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Presentation of Data in Tabular Form — Maths STD 9 — Question

Gujarat BoardEnglish MediumSTD 9MathsPresentation of Data in Tabular Form4 Marks
Question
If $\text{a}=\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$ and $\text{b}=\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}},$ show that $3\text{a}^2+4\text{ab}-3\text{b}^2=4+\frac{56}{3}\sqrt{10}.$
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Answer

$\text{a}=\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$ $\text{b}=\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}$
$=\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}\times\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}+\sqrt{2}}$ $=\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}\times\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}$
$=\frac{\big(\sqrt{5}+\sqrt{2}\big)^2}{\big(\sqrt{5}\big)^2-\big(\sqrt{2}\big)^2}$ $=\frac{\big(\sqrt{5}-\sqrt{2}\big)^2}{\big(\sqrt{5}\big)^2-\big(\sqrt{2}\big)^2}$
$=\frac{5+2\sqrt{10}+2}{5-2}$ $=\frac{5-2\sqrt{10}+2}{5-2}$
$=\frac{7+2\sqrt{10}}{3}$ $=\frac{7-2\sqrt{10}}{3}$
Now,  
$\text{a}^2=\Big(\frac{7+2\sqrt{10}}{3}\Big)^2$ $\text{b}^2=\Big(\frac{7-2\sqrt{10}}{3}\Big)^2$
$=\frac{49+28\sqrt{10}+40}{9}$ $=\frac{49-28\sqrt{10}+40}{9}$
$=\frac{89+28\sqrt{10}}{9}$ $=\frac{89-28\sqrt{10}}{9}$
$\text{L.H.S.}=3\text{a}^2+4\text{ab}-3\text{b}^2$
$=3\times\frac{89+28\sqrt{10}}{9}+4\times\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}\times\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}-3\times\frac{89-28\sqrt{10}}{9}$
$=\frac{89+28\sqrt{10}}{3}+4-\frac{89-28\sqrt{10}}{3}$
$=4-\frac{89+28\sqrt{10}-89+28\sqrt{10}}{3}$
$=4+\frac{56}{3}\sqrt{10}$
$=\text{R.H.S.}$

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