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Css Applied Mathematics2 2009

This document appears to be an exam for Applied Math Paper II. It contains 8 questions testing various math skills across 3 sections: Section A contains differential equations questions; Section B contains tensor and transformation matrix questions; Section C contains numerical methods questions involving Newton's method, Gauss-Seidel iteration, Simpson's rule, Trapezoidal rule, Regula Falsi method, and Lagrange interpolation. Students are instructed to attempt 5 questions total, selecting at least 2 from Section A, 1 from Section B, and 2 from Section C.

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Bakhita Maryam
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61 views2 pages

Css Applied Mathematics2 2009

This document appears to be an exam for Applied Math Paper II. It contains 8 questions testing various math skills across 3 sections: Section A contains differential equations questions; Section B contains tensor and transformation matrix questions; Section C contains numerical methods questions involving Newton's method, Gauss-Seidel iteration, Simpson's rule, Trapezoidal rule, Regula Falsi method, and Lagrange interpolation. Students are instructed to attempt 5 questions total, selecting at least 2 from Section A, 1 from Section B, and 2 from Section C.

Uploaded by

Bakhita Maryam
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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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

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

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