APPLIED MATH, PAPER-I

FEDERAL PUBLIC SERVICE COMMISSION

COMPETITIVE EXAMINATION FOR S.No.
RECRUITMENT TO POSTS IN BPS-17 UNDER
THE FEDERAL GOVERNMENT, 2009
R.No.
APPLIED MATH, PAPER-I

TIME ALLOWED: 3 HOURS MAXIMUM MARKS:100

(i) Attempt FIVE question in all by selecting at least TWO questions from SECTION – A
NOTE: and THREE question from SECTION – B. All questions carry EQUAL marks.
(ii) Use of Scientific Calculator is allowed.

SECTION – A

Q.1. (a) Show that in orthogonal coordinates: (5+5)

∂
( A1 h1 ) − 3 ∂ ( A1 h1 ),
e e
(i) ∇ × ( A1 e1 ) = 2
h1 h3 ∂u 3 h1 h2 ∂u 2
∂
∇ • ( A1 e1 ) = ( A1 h2 h3 ).
1
(ii)
h1 h2 h3 ∂u1
(b) Write Laplace’s equation in parabolic cylindrical coordinates. (10)
ρ ρ ρ
Q.2. (a) Evaluate ∫∫ A • n ds , where A = ziˆ + xˆj − 3 y 2 kˆ and s is the surface of cylinder x2+y2=16
S

included in the first octant between z=0 and z=5. (10)

(b) Verify Green’s theorem in the plane for
∫ (xy + y )dx + x dy
2 2

where C is the closed curve of the region bounded by y = x and y = x2.

Q.3. (a) Find centre of mass of a right circular solid cone of height h. (10)
(b) A light thin rod, 4m long, can turn in a vertical plane about one of its point which is
attached to a pivot. If weights of 3kg and 4kg are suspended from its ends it rests in a
horizontal position. Find the position of the pivot and its reaction on the rod. (10)

SECTION – B

Q.4. (a) Find the radial and transverse components of the velocity of a particle moving along
the curve ax2+by2=1 at any time t if the polar angle θ = ct2. (10)
(b) A particle is projected vertically upwards. After a time t, another particle is sent up
from the same point with the same velocity and meets the first a height h during the
downward flight of the first. Find the velocity of projection. (10)

Q.5. (a) If a point P moves with a velocity v given by

v2=n2 (ax2+bx+c),
show that P executes a simple harmonic motion. Find also, the centre, the amplitude
and the time period of the motion. (10)

Page 1 of 2
APPLIED MATH, PAPER-I
(b) A particle of mass m moves on xy-plane under the force (10)
ϖ k ϖ
F = − 4 r,
r
where r is its distance from the origin O. If it starts on the positive x-axis at a distance “a” from
O with speed vo in a direction making an angle θ with the positive x-direction, prove that at
time t,
ma 2 v o2 sin 2 θ − k
&
r&=
mr 3
Q.6. (a) Define angular momentum and prove that rate of change of angular momentum of a
particle about a point O is equal to the tarque (about O) of the force acting on the particle. (10)
(b) Find the least speed with which a particle must be projected so that it passes through two
points P and Q at heights h1 and h2, respectively. (10)

Q.7. (a) Discuss the polar form of an orbit and prove that when a particle moves under central
force, the areal velocity is constant. (10)
(b) A particle moves under a central repulsive force μ/r3 and is projected from an apse at a
distance “a” with velocity V. Show that the equation to the path is (10)
r cosp θ = a
and that the angle θ described in time t is
1 pVt
tan −1 ,
p a
μ
where p2 =1+ , μ = GM .
a v2
2

Q.8. (a) Define the terms moment of inertia and product of inertia, and find the moment of
inertia of uniform solid sphere of mass m and radius “a”. (10)

(b) Let AB and BC be two equal similar rods freely hinged at B and lie in a straight line
on the smooth table. The end A is struck by a blow P perpendicular to AB. Show
1
that the resulting velocity of A is 3 times that of B. (10)
2
********************

Page 2 of 2
APPLIED MATH, PAPER-II
FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR S.No.
RECRUITMENT TO POSTS IN BPS-17 UNDER
THE FEDERAL GOVERNMENT, 2009
R.No.
APPLIED MATH, PAPER-II

TIME ALLOWED: 3 HOURS MAXIMUM MARKS:100

(i) Attempt FIVE question in all by selecting at least TWO questions from SECTION–A,
ONE question from SECTION–B and TWO questions from SECTION–C. All
NOTE:
questions carry EQUAL marks.
(ii) Use of Scientific Calculator is allowed.

SECTION – A
Q.1. (a) Using method of variation of parameters, find the general solution of the differential equation.
ex
′′ ′
y − 2y + y = . (10)
x
(b) Find the recurrence formula for the power series solution around x = 0 for the differential
equation
y ′′ + xy = e x +1 . (10)
Q.2. (a) Find the solution of the problem (10)
u ′′ + 6u ′ + 9u = 0
u (0) = 2 , u ′ (0) = 0

(b) Find the integral curve of the equation

∂z ∂z
xz + yz = −( x 2 + y 2 ) . (10)
∂x ∂y
Q.3. (a) Using method of separation of variables, solve (10)
∂ 2u ∂ 2u ⎧0 < x < 2
= 900 ⎨ ,
∂t ∂x ⎩t > 0
2 2

subject to the conditions

u (0, t ) = u (2, t ) = 0
∂u
u ( x,0) = 0 t = 0 = 30 sin 4 π x.
∂t
(b) Find the solution of (10)
∂ 2u ∂ 2u ∂ 2u
− 2 + 2 = 4e 3 y + cos x .
∂x 2
∂x∂y ∂y

SECTION – B
Q.4. (a) Define alternating symbol ∈ijk and Kronecker delta δ ij . Also prove that (10)
∈ijk ∈lmk = δ il δ jm − δ im δ jl .

(b) Usingϖtheϖtensor
ϖ notation,
ϖ ϖ prove ϖ thatϖ ϖ ϖ ϖ
∇ × (A × B) = A(∇ • B) − B(∇ • A ) + (B • ∇ ) A − (A • ∇ ) B (10)

Page 1 of 2
APPLIED MATH, PAPER-II

Q.5. (a) Show that the transformation matrix

⎡ 1 1 1⎤
⎢− 2
− ⎥
2⎥
⎢ 2
1 1 ⎥
T= ⎢ 0
⎢ 2 2⎥
⎢ 1 1 1⎥
⎢ − ⎥
⎢⎣ 2 2 2 ⎥⎦
is orthogonal and right-handed. (10)

(b) Prove that (10)

l ik l jk = δ ij
where l ik is the cosine of the angle between ith-axis of the system K ′ and jth-axis of the system
K.
SECTION – C
Q.6. (a) Use Newton’s method to find the solution accurate to within 10-4 for the equation (10)
x3–2x2 – 5 = 0, [1, 4].

(b) Solve the following system of equations, using Gauss-Siedal iteration method (10)
4x1 – x2 + x3 = 8,
2x1 + 5x2 + 2x3 = 3,
x1 + 2x2 + 4x3 = 11.

1
Q.7. (a) Approximate the following integral, using Simpson’s rules (10)
3
1

∫x e − x dx.
2

(b) Approximate the following integral, using Trapezoidal rule (10)

π /4

∫e
3x
sin 2 x dx.
0

Q.8. (a) The polynomial (10)

4 3 2
f(x) = 230 x + 18x + 9x – 221x – 9
has one real zero in [-1, 0]. Attempt approximate this zero to within 10-6, using the Regula Falsi
method.
(b) Using Lagrange interpolation, approximate. (10)
f(1.15), if f(1) = 1.684370, f(1.1) = 1.949477, f(1.2) = 2.199796, f(1.3) = 2.439189,
f(1.4) = 2.670324

********************

Page 2 of 2
 FEDERAL PUBLIC SERVICE COMMISSION
Roll Number
COMPETITIVE EXAMINATION FOR
RECRUITMENT TO POSTS IN BPS-17 UNDER
THE FEDERAL GOVERNMENT, 2010
APPLIED MATH, PAPER-I

TIME ALLOWED: 3 HOURS MAXIMUM MARKS:100

(i) Attempt FIVE question in all by selecting at least TWO questions from SECTION – A
NOTE: and THREE question from SECTION – B. All questions carry EQUAL marks.
(ii) Use of Scientific Calculator is allowed.
SECTION – A
Q.1. Explain the following giving examples and supported by figures: (5+5+5+5)
(a) Gradient
(b) Divergence
(c) Curl
(d) Curvilinear Coordinates
Q.2. Given that A,B,C are vectors having components along axis. Prove that: (10+10)
(a)
i j k
B x C  Bx B y Bz
Cx C y Cz
(b) A x B x C = Ax Bx Cx (i x k) + Ay Bx Cy (j x k)

Q.3. (a) State and prove Stokes Theorem (10)

(b) Given that V=4y i+x j + 2z k, find
(
x V). nd over the hemi sphere x2+y2+z2=a2, z>=0. (10)

SECTION – B

Q.4. Discuss the following systems supported by figures/diagrams:

(a)
 Equilibrium of a System coplanar forces (5)
 Centre of mass of right circular solid cone of height h. (5)
(b) Centre of gravity of a rigid body of any shape. (10)

Q.5. (a) What is Simple Harmonic Motion? Discuss it in detail using Derivatives with respect time. (10)
(b) Describe the Simple Harmonic Motion of a pendulum and Calculate the time period of the
motion. (10)

Q.6. (a) Derive expression for the following:

 Moment of inertia (5)
 Product of inertia (5)
(b) Calculate the moment of inertia of solid sphere of mass m=37 and radius a=15.
Derive the general expression. (10)

Q.7. (a) Explain Kepler’s Laws. (10)

(b) What is Impulsive Motion? Derive its equation. (10)

Q.8. (a) Define Work, Torque, Power and energy. (10)

(b) A cricket ball is thrown vertically upwards, it attained the maximum height h after t
Seconds. Calculate its. (10)
 Velocity of projection in direction vertically upward.
 Acceleration when it returns to the point of projection.
********************
 FEDERAL PUBLIC SERVICE COMMISSION
Roll Number
COMPETITIVE EXAMINATION FOR
RECRUITMENT TO POSTS IN BPS-17 UNDER
THE FEDERAL GOVERNMENT, 2010
APPLIED MATH, PAPER-II

TIME ALLOWED: 3 HOURS MAXIMUM MARKS:100

(i) Attempt FIVE question in all by selecting at least TWO questions from SECTION–A,
ONE question from SECTION–B and TWO questions from SECTION–C. All
NOTE:
questions carry EQUAL marks.
(ii) Use of Scientific Calculator is allowed.

SECTION – A
Q.1. Solve the following equations:
(a) d2y/dx2 + 5 dy/dx + 6y = x (10)
(b) d2y/dx2 + 5 y x = ex (10)
Q.2. (a) Derive Cauchy Rieman partial differential equations. (10)
(b) Derive Lapace Equation. (10)
Q.3. Solve:
(a) ( 2 / x 2   2 / xy   2 / y 2 ) u  4 e 3y (10)
(b) u” + 6u’ + 9=0; Given that u(0)=2 and u’(0)=0. (10)

SECTION – B
Q.4. (a) Discuss the following supported by examples:
 Tensor, (5)
 ijk lmk (5)
 Scaler Fields for a continuously differentiable function f=f(x,y,z) (5)
(b) Can we call a vector as Tensor, discuss.
What is difference between a vector and a tensor?
What happens if we permute the subscripts of a tensor? (5)
Q.5. (a) Discuss the simplest and efficient method of finding the inverse of a square matrix aij
of order 3x3. (10)
(b) Apply any efficient method to compute the inverse of the following matrix A: (10)
25 2 1 
A =  2 10 1
 1 1 4

SECTION – C
Q.6. (a) Develop Gauss Siedal iterative Method for solving a linear system of equations A x = b,
where A is the coefficient matrix. (10)
(b) Apply Gauss Siedal iterative Method to solve the following equations: (10)
25X1 + 2X2 + X3 = 69
2X1 + 10X2 + X3 = 63
X1 + 2X2 + X3 = 43
Q.7. (a) Derive Simpson’s Rule for finding out the integral of a function f(x) from limits x=a to x=b for
n=6 subintervals (i.e. steps). (10)
(b) Apply Simpson’s Rule for n=6 to evaluate: (10)
1

 f(x)dx
0
where f(x)  1/(1  x2).

Q.8. (a) Derive Lagrange Interpolation Formula for 4 points: (10)

(b) A curve passes through the following points:
(0,1),(1,2),(2,5),(3,10). Apply this Lagrange Formula to interpolate the polynomial. (10)

********************
 FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR Roll Number
RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT, 2011

APPLIED MATHEMATICS, PAPER-I

TIME ALLOWED: THREE HOURS MAXIMUM MARKS: 100

NOTE: (i) Attempt FIVE questions in all by selecting THREE questions from SECTION – A and TWO
questions from SECTION – B. All questions carry equal marks.
(ii) Use of Scientific Calculator is allowed.
(iii) Extra attempt of any question or any part of the attempted question will not be considered.

SECTION - A
 
Q.1. (a) Find the divergence and curl f If f  2 xyziˆ  ( x 2 z  2 y ) ˆj  ( x 2 y  3 z 2 )kˆ (10)

(b) Also find a function  such that   f (10)

Q.2. (a) Find the volume R xy dA where R is the region bounded by the line y = x – 1 and the parabola (10)
y2  2x  6 .

(b) Evaluate the following line intergral: (10)

c y dx  xdy where c  c 2 is the line segment joining the points (-5, -3) to (0, 2), and c =
2

c 2 is the arc of the parabola x = 4 - y 2 .

Q.3. (a) Three forces P, Q and R act at a point parallel to the sides of a triangle ABC taken in the same (10)
order. Show that the magnitude of the resultant is
p 2  Q 2  R 2  2QR cos A  2 RP cos B  2 PQ cos C

(b) A hemispherical shell rests on a rough inclined plane whose angle of friction is  . Show that (10)
the inclination of the plane base to the horizontal cannot be greater than arcsin(2 sin  )

Q.4. (a) A uniform square lamina of side 2a rests in a vertical plane with two of its sides in contact (10)
with two smooth pegs distant b apart and in the same horizontal line. Show that if

 b  a , a non symmetric position of equilibrium is possible in which b(sin   cos  )  a
2

(b) Find the centre of mass of a semi circular lamina of radius a whose density varies as the square (10)
of the distance from the centre.

Page 1 of 2
APPLIED MATHEMATICS, PAPER-I
Q.5. (a) (10)

1 x
Evaluate the integral ( x 2  y 2 )dydx
0 x2

also show that the order of integration is immaterial.

(b) Find the directional derivative of the function at the point P along z – axis (10)
f ( x , y )  4 xz 3  3 x 2 y 2 z , P  ( 2,1,2)

SECTION – B
Q.6. (a) A particle is moving along the parabola x 2 = 4ay with constant speed v. Determine the (10)
tangential and the normal components of its acceleration when it reaches the point whose
abscissa is 5a

(b) Find the distance travelled and the velocity attained by a particle moving in a straight line at (10)
any time t, if it starts from rest at t =0 and is subject to an acceleration t 2  sin t  e t

Q.7. (a) A particle moves in the xy – plane under the influence of a force field which is parallel to the (10)
axis of y and varies as the distance from x – axis. Show that, if the force is repulsive, the path
of the particle supposed not straight and then

y = a cosh nx + a sinh nx

where a and b are constants.

(b) Discuss the motion of a particle moving in a straight line with an acceleration x 3 , where x is (10)
the distance of the particle from a fixed point O on the line, if it starts at t = 0 from a point x = c
with the velocity c 2 / 2 .

Q.8. (a) A battleship is steaming ahead with speed V and a gun is mounted on the battleship so as to (10)
point straight backwards and is set at angle of elevation  . If v 0 is the speed of projection
2v
(relative to the gun) show that the range is 0 sin  (v 0 cos   V )
g

(b) Show that the law of force towards the pole of a particle describing the survey r n  a n cos n (10)
( n  1)h 2 a 2 n
is given by f  where h is a constant.
r 2 n 3

*********

Page 2 of 2
 FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR Roll Number
RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT, 2011

APPLIED MATHEMATICS, PAPER-II

TIME ALLOWED: THREE HOURS MAXIMUM MARKS: 100

NOTE: (i) Attempt FIVE questions in all by selecting THREE questions from SECTION – A and TWO
questions from SECTION – B. All questions carry equal marks.
(ii) Use of Scientific Calculator is allowed.
(iii) Extra attempt of any question or any part of the attempted question will not be
considered.

SECTION - A

Q.1. (a) Solve by method of variation of parameter (10)

d2y dy
2
 2  y  xe x 1nx
dx dx
(b) Solve first order non-linear differential equation (10)
dy
x  y  y 2 1nx
dx
Q.2. (a) Solve (10)
c 2u u .
xx tt
u (0, t )  0
u (l , t )  0
 
u ( x,0)   Sin x 
l 
u t ( x,0)  0
z z (10)
(b) Solve x2  y2  ( x  y) z
x y
Q.3. (a) Work out the two dimensional metric tensor for the coordinates p and q given by (10)

 
1 1
p  ( xy ) , q  x 2 / y
3 3

 d  g dc  g 
1 (10)
g g
ab, c 
(b) Prove that
ab 2  ac, b bc, a

Page 1 of 2
APPLIED MATHEMATICS, PAPER-II
Q.4. (a) Work out the Christoffel symbols for the following metric tensor (10)
1 0 
g   
ab  0 r 2 

(b) Work out the covariant derivative of the tensor with components (10)
 r cos  ar sin  ar 
 
 sin  sin  a sin  cos  a
 cos  a sin  0 

Q.5. (a) Find recurrence relations and power series solution of ( x  3) y   2 y  0 (10)
(b) Solve the Cauchy Euler’s equation x 4 y  2 x 3 y  x 2 y  xy  1 (10)

(10)
SECTION – B

Q.6. (a) Find the positive solution of the following equation by Newton Raphson method (10)
2 sin x = x
(b) Solve the following system by Jacobi method: (10)
10x 1 - 8x 2 = - 6
- 8x 1 +10x 2 - x 3 = 9
- x 2 +10x 3 = 28

Q.7. (a) Evaluate the following by using the trapezoidal rule. (10)

0 ( x  1)dx
1

(b) Evaluate the following integral by using Simpson’s rule (10)

4 x
0 e dx
Q.8. (a) Solve the following equation by regular falsi method: (10)
2x 3  x  2  0
(b) Calculate the Lagrange interpolating polynomial using the following table: (10)

x 0 1 2
f(x) 1 0 -1
also calculate f (0.5).

*********

Page 2 of 2
 FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR Roll Number
RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT, 2012
APPLIED MATHS, PAPER-II

TIME ALLOWED: THREE HOURS MAXIMUM MARKS: 100

NOTE:(i) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q. Paper.
(ii) Attempt FIVE questions in all by selecting TWO questions from SECTION-A and ONE
question from SECTION-B and TWO questions from SECTION-C. ALL questions carry
EQUAL marks.
(iii) Extra attempt of any question or any part of the attempted question will not be considered.
(iv) Use of Scientific Calculator is allowed.
SECTION-A

Q. 1. Solve the following differential equations:

ex (10)
(a) y   3 y   2 y  
1  e x
2 xye  x / y 
2

(b) 
y  (10)
y 2  y 2ex / y   2x 2ex / y 
2 2

Q. 2. (a) Find the series solution of the following differential equation:

y  xy  0 (10)
(b) Use the method of Fourier integrals to find the solution of initial value problem
with the partial differential equation.
u 2  u
2
c ;    x   
t x 2
And with initial condition u  x, 0   f  x  (10)

Q. 3. (a) Solve x 2 y  3 xy  5 y  x 2 sin ln x  (10)

(b) Find the solution of wave equation
 2u 2  u
2
 c with boundary and initial conditions
t 2 x 2
u  x, t 
u 0, t   u l , t   0 , u  x, 0   f  x ,  g x  (10)
t

SECTION-B

Q. 4. Discuss the following terms: (5x4=20)

(i) Tensors (ii) Kronecker delta
(iii) Contraction (iv) Metric Tensor
(v) Contravariant tensor of order two

Q. 5. i   (10)
(a) Prove that    i (log g )
ij  x
 m1  m 2  m 3  mi  mj  mk
(b) Prove that    n1  n 2  n 3  mnp and ijk mnp   ni  nj  nk
 p1  p 2  p 3  pi  pj  pk
Hence prove that ijk mnp   im jn   in jm (10)

Page 1 of 2
APPLIED MATHS, PAPER-II
SECTION-C
Q. 6. (a) (i) What is the difference between secant and false position method?
Show also graphically. (5+5=10)
f ( xn)
(ii) Prove that x n 1  x n  2
f xn 
(b) Solve the following system by Jacobi method. (Up to four decimal places).
8x + y – z = 8
2x + y + 9z = 12
x – 8y + 12z = 35

Q. 7. 3 (10)
(a) Evaluate by Simpson’s rule
8
3
 0
x 1  x 2 dx ; with n = 6
Also calculate the absolute error.
(b) The amount A of a substance remaining in a reacting system after an interval of
time t in a certain chemical experiment is given by following data:
A: 94.8 87.9 81.3 68.7
t: 2 5 8 14
Find t when A=80. (10)

Q. 8. (a) If f(x) = x3, show that f(a,b,c) = a + b + c (10)

(b) Solve by trapezoidal rule
2 (10)
 0
x sin x dx ; with n = 8

***********
 FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR
Roll Number
RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT, 2013
APPLIED MATHEMATICS, PAPER-I
TIME ALLOWED: THREE HOURS MAXIMUM MARKS: 100
NOTE: (i) Candidate must write Q.No. in the Answer Book in accordance with Q.No. in the Q.Paper.
(ii) Attempt FIVE questions in all by selecting THREE questions from SECTION-A and TWO
questions from SECTION-B. All questions carry EQUAL marks.
(iii) Use of Calculator is allowed.
(iv) Extra attempt of any question or any part of the attempted question will not be considered.
SECTION-A

Q.1. (10)
(a) Find a function φ such that φ= f

f  x ˆ  2 yˆj  2 kˆ
(b) Prove that (10)
  n  n  n 1 
 
Q.2. (10)
(a) Show that for any vectors a and b
  2   2 2 2
a b  a b  2 a  b 
 
 
(b) Prove that (10)
2
 
 
 
 
 
  

 
 a b    b c    c a    a  b c 
       

Q.3. (a) The greatest result that two forces can have is of magnitude P and the least is of (10)
magnitude Q. Show That when they act an angle α their resultant is of magnitude
P 2 cos 2  / 2  Q 2 sin 2  / 2
(b) A uniform rod of length 2a rests in equilibrium against a smooth vertical wall and upon a (10)
smooth peg at a distance b from the wall. Show that in the position of equilibrium the rod
1

1  b 
3
is inclined to the wall at an angle sin  
a
Q.4. (a) Three forces P, Q and R act along the BC, CA and AB respectively of triangle ABC. (10)
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.

(b) A sphere of weight W and radius a is suspended by a string of length l from a point P (10)
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
 wa 
sin 1  
 (W  w)(a  l) 

Page 1 of 2
APPLIED MATHS, PAPER-I

Q.5.
(a) Find the volume  R
x 3
 4y  dA where R is the region bounded by the
(10)

parabola y  x 2 and the line y = 2x.

(b) Evaluate the following line integral (10)


2
x dy
c
bonded by the triangle having the vertices (-1,0) to (2,0), and (1,1)

SECTION-B

Q.6. (a) The position of a particle moving along an ellipse is given by r  a cos t ˆ  b sin tˆj . If (10)
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.
Q.7. (a) A particle is moving with the uniform speed v along the curve (10)
 a  2
x 2 y  a  x 2  
 5 
10v 2
Show that its acceleration has the maximum value at
9a
(b) An aeroplane is flying with uniform speed v0 in an arc of a vertical circle of radius a, (10)
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
 
KY 2  Y a 2  2hK  K h 2  a 2  0, 
2
ga
where K  h 
2v 2 0
Q.8. (a) Find the tangential and normal components of the acceleration of a particle describing (10)
the ellipse
x2 y2
 1
a2 b2
With uniform speed v when the particle is a a > b
(b) Find the velocity acquired by a block of wood of mass M lb., which is free to recoil when (10)
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
Mm 2
the wood to the bullet, supposed uniform, is lb.wt. and that the time of
2( M  m) ga
2a ma
penetration is sec., during which time the block will move ft.
v mM

*********

Page 2 of 2
 FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR
RECRUITMENT TO POSTS IN BS-17 Roll Number
UNDER THE FEDERAL GOVERNMENT, 2013

APPLIED MATHEMATICS, PAPER-II

TIME ALLOWED: THREE HOURS MAXIMUM MARKS: 100

NOTE: (i) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q. Paper.
(ii) Attempt FIVE questions in all by selecting TWO questions from SECTION-A and ONE
question from SECTION-B and TWO questions from SECTION-C ALL questions carry
EQUAL marks.
(iii) Extra attempt of any question or any part of the attempted question will not be considered.
(iv) Use of Calculator is allowed.

SECTION-A
Q.No.1. Solve the following equations:
d 3 y dy
(a) 3
  Sec 2 x (10)
dx dx
2dy x
(b)   x 3Cos y  0 (10)
dx y

Q.No.2. (a) Find the power series solution of the differential equation (10)
(1  x 2 ) y //  2 xy /  2 y  0 , about the point x= 0.
Z Z
(b) Solve Z(x+y)  Z ( x  y)  (x2  y 2 ) . (10)
x y

Q.No.3. (a) Classify the following equations: (5)

2Z 2  Z
2
1 Z
(i)  x  0
x 2
y 2
x x
2Z 2Z 2  Z
2

(ii) x2  2 xy  y  4x 2
x 2
xy y 2

u  2 u (15)
(b) Solve:  ,  1  x  1, t  0
t x 2
u u
u(-1, t) = u(1, t); (1, t )  (1, t ) for t > 0
x x
u(x, o) = x+1, -1< x <1.

SECTION-B
Q.No.4. (a) Highlight the difference between a vector and a tensor. What happens if we (5)
permute the subscripts of a tensor?
1 0 
Transform g   0 
ab
(b) 1 2  into Cartesian coordinates. (15)
 r 
Page 1 of 2
APPLIED MATHEMATICS, PAPER-II

 a2 0  (10)
Q.No.5. (a) Workout the Christoffel symbols for the metric tensor ab 
g  2 

 0 a2
sin  
(b) Workout the two dimensional metric tensor for the coordinates p and q given by (10)
1 1

p  ( xy ) , q   x 
2 3

y 
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 Sinx = 1 + x3 Using Newton-Raphson method. (10)
x
Q.No.7. (a) Find the root of xe = 3 by regular falsi method correct to three decimal places. (10)
10
dx
(b) Evaluate  using (5+5) (10)
0 1 x
2

(i) Trapezoidal rule and

(ii) Simpson’s rule.

Q.No.8. (a) Find the real root of the equation Cosx = 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
*************

Page 2 of 2
 FEDERAL PUBLIC SERVICE COMMISSION Roll Number
COMPETITIVE EXAMINATION FOR
RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT, 2014
APPLIED MATHEMATICS, PAPER-I
TIME ALLOWED:
MAXIMUM MARKS: 100
THREE HOURS
NOTE:(i) Candidate must write Q.No. in the Answer Book in accordance with Q.No. in the Q.Paper.
(ii) Attempt FIVE questions in all by selecting THREE questions from SECTION-A and TWO
questions from SECTION-B. ALL questions carry EQUAL marks.
(iii) No Page/Space be left blank between the answers. All the blank pages of Answer Book must
be crossed.
(iv) Extra attempt of any question or any part of the attempted question will not be considered.
(v) Use of Calculator is allowed.

SECTION-A
  
Q. No. 1. (a) prove that curl (F )  ( grad ) x F . If F is irrotational and  ( x, y, z ) is a scalar (10)
function.
(b) Determine whether the line integral: (10)

 (2 xyz dx  ( x z  zCosyz )dy  (2 x yz  yCosyz )dz is independent of the

2 2 2 2


path of integration? If so, then compute it from (1,0,1) to (0, ,1).
2

Q. No. 2. (a) State and prove Stoke’s Theorem. (10)

(b) Verify Stoke’s Theorem for the function F  x 2 i  xy j integrated round the (10)
square in the plane z=0 and bounded by the lines x = y = 0, x = y = a.

Q. No. 3. (a) Three forces act perpendicularly to the sides of a triangle at their middle points (10)
and are proportional to the sides. Prove that they are in equilibrium.
(b) Three forces P, Q, R act along the sides BC, CA, AB respectively of a triangle (10)
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.

Q. No. 4. (a) Find the centroid of the surface formed by the revolution of the cardioide (10)
r  a (1  Cos ) about the initial line.
(b) A uniform ladder rests with its upper end against a smooth vertical wall and its (10)
foot on rough horizontal ground. Show that the force of friction at the ground is
1
W tan  , where W is the weight of the ladder and  is its inclination with the
2
vertical.

Q. No. 5. (a) Define briefly laws of friction give atleast one example of each law. (10)
(b) A uniform semi-circular wire hangs on a rough peg, the line joining its (10)
extremities making an angle of 45o with the horizontal. If it is just on the point
of slipping, find the coefficient of friction between the wire and the peg.

Page 1 of 2
APPLIED MATHEMATICS, PAPER-I
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 (10)
executes a simple harmonic motion. Find the center, the amplitude and the
time-period of the motion.
(b) A particle P moves in a plane in such a way that at any time t its distance from a (10)
fixed point O is r= at+bt2 and the line connecting O and P makes an angle
3
  ct 2 with a fixed line OA. Find the radial and transverse components of the
velocity and acceleration of the particle at t =1.

Q. No. 7. (a) A particle of mass m moves under the influence of the force (10)
F  a (i Sin  t  j Cos  t ). If the particles is initially at rest on the origin,
a2
prove that the work done upto time t is given by (1  Cos t ), and that
m 2
a2
the instantaneous power applied is Sin t .
m 2
(b) A battleship is streaming ahead with speed V, and a gun is mounted on the (10)
battleship so as to point straight backwards, and is set an angle of elevation a ,
if vo is the speed of projection relative to the gun, show that the range is
2vo
Sin (vo Cos  V ) . Also prove that the angle of elevation for maximum
g
 V  V 2  8v 2 
range is arcCos 0 .
 4v 0 
 

Q. No. 8. (a) Show that the law of force towards the pole, of a particle describing the curve (10)
(n  1) h 2 a 2 n
r  a Cos n
n n
is given by f  .
r 2 n 3
(b) A bar 2 ft. long of mass 10 Ib., lies on a smooth horizontal table. It is struck (10)
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 Ib. Find the velocities of the ends of the
bar just after it is struck.

*************

Page 2 of 2
 FEDERAL PUBLIC SERVICE COMMISSION Roll Number
COMPETITIVE EXAMINATION FOR
RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT, 2014
APPLIED MATHEMATICS, PAPER-II
TIME ALLOWED:
MAXIMUM MARKS: 100
THREE HOURS
NOTE:(i) Candidate must write Q.No. in the Answer Book in accordance with Q.No. in the Q.Paper.
(ii) Attempt FIVE questions in all by selecting TWO questions from SECTION-A, ONE
question from SECTION-B and TWO questions from SECTION-C. ALL questions carry
EQUAL marks.
(iii) No Page/Space be left blank between the answers. All the blank pages of Answer Book must
be crossed.
(iv) Extra attempt of any question or any part of the attempted question will not be considered.
(v) Use of Calculator is allowed.

SECTION-A
Q. No. 1. (a) Solve the initial-value problem (10)
dy 1
 ; y ( 2)  0 .
dx x  y 2

(b) Initially there were 100 milligrams of a radioactive substance present. After 6 (10)
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.

Q. No. 2. (a) Solve ( x 2  1) y   xy   y  0 . (10)

(b) Obtain the partial differential equation by elimination of arbitrary functions, (10)
a sin x  b cos y  z (take z as dependent variable).

Q. No. 3. (a) Solve the partial differential equation u xx  u yy  ut , (10)

u (0, y, t )  u (a, y, t )  0
subject to the conditions
u ( x, 0, t )  u ( x, a, t )  0
and the initial condition, u ( x, y , 0)   ( x, y ) .
(10)
(b) Solve r  (a  b) s  abt  xy by Monge’s method.

SECTION-B
Q. No. 4. (a) Prove that if Ai , B j , and Ck are three first order tensors, then their product (10)
Ai B j Ck (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.
A ... (10)
(b) If i1i2i3 in is a tensor of order n, then its partial derivative with respect to x p

that is Ai i i ...i is also a tensor of order n+1.
x p 123 n

Page 1 of 2
APPLIED MATHEMATICS, PAPER-II

 x1   3 6 2   x1 
Show that the transformation  x2    2 3 6   x2  is orthogonal and
  1
Q. No. 5. (a) (10)
7
 x3   6 2 3  x3 
right-handed.
A second order tensor Aij is defined in the system Ox1 x2 x3 by
Aij  xi x j i, j  1, 2,3. Evaluate its components at the point P where
x1  0, x2  x3  1. Also evaluate the component A11 of the tensor at P.
m 
(b) The Christofell symbols of the second kind denoted by   are defined (10)
 ij 
m 
   g ij , k  (i, j , k  1, 2,...n).
mk

 
ij
m   m  m 
Prove that (i)      , (ii) [ij , k ]  g mk   ,
 ij   ji   ij 
g ij
 i   i 
(iii)   g im    g jm   .
x k
 km   km 

SECTION-C
Q. No. 6. (a) Apply Newton-Raphson’s method to determine a root of the equation (10)
f ( x)  cos x  xe x  0 such that f ( x* )  10 8 ,where x* is the
approximation to the root.

(b) Consider the system of the equations (10)

2 x1  x2  0 x3  7
 x1  2 x2  x3  1 ,
0 x1  x2  2 x3  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 (10)
3 3

 f ( x)dx   ( x  2 x 2  7 x  5) dx .
3

1 1

(b) Find the approximate root of the equation f ( x)  2 x3  x  2  0 . (10)

Q. No. 8. (a) Find a 5thdegree 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, (10)
Minimize z  3x1  6 x2
subject to 4 x1  x2  20
x1  x2  20
x1  x2  10
x1 , x2  0
************* Page 2 of 2
 FEDERAL PUBLIC SERVICE COMMISSION Roll Number
COMPETITIVE EXAMINATION FOR
RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT, 2015

APPLIED MATHEMATICS, PAPER-I

TIME ALLOWED: THREE HOURS MAXIMUM MARKS = 100

NOTE: (i) Attempt ONLY FIVE questions in all, by selecting THREE questions from SECTION-I and
TWO questions from SECTION-II. ALL questions carry EQUAL marks.
(ii) All the parts (if any) of each Question must be attempted at one place instead of at different
places.
(iii) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
(iv) No Page/Space be left blank between the answers. All the blank pages of Answer Book must
be crossed.
(v) Extra attempt of any question or any part of the attempted question will not be considered.
(vi) Use of Calculator is allowed.

SECTION-I
        
Q. No. 1 (a) Prove that ( A  B)  ( B  C )  (C  A)  2[ A  ( B  C )] . (10)
   x2 y 2
(b) If A  ( x  3 y ) iˆ  ( y  2 x) ˆj , evaluate  A.dr where c is an ellipse  1 (10)
c
9 4
in the xy- plane traversed in the positive direction.

Q. No. 2 (a) Determine the expression for divergence in orthogonal curvilinear coordinates. (10)
(b) Determine the unit vectors in spherical coordinate system. (10)

Q. No. 3 (a) A particle moves from rest at a distance “a” from a fixed point O where the (10)
5

acceleration at distance x is  x 3
. Show that the time taken to arrive at O is given
4
a3
by an equation of the form t  A , where A is a number.


(b) Three forces P, Q, R acting at a point, are in equilibrium, and the angle between (10)
P and Q is double of the angle between P and R. Prove that R 2  Q(Q  P) .

Q. No. 4 (a) AB and AC are similar uniform rods, of length a, smoothly joined at A.BD is a (10)
weightless bar, of length b, smoothly joined at B, and fastened at D to a smooth
ring sliding on AC. The system is hung on a small smooth pin at A. Show that the
b
rod AC makes with the vertical an angle tan 1 .
a  a 2  b2
2 2 2

(b) Find the centroid of the arc of the curve x 3  y 3  a 3 lying in the first quadrant. (10)

Q. No. 5 (a) A hemispherical shell rests on a rough inclined plane whose angle of friction is (10)
 . Show that the inclination of the plane base to the horizontal cannot be greater
than sin 1 (2sin  ) .
(b) A regular octahedron formed of twelve equal rods, each of weight w, freely (10)
jointed together is suspended from one corner. Show that the thrust in each
3
horizontal rod is 2w .
2
Page 1 of 2
APPLIED MATHEMATICS, PAPER-I

SECTION-II
a2
Q. No. 6 (a) A particle is moving with uniform speed v along the curve x 2 y  a( x 2  ). (10)
5
10v 2
Show that its acceleration has the maximum value .
9a
(b) Discuss the motion of a particle moving in a straight line if it starts from rest at a (10)
distance a from a point O and moves with an acceleration equal to  times its
distance from O.

Q. No. 7 (a) Prove that the force field (10)

F  ( y 2  2 xyz 3 ) i  (3  2 xy  x 2 y 3 ) j  (6 z 3  3x 2 yz 2 )k
is conservative, and determine its potential.
(b) The components of velocity along and perpendicular to the radius vector form a (10)
fixed origin are respectively  r 2 and  2 .
Find the polar equation of the path of the particle in terms of r and  .

Q. No. 8 (a) A particle is projected horizontally from the lowest point of a rough sphere of (10)
radius a. After describing an arc less than a quadrant, it returns and comes to rest
2ag (1   2 )
at the lowest point. Show that the initial speed must be (sin  ) ,
(1  2  2 )
Where  is the coefficient of friction and a is the arc through which the
particle moves.
(b) The law of force is Mu 2 and a particle is projected from or apse at distance a. Find (10)
M
the orbit when the velocity of the projection is 2 .
a

Page 2 of 2
 FEDERAL PUBLIC SERVICE COMMISSION Roll Number
COMPETITIVE EXAMINATION FOR
RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT, 2015

APPLIED MATHEMATICS, PAPER-II

TIME ALLOWED:
MAXIMUM MARKS: 100
THREE HOURS
NOTE:(i) Attempt FIVE questions in all by selecting TWO questions from SECTION-A, ONE question
from SECTION-B and TWO questions from SECTION-C. ALL questions carry EQUAL
marks.
(ii) Candidate must write Q.No. in the Answer Book in accordance with Q.No. in the Q.Paper.
(iii) All the parts (if any) of each Question must be attempted at one place instead of at different
places.
(iv) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
(v) No Page/Space be left blank between the answers. All the blank pages of Answer Book must
be crossed.
(vi) Extra attempt of any question or any part of the attempted question will not be considered.
(vii) Use of Calculator is allowed.
SECTION-A
Q. No. 1. (a) Solve the initial value problem. (10)
dy y x
  3 , y (1)  2
dx 2 x y

(b) Solve y   4 y '4 y  e 2 x (10)

Q. No. 2. Solve the following equations: (10)

d2y dy
(a) (1  x 2 ) 2  2 x  2 y  0
dx dx
d 3 y dy
(b)   cos ecx
dx 3 dx (10)

Q. No. 3. (a) Classify the following: (5 each) (10)

(i) x 2U xx  (a 2  y 2 )U yy  0 ,   x  ,  a  y  a

(ii) U xx  6U xy  9U yy  3 y  0
(b) Solve (10)
 2u  2u
 0 x5
 2t  2 x
u (0, t )  u (5, t )  0
u ( x,0)  x 2 ( x  5)
ut ( x,0)  0

SECTION-B
Q. No. 4. (a) Prove that if Ai and B j are two first order tensors, then their product (7)
Ai B j (i, j  1,2,3) is a second order tensor.

(b) If  ( x1 , x 2 , x3 ) is a scalar point function then are the components of a first
xi (7)
order tensor.
(c) Find the invariant of the following second order tensor (6)
2 4  1
6  7 10 
 
3  4 6 

Page 1 of 2
APPLIED MATHEMATICS, PAPER-II

Q. No. 5. (a) Verify that the transformation (7)

1
x1  (5 x1  14 x2  2 x3 )
15
1
x2   (2 x1  x2  2 x3 )
3
1
x3  (10 x1  2 x2  11x3 )
15

Is orthogonal and right handed. A vector field A is defined in the system
Ox1 x 2 x3 by A1  x12 , A2  x 22 , A3  x 32
Evaluate the components Aj of the vector field in the new system Ox1 x 2 x 3 .
(b) Prove that any second order tensor Aij can be written as the sum of a deviator (7)
and an isotropic tensor.
(c) If aij  a ji are constants. Calculate. (6)
2
(aij X i X j )
X k X m

SECTION-C
Q. No. 6. (a) Find the real root of the equation by using Newton – Raphson’s method. (10)
3 x  cos x  1  0

(b) Solve the following system of equations by Gauss-Seidel method.

Take initial approximation as x1  0, x 2  0, x3  0 . Perform 3 Iterations. (10)
20 x1  x 2  2 x3  17
3 x1  20 x 2  x3  18
2 x1  3 x 2  20 x3  25

Q. No. 7. (a) Find the real root of the equation x3  4x  9  0 by Regular falsi method. Take (10)
the interval of the root as (2,3) and perform 4 iterations. .

(b) Find a polynomial which possess through the following points:

(10)
x: 1 0 1 2
f ( x) : 2 1 2 5

Q. No. 8. (a) Use the langrage’s Interpolation formula to find the value f(12) if the values of x (10)
and f(x) are given below
x: 5 7 11 13
f(x) 150 392 1452 2366
(10)
1
(b) Evaluate  30 x 1  x 2 dx using Simpson’s rule and trapezoidal rule for n = 6
3

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 FEDERAL PUBLIC SERVICE COMMISSION
Roll Number
COMPETITIVE EXAMINATION-2016
FOR RECRUITMENT TO POSTS IN BS-17
UNDER THE FEDERAL GOVERNMENT

APPLIED MATHEMATICS

TIME ALLOWED: THREE HOURS MAXIMUM MARKS = 100

NOTE:(i) Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
(ii) All the parts (if any) of each Question must be attempted at one place instead of at different
places.
(iii) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
(iv) No Page/Space be left blank between the answers. All the blank pages of Answer Book must
be crossed.
(vi) Extra attempt of any question or any part of the attempted question will not be considered.
(v) Use of Calculator is allowed.

Q. No. 1. f (r) r⃑ 2 (10)

(a) Prove that . r
= r f ( r ) + f ( r )

(b) Verify Stokes’ theorem for ⃑ = (2x – y) i – y z 2 j – y 2 z k , where S is (10)

the upper half surface of the sphere x 2 + y 2 + z 2 = 1 and C is its
boundary .

Q. No. 2. (a) Forces P, Q, R act at a point parallel to the sides of a triangle ABC taken in the (10)
same order. Show that the magnitude of the resultant force is
2 2 2
P + Q + R – 2 Q R cos A – 2 R P cos B – 2 P Q cos C
(b) Find the distance from the cusp of the centroid of the region bounded by the (10)
cardioid r = a (1 + cos ).

Q. No. 3. (a) A particle describes simple harmonic motion in such a way that its velocity and (10)
acceleration at a point P are u and f respectively and the corresponding quantities
at another point Q are v and g. Find the distance PQ.
(b) Derive the radial and transverse components of velocity and acceleration of a particle. (10)

Q. No. 4. Solve the following differential equations:

dy y 3 4 (10)
(a) + = x y
dx x
2 3 2x
(b) (D – 5D + 6) y = x e (10)

Q. No. 5. (a) Solve the differential equation using the method of variation of parameters (10)
2
d y  
+ y = tan x , –2<x<2
d x2
(b) Solve the Euler – Cauchy differential equation x2 y – 3 x y + 4y = x2 ln x. (10)

Q. No. 6. (a) Find the Fourier series of the following function: (10)
 – x if –  < x < 0
f(x) = 
 x if 0 < x < 
(b) Solve the initial - boundary value problem: (10)
1
APPLIED MATHEMATICS

Q. No. 7. (a) Apply Newton – Raphson method to find the smaller positive root of the equation (10)
x2 – 4x + 2 = 0
(b) Solve the following system of equations by Gauss – Seidel iterative method by (10)
taking the initial approximation as x1 = 0, x2 = 0, x3 = 0:
5x1 + x2 – x3 = 4
x1 + 4x2 + 2x3 = 15
x1 – 2x2 + 5x3 = 12

Q. No. 8. 1
dx
(a) Approximate  1+x
2 using (10)
0
(i) Trapezoidal rule with n = 4 (ii) Simpson’s rule with n = 4
Also compare the results with the exact value of the integral.

(b) Apply the improved Euler method to solve the initial – value problem: (10)
y = x + y, y (0) = 0
by choosing h = 0.2 and computing y1, …, y5.

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