Applied Mathematics KP PMS Paper I 2022

Table of Contents

KHYBER PAKHTUNKHWA PUBLIC SERVICE COMMISSION

Time Allowed: 03 Hours Maximum Marks: 100

Instructions: Attempt total FIVE questions, TWO questions from Section-A and THREE questions from Section-B.

1. Describe the geometrical interpretation of magnitude of $\vec{A} \cdot \vec{B} \times \vec{C}$ (10)
2. Prove that $(\vec{A} + \vec{B}) \cdot (\vec{B} + \vec{C}) \times (\vec{C} + \vec{A}) = 2[\vec{A} \cdot (\vec{B} \times \vec{C})]$ (10)

1. Find the normal derivative (i.e. $\frac{\partial \phi}{\partial n} = \Delta \phi \cdot \hat{n}$) of $\phi = xy^2 + yz^3$ at $(-1, 2, 1)$, where $\hat{n}$ is a unit normal vector to the surface $x \log z – y^3 + 4 = 0$ at $(-1, 2, 1)$. (10)
2. Find constants $a, b, c$ so that $\vec{V} = (x + 2y + az)\hat{i} + (bx – 3y – z)\hat{j} + (4x + cy + 2z)\hat{k}$ is irrotational. Also show that $\vec{V}$ can be expressed as the gradient of a scalar function. (10)

1. State and prove Lami’s theorem. Also if three forces $P, Q, R$ acting at a point are in equilibrium and the angle between $P$ and $Q$ is double of the angle between $P$ and $R$ then prove $R^2 = Q(Q – P)$. (10)
2. $OAB$ is an equilateral triangle of side $a$: $C$ is the mid point of $OA$. Forces $4P, P$ and $P$ act along the sides $OB, BA$ and $AO$ respectively. If $OA$ and $OY$ (parallel to $CB$) are taken as x-axis and y-axis, prove that the resultant of the forces is $3P$ and the equation of its line of action is $3y = \sqrt{3}(3x + a)$. (10)

1. A rod, $4\text{ft.}$ long, rests on a rough floor against the smooth edge of a table of height $3\text{ft.}$ If the rod is on the point of slipping when inclined at an angle of $60^\circ$ to the horizontal, find the coefficient of friction. (10)
2. Two bodies with weights $W_1$ and $W_2$ are placed on an inclined plane and are connected by a light string which coincides with a line of greatest slope of the plane; if the coefficient of friction between the bodies and the plane be respectively $\mu_1$ and $\mu_2$, find the inclination of the plane to the horizon when both bodies are on the point of motion, it being assumed that the smoother body is below the other. (10)

1. The force acting on a particle of mass $m$ at time $t$ is given by $\vec{F} = a \cos \omega t \vec{i} + b \sin \omega t \vec{j}$. If the particle is initially at rest at the origin, show that its position and velocity at any time are given by $\vec{r} = \frac{a}{m\omega^2} (1 – \cos \omega t)\vec{i} + \frac{b}{m\omega^2}(\omega t – \sin \omega t)\vec{j}$,
  $\vec{v} = \frac{a}{m\omega}(\sin \omega t)\vec{i} + \frac{b}{m\omega}(1 – \cos \omega t)\vec{j}$ (10)
2. A particle starts from rest at the highest point of a smooth vertical circle and moves down along the outside of the arc. Discuss the motion and show that the particle leaves contact with the circle after descending a vertical distance equal to one-third of the radius. (10)

1. A point moving in a straight line with uniform acceleration $f$ describes distance $a, b$ metres in successive intervals of time $t_1, t_2$ seconds. Prove that the acceleration is of magnitude $\frac{2(bt_1 – at_2)}{t_1 t_2 (t_1 + t_2)}$. Also prove that if the point describes successive equal distances in times $t_1, t_2, t_3$ then $\frac{1}{t_1} – \frac{1}{t_2} + \frac{1}{t_3} = \frac{3}{t_1 + t_2 + t_3}$. (10)
2. A particle is projected upward with velocity $u$ and the resistance of air produces retardation $kv^2$, when $v$ is the velocity, $k$ being a constant. Show that the particle will return to the point of projection with velocity $\frac{1}{\sqrt{\frac{1}{u^2} + \frac{k}{g}}}$. (10)

1. A particle is projected downward with a speed $80\text{ ms}^{-1}$, and just penetrates a uniform horizontal board of thickness $1\text{ metre}$ whose upper surface is $16\text{ metre}$ below the point of projection. If the same particle were projected vertically upward with a speed of $80\text{ ms}^{-1}$ from a point $16\text{ metre}$ below the lower surface of the board, find how far it would penetrate assuming the resistance offered by the board is constant. (10)
2. The components of velocity along and perpendicular to the radius vector from a fixed origin are respectively $\lambda r^2$ and $\mu \theta^2$. Find the polar equation of the path in terms of $r$ and $\theta$. (10)

1. A projectile of mass $m$ is launched from a point $O$ with velocity $u$ making angle $\alpha$ with the horizontal line $OX$. $\alpha$ will be called the angle of projection. Discuss its motion and the equation of its trajectory. (10)
2. A gun is firing from the sea level out to sea. It is then mounted in a battery $h$ feet higher up and fired at the same elevation $\alpha$. Show that the range is increased by $\frac{1}{2}\left\{\left(1 + \frac{2gh}{u^2 \sin^2 \alpha}\right)^{1/2} – 1\right\}$. (10)

1. Show that the law of force towards the pole of a particle describing the curve $r^n = a^n \cos n\theta$ is given by $f = \frac{(n + 1)h^2 a^{2n}}{r^{2n + 3}}$. (10)
2. $r^n = A \cos n\theta + B \sin n\theta$, prove $F$ varies as $\frac{1}{r^{2n + 3}}$. (10)

2 Views