21.

The function represented by the following graph is

• continuous but not differentiable at x = 1

• differentiable but not continuous at x = 1

• continuous and differentiable at x = 1

• neither continuous nor differentiable at x = 1

If f(x) =  is continuous at x = 0, then the value k is

• $\frac{\mathrm{\pi }}{10}$

• $\frac{3\mathrm{\pi }}{10}$

• $\frac{3\mathrm{\pi }}{2}$

• $\frac{3\mathrm{\pi }}{5}$

If f(x) = $\mathrm{f}\left(\mathrm{\pi } + \mathrm{e} - \mathrm{x}\right) \mathrm{and} {\int }_{\mathrm{e}}^{\mathrm{\pi }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = \frac{2}{\mathrm{e} +\hspace{0.17em}\mathrm{\pi }}, \mathrm{then} {\int }_{\mathrm{e}}^{\mathrm{\pi }}\mathrm{xf}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\mathrm{\pi } - \mathrm{e}$

• $\frac{\mathrm{\pi } + \mathrm{e}}{2}$

• 1

• $\frac{\mathrm{\pi } - \mathrm{e}}{2}$

If linear function f(x) and g(x) satisfy $\int \left[\left(3\mathrm{x} - 1\right)\cos \left(\mathrm{x}\right) + \left(1 - 2\mathrm{x}\right)\sin \left(\mathrm{x}\right)\right]\mathrm{dx}$ = $\mathrm{f}\left(\mathrm{x}\right)\cos \left(\mathrm{x}\right) + \mathrm{g}\left(\mathrm{x}\right)\sin \left(\mathrm{x}\right) + \mathrm{C}$, then

• f(x) = 3(x - 1)

• f(x) = 3x - 5

• g(x) = 3(x - 1)

• g(x) = 3 + x

The value of the integral ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\log \left(\sec \left(\mathrm{\theta }\right) - \tan \left(\mathrm{\theta }\right)\right)\mathrm{d\theta }$ is

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

$\int \frac{\sin \left(2\mathrm{x}\right)}{{\sin }^2\left(\mathrm{x}\right) + 2{\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $- \log \left(1 + {\sin }^2\left(\mathrm{x}\right)\right) + \mathrm{C}$

• $\log \left(1 + {\cos }^2\left(\mathrm{x}\right)\right) + \mathrm{C}$

• $- \log \left(1 + {\cos }^2\left(\mathrm{x}\right)\right) + \mathrm{C}$

• $\log \left(1 + {\tan }^2\left(\mathrm{x}\right)\right) + \mathrm{C}$

The area of the region bounded by the lines y = mx, x = 1, x = 2 and x-axis is 6 sq units, then 'm' is

• 3

• 1

• 2

• 4

The order and degree of the differential equation y = $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{2}{{\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}}$ is

• 1, 2

• 1, 3

• 2, 1

• 1, 1

The general solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{y}}{\mathrm{x}} = 3\mathrm{x}$ is

• $\mathrm{y} = \mathrm{x} - \frac{\mathrm{C}}{\mathrm{x}}$

• $\mathrm{y} = \mathrm{x} + \frac{\mathrm{C}}{\mathrm{x}}$

• $\mathrm{y} = {\mathrm{x}}^2 - \frac{\mathrm{C}}{\mathrm{x}}$

• $\mathrm{y} = {\mathrm{x}}^2 + \frac{\mathrm{C}}{\mathrm{x}}$

The distance of the point P(a, b, c) from the x-axis is

• $\sqrt{{\mathrm{a}}^2 + {\mathrm{b}}^2}$

• $\sqrt{{\mathrm{b}}^2 + {\mathrm{c}}^2}$

• a

• $\sqrt{{\mathrm{a}}^2 + {\mathrm{c}}^2}$