## Class 9 - Mathematics

#### Linear Equations in Two Variables

MCQ
Q&A
Question:
If x = 2 + $\sqrt{3}$ , find the value of $x2+\frac{1}{x2}$

We have, $x=2+\sqrt{3}$

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

Now, $x2+\frac{1}{x2}$ $={\left(x+\frac{1}{x}\right)}^{2}=2$

=> $x2+\frac{1}{x2}$ $={\left(2+\sqrt{3}+2âˆ’\sqrt{3}\right)}^{2}âˆ’2=42âˆ’2=16âˆ’2=14$

ALITER we have, $x=2+\sqrt{3}$

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

Thus, $x2+\frac{1}{x2}$ $={\left(2+\sqrt{3}\right)}^{2}+{\left(2âˆ’\sqrt{3}\right)}^{2}$

=> $x2+\frac{1}{x2}$ $=22+\left(\sqrt{3}\right)2+2Ã—2\sqrt{3}+22+\left(\sqrt{3}\right)2âˆ’2Ã—2\sqrt{3}$

=> $x2+\frac{1}{x2}$ $=4+3+4\sqrt{3}+4+3âˆ’4\sqrt{3}=14$

Linear Equations in Two Variables - Questions
1. If both a and b are rational numbers, find the values of a and b in each of the following equalities :

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

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

2. Rationalise the denominator in each of the following:

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

Rationalise the denominator of : 1 7 + 3 2

4. Prove that :

$=\frac{1}{3âˆ’\sqrt{8}}$ $âˆ’\frac{1}{\sqrt{8}âˆ’\sqrt{7}}$ $+\frac{1}{\sqrt{7}âˆ’\sqrt{6}}$ $âˆ’\frac{1}{\sqrt{6}âˆ’\sqrt{5}}$ $+\frac{1}{\sqrt{5}âˆ’2}=5$
5. If $x=1âˆ’\sqrt{2}$ , find the value of ${\left(xâˆ’\frac{1}{x}\right)}^{3}$
6. If x = 3 - $2\sqrt{2}$ , find the value of $x2+\frac{1}{x2}$
7. Simplify:
(i) ${\left(\sqrt{5}+\sqrt{2}\right)}^{2}$ [NCERT] (ii) ${\left(\sqrt{11}-\sqrt{5}\right)}^{2}$
8. Simply each of the following: (i) $5\sqrt{16}×5\sqrt{2}$ (ii) $\frac{4\sqrt{243}}{4\sqrt{3}}$
9.

Rationalise the denominator of :
2 3

10. Simply the following expression:
(i) $\left(3+\sqrt{3}\right)\left(2+\sqrt{2}\right)$ [NCERT] (ii) $\left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right)$
11.

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

12. 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
13. if $x=\frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}âˆ’\sqrt{2}}$ and $y=\frac{\sqrt{3}âˆ’\sqrt{2}}{\sqrt{3}+\sqrt{2}}$ , find $x2+y2$

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

$\frac{1}{1+\sqrt{2}}$ $+\frac{1}{\sqrt{2}+\sqrt{3}}$ $+\frac{1}{\sqrt{3}+\sqrt{4}}$ $+\frac{1}{\sqrt{4}+\sqrt{5}}$ $+\frac{1}{\sqrt{5}+\sqrt{6}}$ $+\frac{1}{\sqrt{6}+\sqrt{7}}$ $+\frac{1}{\sqrt{7}+\sqrt{8}}$ $+\frac{1}{\sqrt{8}+\sqrt{9}}=2$
16. Simplify each of the following :

(i) $\frac{3}{5âˆ’\sqrt{3}}$ $+\frac{2}{5+\sqrt{3}}$ (ii) $\frac{4+\sqrt{5}}{4âˆ’\sqrt{5}}$ $+\frac{4âˆ’\sqrt{5}}{4+\sqrt{5}}$ (iii) $\frac{\sqrt{5}âˆ’2}{\sqrt{5}+2}$ $âˆ’\frac{\sqrt{5}+2}{\sqrt{5}âˆ’2}$
17. if $a=\frac{2âˆ’\sqrt{5}}{2+\sqrt{5}}$and $b=\frac{2+\sqrt{5}}{2âˆ’\sqrt{5}}$, find $a2âˆ’b2$
18. Simplify:
(i) $\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)$ (ii) $\left(3+2\sqrt{2}\right)\left(3-2\sqrt{2}\right)$
19. 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)$\frac{1}{\sqrt{10}}$

20. if $x=\frac{1}{2âˆ’\sqrt{3}}$, find the value of $x3âˆ’2x2âˆ’7x+5$
21. 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) $\frac{2\sqrt{3}âˆ’\sqrt{5}}{2\sqrt{3}+3\sqrt{3}}$
22. If x = 2 + $\sqrt{3}$ , find the value of $x2+\frac{1}{x2}$
23.

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

24. 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) $\frac{2\sqrt{3}âˆ’\sqrt{5}}{2\sqrt{3}+3\sqrt{3}}$
25.

Rationalise the denominator of : 1 3 + 2

26.

Rationalise the denominator of : 5 3 - 5

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