## ML Aggarwal Class 8 Solutions for ICSE Maths Chapter 10 Algebraic Expressions and Identities Ex 10.2

Question 1.

Find the product of:

(i) 4x^{3} and -3xy

(ii) 2xyz and 0

(iii) –\(\frac{2}{3}\)p^{2}q, \(\frac{3}{4}\)pq^{2} and 5pqr

(iv) -7ab,-3a^{3} and –\(\frac{2}{7}\)ab^{2}

(v) –\(\frac{1}{2}\)x^{2} – \(\frac{3}{5}\)xy, \(\frac{2}{3}\)yz and \(\frac{5}{7}\)xyz

Solution:

Product of

(i) 4x^{3} and -3xy = 4x^{3} × (-3xy) = -12x^{3+1} y = -12x^{4}y

(ii) 2xyz and 0 = 2xyz × 0 = 0

Question 2.

Multiply:

(i) (3x – 5y + 7z) by – 3xyz

(ii) (2p^{2} – 3pq + 5q^{2} + 5) by – 2pq

(iii) (\(\frac{2}{3}\)a^{2}b – \(\frac{4}{5}\)ab^{2} + \(\frac{2}{7}\)ab + 3) by 35ab

(iv) (4x^{2} – 10xy + 7y^{2} – 8x + 4y + 3) by 3xy

Solution:

(i) – 3xyz (3x – 5y + 7z)

= (- 3xyz) × 3x + (- 3xyz) × (- 5y) + (- 3xyz) × (7z)

= – 9x^{2}yz + 15xyz^{2} – 21xyz^{2}

(ii) -2pq (2p^{2} – 3pq + 5q^{2} + 5)

= (-2pq) × 2p^{2} + (-2pq) × (-3pq) + (- 2pq) × (5q^{2}) + (-2pq) × 5

= -4p^{3}q + 6p^{2}q^{2} – 10pq^{3} – 10pq

(iii) \(\left(\frac{2}{3} a^{2} b-\frac{4}{5} a b^{2}+\frac{2}{7} a b+3\right)\) by 35ab

= \(\frac{2}{3}\)a^{2}b × 35ab – \(\frac{4}{5}\)ab^{2} × 35ab = \(\frac{2}{7}\)ab × 35ab + 3 × 35ab

= \(\frac{70}{3}\)a^{3}b^{2} – 28a^{2}b^{3} + 10a^{2}b^{2} + 105ab

(iv) (4x^{2} – 10xy + 7y^{2} – 8x + 4y + 3) by 3xy

4x^{2} × 3xy – 10xy × 3xy + 7y^{2} × 3xy – 8x × 3xy + 4y × 3xy + 3 × 3xy

= 12x^{3}y – 30x^{2}y^{2} + 21xy^{3} – 24x^{2}y + 12xy^{2} + 9xy

Question 3.

Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively:

(i) (p^{2}q, pq^{2})

(ii) (5xy, 7xy^{2})

Solution:

(i) Sides of a rectangle are p^{2}q and pq^{2}

Area = p^{2}q × pq^{2} = p^{2+1}q^{2+1} = p^{3}q^{3}

(ii) Sides are 5xy and 7xy^{2}

Area = 5xy × 7xy^{2} = 35x^{1+1} × y^{1+2 }= 35x^{2}y^{3}

Question 4.

Find the volume of rectangular boxes with the following length, breadth and height respectively:

(i) 5ab, 3a^{2}b, 7a^{4}b^{2}

(ii) 2pq, 4q^{2}, 8rp

Solution:

Length, breadth and height of a rectangular box are

(i) 5ab, 3a^{2}b, 7a^{4}b^{2}

∴ Volume = Length × breadth × height

= 5ab × 3a^{2}b × 7a^{4}b^{2}

= 5 × 3 × 7 × a^{1+2+4} × b^{1+1+2}

= 105a^{7}b^{4}

(ii) 2pq, 4q^{2}, 8rp

∴ Volume = 2pq × 4q^{2} × 8rp

= 2 × 4 × 8 × p^{1+1} × q^{1+2} × r

= 64p^{2}q^{3}r

Question 5.

Simplify the following expressions and evaluate them as directed:

(i) x^{2}(3 – 2x + x^{2}) for x = 1; x = -1; x = \(\frac{2}{3}\) and x = –\(\frac{1}{2}\)

(ii) 5xy(3x + 4y – 7) – 3y(xy – x^{2} + 9) – 8 for x = 2, y = -1

Solution:

(i) x^{2}(3 – 2x + x^{2})

for x = 1; x = -1; x = – 1 x = \(\frac{2}{3}\) and x = –\(\frac{1}{2}\)

x^{2}(3 – 2x + x^{2}) = 3x^{2} – 2x^{3} + x^{4}

(a) x = 1, then

3x^{2} – 2x^{3} + x^{4} = 3(1)^{2} – 2(1 )^{3} + (1)^{4}

= 3 × 1 – 2 × 1 + l

=3 – 2 + 1 = 2

(b) x = -1

3x^{2} – 2x^{3} + x^{4} = 3(-1)^{2} – 2(-1)^{3} + (-1)^{4}

= 3 × 1 – 2 × (-1) + 1 = 3 + 2 + 1 = 6

(c) x = \(\frac{2}{3}\)

(ii) 5xy(3x + 4y – 7) – 3y(xy – x^{2} + 9) – 8

= 15x^{2}y + 20xy^{2} – 35xy – 3xy^{2} + 3 x^{2}y – 21y – 8

= 18x^{2}y + 17xy^{2} – 35xy – 27y – 8

When x = 2, y = -1

= 18(2)^{2} × (-1) + 17(2) (-1)^{2} – 35(2) (-1) – 27(-1) – 8

= 18 × 4 × (-1) + 17 × 2 × 1 – 35 × 2 × (-1) – 27 × (-1) – 8

= -74 + 34 + 70 + 27 – 8

= 131 – 80 = 51

Question 6.

Add the following:

(i) 4p(2 – p^{2}) and 8p^{3} – 3p

(ii) 7xy(8x + 2y – 3) and 4xy^{2}(3y – 7x + 8)

Solution:

Add

(i) 4p(2 – p^{2}) and 8p^{3} – 3p

= 8p – 4p^{3} + 8p^{3} – 3p

= 5p + 4p^{3}

= 4p^{3} + 5p

(ii) 7xy(8x + 2y – 3) and 4xy^{2}(3y – 7x + 8)

= 56x^{2}y + 14xy^{2} – 21xy + 12xy^{3} – 28x^{2}y^{2} + 32xy^{2}

= 12xy^{3} – 28x^{2}y^{2} + 56x^{2}y +46xy^{2} = 21xy

Question 7.

Subtract:

(i) 6x(x – y + z)- 3y(x + y – z) from 2z(-x + y + z)

(ii) 7xy(x^{2} -2xy + 3y^{2}) – 8x(x^{2}y – 4xy + 7xy^{2}) from 3y(4x^{2}y – 5xy + 8xy^{2})

Solution:

(i) 6x(x – y + z) – 3y(x + y – z) from 2z(-x + y + z)

6x^{2} – 6xy + 6xz – 3xy – 3y^{2} + 3yz from – 2xz + 2yz + 2z^{2}

= (-2xz + 2yz + 2z^{2}) – (6x^{2} – 6xy + 6xz – 3xy – 3y^{2} + 3yz)

= – 2xz + 2yz + 2z^{2} – 6x^{2} + 6xy – 6xz + 3xy + 3y^{2} – 3yz

= 9xy – yz – 8zx – 6x^{2} + 3y^{2} + 2z^{2}

(ii) 7xy(x^{2} – 2xy + 3y^{2}) – 8x(x^{2}y – 4xy + 7xy^{2}) from 3y(4x^{2}y – 5xy + 8xy^{2})

7x^{3}y – 14x^{2}y^{2} + 21xy^{3} – 8x^{3}y + 32x^{2}y – 56x^{2}y^{2} from 12x^{2}y^{2} – 15xy^{2} + 24xy^{3}

= (12x^{2}y^{2} – 15xy^{2} + 24xy^{3}) – (7x^{3}y – 14x^{2}y^{2} + 21xy^{3} – 8x^{3}y + 32x^{2}y – 56x^{2}y^{2}

= 12x^{2}y^{2} – 15xy^{2} + 24xy^{3} – 7x^{3}y + 14x^{2}y^{2} – 12xy^{3} + 8x^{3}y – 32x^{2}y + 56x^{2}y^{2}

= 82x^{2}y^{2} + 3xy^{3} + x^{3}y – 15xy^{2} – 32x^{2}y