51.

• 3u

• 4u

• 3sin(u)

• 3tan(u)

$\mathrm{If} \frac{3{\mathrm{x}}^2 + \mathrm{x} + 1}{{\left(\mathrm{x} - 1\right)}^4} = \frac{\mathrm{a}}{\left(\mathrm{x} - 1\right)} + \frac{\mathrm{b}}{{\left(\mathrm{x} - 1\right)}^2} +\hspace{0.17em}\frac{\mathrm{c}}{{\left(\mathrm{x} - 1\right)}^3} + \frac{\mathrm{d}}{{\left(\mathrm{x} - 1\right)}^4}, \mathrm{then} \left[\begin{array}{cc}\mathrm{a}& \mathrm{b}\\ \mathrm{c}& \mathrm{d}\end{array}\right] = ?$

• $\left[\begin{array}{cc}3& 7\\ 5& 0\end{array}\right]$

• $\left[\begin{array}{cc}0& 3\\ 7& 5\end{array}\right]$

• $\left[\begin{array}{cc}0& 7\\ 3& 5\end{array}\right]$

• $\left[\begin{array}{cc}3& 5\\ 7& 0\end{array}\right]$

$\mathrm{If} \left|\begin{array}{ccc}\mathrm{x}& {\mathrm{x}}^2& 1 + {\mathrm{x}}^3\\ \mathrm{y}& {\mathrm{y}}^2& 1 + {\mathrm{y}}^3\\ \mathrm{z}& {\mathrm{z}}^2& 1 + {\mathrm{z}}^3\end{array}\right| = 0, \mathrm{x}\ne \mathrm{y} \ne \mathrm{z}, \mathrm{then} 1 + \mathrm{xyz} = ?$

• 0

• - 1

• 1

• 2

$$\begin{gathered} \mathrm{If} \mathrm{the} \mathrm{system} \mathrm{of} \mathrm{equations} \\ {\left(\mathrm{k} + 1\right)}^3\mathrm{x} + {\left(\mathrm{k} + 2\right)}^3\mathrm{y} = {\left(\mathrm{k} + 3\right)}^3 \\ \left(\mathrm{k} + 1\right)\mathrm{x} + \left(\mathrm{k} + 2\right)\mathrm{y} = \mathrm{k} +\hspace{0.17em}3 \\ \mathrm{x} + \mathrm{y} = 1 \\ \mathrm{is} \mathrm{consistent}, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{k} \mathrm{is} \end{gathered}$$

• 2

• - 2

• - 1

• 1

If A is a non-zero square matrix of order n with det (I + A) $\ne$ 0 and A³ = 0, where I, 0 are unit and null matrices of order n x n respectively, then (I + A)^{- 1} is equal to

• I - A + A²

• I + A + A²

• I + A ^{- 1}

• I + A

$\mathrm{If} {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) + {\tan }^{-1}\left(\mathrm{z}\right) = \frac{\mathrm{\pi }}{2}, \mathrm{then} 1 - \mathrm{xy} - \mathrm{yz} - \mathrm{zx} = ?$

• 1

• 0

• - 1

• 2

$\mathrm{If} {\tanh }^{-1}\left(\mathrm{x}\right) = \mathrm{alog}\left(\frac{1 + \mathrm{x}}{1 - \mathrm{x}}\right), \left|\mathrm{x}\right| < 1, \mathrm{then} \mathrm{a} = ?$

• 1

• 2

• $\frac{1}{2}$

• $\frac{1}{4}$

$$\begin{gathered} \mathrm{If} {\mathrm{m}}_1 \mathrm{and} {\mathrm{m}}_2 \mathrm{are} \mathrm{the} \mathrm{roots} \mathrm{of} \mathrm{the} \mathrm{equation} {\mathrm{x}}^2 + \left(\sqrt{3} + 2\right)\mathrm{x} + \left(\sqrt{3} - 1\right) = 0, \\ \mathrm{then} \mathrm{the} \mathrm{area} \mathrm{of} \mathrm{the} \mathrm{triangle} \mathrm{formed} \mathrm{by} \mathrm{the} \mathrm{lines} \\ \mathrm{y} = {\mathrm{m}}_1\mathrm{x}, \mathrm{y} = {\mathrm{m}}_2\mathrm{x} \mathrm{and} \mathrm{y} = \mathrm{c} \end{gathered}$$

• $\left(\frac{\sqrt{33} - \sqrt{11}}{4}\right){\mathrm{c}}^2$

• $\left(\frac{\sqrt{33} + \sqrt{11}}{4}\right){\mathrm{c}}^2$

• $\left(\frac{\sqrt{11} - \sqrt{33}}{2}\right){\mathrm{c}}^2$

• $\frac{\sqrt{33}}{2}{\mathrm{c}}^2$

$\mathrm{Let} \overrightarrow{\mathrm{a}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}} = 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} - \hat{\mathrm{k}}, \overrightarrow{\mathrm{c}} = \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 3\hat{\mathrm{k}}$. If $\overrightarrow{\mathrm{c}}$ is parallel to the plane containing $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}$, then $\mathrm{\lambda }$ is equal to

• 0

• 1

• - 1

• 2

If A_i (i = 1, 2, 3, ... , n) are n independent events with P(A_i) = $\frac{1}{1 + \mathrm{i}}$ for each i, then the probability that none of A_i occurs is

• $\frac{\mathrm{n} - 1}{\mathrm{n} + 1}$

• $\frac{\mathrm{n}}{\mathrm{n} + 1}$

• $\frac{\mathrm{n}}{\mathrm{n} + 2}$

• $\frac{1}{\mathrm{n} + 1}$