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

In which of the following pairs, the two physical quantities have different dimensions ?

• Planck's constant and angular momentum

• Impulse and linear momentum

• Moment of inertia and moment of a force

• Energy and torque

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

A smooth massless string passes over a smooth fixed pulley. Two masses m₁ and m₂, (m₁ > m₂) are tied at the two ends of the string. The masses are allowed to move under gravity starting from rest. The total external force acting on the two masses is

• (m₁ + m₂) g

• $\frac{{\left({\mathrm{m}}_1 - {\mathrm{m}}_2\right)}^2}{{\mathrm{m}}_1 + {\mathrm{m}}_2} \mathrm{g}$

• (m₁ − m₂) g

• $\frac{{\left({\mathrm{m}}_1 + {\mathrm{m}}_2\right)}^2}{{\mathrm{m}}_1 - {\mathrm{m}}_2} \mathrm{g}$

To determine the coefficient of friction between a rough surface and a block, the surface is kept inclined at 45° and the block is released from rest. The block takes a time t in moving a distance d. The rough surface is then replaced by a smooth surface and the same experiment is repeated. The block now takes a time t/2 in moving down the same distance d. The coefficient of friction is

• 3/4

• 5/4

• 1/2

• $1/\sqrt{2}$

7.

A particle is moving uniformly in a circular path of radius r. When it moves through an angular displacement θ, then the magnitude of the corresponding linear displacement will be

• $2\mathrm{r} \cos \left(\frac{\mathrm{\theta }}{2}\right)$

• $2\mathrm{r} \cot \left(\frac{\mathrm{\theta }}{2}\right)$

• $2\mathrm{r} \tan \left(\frac{\mathrm{\theta }}{2}\right)$

• $2\mathrm{r} \sin \left(\frac{\mathrm{\theta }}{2}\right)$

A uniform rod is suspended horizontally from its mid- point. A piece of metal whose weight is w is suspended at a distance l from the mid-point. Another weight w₁ is suspended on the other side at a distance l₁ from the mid-point to bring the rod to a horizontal position. When w is completely immersed in water, w₁ needs to be kept at a distance l₁ from the mid-point to get the rod back into horizontal position. The specific gravity of the metal piece is

• $\frac{\mathrm{w}}{{\mathrm{w}}_1}$

• $\frac{{\mathrm{wl}}_1}{\mathrm{wl} - {\mathrm{w}}_1{\mathrm{l}}_2}$

• $\frac{{\mathrm{l}}_1}{{\mathrm{l}}_1 - {\mathrm{l}}_2}$

• $\frac{{\mathrm{l}}_1}{{\mathrm{l}}_2}$

A particle moves with constant acceleration along a straight line starting from rest. The percentage increase in its displacement during the 4th second compared to that in the 3rd second is

• 33 %

• 40 %

• 66 %

• 77 %

An artificial satellite moves in a circular orbit around the earth. Total energy of the satellite is given by E. The potential energy of the satellite is 

• - 2E

• 2E

• $\frac{2\mathrm{E}}{3}$

• $- \frac{2\mathrm{E}}{3}$