UPSC IFS 2017 Question Paper for Mathematics Paper-II

q'ql s.rq DETACHABLE
LR S. EXAM.-(M)2017
MATHEMATICS
Paper II

Time Allowed : Three Hours Maximum Marks: 200

QUESTION PAPER SPECIFIC INSTRUCTIONS

Please read each of the following instructions carefully before attempting questions.

There are EIGHT questions in all, out of which FIVE are to be attempted.

Question Nos. 1 and 5 are compulsory. Out of the remaining SIX questions, THREE are to be
attempted selecting at least ONE question from each of the two Sections A and B.

Attempts of questions shall be counted in sequential order. Unless struck off, attempt of a question
shall be counted even if attempted partly. Any page or portion of the page left blank in the
Question-cum-Answer Booklet must be clearly struck off.

All questions carry equal marks. The number of marks carried by a question /part is indicated
against it.

Answers must be written in ENGLISH only.

Unless otherwise mentioned, symbols and notations have their usual standard meanings.

Assume suitable data, if necessary and indicate the same clearly.

SECTION 'A'

1. (a) Prove that every group of order four is Abelian. 8

1. (b) A funct on f: IR—>R is defined as below:
Ix if x is rational
f (x)=
1— x if x is irrational

Prove that f is continuous at x = but discontinuous at all other points in IR. 10

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1. (c) If f(z)= u(x,y)+ iv(x,y) is an analytic function of z = x+ zy and
u + 2v = x3 - 2y3 + 3xy(2x -y) then find f(z) in terms of z. 8

1. (d) Solve by simplex method the following LPP :
Minimize Z = x1 -3x2 + 2x3
subject to the constraints
3x1 - x2 ± 2x3 'S. 7
2x + 4x2 s. 12
4x1 + 3x2 + 8x3 E. 0
and x1' x2' X3 0 14

2. (a) Let G be the set of all real numbers except -1 and define wob=a+b+ab
v a, beG. Examine if G is an Abelian group under *. 10
2. (b) Let H and K are two finite normal subgroups of co-prime order of a group G. Prove
that hk= kh vheH and keK. 10
2. (c) Let A be an ideal of a commutative ring R and
B = {xeR : xn EA for some positive integer n}.
Is B an ideal of R ? Justify your answer. 10
2. (d) Prove that the ring
Z[i] = fa + lb : a, b eZ, i = Vq} of Gaussian integers is a Euclidean domain. 10
3. (a) Evaluate 410, 0) and fyx(0, 0) given that

x2 tan-1 X - y2 tan-1 —
x if xy #0
f (x, .0= x Y 10
0 , otherwise

3.(b) Find the maximum and minimum values of x2 + y2 + z2 subject to the condition
2 2 2
x y Z
10
4 5 25

3. (c) Prove that smx dx is convergent but not absolutely convergent. 12
x
(d) Find the volume of the region common to the cylinders x2 + y2 = a2 and xz az.
8

(a) 1+2 cosO
Prove by the method of contour integration that dt9 = 0. 12
5+4cos0

4. (b) Find the sum of residues of f (z)= snz at its poles inside the circle lz I = 2. 8
cosz

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• Evaluate
co
r rx
ixe dy cbc 8
x=o y=o
A computer centre has four expert programmers. The centre needs four application
programs to be developed. The head of the centre after studying carefully the
programs to be developed, estimates the computer times in hours required by the
experts to the application programs as follows:
Programs
ABCD
5 3 2 8
7 9 2 6
6 4 5 7
5 7 7 8

Assign the programs to the programmers in such a way that total computer time is
least. 12
SECTION '13'
Form the partial differential equation by eliminating arbitrary functions yo and
from the relation z = (x2 - y)-+ (x2 + y). 8
Write a BASIC program to compute the multiplicative inverse of a non-singular
square matrix. 12
A uniform rectangular parallelopiped of mass M has edges of lengths 2a, 2b, 2c. Find
the moment of inertia of this rectangular parallelopiped about the line through its
centre parallel to the edge of length 2a. 10

2
Evaluate eThr dx using the composite trapezoidal rule with four decimal precision,

i.e., with the absolute value of the error not exceeding 5x10-5. 10

6.(a) Solve the partial differential equation:
az az
(x - y)-- + (x + y)—= 2xz 8
ax ay
6. (b) Find the surface which is orthogonal to the family of surfaces z(x +y)= c(3z + 1)
and which passes through the circle x2 + y2 = 1, z = 1. 8
az az
6. (c) Find complete integral of xp - yq = xq f(z - px - qy) where p = — , q = —. 12
ax ay

6.(d) A tightly stretched string with fixed end points x = 0 and x = 1 is initially in a

position given by y = yo sin3( . It is released from rest from this position, find
/
the displacement y(x, t). 12

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7. (a) Find the real root of the equation x3 + x2 + 3x + 4 = 0 correct up to five places of
decimal using Newton-Raphson method. 10
7. (b) A river is 80 metre wide, the depth y, in metre, of the river at a distance x from one
bank is given by the following table:
x 0 10 20 30 40 50 60 70 80
y 0 4 7 9 12 15 14 8 3
Find the area of cross-section of the river using Simpson's rd rule. 10

7. (c) Find y for x = 0.2 taking h= 0.1 by modified Euler's method and compute the error,
dy
given that: — = x+ y, y(0)=1. 10
dx

(d) Assuming a 32 bit computer representation of signed integers using 2's complement
representation, add the two numbers —1 and —1024 and give the answer in 2's
complement representation. 10
(a) Consider a mass m on the end of a spring of natural length 1 and spring constant k.
Let y be the vertical coordinate of the mass as measured from the top of the spring.
Assume that the mass can only move up and down in the Vertical direction.
Show that
= my,2 02 ±ingy
Also determine and solve the corresponding Euler-Lagrange equations of motion.
12
8.(b) Find the streamlines and pathlines of the two dimensional velocity field:
u— ,v y, w = 0. 8
1+t

8. (c) The velocity vector in the flow field is given by
= (az — by)i +(bx cz); + (cy — ax)k
where a, b, c are non-zero constants. Determine the equations of vortex lines. 8
a2u a2u
8. (d) Solve Laplace's equation + 0 subject to the conditions
5x2 83)2 —
in;rx).
u(0 , y)= u(1, y)= u(x, 0) = 0 and u(x, a) = sin 12
1)

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