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

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

Question 1.
Find the product of:
(i) 4x³ and -3xy
(ii) 2xyz and 0
(iii) –\(\frac{2}{3}\)p²q, \(\frac{3}{4}\)pq² and 5pqr
(iv) -7ab,-3a³ and –\(\frac{2}{7}\)ab²
(v) –\(\frac{1}{2}\)x² – \(\frac{3}{5}\)xy, \(\frac{2}{3}\)yz and \(\frac{5}{7}\)xyz
Solution:
Product of
(i) 4x³ and -3xy = 4x³ × (-3xy) = -12x³⁺¹ y = -12x⁴y
(ii) 2xyz and 0 = 2xyz × 0 = 0

Question 2.
Multiply:
(i) (3x – 5y + 7z) by – 3xyz
(ii) (2p² – 3pq + 5q² + 5) by – 2pq
(iii) (\(\frac{2}{3}\)a²b – \(\frac{4}{5}\)ab² + \(\frac{2}{7}\)ab + 3) by 35ab
(iv) (4x² – 10xy + 7y² – 8x + 4y + 3) by 3xy
Solution:
(i) – 3xyz (3x – 5y + 7z)
= (- 3xyz) × 3x + (- 3xyz) × (- 5y) + (- 3xyz) × (7z)
= – 9x²yz + 15xyz² – 21xyz²

(ii) -2pq (2p² – 3pq + 5q² + 5)
= (-2pq) × 2p² + (-2pq) × (-3pq) + (- 2pq) × (5q²) + (-2pq) × 5
= -4p³q + 6p²q² – 10pq³ – 10pq

(iii) \(\left(\frac{2}{3} a^{2} b-\frac{4}{5} a b^{2}+\frac{2}{7} a b+3\right)\) by 35ab
= \(\frac{2}{3}\)a²b × 35ab – \(\frac{4}{5}\)ab² × 35ab = \(\frac{2}{7}\)ab × 35ab + 3 × 35ab
= \(\frac{70}{3}\)a³b² – 28a²b³ + 10a²b² + 105ab

(iv) (4x² – 10xy + 7y² – 8x + 4y + 3) by 3xy
4x² × 3xy – 10xy × 3xy + 7y² × 3xy – 8x × 3xy + 4y × 3xy + 3 × 3xy
= 12x³y – 30x²y² + 21xy³ – 24x²y + 12xy² + 9xy

Question 3.
Find the areas of rectangles with the following pairs of monomials as their lengths and breadths respectively:
(i) (p²q, pq²)
(ii) (5xy, 7xy²)
Solution:
(i) Sides of a rectangle are p²q and pq²
Area = p²q × pq² = p²⁺¹q²⁺¹ = p³q³

(ii) Sides are 5xy and 7xy²
Area = 5xy × 7xy² = 35x¹⁺¹ × y¹⁺² = 35x²y³

Question 4.
Find the volume of rectangular boxes with the following length, breadth and height respectively:
(i) 5ab, 3a²b, 7a⁴b²
(ii) 2pq, 4q², 8rp
Solution:
Length, breadth and height of a rectangular box are
(i) 5ab, 3a²b, 7a⁴b²
∴ Volume = Length × breadth × height
= 5ab × 3a²b × 7a⁴b²
= 5 × 3 × 7 × a¹⁺²⁺⁴ × b¹⁺¹⁺²
= 105a⁷b⁴

(ii) 2pq, 4q², 8rp
∴ Volume = 2pq × 4q² × 8rp
= 2 × 4 × 8 × p¹⁺¹ × q¹⁺² × r
= 64p²q³r

Question 5.
Simplify the following expressions and evaluate them as directed:
(i) x²(3 – 2x + x²) for x = 1; x = -1; x = \(\frac{2}{3}\) and x = –\(\frac{1}{2}\)
(ii) 5xy(3x + 4y – 7) – 3y(xy – x² + 9) – 8 for x = 2, y = -1
Solution:
(i) x²(3 – 2x + x²)
for x = 1; x = -1; x = – 1 x = \(\frac{2}{3}\) and x = –\(\frac{1}{2}\)
x²(3 – 2x + x²) = 3x² – 2x³ + x⁴
(a) x = 1, then
3x² – 2x³ + x⁴ = 3(1)² – 2(1 )³ + (1)⁴
= 3 × 1 – 2 × 1 + l
=3 – 2 + 1 = 2

(b) x = -1
3x² – 2x³ + x⁴ = 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 – x² + 9) – 8
= 15x²y + 20xy² – 35xy – 3xy² + 3 x²y – 21y – 8
= 18x²y + 17xy² – 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 – p²) and 8p³ – 3p
(ii) 7xy(8x + 2y – 3) and 4xy²(3y – 7x + 8)
Solution:
Add
(i) 4p(2 – p²) and 8p³ – 3p
= 8p – 4p³ + 8p³ – 3p
= 5p + 4p³
= 4p³ + 5p

(ii) 7xy(8x + 2y – 3) and 4xy²(3y – 7x + 8)
= 56x²y + 14xy² – 21xy + 12xy³ – 28x²y² + 32xy²
= 12xy³ – 28x²y² + 56x²y +46xy² = 21xy

Question 7.
Subtract:
(i) 6x(x – y + z)- 3y(x + y – z) from 2z(-x + y + z)
(ii) 7xy(x² -2xy + 3y²) – 8x(x²y – 4xy + 7xy²) from 3y(4x²y – 5xy + 8xy²)
Solution:
(i) 6x(x – y + z) – 3y(x + y – z) from 2z(-x + y + z)
6x² – 6xy + 6xz – 3xy – 3y² + 3yz from – 2xz + 2yz + 2z²
= (-2xz + 2yz + 2z²) – (6x² – 6xy + 6xz – 3xy – 3y² + 3yz)
= – 2xz + 2yz + 2z² – 6x² + 6xy – 6xz + 3xy + 3y² – 3yz
= 9xy – yz – 8zx – 6x² + 3y² + 2z²

(ii) 7xy(x² – 2xy + 3y²) – 8x(x²y – 4xy + 7xy²) from 3y(4x²y – 5xy + 8xy²)
7x³y – 14x²y² + 21xy³ – 8x³y + 32x²y – 56x²y² from 12x²y² – 15xy² + 24xy³
= (12x²y² – 15xy² + 24xy³) – (7x³y – 14x²y² + 21xy³ – 8x³y + 32x²y – 56x²y²
= 12x²y² – 15xy² + 24xy³ – 7x³y + 14x²y² – 12xy³ + 8x³y – 32x²y + 56x²y²
= 82x²y² + 3xy³ + x³y – 15xy² – 32x²y

ML Aggarwal Class 8 Solutions for ICSE Maths