ML AGGARWAL CLASS 8 CHAPTER 10 Algebraic Expressions and Identities Exercise 10.5
Exercise 10.5
Question 1
Sol :
(i) (3 x+5)(3 x+5)
\begin{aligned}(3 x+5)^{2} &=(3 x)^{2}+2(3 x) .5+5^{2}\left(\therefore(a+b)^{2}=a^{2}+2 a b+b^{2}\right) \\ &=9 x^{2}+30 x+25 \end{aligned}
(ii) \begin{aligned}(9 y-5) &(9 y-5) \\(9 y-5)^{2} &=(9 y)^{2}-2(9 y) 5+(+5)^{2}\left(\because(a-b)^{2}=a^{2}-2 a b+b^{2}\right) \\ &=81 y^{2}-90 y+25 \end{aligned}
(iii) (4 x+11 y)(4 x-11 y)
=(4 x)^{2}-(11 y)^{2}\left(\because(a+b)(a-b)=a^{2}-b^{2}\right)
=16 x^{2}-121 y^{2}
(v) \left(\frac{2}{a}+\frac{5}{b}\right)\left(\frac{2}{a}+\frac{5}{b}\right)
\begin{aligned}\left(\frac{2}{a}+\frac{5}{b}\right)^{2} &=\left(\frac{2}{a}\right)^{2}+2 \cdot \frac{2}{a} \cdot \frac{5}{b}+\left(\frac{5}{b}\right)^{2}\left(\because \cdot(a+b)^{2}=a^{2}+2 a b+b^{2}\right) \\ &=\frac{4}{a^{2}}+\frac{20}{a b}+\frac{25}{b^{2}} \end{aligned}
(vi) \left(\frac{p^{2}}{2}+\frac{2}{q^{2}}\right)\left(\frac{p^{2}}{2}-\frac{2}{q^{2}}\right)
\left(\frac{p^{2}}{2}\right)^{2}-\left(\frac{2}{q^{2}}\right)^{2} \quad\left(\because(a+b)(a-b)=a^{2}-b^{2}\right.
\frac{p^{4}}{4}-\frac{4}{q^{4}}
Question 2
Sol :
(i) \begin{aligned} 81^{2} &=(80+1)^{2} \\ &=80^{2}+2.80 .1+1^{2}\left(\because(a+b)^{2}=a^{2}+2 a b+b^{2}\right) \\ &=6400+160+1 \\ 81^{2} &=6561 \end{aligned}
(ii)
\begin{aligned} 97^{2} &=(100-3)^{2} \\ &=(100)^{2}-2 \cdot 100 \cdot 3+3^{2}\left(\because(a-b)^{2}=a^{2}-2 a b+b^{2}\right] \\ &=10000-600+9 \\ &=9409^{} \end{aligned}
(iii)
\begin{aligned} 105^{2} &=(100+5)^{2} \\ &=100^{2}+2.100 \cdot 5+5^{2}\left(\cdots(a+b)^{2}=a^{2}+2 a b+b^{2}\right) \\ &=10000+1000+25 \\ &=11025 \end{aligned}
(iv)
\begin{aligned} 997^{2} &=(1000-3)^{2} \\ &=(1000)^{2}-2 \cdot 1000 \cdot 3+3^{2}\left(\because(a-b)^{2}=a^{2}-2 a b+b^{2}\right) \\ &=1000000-6000+9 \\ 997^{2} &=994009 \end{aligned}
(v)
\begin{aligned} 6.1^{2} &=(6+0.1)^{2} \\ &=6^{2}+2 \cdot 6 \cdot(0.1)+0.1^{2}\left(\because(a+b)^{\gamma}=a^{2}+2 a b+b^{2}\right) \\ &=36+1.2+0.01 \\ 6.1^{2} &=31.21 \end{aligned}
(vi)
\begin{aligned} 496 \times 504 &=(500-4)(500+4) \\ &=(500)^{2}-4^{2}\left(\because \cdot(a+b)(a-b)=a^{2}-b^{2}\right) \\ &=250000-16 \\ 496 \times 504 &=249984 \end{aligned}
(vii)
\begin{aligned} 20.5 \times 19.5 &=(20+0.5)(20-0.5) \\ & \left.=20^{2}-0.5^{2}(19+b)(a-b)=a^{2}-b^{2}\right) \\ &=400-0.25 \\ 20.5 \times 19.5 &=399.75 \end{aligned}
(viii)
\begin{aligned} 9.6^{2} &=(10-0.4)^{2} \\ &=10^{2}-2.10 \cdot(0.4)+(0.4)^{2}\left(\because \cdot(a-b)^{2}=a^{2}-2 a b+b^{2}\right) \\ &=100-8+0.16 \\ 9.6^{2} &=92.16 \end{aligned}
Question 3
Sol :
(i) \begin{aligned}(p q+5 r)^{2} &=(P q)^{2}+2 \cdot p q \cdot 5 r+(5 r)^{2}\left(-(a+b)^{2}=a^{2}+b^{2}+2 a b\right) \\ &=p^{2} q^{r}+10 p q r+25 2^{2} \end{aligned}
(ii)
\begin{aligned}\left(\frac{5}{2} a-\frac{3}{5} \cdot b\right)^{2} &=\left(\frac{5}{2} \cdot a\right)^{2}-2 \cdot \frac{5}{2} \cdot a \cdot \frac{3}{5} \cdot b+\left(\frac{3}{5} b\right)^{2} \\ &\left(\because(a-b)^{2}=a^{2}-2 a b+b^{2}\right) \\ &=\frac{25}{4} a^{2}-3 a b+\frac{9}{25} b^{2} \end{aligned}
(iii)
\begin{aligned}(\sqrt{2} \cdot a+\sqrt{3} \cdot b)^{2} &=(\sqrt{2} \cdot a)^{2}+2 \sqrt{2} \cdot a \cdot \sqrt{3} \cdot b+(\sqrt{3} \cdot b)^{2} \\(\therefore& \left.(a+b)^{2}=a^{2}+2 a b+b^{2}\right) \\ &=2 a^{2}+2 \sqrt{6} a b+3 b^{2} \end{aligned}
(iv)
\left(\frac{2 x}{3 y}-\frac{3 y}{2 x}\right)^{2}=\left(\frac{2 x}{3 y}\right)^{2}-2 \cdot \frac{2 x}{3 y} \cdot \frac{3 y}{2 x}+\left(\frac{3 y}{2 x}\right)^{2} \left(∴(a+b)^{2}=a^{2}+b^{2}+2 a b\right)
=\frac{4 x^{2}}{9 y^{2}}-2+\frac{9 y^{2}}{4 x^{2}}
Question 4
Sol :
(i)
\begin{aligned}(x+7)(x+3) &=x^{2}+(7+3) x+7 \times 3\left(:(x+a)(x+b)=x^{2}+(a+b) x+a b\right) \\ &=x^{2}+10 x+21 \end{aligned}
(ii) (3 x+4)(3 x-5)=(3 x)^{v}+(4+(-5))(3 x)+4 x-5
\left(\because(x+a)(x+b)=x^{2}+(a+b) x+a b\right)
=9 x^{2}-3 x-20
(iii) \left(p^{2}+2 q\right)\left(p^{2}-3 q\right)=\left(p^{2}\right)^{2}+(2 q+(-3 q)) p^{2}+2 q x-3 q
=p^{4}-^{2} q^{2}-6 q^{2}
=p^{4}-p_{q}^{2}-6 q^{2}
(iv) (a b c+3)(a b c-5)=(a b c)^{2}+(3+(-5)) \cdot a b c+3 x-5
\left(\because(x+a)(x+b)=x^{2}+(a+b) \cdot x+a b\right)
=(a b c)^{2}-2 a b c-15
Question 5
Sol :
(i) 203 \times 204=(200+3)(200+4)
=(200)^{2}+(3+4) 200+3 \times 4\left(\because(x+a)(x+b)=x^{2}+(a+b)+x+ab\right.
=40000+1400+12
=41412
(ii) 8.2 \times 8.7=(8+0.2)(8+0.7)
=8^{2}+(0.2+0.7) 8+0.2 \times 0.7
=64+7.2+0.14
=71.34
(iii) 107 \times 93=(100+7)(100-7)
=(100)^{2}+(7+(-7)) \cdot 100+7 x-7
\left(\because(x+a)(x+b)=x^{2}+(a+b) \cdot x+a b\right)=10000+0.100=49
=9951
Question 6
Sol :
(i) 53^{2}-47^{2}=(53+47)(53-47) \left(∴ a^{2}-b^{2}=(a+b)(a-b)\right)
=(100)(6)
=600
(ii) (2.05)^{2}-(0.95)^{2}=(2.05+0.95)(2.05-0.95)
=3 \times 0.1
=0.3
Question 7
Sol :
i. (2 x+5 y)^{2}+(2 x-5 y)^{2}
(2 x)^{2}+(5 y)^{2}+2 \cdot 2 x \cdot 5 y+(2 x)^{2}+(5 y)^{2}-2 \cdot 2 x \cdot 5 y
\because(a+b)^{2}=a^{2}+b^{2}+2 a b
2(2 x)^{2}+2(5 y)^{2}
2 \cdot\left[4 x^{-1}+25 y^{2}\right]
8 x^{2}+50 y^{2}
(ii) \left(\frac{7}{2} a-\frac{5}{2} b\right)^{2}-\left(\frac{5}{2} a-\frac{z}{2} b\right)^{2}
=\left(\frac{7}{2} a\right)^{2}+\left(\frac{5}{2} b\right)^{2}-2 \cdot \frac{7}{2} a \cdot \frac{5}{2} \cdot b-\left[\left(\frac{5}{2} a\right)^{n}+\left(\frac{7}{2} b\right)^{2}-2 \cdot \frac{5}{2} \cdot a \cdot \frac{7}{2} b\right]
=\frac{49}{4} a^{2}+\frac{25}{4} b^{2}-2 \cdot \frac{7}{2} \cdot \frac{5}{2} \cdot b-\frac{25}{4} a^{2}-\frac{49}{4} b^{2}+2 \cdot \frac{5}{2} a\cdot \frac{7}{2} \cdot b
=\left(\frac{49}{4}-\frac{25}{4}\right) a^{2}+\left(\frac{25}{4}-\frac{49}{4}\right) b^{2}
=\frac{24}{4} \cdot a^{2}-\frac{24}{4} \cdot b^{2}
=6\left(a^{2}-b^{2}\right)
(iii) \left(p^{2}-qr^{2} \cdot \right)^{2}+2 p^{2} q^{2} 2
\left(p^{2}\right)^{2}-2 \cdot p^{2} \cdot q^{2} \cdot 2+\left(q^{2} \cdot 2\right)^{2}+2 p^{2} q^{2} \cdot r \left(\therefore(a+b)^{2}=a^{2}+2 a b+b^{2}\right.
p^{4}-2 p^{2} q^{2} \cdot r+q^{4} \cdot r^{2}+2 p^{2} q^{2} \cdot r
p^{4}+q^{4} \cdot r^{2}
Question 8
Sol :
L H S
(i) (4 x+7 y)^{2}-(4 x-7 y)^{n}
(4 x)^{2}+(7 y)^{2}+2 \cdot 4 x \cdot 7 y-\left[(4 x)^{2}+(7 y)^{2}-2.4 x-7 y\right]
\left(\because(a-b)^{2}=a^{2}+b^{2}-2 a b\right)
(4 x)^{2}+(7 y)^{2}+2.4 x \cdot 7 y-(4 x)^{2}-(7 y)^{2}+2 \cdot 4 x \cdot 7 y
4.4 x \cdot 7 y
112 x y= R.H.S
(ii) \left(\frac{3}{7} p-\frac{7}{6} q\right)^{2}+p q
\left(\frac{3}{7} p\right)^{2}+\left(\frac{7}{6} q\right)^{2}-2 \cdot \frac{3}{7} p \cdot \frac{1}{6} \cdot q+p q
\left(\because(a-b)^{2}=a^{2}+b^{2}-2 a b\right)
\frac{9}{49} p^{2}+\frac{49}{36} q^{2}-p q+p q
\begin{aligned} \frac{9}{49} \cdot p^{2}+\frac{49}{36} q^{2} &=R \cdot H \cdot S \\ \therefore \quad L . H \cdot S &=R \cdot H \cdot S \end{aligned}
(iii) L \cdot H \cdot s=(p-q)(p+q)+(p-r)(q+r)+(r-p)(r+p)
=p^{2}-q^{2}+q^{2}-r^{2}+r^{2}-p^{2}
\left(\because+(a+b)(a-b)=a^{2}-b^{2}\right]
=0=RHS
\therefore \quad L H S=R H S
Question 9
Sol :
Given : \left(x+\frac{1}{x}\right)=2
Squaring on both sides
(i) \left(x+\frac{1}{x}\right)^{2}=2^{2}
x^{2}+2 \cdot x \cdot \frac{1}{x}+\left(\frac{1}{x}\right)^{2}=4\left(\because(a+b)^{2}=a^{2}+b^{2}+2 a b\right)
x^{2}+2+\frac{1}{x^{2}}=4
x^{2}+\frac{1}{x^{2}}=4-2
x^{2}+\frac{1}{x^{2}}=2
(ii) Again squaring on both sides
\left(x^{2}+\frac{1}{x^{2}}\right)^{2}=2^{2}
\left(x^{2}\right)^{n}+2 \cdot x^{2}-\frac{1}{x^{2}}+\left(\frac{1}{x^{2}}\right)^{2}=4
x^{4}+2+\frac{1}{x^{4}}=4
x^{4}+\frac{1}{x^{4}}=4-2
x^{4}+\frac{1}{x^{4}}=2n
Question 10
Sol :
(i) x-\frac{1}{x}=7
Squaring on both sides
\left(x-\frac{1}{x}\right)^{2}=7^{2}
x^{2}-2 \cdot x \cdot \frac{1}{x}+\left(\frac{1}{x}\right)^{2}=49
\left(\because(a-b)^{2}=a^{2}-2 a b+b^{2}\right)
x^{2}-2+\frac{1}{x^{2}}=49
x^{2}+\frac{1}{x^{2}}=49+2
x^{2}+\frac{1}{x^{2}}=51
(ii) x^{2}+\frac{1}{x^{2}}=51
squaring on both sides
\left(x^{2}+\frac{1}{x^{2}}\right)^{2}=51^{2}
\left(x^{2}\right)^{2}+2 \cdot x^{2} \cdot \frac{1}{x^{2}}+\left(\frac{1}{x^{2}}\right)^{2}=2601
x^{4}+2+\frac{1}{x^{4}}=2601
x^{4}+\frac{1}{x^{4}}=2601-2
x^{4}+\frac{1}{x^{4}}=2599
Question 11
Sol :
x^{2}+\frac{1}{x^{2}}=23
(i) x^{2}+\frac{1}{x^{2}}=23
Adding 2 on both sides
x^{2}+\frac{1}{x^{2}}+2=23+2
(x)^{2}+\left(\frac{1}{x}\right)^{2}+2 \cdot x \cdot \frac{1}{x}=25\left(\because a^{2}+b^{2}+2 a b=(a+b)^{2}\right)
\left(x+\frac{1}{x}\right)^{2}=25
x+\frac{1}{x}=5
(ii) x+\frac{1}{x^{2}}=25^{2}
Subtract '2' on both sides
x^{2}+\frac{1}{x^{2}}-2=23-2
(x)^{2}+\left(\frac{1}{x}\right)^{n}-2 \cdot x \cdot \frac{1}{x}=21
\left(x-\frac{1}{x}\right)^{2}=21\left(\because \quad a^{2}+b^{2}-2 a b=(a-b)^{2}\right)
x-\frac{1}{x}=\sqrt{21}
x-\frac{1}{x}=3 \sqrt{3}
Question 12
Sol :
given a+b=9, a b=10
Squaring on both sides
(a+b)^{2}=9^{2}
a^{2}+b^{2}+2 a b=81
a^{2}+b^{2}+2 \times 10=81 \quad(\because given a b=10)
a^{2}+b^{2}+20=81
a^{2}+b^{2}=61
Question 13
Sol :
given a-b=6, a^{2}+b^{2}=42
a-b=6
Squaring on both Sides
(a-b)^{2}=6^{2}
a^{2}+b^{2}-2 a b=36
42-2 a b=36\left(\because a^{2}+b^{2}=42\right)
42-36=2 a b
2 a b=6
a b=3
Question 14
Sol :
given a^{2}+b^{2}=41, a b=4
(i) Consider
\begin{aligned}(a+b)^{2} &=a^{2}+b^{2}+2 a b \\(a+b)^{2} &=41+2 \times u=41+8 \\(a+b)^{2} &=49 \\ a+b &=7 \end{aligned}
(ii) Consider
\begin{aligned}(a-b)^{2} &=a^{2}+b^{2}-2 a b \\ &=41-2 \times 4=41-8=33 . \\(a-b)^{2} &=33 \\ a-b &=\sqrt{33} \end{aligned}
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