$\require{\cancel}$ $\require{\stix[upint]}$

Qtn No.	1	2	3	4	5	Total
Marks	6	6	8	7	8	35
Score

Question 1 Code: 9709/51/M/J/10/3, Topic: -

 

A particle of mass $0.24 \mathrm{~kg}$ is attached to one end of a light inextensible string of length $2 \mathrm{~m}$. The other end of the string is attached to a fixed point. The particle moves with constant speed in a horizontal circle. The string makes an angle $\theta$ with the vertical (see diagram), and the tension in the string is $T \mathrm{~N}$. The acceleration of the particle has magnitude $7.5 \mathrm{~m} \mathrm{~s}^{-2}$.

$\text{(i)}$ Show that $\tan \theta=0.75$ and find the value of $T$. $[4]$

$\text{(ii)}$ Find the speed of the particle. $[2]$

Question 2 Code: 9709/52/M/J/10/3, Topic: -

Question 3 Code: 9709/53/M/J/10/3, Topic: -

Two particles $P$ and $Q$ are projected simultaneously with speed $40 \mathrm{~m} \mathrm{~s}^{-1}$ from a point $O$ on a horizontal plane. Both particles subsequently pass at different times through the point $A$ which has horizontal and vertically upward displacements from $O$ of $40 \mathrm{~m}$ and $15 \mathrm{~m}$ respectively.

$\text{(i)}$ By considering the equation of the trajectory of a projectile, show that each angle of projection satisfies the equation $\tan ^{2} \theta-8 \tan \theta+4=0$. $[3]$

$\text{(ii)}$ Calculate the distance between the points at which $P$ and $Q$ strike the plane. $[5]$

Question 4 Code: 9709/52/O/N/10/3, Topic: -

Question 5 Code: 9709/53/O/N/10/3, Topic: -

 

Particles $P$ and $Q$ have masses $0.8 \mathrm{~kg}$ and $0.4 \mathrm{~kg}$ respectively. $P$ is attached to a fixed point $A$ by a light inextensible string which is inclined at an angle $\alpha^{\circ}$ to the vertical. $Q$ is attached to a fixed point $B$, which is vertically below $A$, by a light inextensible string of length $0.3 \mathrm{~m}$. The string $B Q$ is horizontal. $P$ and $Q$ are joined to each other by a light inextensible string which is vertical. The particles rotate in horizontal circles of radius $0.3 \mathrm{~m}$ about the axis through $A$ and $B$ with constant angular speed $5 \mathrm{rad} \mathrm{s}^{-1}$ (see diagram).

$\text{(i)}$ By considering the motion of $Q$, find the tensions in the strings $P Q$ and $B Q$. $[3]$

$\text{(ii)}$ Find the tension in the string $A P$ and the value of $\alpha$. $[5]$