Trigonometric Functions

The most general solutions of the equation

$\sec \left(\mathrm{x}\right) - 1 = \left(\sqrt{2} - 1\right)\tan \left(\mathrm{x}\right)$ are given by

• $\mathrm{n\pi } + \frac{\mathrm{\pi }}{8}$

• $2\mathrm{n\pi }, 2\mathrm{n\pi } + \frac{\mathrm{\pi }}{4}$

• $2\mathrm{n\pi }$

• None of these

If $\mathrm{\alpha } + \mathrm{\beta } - \mathrm{\gamma } = \mathrm{\pi }, \mathrm{then} {\sin }^2\left(\mathrm{\alpha }\right) + {\sin }^2\left(\mathrm{\beta }\right) - {\sin }^2\left(\mathrm{\gamma }\right)$ is equal to

• $2\sin \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\beta }\right)\cos \left(\mathrm{\gamma }\right)$

• $2\cos \left(\mathrm{\alpha }\right)\cos \left(\mathrm{\beta }\right)\cos \left(\mathrm{\gamma }\right)$

• $2\sin \left(\mathrm{\alpha }\right)\sin \left(\mathrm{\beta }\right)\sin \left(\mathrm{\gamma }\right)$

• None of the above

The range of the function f (x) = ${}^{7 - \mathrm{x}}\mathrm{P}_{\mathrm{x} - 3}$ is

• {1, 2, 3}

• {1, 2, 3, 4, 5, 6}

• {1, 2, 3, 4}

• {1, 2, 3, 4, 5}

The principal amplitude of ${\left(\sin \left(40°\right) + \mathrm{icos}\left(40°\right)\right)}^5$ is

• $70°$

• $- 110°$

• $110°$

• $- 70°$

If the relation between direction ratios of two lines are given by a + b + c = 0 and 2ab + 2ac - bc = 0, then the angle between the lines is

• $\mathrm{\pi }$

• $\frac{2\mathrm{\pi }}{3}$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

The value of ${\left[\sqrt{2}\left\{\cos \left(56°15\text{'}\right) +\hspace{0.17em}\mathrm{isin}\left(56°15\text{'}\right)\right\}\right]}^8$

• - 16i

• 16i

• 8i

• 4i

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

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