ML AGGARWAL CLASS 8 CHAPTER 10 Algebraic Expressions and Identities Exercise 10.2

 Exercise 10.2

Question 1

Sol :

(i) $4 x^{3}$ and $-3 x y$

=$4 x^{3} x-3 x y$

=$(4 x-3) \times\left(x^{3} \times x y\right)$

=$-12 x^{4} y$


(ii)  2xyz and 0

=$(2 x y z) \times 0$

=0


(iii) $-\frac{2}{3} P^{2} q \cdot \frac{3}{4} p q^{2}$ and $5 p q r$

=$\left(\frac{-2}{3} p^{2} q\right) \times\left(\frac{3}{4} p q^{2}\right)(5 p q r)$

=$\frac{-5}{2} p^{4} q^{4} r$



(iv) $-7 a b,-3 a^{3}$ and $\frac{-2}{7} a b^{2}$

$(-7 a b) \times\left(-3 a^{3}\right) \times\left(\frac{-2}{7} a b^{2}\right)$

$\left(-7 x-3 x \times \frac{-2}{7}\right) \times a b \times a^{3} \times a b^{2}$

$-6 a^{5} b^{3}$



(v) $\frac{-1}{2} x^{2},-\frac{3}{5} x y, \frac{2}{3} y z \operatorname{and} \frac{5}{7} x y z$

$\left(-\frac{1}{2} x^{2}\right) \times\left(\frac{-3}{5} x y\right) \times\left(\frac{2}{3} y z\right) \times\left(\frac{5}{7} x y z\right)$

$\left(\frac{-1}{2} \times-\frac{3}{5} \times \frac{2}{3} \times \frac{5}{7}\right) \times x^{2} \times x y \times y z \times x y z$

$\frac{1}{7} x^{4} y^{3} z^{2}$


Question 2

Sol :

(i) $(3 x-5 y+7 z) \times(-3 x y z)$

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

=$-9 x^{2} y z+15 x y^{2} z-21 x y z^{2}$



(ii) $\left(2 p^{2}-3 p q+5 q^{2}+5\right) \times(-2 p q)$

=$\left(2 p^{2} x-2 p q\right)+(-3 p q x-2 p q)+\left(5 q^{2} x-2 p q\right)+$$(5 x-2 p q)$

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



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

$\left(\frac{2}{3} a^{2} b \times 35 a b\right)+\left(\frac{-4}{5} a b^{2} \times 35 a b\right)+\left(\frac{2}{7} a b \times 35 a b\right)+(3 \times 35 a b)$

$\frac{70}{3} a^{3} b-28 a^{2} b^{3}+10 a^{2} b^{2}+105 a b$



(iv) $\left(4 x^{2}-10 x y+7 y^{2}-8 x+4 y+3\right) \times(3 x y)$

$\left(4 x^{2} \times 3 x y\right)+(-10 x y \times 3 x y)+\left(7 y^{2} \times 3 x y\right)+(-8 x \times 3 x y)+(4 y \times 3 x y)+(3 \times 3 x y)$

$12 x^{3} y-30 x^{2} y^{2}+21 x y^{3}-24 x^{2} y+12 x y^{2}+9 x y$

Question 3

Sol :
(i) Given length (l) = $P^{2} q$

Breadth $(b)=p q^{2}$

$\begin{aligned} \text { Rectangle - Area } &=l \times b \\ &=\left(p^{2} q\right) \times\left(p q^{2}\right) \end{aligned}$

Area $=p^{3} q^{3}$


(ii) Given  length (l) = 5xy 

breadth (b) =$7 x y^{2}$

Rectangle area = L $\times $ b 

$\begin{aligned} &=(5 x y) \times\left(7 x y^{2}\right) \\ \text { Area } &=35 x^{2} y^{3} \end{aligned}$

Question 4

Sol :
(i) Given  length (l) = 5ab 

breadth(b) = $3 a^{2} b$

height (h) = $7 a^{4} b^{2}$

Volume of rectangular box $=\operatorname{lx} b \times h$

$=5 a b \times 3 a^{2} b \times 7 a^{4} b^{2}$

$=(3 \times 5 \times 7) \times a b \times a^{2} b \times a^{4} b^{2}$

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



(ii) Given  length (l) = 2pq
breadth (b) = $4 q^{2}$
height $(h)=8 r p$
Volume of rectangular box $=\operatorname{lxb} \times h$

$=(2 p q) \times\left(4 q^{2}\right) \times(8 rp)$

$=(2 \times 4 \times 8) P q \times q^{2}+r p$

volume of rectangular box = $64 p^{2} q^{3}r =64 p^{2} q^{3}r $

Question 5

Sol :
(i) $x^{2}\left(3-2 x+x^{2}\right)$

=$3 x^{2}-2 x^{3}+x^{4}$

For x = 1 

=$3 \times 1^{2}-2 \times 1^{3}+14$
=$3 \times 1-2 \times 1+1$
=3-2+1
=4-2

2 for x=1

=$3 x^{2}-2 x^{3}+x^{4}$
=$3 x(-1)^{2}+(-2)(-1)^{3}+(-1)^{4}$
=$3 \times 1+(-2 x-1)+1$
=3+2+1
{6} for x=-1

For $x=\frac{2}{3}$
$3 x^{2}-2 x^{3}+x^{4}$
$3 x\left(\frac{2}{3}\right)^{2}-2\left(\frac{2}{3}\right)^{3}+\left(\frac{2}{3}\right)^{4}$
$3 \times \frac{4}{9}-2 \times \frac{8}{27}+\frac{16}{81}$
$\frac{12}{9}-\frac{16}{27}+\frac{16}{81}$
$\frac{108-48+16}{81}$
$\frac{76}{81}$ for $x=\frac{2}{3}$
$\frac{76}{81}$ for $x=\frac{2}{3}$

For $x=\frac{-1}{2}$
$3 x^{2}-2 x^{3}+x^{4}$
$3 x\left(\frac{-1}{2}\right)^{2}-2 \times\left(-\frac{1}{2}\right)^{3}+\left(-\frac{1}{2}\right)^{4}$
$3 \times \frac{1}{4}-2 \times\left(-\frac{1}{8}\right)+\frac{1}{16}$
$\frac{3}{4}+\frac{2}{8}+\frac{1}{16}$
$\frac{12+4+1}{16}$
$\frac{17}{16}$ for $x=\frac{-1}{2}$


(ii) $5 x y(3 x+4 y-7)-3 y\left(x y-x^{2}+9\right)-8$
$(5 x y \times 3 x)+(5 x y \times 4 y)-5 x y \times 7+(-3 y \times x y)+\left(-3 y x-x^{2}\right)+(-3 y \times 9)-8$
$15 x^{2} y+20 x y^{2}-35 x y-3 x y^{2}+3 x^{2} y-27 y-8$
$18 x^{2} y+17 x y^{2}-62 x y-8$


For x=2,  y=-1
=$18\left(2^{2}\right)(-1)+\left(17 \times 2 \times(-1)^{2}\right)-(62 \times 2 \times-1)-8$
=-72+34+124-8
=78

Question 6

Sol :
(i) First expression $=4 p\left(2-p^{2}\right)=8 p-4 p^{3}$

Second expression $=8 p^{3}-3 p$

$\quad$ Required $\quad s u m=\left(8 p-4 p^{3}\right) p+\left(8 p^{3}-3 p\right)$

$=4 p^{3}+5 P$


(ii) First expression = $7 x y(8 x+2 y-3)=56 x^{2} y+14 x y^{2}-21 x y $
second expression $=3 y\left(4 x^{2} y-5 x y+8 x y^{2}\right)$
$=12 x^{2} y^{2}-15 x y^{2}+24 x y 3$

Required sum $=\left(56 x^{2} y+14 x y^{2}-21 x y\right)+\left(12 x^{2} y^{2}-15 x y^{2}+\right.$
$\left.24 x y^{3}\right)$

(iii) First expression $=7 x y(8 x+2 y-3)=56 x^{2} y+14 x y^{2}-21 x y$
Second expression $=4 x y^{2}(3 y-7 x+8)=12 x y^{3}-28 x^{2} y^{2}+32 x y^{2}$
Required sum = $\left(56 x^{2} y+14 x y^{2}-21 x y\right)+\left(12 x y^{3}-28 x^{2} y^{2}+\right.$
$\left.32 x y^{+}\right)$
$=12 x y^{3}-28 x^{2} y^{2}+56 x^{2} y+46 x y^{2}-21 x y$

Question 7

Sol :
(i) First expression = 6 x(x-y+z)-3 y(x+y-z)
$=6 x^{2}-6 x y+6 x z-3 x y-3 y^{2}+3 y z$
$6 x^{2}+6 x z-6 x y+3 y z-3 y^{2} \rightarrow 1$


Second expression =2 z(-x+y+z)
$=-2 x z-2 y z+2 z^{2}-x(2) \rightarrow$

②-①
(-2xz - 2yz + 2$z^{2}$ $-6 x^{2}-6 x z+6 x y-3 y z$ $+3 y 2$) 

$=-6 x^{2}+3 y^{2}+2 z^{2}+6 x y-5 y z-8 x z$


(ii) First expression = $7 x y\left(x^{2}-2 x y+3 y^{2}\right)-8 x\left(x^{2} y-4 x y+\right.$ $7 x y^{2}$

$=7 x^{3} y-14 x^{2} y^{2}+21 x y^{3}-8 x^{3} y+32 x^{2} y$ $-56 x^{2} y^{2}$

First expression = $-x^{3} y$ $+21 x y 3$ $-70 x^{2} y^{2}$ $+32 x^{2} y$------①


Second expression = $3 y\left(4 x^{2} y-5 x y+8 x y^{2}\right)$

$=12 x^{2} y^{2}-15 x y^{2}+24 x y^{3}---(2)$


②-①

$=12 x^{2} y^{2}-15 x y^{2}+24 x y^{3}-\left(-x^{3} y+21 x y^{3}-70 x^{2} y^{2}\right.$ $+32 x^{2} y$

=$12 x^{2} y^{2}-15 x y^{2}+24 x y^{3}+x^{3} y-21 x y^{3}+70 x^{2} y^{2}$ $-32 x^{2} y$

=$82 x^{2} y^{2}+x^{3} y+3 x y^{3}-15 x y^{2}-32 x^{2} y$

$=x^{3} y+3 x y^{2}+82 x^{2} y^{2}-15 x y^{2}-32 x^{2} y$

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