 # Linear Equations in Two Variables

### Question:

if $x=\frac{1}{2-\sqrt{3}}$, find the value of $x3-2x2-7x+5$

We have,

$x=\frac{1}{2-\sqrt{3}}$ $=\frac{1}{2-\sqrt{3}}$ $×\frac{2+\sqrt{3}}{2+\sqrt{3}}$ $=\frac{2+\sqrt{3}}{22-\left(\sqrt{3}\right)2}$ $=\frac{2+\sqrt{3}}{4-3}$ $=2+\sqrt{3}$

=>$x-2=\sqrt{3}$

=>$\left(x-2\right)2=\left(\sqrt{3}\right)2$

=>$x2-4x+4= 3=>x2- 4x + 1 = 0$

$x3-2x2-7x+5= x\left(x2- 4x + 1\right)+ 2\left(x2- 4x + 1\right)+ 3= x × 0 + 2 × 0 + 3 = 3$

### Linear Equations in Two Variables

#### Q 1.

Rationalise the denominator of :
2 3

#### Q 2.

if $a=\frac{2-\sqrt{5}}{2+\sqrt{5}}$and $b=\frac{2+\sqrt{5}}{2-\sqrt{5}}$, find $a2-b2$

#### Q 3.

If both a and b are rational numbers, find the values of a and b in each of the following equalities :

(i)$\frac{\sqrt{3}-1}{\sqrt{3}+1}=a+b\sqrt{3}$ (ii) $\frac{3+\sqrt{7}}{3-\sqrt{7}}=a+b\sqrt{7}$ (iii) $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$

(iv) $\frac{5+\sqrt{3}}{7-4\sqrt{3}}=47a+\sqrt{3}b$ (v) $\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}=a+b\sqrt{15}$ (vi) $\frac{\sqrt{2}+\sqrt{3}}{3\sqrt{2}-2\sqrt{3}}=a-b\sqrt{6}$

#### Q 4.

Simplify:
(i) ${\left(\sqrt{5}+\sqrt{2}\right)}^{2}$ [NCERT] (ii) ${\left(\sqrt{11}-\sqrt{5}\right)}^{2}$

#### Q 5.

Find the value to three places of decimals, of each of the following: It is given that $\sqrt{2}=1.414 ,\sqrt{3}=1.732,\sqrt{10}=3.162 and\sqrt{5}=2.236 \left(approx.\right)$

(i)$\frac{1}{\sqrt{2}}$ (ii)$\frac{1}{\sqrt{3}}$ (iii)$\frac{1}{\sqrt{10}}$

#### Q 6.

Simplify:
(i) $\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)$ (ii) $\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)$

#### Q 7.

Simply the following expression:
(i) $\left(3+\sqrt{3}\right)\left(2+\sqrt{2}\right)$ [NCERT] (ii) $\left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right)$

#### Q 8.

Rationalise the denominator of : 1 3 + 2

#### Q 9.

Prove that :

$\frac{1}{1+\sqrt{2}}$ $+\frac{1}{\sqrt{2}+\sqrt{3}}$ $+\frac{1}{\sqrt{3}+\sqrt{4}}$ $+\frac{1}{\sqrt{4}+\sqrt{5}}$ $+\frac{1}{\sqrt{5}+\sqrt{6}}$ $+\frac{1}{\sqrt{6}+\sqrt{7}}$ $+\frac{1}{\sqrt{7}+\sqrt{8}}$ $+\frac{1}{\sqrt{8}+\sqrt{9}}=2$

#### Q 10.

if $x=\frac{1}{2-\sqrt{3}}$, find the value of $x3-2x2-7x+5$

#### Q 11.

Rationalise the denominator in each of the following:

(i) $\frac{2}{\sqrt{7}}$ [NCERT] (ii) $\frac{2}{3\sqrt{3}}$ (iii) $\frac{2\sqrt{7}}{\sqrt{11}}$

#### Q 12.

If 2 = 1.414 , find the value of 3 ÷ 6 upto three places of decimals.

#### Q 13.

if $x=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}}$ and $y=\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ , find $x2+y2$

#### Q 14.

Simplify each of the following :

(i) $\frac{3}{5-\sqrt{3}}$ $+\frac{2}{5+\sqrt{3}}$ (ii) $\frac{4+\sqrt{5}}{4-\sqrt{5}}$ $+\frac{4-\sqrt{5}}{4+\sqrt{5}}$ (iii) $\frac{\sqrt{5}-2}{\sqrt{5}+2}$ $-\frac{\sqrt{5}+2}{\sqrt{5}-2}$

#### Q 15.

Prove that :

$=\frac{1}{3-\sqrt{8}}$ $-\frac{1}{\sqrt{8}-\sqrt{7}}$ $+\frac{1}{\sqrt{7}-\sqrt{6}}$ $-\frac{1}{\sqrt{6}-\sqrt{5}}$ $+\frac{1}{\sqrt{5}-2}=5$

#### Q 16.

Simplify each of the following by rationalising the denominator:

(i) $\frac{1}{5+\sqrt{2}}$ (ii) $\frac{5+\sqrt{6}}{5-\sqrt{6}}$ (iii) $\frac{7+3\sqrt{5}}{7-3\sqrt{5}}$ (iv) $\frac{2\sqrt{3}-\sqrt{5}}{2\sqrt{3}+3\sqrt{3}}$

#### Q 17.

If x = 2 + $\sqrt{3}$ , find the value of $x2+\frac{1}{x2}$

#### Q 18.

Given that $\sqrt{5}=2.236$ approximately, find to three places of decimals the value of $\frac{2}{\sqrt{5}}$

#### Q 19.

Simplify each of the following by rationalising the denominator:

(i) $\frac{1}{5+\sqrt{2}}$ (ii) $\frac{5+\sqrt{6}}{5-\sqrt{6}}$ (iii) $\frac{7+3\sqrt{5}}{7-3\sqrt{5}}$ (iv) $\frac{2\sqrt{3}-\sqrt{5}}{2\sqrt{3}+3\sqrt{3}}$

#### Q 20.

Rationalise the denominator of : 1 7 + 3 2

#### Q 21.

Simply each of the following: (i) $5\sqrt{16}×5\sqrt{2}$ (ii) $\frac{4\sqrt{243}}{4\sqrt{3}}$

#### Q 22.

Rationalise the denominator of : 5 3 - 5

#### Q 23.

If x = 3 - $2\sqrt{2}$ , find the value of $x2+\frac{1}{x2}$

#### Q 24.

If $x=1-\sqrt{2}$ , find the value of ${\left(x-\frac{1}{x}\right)}^{3}$

#### Q 25.

Evaluate $\frac{15}{\sqrt{10}+\sqrt{20}+\sqrt{40}-\sqrt{5}-\sqrt{80}}$ , is being given that $\sqrt{5}$ = 2.236 and $\sqrt{10}$ = 3.162

#### Q 26.

Find the value to three places of decimals, of each of the following:
It is given that 2 = 1.414 , 3 = 1.732, 10 = 3.162 and 5 = 2.236 (approx.).
(i) 2 + 1 5 (ii) 2 - 3 3 (iii) 10 - 5 2