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Qtn No.	1	2	3	4	5	Total
Marks	11	11	10	11	10	53
Score

Question 1 Code: 9709/51/M/J/16/7, Topic: -

A particle $P$ is attached to one end of a light elastic string of natural length $1.2 \mathrm{~m}$ and modulus of elasticity $12 \mathrm{~N}$. The other end of the string is attached to a fixed point $O$ on a smooth plane inclined at an angle of $30^{\circ}$ to the horizontal. $P$ rests in equilibrium on the plane, $1.6 \mathrm{~m}$ from $O$.

$\text{(i)}$ Calculate the mass of $P$. $[2]$

A particle $Q$, with mass equal to the mass of $P$, is projected up the plane along a line of greatest slope. When $Q$ strikes $P$ the two particles coalesce. The combined particle remains attached to the string and moves up the plane, coming to instantaneous rest after moving $0.2 \mathrm{~m}$.

$\text{(ii)}$ Show that the initial kinetic energy of the combined particle is $1 \mathrm{~J}$. $[4]$

The combined particle subsequently moves down the plane.

$\text{(iii)}$ Calculate the greatest speed of the combined particle in the subsequent motion. $[5]$

Question 2 Code: 9709/53/M/J/16/7, Topic: -

1 51 Question

Question 3 Code: 9709/51/O/N/16/7, Topic: -

A particle $P$ is projected with speed $35 \mathrm{~ms}^{-1}$ from a point $O$ on a horizontal plane. In the subsequent motion, the horizontal and vertically upwards displacements of $P$ from $O$ are $x \mathrm{~m}$ and $y \mathrm{~m}$ respectively. The equation of the trajectory of $P$ is

$$ y=k x-\frac{\left(1+k^{2}\right) x^{2}}{245} $$

where $k$ is a constant. $P$ passes through the points $A(14, a)$ and $B(42,2 a)$, where $a$ is a constant.

$\text{(i)}$ Calculate the two possible values of $k$ and hence show that the larger of the two possible angles of projection is $63.435^{\circ}$, correct to 3 decimal places. $[5]$

For the larger angle of projection, calculate

$\text{(ii)}$ the time after projection when $P$ passes through $A$, $[2]$

$\text{(iii)}$ the speed and direction of motion of $P$ when it passes through $B$. $[4]$

Question 4 Code: 9709/52/O/N/16/7, Topic: -

 

A small ball $B$ of mass $0.5 \mathrm{~kg}$ moves in a horizontal circle with centre $O$ and radius $0.4 \mathrm{~m}$ on the smooth inner surface of a hollow cone fixed with its vertex down. The axis of the cone is vertical and the semi-vertical angle is $60^{\circ}$ (see diagram).

$\text{(i)}$ Show that the magnitude of the force exerted by the cone on $B$ is $5.77 \mathrm{~N}$, correct to 3 significant figures, and calculate the angular speed of $B$. $[4]$

One end of a light elastic string of natural length $0.45 \mathrm{~m}$ and modulus of elasticity $36 \mathrm{~N}$ is attached to $B$. The other end of the string is attached to the point on the axis $0.3 \mathrm{~m}$ above $O$. The ball $B$ again moves on the surface of the cone in the same horizontal circle as before.

$\text{(ii)}$ Calculate the speed of $B$. $[6]$

Question 5 Code: 9709/53/O/N/16/7, Topic: -