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. No. 1. (a) Prove that curl(ϕF⃗) = (grad ϕ) × F⃗. If F⃗ is irrotational and ϕ(x, y, z) is a scalar function. (10)
(b) Determine whether the line integral:
∫_C (2xyz + z²) dx + (x²z + z² Cos yz) dy + (x²y + 2xz + yCos yz) dz
is independent of the path of integration? If so, then compute it from (1,0,1) to (0, π, 2). (10)
Q. No. 2. (a) State and prove Stoke’s Theorem. (10)
(b) Verify Stoke’s Theorem for the function F⃗ = x² î – xy ĵ integrated round the square in the plane z = 0 and bounded by the lines x = y = 0, x = y = a. (10)
Q. No. 3. (a) Three forces act perpendicularly to the sides of a triangle at their middle points and are proportional to the sides. Prove that they are in equilibrium. (10)
(b) Three forces P, Q, R act along the sides BC, CA, AB respectively of a triangle ABC. Prove that, if P Sec A + Q Sec B + R Sec C = 0, then the line of action of the resultant passes through the orthocentre of the triangle. (10)
Q. No. 4. (a) Find the centroid of the surface formed by the revolution of the cardioid r = a(1 + Cos θ) about the initial line. (10)
(b) A uniform ladder rests with its upper end against a smooth vertical wall and its foot on rough horizontal ground. Show that the force of friction at the ground is (1/2) W tan θ, where W is the weight of the ladder and θ is its inclination with the vertical. (10)
Q. No. 5. (a) Define briefly laws of friction. Give at least one example of each law. (10)
(b) A uniform semi-circular wire hangs on a rough peg, the line joining its extremities making an angle of 45° with the horizontal. If it is just on the point of slipping, find the coefficient of friction between the wire and the peg. (10)
SECTION-B
Q. No. 6. (a) If a point P moves with a velocity v given by v² = n²(ax² + 2bx + c), show that P executes a simple harmonic motion. Find the center, the amplitude and the time-period of the motion. (10)
(b) A particle P moves in a plane in such a way that at any time t its distance from a fixed point O is r = at + bt² and the line connecting O and P makes an angle θ = ct^(3/2) with a fixed line OA. Find the radial and transverse components of the velocity and acceleration of the particle at t = 1. (10)
Q. No. 7. (a) A particle of mass m moves under the influence of the force F⃗ = a(î Sin ωt + ĵ Cos ωt). If the particle is initially at rest on the origin, prove that the work done up to time t is given by (a² / (mω²)) (1 – Cos ωt), and that the instantaneous power applied is (a² / (mω)) Sin ωt. (10)
(b) A battleship is streaming ahead with speed V, and a gun is mounted on the battleship so as to point straight backwards, and is set at an angle of elevation α. If v₀ is the speed of projection relative to the gun, show that the range is (2 / g) v₀ Sin α (v₀ Cos α – V). Also prove that the angle of elevation for maximum range is arcCos((V + √(V² + 8v₀²)) / (4v₀)). (10)
Q. No. 8. (a) Show that the law of force towards the pole, of a particle describing the curve rⁿ = aⁿ Cos nθ is given by f = (h² a²ⁿ (n + 1)) / r^(2n + 3). (10)
(b) A bar 2 ft. long of mass 10 lb., lies on a smooth horizontal table. It is struck horizontally at a distance of 6 inches from one end, the blow being perpendicular to the bar. The magnitude of the blow is such that it would impart a velocity of 3 ft./sec. to a mass of 2 lb. Find the velocities of the ends of the bar just after it is struck. (10)
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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
No. 1. (a) Solve the initial-value problem
dy/dx = 1 / (x + y²) ; y(-2) = 0 (10)
(b) Initially there were 100 milligrams of a radioactive substance present. After 6 hours the mass decreased by 3%. If the rate of decay is proportional to the amount of the substance present at any time, find the amount remaining after 24 hours. (10)
No. 2. (a) Solve (x² + 1)y” + xy’ – y = 0. (10)
(b) Obtain the partial differential equation by elimination of arbitrary functions,
a sin x + b cos y = z (take z as dependent variable). (10)
No. 3. (a) Solve the partial differential equation u_xx + u_yy = u_t ,
subject to the conditions:
u(0, y, t) = u(a, y, t) = 0
u(x, 0, t) = u(x, a, t) = 0
and the initial condition, u(x, y, 0) = ϕ(x, y). (10)
(b) Solve r + (a + b)s + abt = xy by Monge’s method. (10)
SECTION-B
No. 4. (a) Prove that if A_i , B_j , and C_k are three first order tensors, then their product A_i B_j C_k (i, j, k = 1, 2, 3) is a tensor of order 3, while A_i B_j C_k (i, j = 1, 2, 3) form a first order tensor. (10)
(b) If A_{i₁i₂i₃…i_n} is a tensor of order n, then its partial derivative with respect to x_p , that is ∂/∂x_p (A_{i₁i₂i₃…i_n}) is also a tensor of order n+1. (10)
Q. No. 5. (a) Show that the transformation ⎡x₁’⎤ 1 ⎡-3 -6 -2⎤ ⎡x₁⎤
⎢x₂’⎥ = ━ ⎢-2 3 -6⎥ ⎢x₂⎥ is orthogonal and
⎣x₃’⎦ 7 ⎣ 6 -2 -3⎦ ⎣x₃⎦
right-handed.
A second order tensor A_ij is defined in the system Ox₁x₂x₃ by
A_ij = x_i x_j i, j = 1,2,3. Evaluate its components at the point P where
x₁ = 0, x₂ = x₃ = 1. Also evaluate the component A’₁₁ of the tensor at P. (10)
(b) The Christoffel symbols of the second kind denoted by ⎧ m ⎫ are defined
⎩i j⎭
⎧ m ⎫ = g^mk [i j, k] (i, j, k = 1,2,…n).
⎩i j⎭
Prove that (i) ⎧ m ⎫ = ⎧ m ⎫, (ii) [i j, k] = g_mk ⎧ m ⎫,
⎩i j⎭ ⎩j i⎭ ⎩i j⎭
(iii) ∂g^ij/∂x^k = -g^im ⎧ j ⎫ – g^jm ⎧ i ⎫. (10)
⎩k m⎭ ⎩k m⎭
SECTION-C
Q. No. 6. (a) Apply Newton-Raphson’s method to determine a root of the equation
f(x) = cos x – xe^x = 0 such that |f(x^*)| < 10⁻⁵ , where x^* is the
approximation to the root. (10)
(b) Consider the system of the equations (10)
2x₁ – x₂ + 0x₃ = 7
-x₁ + 2x₂ – x₃ = 1
0x₁ – x₂ + 2x₃ = 1
Solve the system by using Gauss-Seidel iterative method and perform three
iterations.
Q. No. 7. (a) Use the trapezoidal and Simpson’s rules to estimate the integral
∫₁² f(x)dx = ∫₁² (x³ – 2x² + 7x – 5)dx . (10)
(b) Find the approximate root of the equation f(x) = 2x³ + x – 2 = 0 . (10)
Q. No. 8. (a) Find a 5ᵗʰ degree polynomial which passes through the 6 points given below. (10)
| x | 1.0 2.0 4.0 5.0 7.0 8.0 |
| f(x) | -9 -41 -189 -173 9 523 |
(b) Determine the optimal solution graphically to the linear programming problem,
Minimize z = 3x₁ + 6x₂
subject to 4x₁ + x₂ ≥ 20
x₁ + x₂ ≤ 20
x₁ + x₂ ≥ 10
x₁, x₂ ≥ 0 (10)
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