Which one of the following statements is incorrect?

A block of mass $m$ is placed on a smooth inclined wedge $ABC$ of inclination $\theta$ as shown in the figure. The wedge is given an acceleration '$a$' towards the right. The relation between $a$ and $\theta$ for the block to remain stationary on the wedge is:
          
1. $a = \dfrac{g}{\mathrm{cosec }~ \theta}$
2. $a = \dfrac{g}{\sin\theta}$
3. $a = g\cos\theta$
4. $a = g\tan\theta$

One end of the string of length $l$ is connected to a particle of mass $m$ and the other end is connected to a small peg on a smooth horizontal table. If the particle moves in a circle with speed $v$, the net force on the particle (directed towards the centre) will be: ($T$ represents the tension in the string)

A massless and inextensible string connects two blocks $\mathrm{A}$ and $\mathrm{B}$ of masses $3m$ and $m,$ respectively. The whole system is suspended by a massless spring, as shown in the figure. The magnitudes of acceleration of $\mathrm{A}$ and $\mathrm{B}$ immediately after the string is cut, are respectively:
         

A spring of force constant k is cut into lengths of ratio 1:2:3. They are connected in series and the new force constant is $k^{\text{'}}$. If they are connected in parallel and force constant is $k^{\text{'}\text{'}}, then k^{\text{'}}:k^{\text{'}\text{'}}$ is 

(1) 1:6

(2) 1:9

(3) 1:11

(4) 6:11

A bullet of mass 10g moving horizontal with a velocity of 400 m/s strikes a wood block of mass 2 kg which is suspended by light inextensible string of length 5 m. As result, the centre of gravity of the block found to rise a vertical distance of 10 cm. The speed of the bullet after it emerges of horizontally from the block wiil be 

1. 100 m/s 

2. 80 m/s

3. 120 m/s 

4. 160 m/s 

A car is negotiating a curved road of radius R. The road is banked at angle $\mathrm{\theta }$. The coefficient of friction between the tyres of the car and the road is ${\mathrm{\mu }}_{\mathrm{s}}$. The maximum safe velocity on this road is

1. $\sqrt{\mathrm{gR}\left(\frac{{\mathrm{\mu }}_{\mathrm{s}}+\mathrm{tan\theta }}{1-{\mathrm{\mu }}_{\mathrm{s}}\mathrm{tan\theta }}\right)}$

2. $\sqrt{\frac{\mathrm{g}}{\mathrm{R}}\left(\frac{{\mathrm{\mu }}_{\mathrm{s}}+\mathrm{tan\theta }}{1-{\mathrm{\mu }}_{\mathrm{s}}\mathrm{tan\theta }}\right)}$

3. $\sqrt{\frac{\mathrm{g}}{{\mathrm{R}}^2}\left(\frac{{\mathrm{\mu }}_{\mathrm{s}}+\mathrm{tan\theta }}{1-{\mathrm{\mu }}_{\mathrm{s}}\mathrm{tan\theta }}\right)}$

4. $\sqrt{{\mathrm{gR}}^2\left(\frac{{\mathrm{\mu }}_{\mathrm{s}}+\mathrm{tan\theta }}{1-{\mathrm{\mu }}_{\mathrm{s}}\mathrm{tan\theta }}\right)}$

A rigid ball of mass $M$ strikes a rigid wall at $60^{\circ}$ and gets reflected without loss of speed, as shown in the figure. The value of the impulse imparted by the wall on the ball will be:
         

A car is negotiating a curved road of radius $R$. The road is banked at an angle $\theta$. The coefficient of friction between the tyre of the car and the road is $\mu_s$. The maximum safe velocity on this road is:
1. $\sqrt{\operatorname{gR}\left(\dfrac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\mathrm{s}} \tan \theta}\right)}$
2. $\sqrt{\frac{\mathrm{g}}{\mathrm{R}}\left(\dfrac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\mathrm{s}} \tan \theta}\right)}$
3. $\sqrt{\frac{\mathrm{g}}{\mathrm{R}^2}\left(\dfrac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\operatorname{s}} \tan \theta}\right)}$
4. $\sqrt{\mathrm{gR}^2\left(\dfrac{\mu_{\mathrm{s}}+\tan \theta}{1-\mu_{\mathrm{s}} \tan \theta}\right)}$