APPLIED MATH, PAPER-II
FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION FOR
RECRUITMENT TO POSTS IN BPS-17 UNDER
THE FEDERAL GOVERNMENT, 2009
APPLIED MATH, PAPER-II
TIME ALLOWED: 3 HOURS MAXIMUM MARKS:100
NOTE:
(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
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.
x
y y y e
x
′′ − 2 ′ + = . (10)
(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)
(0) 2, (0) 0
6 9 0
= ′ =
′′ + ′ + =
u u
u u u
(b) Find the integral curve of the equation
(x2 y 2 )
y
yz z
x
xz z = − +
∂
∂
+
∂
∂
. (10)
Q.3. (a) Using method of separation of variables, solve (10)
900 2
2
2
2
x
u
t
u
∂
∂
=
∂
∂
⎩ ⎨ ⎧
>
< <
0
0 2
t
x
,
subject to the conditions
( ,0) 0 30 s 4 .
(0, ) (2, ) 0
0 in x
t
u x u
u t u t
t = π
∂
∂
=
= =
=
(b) Find the solution of (10)
e x
y
u
x y
u
x
u 2 4 3 y cos
2
2 2
2
2
= +
∂
∂
+
∂ ∂
∂
−
∂
∂
.
SECTION – B
Q.4. (a) Define alternating symbol ijk ∈ and Kronecker delta ij δ . Also prove that (10)
ijk ∈ . lmk il jm im jl ∈ = δ δ − δ δ
(b) Using the tensor notation, prove that
(A B) A( B) B( A) (B )A (A )B
ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ
∇× × = ∇• − ∇• + •∇ − •∇ (10)
S.No.
R.No.
APPLIED MATH, PAPER-II
Page 2 of 2
Q.5. (a) Show that the transformation matrix
T =
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
⎦
⎤
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎣
⎡
−
− −
2
1
2
1
2
1
2
1
2
0 1
2
1
2
1
2
1
is orthogonal and right-handed. (10)
(b) Prove that (10)
ik jk ij l l = δ
where ik l 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.
Q.7. (a) Approximate the following integral, using Simpson’s
3
1 rules (10)
∫ −
1
0
x2e xdx.
(b) Approximate the following integral, using Trapezoidal rule (10)
∫
/ 4
0
3 sin 2 .
π
e x x dx
Q.8. (a) The polynomial (10)
f(x) = 230 x4 + 18x3 + 9x2 – 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
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Service commission Papers, Combined Competitive Exam papers, CSS past papers, Combined Competitive exams, FPCs papers, Fedral Public
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