A small spherical ball of mass m slides without friction from the top of a hemisphere of radius R. At what height will the ball lose contact with surface of the sphere ?

Given $\overrightarrow{\mathrm{A}} = 2 \hat{\mathrm{i}} + 3 \hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{B}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}$. The component of vector $\overrightarrow{\mathrm{A}}$ along vector $\overrightarrow{\mathrm{B}}$ is

• $\frac{1}{\sqrt{2}}$

• $\frac{3}{\sqrt{2}}$

• $\frac{5}{\sqrt{2}}$

• $\frac{7}{\sqrt{2}}$

f A = B + C have scalar magnitudes of 5, 4, 3 units respectively, then the angle between A and C is

• cos^-1 (3/5)

• cos^-1 (4/5)

• π/2

• sin^-1 (3/4)

A particle is projected from the ground with a kinetic energy E at an angle of 60° with the horizontal. Its kinetic energy at the highest point of its motion will be

• $\mathrm{E}/\sqrt{2}$

• E/2

• E/4

• E/8

A bullet on penetrating 30 cm into its target loses its velocity by 50%. What additional distance will it penetrate into the target before it comes to rest ?

• 30 cm

• 20 cm

• 10 cm

• 5 cm

The velocity of a projectile at the initial point A is (2i + 3j) m/s. Its velocity (in m/s) at point B is

• − 2i − 3j

• − 2i + 3j

• 2i − 3j

• 2i + 3j

A small object of uniform density rolls up a curved surface with an initial velocity v'. It reaches up to a maximum height of $\frac{3{\mathrm{v}}^2}{4\mathrm{g}}$ with respect to the initial position. The object is

• ring

• solid sphere

• hollow sphere

• disc

Consider three vectors $\mathrm{A} = \hat{\mathrm{i}} + \hat{\mathrm{j}} - 2\hat{\mathrm{k}}$, $\mathrm{B} = \hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and $\mathrm{C} = 2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + 4\hat{\mathrm{k}}$. A vector X of the form $\mathrm{\alpha A} + \mathrm{\beta B}$ (α and β are numbers) is perpendicular to C. The ratio of α and β is

• 1 : 1

• 2 : 1

• − 1 : 1

• 3 : 1

A cricket ball thrown across a field is at heights h₁ and h₂ from the point of projection at times t₁ and t₂ respectively after the throw. The ball is caught by a fielder at the same height as that of projection. The time of flight of the hall in this journey is

• $\frac{{\mathrm{h}}_1{{\mathrm{t}}_2}^2 - {\mathrm{h}}_2{{\mathrm{t}}_1}^2}{{\mathrm{h}}_1{\mathrm{t}}_2 - {\mathrm{h}}_2{\mathrm{t}}_1}$

• $\frac{{\mathrm{h}}_1{{\mathrm{t}}_1}^2 + {\mathrm{h}}_2{{\mathrm{t}}_2}^2}{{\mathrm{h}}_2{\mathrm{t}}_1 + {\mathrm{h}}_1{\mathrm{t}}_2}$

• $\frac{{\mathrm{h}}_1{{\mathrm{t}}_2}^2 + {\mathrm{h}}_2{{\mathrm{t}}_1}^2}{{\mathrm{h}}_1{\mathrm{t}}_2 + {\mathrm{h}}_2{\mathrm{t}}_1}$

• $\frac{{\mathrm{h}}_1{{\mathrm{t}}_1}^2 - {\mathrm{h}}_2{{\mathrm{t}}_2}^2}{{\mathrm{h}}_1{\mathrm{t}}_1 - {\mathrm{h}}_2{\mathrm{t}}_2}$

A wooden block is floating on water kept in a beaker. 40% of the block is above the water surface. Now the beaker is kept inside a lift that starts going upward with acceleration equal to g/2. The block will then

• sink

• float with 10% above the water surface

• float with 40% above the water surface

• float with 70% above the water surface