If a = $\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ and b = $2\hat{\mathrm{i}} - \hat{\mathrm{j}} +\hspace{0.17em}\hat{\mathrm{k}}$, then the angle $\mathrm{\theta }$ between a and b is given by

• ${\tan }^{-1}\left(1\right)$

• ${\sin }^{-1}\left(\frac{1}{2}\right)$

• ${\sec }^{-1}\left(1\right)$

• ${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$

At t = 0, the function f(t) = $\frac{\sin \left(\mathrm{t}\right)}{\mathrm{t}}$ has

• a minimum

• a discontinuity

• a point of inflexion

• a maximum

113.

If ${∆}_{\mathrm{r}} = \left|\begin{array}{ccc}2\mathrm{r} - 1& {}^{\mathrm{m}}\mathrm{C}_{\mathrm{r}}& 1\\ {\mathrm{m}}^2 - 1& 2^{\mathrm{m}}& \mathrm{m} + 1\\ {\sin }^2\left({\mathrm{m}}^2\right)& {\sin }^2\left(\mathrm{m}\right)& {\sin }^2\left(\mathrm{m} + 1\right)\end{array}\right|$, then the value of $\sum _{\mathrm{r} = 0}^{\mathrm{m}}{∆}_{\mathrm{r}}$

• 1

• 0

• 2

• None of these

If $\cos \left(\mathrm{\alpha }\right) + \mathrm{isin}\left(\mathrm{\alpha }\right), \mathrm{b} = \cos \left(\mathrm{\beta }\right) + \mathrm{isin}\left(\mathrm{\beta }\right), \mathrm{c} = \cos \left(\mathrm{\gamma }\right) + \mathrm{isin}\left(\mathrm{\gamma }\right)$ and $\frac{\mathrm{b}}{\mathrm{c}} + \frac{\mathrm{c}}{\mathrm{a}} + \frac{\mathrm{a}}{\mathrm{b}} = 1$, then $\cos \left(\mathrm{\beta } - \mathrm{\gamma }\right) + \cos \left(\mathrm{\gamma } - \mathrm{\alpha }\right) + \cos \left(\mathrm{\alpha } - \mathrm{\beta }\right)$ is equal to

• $\frac{3}{2}$

• - $\frac{3}{2}$

• 0

• 1

The maximum value of 4 sin²(x) - 12sin(x) + 7 is

• 25

• 4

• does not exist

• None of the above

A line making angles 45° and 60° with the positive directions of the axes of x and y makes with the positive direction of z-axis, an angle of

• 60°

• 120°

• 60° or 120°

• None of these

If I = $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, J = $\left[\begin{array}{cc}0& 1\\ - 1& 0\end{array}\right]$ and $\mathrm{B} = \left[\begin{array}{cc}\cos \left(\mathrm{\theta }\right)& \sin \left(\mathrm{\theta }\right)\\ - \sin \left(\mathrm{\theta }\right)& \cos \left(\mathrm{\theta }\right)\end{array}\right]$, then B is equal to

• $\mathrm{I} \cos \left(\mathrm{\theta }\right) + \mathrm{Jsin}\left(\mathrm{\theta }\right)$

• $\mathrm{I} \sin \left(\mathrm{\theta }\right) + \mathrm{Jcos}\left(\mathrm{\theta }\right)$

• $\mathrm{I} \cos \left(\mathrm{\theta }\right) - \mathrm{Jsin}\left(\mathrm{\theta }\right)$

• $- \mathrm{I} \cos \left(\mathrm{\theta }\right) + \mathrm{Jsin}\left(\mathrm{\theta }\right)$

Find the value of $\sin \left(12°\right)\sin \left(48°\right)\sin \left(54°\right)$.

• $\frac{1}{2}$

• $\frac{1}{4}$

• $\frac{1}{6}$

• $\frac{1}{8}$

If $3\sin \left(\mathrm{\theta }\right) + 5\cos \left(\mathrm{\theta }\right)$, then the value of $5\sin \left(\mathrm{\theta }\right) - 3\cos \left(\mathrm{\theta }\right)$ is equal to

• 5

• 3

• 4

• None of these

Domain of the function f(x) = log_x(cos(x)), is

• $\left(- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right) - \left\{1\right\}$

• $\left[- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right] - \left\{1\right\}$

• $\left(- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right)$

• None of these