Advanced level 2026 BAEBOC pure mathematics with mechanics 3

1. (i) A force $F$, where $F = (18i + 18tj – 36tk)\text{ N}$, acts at time $t$ seconds on a particle of mass $3\text{kg}$. Initially the particle is at the origin moving with velocity $(2i + j – 3k)\text{ ms}^{-1}$.

   Find,

   a) the acceleration of the particle at time $t$.

   b) the velocity of the particles at time $t$.

   c) when $t=2$, the kinetic energy of the particle. (2, 3, 3 marks)

(ii) A particle start from rest and moves with an acceleration of $(2v^2 + 1)\text{ ms}^{-1}$ where $v$ is the speed of the particle. Find the distance taken by the particle to attain a speed of $3\text{mls}$. (5 marks)

2. (i) A particles $p$, of mass $40\text{kg}$ placed on a rough horizontal plane, is connected to another particle $Q$, of mass $50\text{kg}$ hanging freely, by the means of a light inextensible string which passes over a smooth pulley at the top edge of the plane. Given that the plane is inclined at $30^\circ$ to the horizontal and that the coefficient of friction between the plane and particle is $\frac{\sqrt{3}}{4}$. Find,

   a) the acceleration of the particle and the tension in the string.

   b) the magnitude and direction of the force exerted on the string by the pulley.

   The particle $Q$ hits the ground after travelling for $2$ seconds and does not rebound.

   c) Find the further distance which $P$ covers before momentarily coming to rest. (take $g$ as $10\text{mls}^2$) (6, 3, 4 marks)

3. A particle is projected with initial velocity $(20\sqrt{3}i + 20j)\text{ mls}$. Find

   a) The speed and direction of the particle after $1$ second.

   b) The time of flight of the particle.

   C) The range of the particle.

   d) The maximum height attained by the particle.

   e) The Cartesian equation of the path of the particle. (4, 2, 2, 2, 3 marks)

4. i) A force $F_1 = (2i + 3i – k)\text{ N}$, acts through a point with position vector $r_1 = (3i + 6j + k)\text{m}$. Another force $F_2 = (i – 2j + k)\text{N}$ acts through a point with position vector $r_2 = (3i – j + 4k)\text{m}$.

   a) Show that these forces are concurrent starting the position vector of their point of intersection.

   b) Find also the line of action of the resultant of these forces. (4, 3 marks)

ii) A uniform ladder rest with one end against a rough vertical wall and the other end on rough horizontal ground. When the ladder is inclined at $\theta^\circ$ to the horizontal, it is at the point of slipping. Given that the coefficient of friction between the ladder and the ground and between the ladder and the wall is $\frac{1}{2}$, show that $\tan \theta = \frac{3}{4}$. (6 marks)

[Diagram labeled FIG 1: A square ABCD with internal points E, F, H, J and a square section HECJ indicated by dashed lines.]

FIG 1

5. i) A uniform lamina is in the form of a square ABCD where AB = $4a$ and BC = $4a$. E and F are mid-point of BC and AD respectively while H and J are the mid points of EF and CD. A cut is made through the lamina along the line EH and the square section HECJ is folded along HJ so as to… top of FHJD. Find the center of gravity of the resulting lamina from AB and AD. (7 marks)