51.

If x, y, z are all positive and are the pth, qth and rth terms of a geometric progression respectively, then the value of the determinant

$\left|\begin{array}{ccc}\log \left(\mathrm{x}\right)& \mathrm{p}& 1\\ \log \left(\mathrm{y}\right)& \mathrm{q}& 1\\ \log \left(\mathrm{z}\right)& \mathrm{r}& 1\end{array}\right|$ equals

• log(xyz)

• (p - 1)(q - 1)(r - 1)

• pqr

• 0

If $\left[\begin{array}{ccc}1& - 1& \mathrm{x}\\ 1& \mathrm{x}& 1\\ \mathrm{x}& - 1& 1\end{array}\right]$ has no inverse, then the real value of x is

• 2

• 3

• 0

• 1

A binary sequence is an array of O's and 1's. The number of n-digit binary sequences which contain even number of 0's is

• 2^{n - 1}

• 2ⁿ - 1

• 2^{n - 1} - 1ⁿ

• 2ⁿ

${\cos }^{-1}\left(\frac{- 1}{2}\right) - 2{\sin }^{-1}\left(\frac{1}{2}\right) +3{\cos }^{-1}\left(\frac{- 1}{\sqrt{2}}\right) - 4{\tan }^{-1}\left(- 1\right) = ?$

• $\frac{19\mathrm{\pi }}{12}$

• $\frac{35\mathrm{\pi }}{12}$

• $\frac{47\mathrm{\pi }}{12}$

• $\frac{43\mathrm{\pi }}{12}$

${\sinh }^{-1}\left(2\right) + {\sinh }^{-1}\left(3\right) = \mathrm{x} \Rightarrow \cosh \left(\mathrm{x}\right) = ?$

• $\frac{1}{2}\left(3\sqrt{5} + 2\sqrt{10}\right)$

• $\frac{1}{2}\left(3\sqrt{5} - 2\sqrt{10}\right)$

• $\frac{1}{2}\left(12 + 2\sqrt{50}\right)$

• $\frac{1}{2}\left(12 - 2\sqrt{50}\right)$

$\mathrm{In} \mathrm{any} ∆\mathrm{ABC}, \mathrm{a}\left(\mathrm{bcos}\left(\mathrm{C}\right) - \mathrm{ccos}\left(\mathrm{B}\right)\right) = ?$

• b² + c²

• b² - c²

• $\frac{1}{\mathrm{b}} + \frac{1}{\mathrm{c}}$

• $\frac{1}{{\mathrm{b}}^2} - \frac{1}{{\mathrm{c}}^2}$

$$\begin{gathered} \mathrm{If} \mathrm{f} : \mathrm{R} \to \mathrm{R} \mathrm{is} \mathrm{defined} \mathrm{by} \\ \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{2\sin \left(\mathrm{x}\right) - \sin \left(2\mathrm{x}\right)}{2\mathrm{xcos}\left(\mathrm{x}\right)}, \mathrm{if} \mathrm{x} \ne 0, \\ \mathrm{a} , \mathrm{if} \mathrm{x} = 0\right\}, \\ \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{a} \mathrm{so} \mathrm{that} \mathrm{f} \mathrm{is} \mathrm{continuous} \mathrm{at} 0 \mathrm{is} \end{gathered}$$

• 2

• 0

• 1

• 2

${\text{x = cos}}^{-1}\left(\frac{1}{\sqrt{1 +\hspace{0.17em}{\mathrm{t}}^2}}\right), \mathrm{y} = {\sin }^{-1}\left(\frac{\mathrm{t}}{\sqrt{1 +\hspace{0.17em}{\mathrm{t}}^2}}\right) \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = ?$

• 0

• tan(t)

• 1

• sin(t)cos(t)

$\frac{\mathrm{d}}{\mathrm{dx}}\left[\mathrm{a} {\tan }^{-1}\left(\mathrm{x}\right) + \mathrm{blog}\left(\frac{\mathrm{x} - 1}{\mathrm{x} + 1}\right)\right] = \frac{1}{{\mathrm{x}}^4 - 1} \Rightarrow \mathrm{a} - 2\mathrm{b} = ?$

• 1

• 0

• - 1

• 2

$\mathrm{y} = {\mathrm{e}}^{{\mathrm{asin}}^{-1}\left(\mathrm{x}\right)} \Rightarrow \left(1 - {\mathrm{x}}^2\right){\mathrm{y}}_{\mathrm{n} +\hspace{0.17em}2} - \left(2\mathrm{n} +\hspace{0.17em}1\right){\mathrm{xy}}_{\mathrm{n} +\hspace{0.17em}1} \mathrm{is}$equal to

• $-\left({\mathrm{n}}^2 +\hspace{0.17em}{\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $\left({\mathrm{n}}^2 -\hspace{0.17em}{\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $\left({\mathrm{n}}^2 +\hspace{0.17em}{\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$

• $-\left({\mathrm{n}}^2 -\hspace{0.17em} {\mathrm{a}}^2\right){\mathrm{y}}_{\mathrm{n}}$