions:**

(– qrxy + pryz – rxyz) ÷ (– xyz)

**Factorise the expressions and divide them as directed:**

(x^{2} – 22x + 117) ÷ (x – 13)

**Factorise the expressions and divide them as directed:**

(x^{3} + x^{2} – 132x) ÷ x(x – 11)

**Factorise the expressions and divide them as directed:**

(2x^{3} – 12x^{2} + 16x) ÷ (x – 2)(x – 4)

**Factorise the expressions and divide them as directed:**

(9x^{2} – 4) ÷ (3x + 2)

**Factorise the expressions and divide them as directed:**

(3x^{2} – 48) ÷ (x – 4)

**Factorise the expressions and divide them as directed:**

(x^{4} – 16) ÷ x^{3} + 2x^{2} + 4x + 8

**Factorise the expressions and divide them as directed:**

(3x^{4} – 1875) ÷ (3x^{2} – 7)

The area of a square is given by 4x^{2} + 12xy + 9y^{2}. Find the side of the square.

The area of a square is 9x^{2} + 24xy + 16y^{2}. Find the side of the square.

The area of a rectangle is x^{2} + 7x + 12. If its breadth is (x + 3), then find its length.

The curved surface area of a cylinder is 2π(y^{2} – 7y + 12) and its radius is (y – 3). Find the height of the cylinder (C.S.A. of cylinder = 2πrh).

The area of a circle is given by the expression πx^{2} + 6πx + 9π. Find the radius of the circle.

The sum of first n natural numbers is given by the expression `n^2/2 + n/2`. Factorise this expression.

The sum of (x + 5) observations is x^{4} – 625. Find the mean of the observations.

The height of a triangle is x^{4} + y^{4} and its base is 14xy. Find the area of the triangle.

The cost of a chocolate is Rs (x + y) and Rohit bought (x + y) chocolates. Find the total amount paid by him in terms of x. If x = 10, find the amount paid by him.

The base of a parallelogram is (2x + 3 units) and the corresponding height is (2x – 3 units). Find the area of the parallelogram in terms of x. What will be the area of parallelogram of x = 30 units?

The radius of a circle is 7ab – 7bc – 14ac. Find the circumference of the circle. `(pi = 22/7)`

If p + q = 12 and pq = 22, then find p^{2} + q^{2}.

If a + b = 25 and a^{2} + b^{2} = 225, then find ab.

If x – y = 13 and xy = 28, then find x^{2} + y^{2}.

If m – n = 16 and m^{2} + n^{2} = 400, then find mn.

If a^{2} + b^{2} = 74 and ab = 35, then find a + b.

**Verify the following:**

(ab + bc)(ab – bc) + (bc + ca)(bc – ca) + (ca + ab)(ca – ab) = 0

**Verify the following:**

(a + b + c) (a^{2} + b^{2} + c^{2} – ab – bc – ca) = a^{3} + b^{3}+ c^{3} – 3abc

**Verify the following:**

(p – q)(p^{2} + pq + q^{2}) = p^{3} – q^{3}

**Verify the following:**

(m + n) (m^{2} – mn + n^{2}) = m^{3} + n^{3}

**Verify the following:**

(a + b)(a + b)(a + b) = a^{3} + 3a^{2}b + 3ab^{2} + b^{3}

**Verify the following:**

(a – b)(a – b)(a – b) = a^{3} – 3a^{2}b + 3ab^{2} – b^{3}

**Verify the following:**

(a^{2} – b^{2})(a^{2} + b^{2}) + (b^{2} – c^{2})(b^{2} + c^{2}) + (c^{2} – a^{2}) + (c^{2} + a^{2}) = 0

**Verify the following:**

(5x + 8)^{2} – 160x = (5x – 8)^{2}

**Verify the following:**

(7p – 13q)^{2} + 364pq = (7p + 13q)^{2}

**Verify the following:**

`((3p)/7 + 7/(6p))^2 - (3/7p + 7/(6p))^2` = 2

Find the value of a, if 8a = 35^{2} – 27^{2}

Find the value of a, if 9a = (76)^{2} – (67)^{2}

Find the value of a, if pqa = (3p + q)^{2} – (3p – q)^{2}

Find the value of a, if pq^{2}a = (4pq + 3q)^{2} – (4pq – 3q)^{2 }

What should be added to 4c(– a + b + c) to obtain 3a(a + b + c) – 2b(a – b + c)?

Subtract b(b^{2} + b – 7) + 5 from 3b^{2} – 8 and find the value of expression obtained for b = – 3.

If `x - 1/x = 7` then find the value of `x^2 + 1/x^2`.

Factorise `x^2 + 1/x^2 + 2 - 3x - 3/x`

Factorise p^{4} + q^{4} + p^{2}q^{2}.

Find the value of `(6.25 xx 6.25 - 1.75 xx 1.75)/(4.5)`

Find the value of `(198 xx 198 - 102 xx 102)/96`

The product of two expressions is x^{5} + x^{3} + x. If one of them is x^{2} + x + 1, find the other.

Find the length of the side of the given square if area of the square is 625 square units and then find the value of x.

Take suitable number of cards given in the adjoining diagram [G(x × x) representing x^{2}, R(x × 1) representing x and Y(1 × 1) representing 1] to factorise the following expressions, by arranging the cards in the form of rectangles: (i) 2x^{2} + 6x + 4. Factorise 2x^{2} + 6x + 4 by using the figure.

Calculate the area of figure.

Take suitable number of cards given in the adjoining diagram [G(x × x) representing x^{2}, R(x × 1) representing x and Y(1 × 1) representing 1] to factorise the following expressions, by arranging the cards in the form of rectangles: x^{2} + 4x + 4. Factorise 2x^{2} + 6x + 4 by using the figure.

Calculate the area of figure.

The figure shows the dimensions of a wall having a window and a door of a room. Write an algebraic expression for the area of the wall to be painted.

#### Match the expressions of column I with that of column II:

Column I |
Column II |

(1) `(21x + 13y)^2` | (a) `441x^2 - 169y^2` |

(2) `(21x - 13y)^2` | (b) `441x^2 + 169y^2 + 546xy` |

(3) `(21x - 13y)(21x + 13y)` | (c) `441x^2 + 169y^2 - 546xy` |

(d) `441x^2 - 169y^2 + 546xy` |

## Chapter 7: Algebraic Expression, Identities and Factorisation

## NCERT solutions for Mathematics Exemplar Class 8 chapter 7 - Algebraic Expression, Identities and Factorisation

NCERT solutions for Mathematics Exemplar Class 8 chapter 7 (Algebraic Expression, Identities and Factorisation) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics Exemplar Class 8 solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. NCERT textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Mathematics Exemplar Class 8 chapter 7 Algebraic Expression, Identities and Factorisation are Algebraic Expressions, Terms, Factors and Coefficients of Expression, Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials, Addition of Algebraic Expressions, Multiplication of Algebraic Expressions, Multiplying Monomial by Monomials, Multiplying a Monomial by a Binomial, Like and Unlike Terms, Subtraction of Algebraic Expressions, Multiplying a Monomial by a Trinomial, Multiplying a Binomial by a Binomial, Multiplying a Binomial by a Trinomial, Concept of Identity, Expansion of (a + b)2 = a2 + 2ab + b2, Expansion of (a - b)2 = a2 - 2ab + b2, Expansion of (a + b)(a - b), Expansion of (x + a)(x + b), Factors and Multiples, Factorisation by Taking Out Common Factors, Factorising Algebraic Expressions, Factorisation by Regrouping Terms, Factorisation Using Identities, Factors of the Form (x + a)(x + b), Dividing a Monomial by a Monomial, Dividing a Polynomial by a Monomial, Dividing a Polynomial by a Polynomial, Concept of Find the Error.

Using NCERT Class 8 solutions Algebraic Expression, Identities and Factorisation exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in NCERT Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 8 prefer NCERT Textbook Solutions to score more in exam.

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