Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
APPLIED MATHEMATICS, PAPER-I
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A and THREE Questions from SECTION-B, All questions carry EQUAL marks.
(iii)Use of Scientific Calculator is Allowed.
SECTION-A
Q.1. (a) Find a function φ such that ∇φ = f⃗
f⃗ = x î + 2y ĵ + 2 k̂ (10)
(b) Prove that
∇ φⁿ = n φⁿ⁻¹ ∇φ (10)
Q.2. (a) Show that for any vectors a⃗ and b⃗
|a⃗ + b⃗|² + |a⃗ – b⃗|² = 2(|a⃗|² + |b⃗|²) (10)
(b) Prove that
(a⃗ × b⃗) · (b⃗ × c⃗) × (c⃗ × a⃗) = (a⃗ · b⃗ × c⃗)² (10)
Q.3. (a) The greatest resultant that two forces can have is of magnitude P and the least is of magnitude Q. Show that when they act at an angle α their resultant is of magnitude
√(P² cos²(α/2) + Q² sin²(α/2)) (10)
(b) A uniform rod of length 2a rests in equilibrium against a smooth vertical wall and upon a smooth peg at a distance b from the wall. Show that in the position of equilibrium the rod is inclined to the wall at an angle sin⁻¹(b/a)^(1/3) (10)
Q.4. (a) Three forces P, Q and R act along the BC, CA and AB respectively of triangle ABC. Prove that if P cos A + Q cos B + R cos C = 0, then the line of action of the resultant passes through the circum center of the triangle. (10)
(b) A sphere of weight W and radius a is suspended by a string of length l from a point P and a weight w is also suspended from P by a string sufficiently long for the weight to hang below the sphere. Show that the inclination of the first string to the vertical is
sin⁻¹(wa / ((W + w)(a + l))) (10)
Q.5. (a) Find the volume ∬_R (x³ + 4y) dA where R is the region bounded by the parabola y = x² and the line y = 2x. (10)
(b) Evaluate the following line integral
∫_C x² dy
bounded by the triangle having the vertices (-1,0) to (2,0), and (1,1) (10)
SECTION-B
Q.6. (a) The position of a particle moving along an ellipse is given by r⃗ = a cos t î + b sin t ĵ. If a > b, find the position of the particle where its velocity has maximum or minimum magnitude. (10)
(b) Prove that the speed at any point of a central orbit is given by
vp = h,
When h is the areal speed and p is the perpendicular distance from the centre of force, of the tangent at the point. Find the expression for v when a particle subject to the inverse square law of force describes an ellipse, a parabolic and hyperbolic orbit. (10)
Q.7. (a) A particle is moving with the uniform speed v along the curve
x³y = a(x³ + a² / √5)
Show that its acceleration has the maximum value at 10v² / 9a. (10)
(b) An aeroplane is flying with uniform speed v₀ in an arc of a vertical circle of radius a, whose centre is a height h vertically above a point O of the ground. If a bomb is dropped from the aeroplane when at a height Y and strikes the ground at O, show that Y satisfies the equations
KY² + Y(a² – 2hK) + K(h² – a²) = 0,
where K = h + ga² / 2v₀². (10)
Q.8. (a) Find the tangential and normal components of the acceleration of a particle describing the ellipse
x²/a² + y²/b² = 1
With uniform speed v when the particle is a a > b. (10)
(b) Find the velocity acquired by a block of wood of mass M lb., which is free to recoil when it is struck by a bullet of mass m lb. moving with velocity v in a direction passing through the centre of gravity. If the bullet is embedded a ft., show that the resistance of the wood to the bullet, supposed uniform, is Mmv² / (2(M + m)ga) lb.wt. and that the time of penetration is 2a / v sec., during which time the block will move ma / (m + M) ft. (10)
APPLIED MATHEMATICS, PAPER-II
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A and THREE Questions from SECTION-B, All questions carry EQUAL marks.
SECTION-A
Q.No.1. Solve the following equations:
(a) d³y/dx³ + dy/dx = Sec²x (10)
(b) 2dy/dx – x/y + x³Cos y = 0 (10)
Q.No.2. (a) Find the power series solution of the differential equation
(1 – x²)y” – 2xy’ + 2y = 0, about the point x = 0. (10)
(b) Solve Z(x + y) ∂Z/∂x + Z(x – y) ∂Z/∂y = (x² + y²). (10)
Q.No.3. (a) Classify the following equations: (5)
(i) ∂²Z/∂x² + x² ∂²Z/∂y² – 1/x ∂Z/∂x = 0
(ii) x² ∂²Z/∂x² + 2xy ∂²Z/∂x∂y + y² ∂²Z/∂y² = 4x²
(b) Solve: ∂u/∂t = ∂²u/∂x², -1 < x < 1, t > 0 (15)
u(-1, t) = u(1, t); ∂u/∂x(-1, t) = ∂u/∂x(1, t) for t > 0
u(x, 0) = x + 1, -1 < x < 1.
SECTION-B
Q.No.4. (a) Highlight the difference between a vector and a tensor. What happens if we permute the subscripts of a tensor? (5)
(b) Transform g^(αβ) = ( 1 0 ) into Cartesian coordinates. (15)
( 0 1/r² )
Q.No.5. (a) Workout the Christoffel symbols for the metric tensor g_ab = ( a² 0 ) (10)
( 0 a² sin² θ )
(b) Workout the two dimensional metric tensor for the coordinates p and q given by (10)
p = (xy)^(1/3), q = (x²/y)^(1/3)
SECTION-C
Q.No.6. (a) Solve the following system of equations by Jacobi iteration method: (10)
10x + y – 2z = 7.74
x + 12y + 3z = 39.66
3x + 4y + 15z = 54.8
(b) Solve Sin x = 1 + x³ Using Newton-Raphson method. (10)
Q.No.7. (a) Find the root of x eˣ = 3 by regular falsi method correct to three decimal places. (10)
(b) Evaluate ∫₀¹⁰ dx / (1 + x²) using (5+5) (10)
(i) Trapezoidal rule and
(ii) Simpson’s rule.
Q.No.8. (a) Find the real root of the equation Cos x = 3x – 1 correct to seven decimal places (10)
by the iterative method.
(b) Use Lagrange’s interpolation formula to find the value of y when x = 10, if the (10)
values of x and y are given below:
| X | 5 | 6 | 9 | 11 |
|—|—-|—-|—-|—-|
| Y | 12 | 13 | 14 | 16 |
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