If function , then the number of points at which f(x) is continuous, is

• 1

• 0

• None of these

If x^myⁿ = (x + y)^{m + n}, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is

• $\frac{\mathrm{x} +\hspace{0.17em}\mathrm{y}}{\mathrm{xy}}$

• xy

• $\frac{\mathrm{x}}{\mathrm{y}}$

• $\frac{\mathrm{y}}{\mathrm{x}}$

If ABCDEF is a regular hexagon with $\overrightarrow{\mathrm{AB}} = \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{BC}} = \overrightarrow{\mathrm{b}}$, then $\overrightarrow{\mathrm{CE}}$ equals

• $\overrightarrow{\mathrm{b}} - \overrightarrow{\mathrm{a}}$

• $- \overrightarrow{\mathrm{b}}$

• $\overrightarrow{\mathrm{b}} - 2\overrightarrow{\mathrm{a}}$

• None of these

Magnitudes of vectors $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are 3, 4, 5 respectively. If $\overrightarrow{\mathrm{a}} \mathrm{and} \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}},$ $\overrightarrow{\mathrm{b}} \mathrm{and} \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{c}} \mathrm{and} \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}}$  are mutually perpendicular, then magnitude of $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}$ is

• $4\sqrt{2}$

• $3\sqrt{2}$

• $5\sqrt{2}$

• $3\sqrt{3}$

The differential equation of all circles which passes through the origin and whose centre lies on y-axis is

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} - 2\mathrm{xy} = 0$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{xy} = 0$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{xy} = 0$

• $\left({\mathrm{x}}^2 - {\mathrm{y}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = 0$

If forces $\overrightarrow{\mathrm{P}}, \overrightarrow{\mathrm{Q}}, \overrightarrow{\mathrm{R}}$ acting at a point can be represented by the sides of a triangle taken in order, then

• $\overrightarrow{\mathrm{P}} + \overrightarrow{\mathrm{Q}} + \overrightarrow{\mathrm{R}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{P}} - \overrightarrow{\mathrm{Q}} + \overrightarrow{\mathrm{R}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{P}} + \overrightarrow{\mathrm{Q}} - \overrightarrow{\mathrm{R}} = \overrightarrow{0}$

• $\overrightarrow{\mathrm{P}} - \overrightarrow{\mathrm{Q}} - \overrightarrow{\mathrm{R}} = \overrightarrow{0}$

The area bounded by the curve y²(2a - x) = x³ and the line x = 2a is

• $3{\mathrm{\pi a}}^2$ sq unit

• $\frac{3{\mathrm{\pi a}}^2}{2} \mathrm{sq} \mathrm{unit}$

• $\frac{3{\mathrm{\pi a}}^2}{4} \mathrm{sq} \mathrm{unit}$

• $\frac{6{\mathrm{\pi a}}^2}{5} \mathrm{sq} \mathrm{unit}$

18.

A sphere S₁ impings directly on an equal sphere S₂ at rest. If the coefficient of restitution is e, then the velocities of S₁ and S₂ are in the ratio

• $\frac{1 + \mathrm{e}}{1 - \mathrm{e}}$

• $\frac{1 - \mathrm{e}}{1 + \mathrm{e}}$

• $\frac{\mathrm{e} - 1}{\mathrm{e} + 1}$

• $\frac{\mathrm{e} + 1}{\mathrm{e} - 1}$

If $\left|\begin{array}{ccc}\mathrm{a}& {\mathrm{a}}^2& 1 + {\mathrm{a}}^3\\ \mathrm{b}& {\mathrm{b}}^2& 1 + {\mathrm{b}}^3\\ \mathrm{c}& {\mathrm{c}}^2& 1 + {\mathrm{c}}^3\end{array}\right|$ = 0 and the vectors $\overrightarrow{\mathrm{A}} = \left(1, \mathrm{a}, {\mathrm{a}}^2\right), \overrightarrow{\mathrm{B}} = \left(1, \mathrm{b}, {\mathrm{b}}^2\right) \mathrm{and} \overrightarrow{\mathrm{C}} = \left(1, \mathrm{c}, {\mathrm{c}}^2\right)$ are non- coplana, then abc is equal to

• 0

• - 1

• 1

• None of these

The magnitude of the two forces forming a couple is 36 N each and the arm of the couple is 4m. The magnitude of each force of an equivalent couple whose arm is 9m, is (in Newtons)

• 18

• 26

• 16

• 15