The maximum value of $3 \cos \left(\mathrm{\theta }\right) + 4 \sin \left(\mathrm{\theta }\right)$ is

• 3

• 4

• 5

• None of these

$\left[\begin{array}{c}1\\ - 1\\ 2\end{array}\right]\left[\begin{array}{ccc}2& 1& - 1\end{array}\right]$ is equal to

• $\left[\begin{array}{c}2\\ - 1\\ - 2\end{array}\right]$

• $\left[\begin{array}{ccc}2& 1& - 1\\ - 2& - 1& 1\\ 4& 2& - 2\end{array}\right]$

• [- 1]

• Not defined

The domain of the function ${\sin }^{-1}\left({\log }_2\left(\frac{{\mathrm{x}}^2}{2}\right)\right)$ is

• [- 1, 2] - {0}

• [- 2, 2] - (- 1, 1)

• [- 2, 2] - {0}

• [1, 2]

Function f(x) = $$\begin{gathered} \left\{\mathrm{x} - 1, \mathrm{x} <\hspace{0.17em}2 \\ 2\mathrm{x} - 3, \mathrm{x} \ge 2\right\} \end{gathered}$$ is a continuous function

• for x = 2 only

• for all real values of x such that x $\ne$ 2

• for all real values of x

• for all integral values of x only

Differential coefficient of $\sqrt{\sec \left(\sqrt{\mathrm{x}}\right)}$ is

• $\frac{1}{4\sqrt{\mathrm{x}}}\sec \left(\sqrt{\mathrm{x}}\right)\sin \left(\sqrt{\mathrm{x}}\right)$

• $\frac{1}{4\sqrt{\mathrm{x}}}{\left(\sqrt{\sec \left(\sqrt{\mathrm{x}}\right)}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} . \sin \left(\sqrt{\mathrm{x}}\right)$

• $\frac{1}{2}\sqrt{\mathrm{x}}\sqrt{\sec \left(\sqrt{\mathrm{x}}\right)} . \sin \left(\sqrt{\mathrm{x}}\right)$

• $\frac{1}{2}\sqrt{\mathrm{x}}{\left(\sqrt{\sec \left(\sqrt{\mathrm{x}}\right)}\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} . \sin \left(\sqrt{\mathrm{x}}\right)$

The function x⁵ - 5x⁴ + 5x³ - 1 is

• neither maximum nor minimum at x = 0

• maximum at x = 0

• maximum at x = 1 and minimum at x = 3

• minimum at x = 0

7.

If the radius of a circle be increasing at a uniform rate of 2 cm/s. The area of increasing of area of circle, at the instant when the radius is 20 cm, is

• $70 \mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $70 {\mathrm{cm}}^2/\mathrm{s}$

• $80 \mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $80 {\mathrm{cm}}^2/\mathrm{s}$

The equation of normal at the point (0, 3) of the ellipse 9x² + 5y² = 45 is

• x-axis

• y-axis

• y + 3 = 0

• y - 3 = 0

Let the functions f, g, h are defined from the set of real numbers R to R such that

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = {\mathrm{x}}^2 - 1, \mathrm{g}\left(\mathrm{x}\right) = \sqrt{\left({\mathrm{x}}^2 + 1\right)} \mathrm{and} \\ \mathrm{h}\left(\mathrm{x}\right) = \left\{0, \mathrm{if} \mathrm{x} < 0 \\ \mathrm{x}, \mathrm{if} \mathrm{x} \ge 0\right\} \end{gathered}$$

then ho(fog)(x) is defined by

• x

• x²

• 0

• None of these

$3{\tan }^{-1}\left(\mathrm{a}\right)$ is equal to

• ${\tan }^{-1}\left(\frac{3\mathrm{a} + {\mathrm{a}}^3}{1 + 3{\mathrm{a}}^2}\right)$

• ${\tan }^{-1}\left(\frac{3\mathrm{a} - {\mathrm{a}}^3}{1 + 3{\mathrm{a}}^2}\right)$

• ${\tan }^{-1}\left(\frac{3\mathrm{a} + {\mathrm{a}}^3}{1 - 3{\mathrm{a}}^2}\right)$

• ${\tan }^{-1}\left(\frac{3\mathrm{a} - {\mathrm{a}}^3}{1 - 3{\mathrm{a}}^2}\right)$