Advanced level 2025 Littoral Regional Mock further mathematics 2

Advanced level 2025 Littoral Regional Mock further mathematics 2

Advanced level 2025 Littoral Regional Mock further mathematics 2

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1. A particle of mass $m$ is falling vertically under gravity in a resisting medium. The speed $v$ of the particle at a distance $x$ from rest is given by v2=kmg​(1−e−2kx), where $k$ is a positive constant.

(i) Show that the resistance is mg(1−g+kv2v2​). (3 marks)

The particle is projected vertically upwards in the same medium with speed

k3g​​.

(ii) Show that the maximum height reached by the particle is 2k1​ln2. (3 marks) (iii) Find the time taken by the particle to reach maximum height above its point of projection. (4 marks)

2. (a) A particle P of mass 4 kg is moving in a horizontal straight line. At time $t$ seconds, the velocity of P is (4t−t2)ms−1 and the displacement of P from a fixed point O on the line is $xm$. The only force acting on P is a resistance of magnitude $(4v+12)N$.

When $t=0$, $x=0$ and $v=4$.

(i) Show that v=1+3et4​. (4 marks) (ii) Find an expression for $x$ in terms of $t$. (4 marks)

(b) A particle moves in a straight line with simple harmonic motion such that its displacement $x$ at time $t$ seconds relative to a fixed origin on this line is $x$ metres. The motion of the particle satisfies the differential equation dt2d2x​</0>=−4x.

(i) Verify that $x=A\cos 2t+B\sin 2t$, where A and B are constants, is a solution to this differential equation. (2 marks)

When $t=0$, the particle is momentarily at rest.

(ii) Show that $B=0$. (1 mark) (iii) Given that $x=h$ (h > 0) when t=12π​, find A in terms of h. (1 mark) (iv) Find the maximum speed of the particle in terms of h. (2 marks) (v) The mass of the particle is 2 kg. Find the magnitude of the maximum force acting on the particle during the motion. Give your answer in terms of h and m. (2 marks)  

3. A uniform circular disc has diameter AB, mass 2m and radius a. A particle of mass $m$ is attached to the disc at B. The disc can rotate about a smooth fixed horizontal axis through A. The axis is tangential to the disc.  

(i) Show that the moment of inertia of the system is 213​ma2. (4 marks)

The disc is held with AB horizontal and released.

(ii) Show that the angular speed of the disc when AB is vertical is 13a12g​​. (4 marks)

4. A random variable X has probability generating function GX​(t), given by GX​(t)=k(1+3t)2, where k is a constant.

(i) Find the value of the constant k. (1 mark) (ii) Show that E(X)=811​. (3 marks)