If x - 3 - 2-2 - find the value of-i 1 xii x - 1 xiii x - 1 xiv -x - 1 -xv -x - 1 -xvi x2 - 1 x2vii x2 - 1 x2

I) 1x =13 2√(2) =1·(3+2√(2))(3 2√(2))(3+2√(2)) =3+2√(2) ii) x+1x =3 2√(2)+13 2√(2) =3 2√(2)+3+2√(2) =6 iii) x 1x ...

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Byju's Answer

Standard XII

Mathematics

Functions

If x = 3 - ...

Question

If $x = 3 - 2 \sqrt{2}$ , find the value of:
i) $\frac{1}{x}$
ii) $x + \frac{1}{x}$
iii) $x - \frac{1}{x}$
iv) $\sqrt{x} + \frac{1}{\sqrt{x}}$
v) $\sqrt{x} - \frac{1}{\sqrt{x}}$
vi) $x^{2} + \frac{1}{x^{2}}$
vii) $x^{2} - \frac{1}{x^{2}}$

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Solution

i) $\frac{1}{x}$
$= \frac{1}{3 - 2 \sqrt{2}}$
$= \frac{1 \cdot (3 + 2 \sqrt{2})}{(3 - 2 \sqrt{2}) (3 + 2 \sqrt{2})}$
$= 3 + 2 \sqrt{2}$

ii) $x + \frac{1}{x}$
$= 3 - 2 \sqrt{2} + \frac{1}{3 - 2 \sqrt{2}}$
$= 3 - 2 \sqrt{2} + 3 + 2 \sqrt{2}$
$= 6$

iii) $x - \frac{1}{x}$
$= 3 - 2 \sqrt{2} - \frac{1}{3 - 2 \sqrt{2}}$
$= 3 - 2 \sqrt{2} - 3 - 2 \sqrt{2}$
$= - 4 \sqrt{2}$

iv) $\sqrt{x} + \frac{1}{\sqrt{x}}$
$\sqrt{3 - 2 \sqrt{2}} + \frac{1}{\sqrt{3 - 2 \sqrt{2}}}$
$= \sqrt{2} - 1 + \frac{1}{\sqrt{2} - 1}$
$= \sqrt{2} - 1 + \sqrt{2} + 1$
$= 2 \sqrt{2}$

v) $\sqrt{x} - \frac{1}{\sqrt{x}}$
$\sqrt{3 - 2 \sqrt{2}} - \frac{1}{\sqrt{3 - 2 \sqrt{2}}}$
$= \sqrt{2} - 1 - \frac{1}{\sqrt{2} - 1}$
$= \sqrt{2} - 1 - \sqrt{2} - 1$
$= - 2$

vi) $x^{2} + \frac{1}{x^{2}}$
${(3 - 2 \sqrt{2})}^{2} + \frac{1}{{(3 - 2 \sqrt{2})}^{2}}$
$= 17 - 12 \sqrt{2} + \frac{1}{17 - 12 \sqrt{2}}$
$= 17 - 12 \sqrt{2} + 17 + 12 \sqrt{2}$
$= 34$

vii) $x^{2} - \frac{1}{x^{2}}$
${(3 - 2 \sqrt{2})}^{2} - \frac{1}{{(3 - 2 \sqrt{2})}^{2}}$
$= 17 - 12 \sqrt{2} - \frac{1}{17 - 12 \sqrt{2}}$
$= 17 - 12 \sqrt{2} - 17 - 12 \sqrt{2}$
$= - 24 \sqrt{2}$

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If x=3−2√2 , find the value of:i) 1xii) x+1xiii) x−1xiv) √x+1√xv) √x−1√xvi) x2+1x2vii) x2−1x2

If $x = 3 - 2 \sqrt{2}$ , find the value of:
i) $\frac{1}{x}$
ii) $x + \frac{1}{x}$
iii) $x - \frac{1}{x}$
iv) $\sqrt{x} + \frac{1}{\sqrt{x}}$
v) $\sqrt{x} - \frac{1}{\sqrt{x}}$
vi) $x^{2} + \frac{1}{x^{2}}$
vii) $x^{2} - \frac{1}{x^{2}}$