# NIELIT 2019 Feb Scientist D - Section D: 19

1 vote
120 views

If $x= \frac{\sqrt{p^{2}+q^{2}}+\sqrt{p^{2}-q^{2}}}{{\sqrt{p^{2}+q^{2}}-\sqrt{p^{2}-q^{2}}}}$ then $q^{2}x^{2}-2p^{2}x+q^{2}$ equals to :

1. $3$
2. $-1$
3. $-2$
4. $0$

recategorized

\begin{align} &\frac{x}{1}= \frac{\sqrt{p^{2}+q^{2}}+\sqrt{p^{2}-q^{2}}}{{\sqrt{p^{2}+q^{2}}-\sqrt{p^{2}-q^{2}}}} \\ \implies &\frac{x+1}{x-1} = \frac{\sqrt{p^2 + q^2} }{\sqrt{p^2-q^2}} \qquad \qquad \rightarrow \text{ apply Componendo and Dividendo} \\ \implies & \frac{(x+1)^2}{(x-1)^2} = \frac{p^2+q^2}{p^2-q^2} \qquad \qquad \rightarrow \text{take square on both side} \\ \implies & \frac{(x+1)^2 + (x-1)^2}{(x+1)^2 - (x-1)^2} = \frac{p^2}{q^2} \qquad \rightarrow \text{ apply Componendo and Dividendo} \\ \implies & \frac{x^2 + 1}{2x} = \frac{p^2}{q^2} \\ \implies& q^2(x^2+1) = 2xp^2 \\ \implies & q^2x^2 -2p^2x +q^2 = 0 \end{align}

Option D.

518 points 1 2 12

## Related questions

1
92 views
sum of roots of the equation $\dfrac{3x^{3}-x^{2}+x-1}{3x^{3}-x^{2}-x+1}=\dfrac{4x^{3}-7x^{2}+x+1}{4x^{3}+7x^{2}-x-1}$ is : $0$ $1$ $-1$ $2$
2
73 views
If ${m_1}$ and ${m_2}$ are the roots of equation $x^{2}+(\sqrt{3}+2)x+\sqrt{3}-1=0$ then area of the triangle formed by the lines $y={m_1}x, \: \: y={m_2}x, \: \: y=c$ is: $\bigg(\dfrac{\sqrt{33}+\sqrt{11}}{4}\bigg) c^{2}$ $\bigg( \dfrac{\sqrt{32}+\sqrt{11}}{16}\bigg ) c$ $\bigg (\dfrac{\sqrt{33}+\sqrt{10}}{4} \bigg ) c^{2}$ $\bigg( \dfrac{\sqrt{33}+\sqrt{21}}{4} \bigg) c^{3}$
A man invests some money partly in $3\%$ stock at $96$ and partly in $4\%$ stock at $120$. To get equal dividends from both, he must invest the money in the ratio : $16 : 25$ $4 : 3$ $4 : 5$ $3 : 5$
In a swimming-pool $90$ m by $40$ m, $150$ men take a dip. If the average displacement of water by a man is $8$ cubic metres, what will be rise in water level ? $30$ cm $33.33$ cm $20.33$ cm $25$ cm
A conical tent is to accommodate $10$ persons. Each person must have $6$ $m$^{2}$space to sit and$30m$^{2}$ of air to breath. What will be height of cone ? $37.5$ $m$ $150$ $m$ $75$ $m$ $15$ $m$