The function f(x) = $\left|\mathrm{x} - 2\right| +\hspace{0.17em}\mathrm{x}$ is

• differentiable at both x = 2 and x = 0

• differentiable at x = 2 but not at x = 0

• continuous at x = 2 but not at x = 0

• continuous at both x = 2and x = 0

A wire of length 20 cm is bent in the form of a sector of a circle. The maximum area that can be enclosed by the wire is

• 20 sq cm

• 25 sq cm

• 10 sq cm

• 30 sq cm

The condition for the line y = mx + c to be a normal to the parabola y² = 4ax is

• c = - 2am - am³

• $\mathrm{c} = - \frac{\mathrm{a}}{\mathrm{m}}$

• $\mathrm{c} = \frac{\mathrm{a}}{\mathrm{m}}$

• c = 2am + am³

$\mathrm{If} \mathrm{y} = {\tan }^{-1}\left(\sqrt{{\mathrm{x}}^2 - 1}\right), \mathrm{then} \mathrm{the} \mathrm{ratio} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} : \frac{\mathrm{dy}}{\mathrm{dx}}$ is

• $\frac{\mathrm{x}\left({\mathrm{x}}^2 - 1\right)}{1 - 2{\mathrm{x}}^2}$

• $\frac{1 - 2{\mathrm{x}}^2}{\mathrm{x}\left({\mathrm{x}}^2 - 1\right)}$

• $\frac{1 + 2{\mathrm{x}}^2}{\mathrm{x}\left({\mathrm{x}}^2 + 1\right)}$

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

P is the point of contact of the tangent form the origin to the curve y = log_e(x). The length of the perpendicular drawn form the origin to the normal at P is

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{\mathrm{e}}$

• $2\sqrt{{\mathrm{e}}^2 + 1}$

• $\sqrt{{\mathrm{e}}^2 +\hspace{0.17em}1}$

16.

For the curve 4x⁵ = 5y⁴, the ratio of the cube of the subtangent at a point on the curve the square of the subnormal at the same point is

• $\frac{4^4}{5}$

• $\frac{5^4}{4}$

• $\frac{4^4}{5^4}$

• ${\left(\frac{5}{4}\right)}^4$

The set of real values of x for which f(x) = $\frac{\mathrm{x}}{\log \left(\mathrm{x}\right)}$ is increasing is

• $\left\{\mathrm{x} :\hspace{0.17em}\mathrm{x} \ge \mathrm{e}\right\}$

• empty

• {x : x < e}

• {1}

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are non-zero coplanar vectors, then $\left[2\overrightarrow{\mathrm{a}} - \overrightarrow{\mathrm{b}} 3\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{c}} 4\overrightarrow{\mathrm{c}} - \overrightarrow{\mathrm{a}}\right]$ is

• 25

• 0

• 27

• 9

A space vector makes the angles 150° and 60° with the positive direction of x-and y-axes. The angle made by the vector with the positive direction z-axis is

• 90°

• 60°

• 180°

• 120°

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}}$ are unit vectors, such that $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$, then $3\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{b}} . \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} . \overrightarrow{\mathrm{a}}= \overrightarrow{0}$ is

• - 1

• 1

• - 3

• 3