If $\left|\begin{array}{ccc}1 + {\sin }^2\left(\mathrm{\theta }\right)& {\cos }^2\left(\mathrm{\theta }\right)& 4\sin \left(2\mathrm{\theta }\right)\\ {\sin }^2\left(\mathrm{\theta }\right)& 1 + {\cos }^2\left(\mathrm{\theta }\right)& 4\sin \left(2\mathrm{\theta }\right)\\ {\sin }^2\left(\mathrm{\theta }\right)& {\cos }^2\left(\mathrm{\theta }\right)& 4\sin \left(2\mathrm{\theta }\right) - 1\end{array}\right| = 0$ and $0 < \mathrm{\theta } < \frac{\mathrm{\pi }}{2}$, then $\cos \left(4\mathrm{\theta }\right)$ is equal to

• $\frac{\sqrt{3}}{2}$

• 0

• $\frac{- 1}{2}$

• $\frac{1}{2}$

Th function f(x) = $\frac{\log \left(1 +\hspace{0.17em}\mathrm{ax}\right) - \log \left(1 - \mathrm{bx}\right)}{\mathrm{x}}$is not defined at x = 0. The value which should be assigned to f at x = 0 so that it is continuousat x = 0 is 

• a - b

• a + b

• log(a) + log(b)

• 0

If f(x) = 1 + nx + $\frac{\mathrm{n}\left(\mathrm{n} - 1\right)}{2}{\mathrm{x}}^2$ + $\frac{\mathrm{n}\left(\mathrm{n} - 1\right)\left(\mathrm{n} - 2\right)}{6}{\mathrm{x}}^3$ + ... + xⁿ, then f''(1) is equal to

• n(n - 1)2^{n - 1}

• (n - 1)2^{n - 1}

• n(n - 1)2^{n - 2}

• n(n - 1)2ⁿ

If f(x) = ${\log }_{{\mathrm{x}}^2}\left({\log }_{\mathrm{e}}\left(\mathrm{x}\right)\right)$, then f'(x) at x = e is

• 1

• $\frac{1}{\mathrm{e}}$

• $\frac{1}{2\mathrm{e}}$

• 0

If f(x) = $\frac{\mathrm{g}\left(\mathrm{x}\right) + \mathrm{g}\left(- \mathrm{x}\right)}{2} + \frac{2}{{\displaystyle {\left[\mathrm{h}\left(\mathrm{x}\right) + \mathrm{h}\left(- \mathrm{x}\right)\right]}^{- 1}}}$ where g and h are differentiable function, then f'(0)

• 1

• $\frac{1}{2}$

• $\frac{3}{2}$

• 0

The tangent to a given curve y = f(x) is perpendicular to the x-axis, if

• $\frac{\mathrm{dy}}{\mathrm{dx}} - 1$

• $\frac{\mathrm{dx}}{\mathrm{dy}} = 0$

• $\frac{\mathrm{dx}}{\mathrm{dy}} = 1$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

The minimum value of ${27}^{\cos \left(2\mathrm{x}\right)} {81}^{\sin \left(2\mathrm{x}\right)}$ is

• - 5

• $\frac{1}{5}$

• $\frac{1}{243}$

• $\frac{1}{27}$

A stone is thrown vertically upwards from the top of a tower 64 m high according to the law s = 48t - 16t². The greatest height attained by the stone above ground is

• 36 m

• 32 m

• 100  m

• 64 m

The length of the subtangent at t on the curve $\mathrm{x} = \mathrm{a}\left(\mathrm{t} + \sin \left(\mathrm{t}\right)\right)$, y = $\mathrm{a}\left(1 - \cos \left(\mathrm{t}\right)\right)$

• $\mathrm{asin}\left(\mathrm{t}\right)$

• $2\mathrm{asin}\left(\frac{\mathrm{t}}{2}\right)\tan \left(\frac{\mathrm{t}}{2}\right)$

• $2\mathrm{asin}\left(\frac{\mathrm{t}}{2}\right)$

• $2{\mathrm{asin}}^3\left(\frac{\mathrm{t}}{2}\right)\sec \left(\frac{\mathrm{t}}{2}\right)$

On the set of all natural numbers N, which one of the following * is a binary operation?

• $\mathrm{a} * \mathrm{b} = \sqrt{\mathrm{ab}}$

• $\mathrm{a} * \mathrm{b} = \frac{\mathrm{a} - \mathrm{b}}{\mathrm{a} + \mathrm{b}}$

• a * b = a + 3b

• a * b = 3a - 4b