51.

${\cos }^2\left(\frac{\mathrm{\pi }}{6} + \mathrm{\theta }\right) - {\sin }^2\left(\frac{\mathrm{\pi }}{6} + \mathrm{\theta }\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{2}\cos \left(2\mathrm{\theta }\right)$

• 0

• $- \frac{1}{2}\cos \left(2\mathrm{\theta }\right)$

• $\frac{1}{2}$

$\mathrm{In} ∆\mathrm{ABC}, \frac{\cos \left(\mathrm{C}\right) + \cos \left(\mathrm{A}\right)}{\mathrm{c} + \mathrm{a}} + \frac{\cos \left(\mathrm{B}\right)}{\mathrm{b}} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{\mathrm{a}}$

• $\frac{1}{\mathrm{b}}$

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

• 1

$\mathrm{In} ∆\mathrm{ABC}, \frac{\mathrm{a}}{{\mathrm{b}}^2 - {\mathrm{c}}^2} + \frac{\mathrm{c}}{{\mathrm{b}}^2 - {\mathrm{a}}^2} = 0, \mathrm{then} \mathrm{B} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{3}$

$\mathrm{In} ∆\mathrm{ABC},{\mathrm{a}}^2\sin \left(2\mathrm{C}\right) + {\mathrm{c}}^2\sin \left(2\mathrm{A}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $∆$

• $2∆$

• $3∆$

• $4∆$

$$\begin{gathered} \mathrm{The} \mathrm{shadow} \mathrm{of} \mathrm{a} \mathrm{tower} \mathrm{standing} \mathrm{on} \mathrm{a} \mathrm{level} \mathrm{ground} \mathrm{is} \mathrm{found} \mathrm{to} \mathrm{be} 60\mathrm{m} \mathrm{longer} \mathrm{when} \mathrm{the} \mathrm{sun}\text{'}\mathrm{s} \\ \mathrm{altitude} \mathrm{is} 30° \mathrm{than} \mathrm{when} \mathrm{it} \mathrm{is} 45°.\mathrm{The} \mathrm{height} \mathrm{of} \mathrm{the} \mathrm{tower} \mathrm{is} \end{gathered}$$

• $30\mathrm{m}$

• $90\mathrm{m}$

• $60\sqrt{3}\mathrm{m}$

• $30\left(\sqrt{3} + 1\right)\mathrm{m}$

$\mathrm{If} \mathrm{cosec\theta } = \frac{\mathrm{p} + \mathrm{q}}{\mathrm{p} - \mathrm{q}},\mathrm{then} \cot \left(\frac{\mathrm{\pi }}{4} + \frac{\mathrm{\theta }}{4}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{\sqrt{\mathrm{p}}}{\mathrm{q}}$

• $\sqrt{\frac{\mathrm{q}}{\mathrm{p}}}$

• $\sqrt{\mathrm{pq}}$

• $\mathrm{pq}$

If f(x) = $\frac{{\mathrm{x}}^2 - 10\mathrm{x} + 25}{{\mathrm{x}}^2 - 7\mathrm{x} + 10}$ and f is continuous at x = 5, then f(5) is equal to

• 0

• 5

• 10

• 25

If h(x) = ${\mathrm{x}}^{{\mathrm{x}}^{\mathrm{x}}}$, then at x = 1, $\frac{\mathrm{h}\text{'}\left(\mathrm{x}\right)}{\mathrm{h}\left(\mathrm{x}\right)}$ is equal to

• h(x)

• $\frac{1}{\mathrm{h}\left(\mathrm{x}\right)}$

• $1 + \log \left(\mathrm{h}\left(\mathrm{x}\right)\right)$

• $- \log \left(\mathrm{h}\left(\mathrm{x}\right)\right)$

$\frac{\mathrm{d}}{\mathrm{dx}}\left({\sin }^{-1}\left(3\mathrm{x} - 4{\mathrm{x}}^3\right)\right)$ is equal to

• $\frac{3}{\sqrt{4 - {\mathrm{x}}^2}}$

• $\frac{3}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{1}{\sqrt{4 - {\mathrm{x}}^2}}$

• $- \frac{1}{\sqrt{4 - {\mathrm{x}}^2}}$

If f(x) = $\frac{{\mathrm{x}}^2}{\mathrm{x} + \mathrm{a}}$, then f''(a) is equal to

• 4a

• $\frac{1}{8\mathrm{a}}$

• $\frac{1}{4\mathrm{a}}$

• 8a