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INDIAN FOREST SERVICE (MAINJEXAM. 2023
BJKE-B-MTH
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.
Questions no. 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.
Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Assume suitable data, if necessary, and indicate the same clearly.
Answers must be written in ENGLISH only.
BJKE-B-MTH 1
Ql. (a)
(b)
(c)
(d)
(e)
Q2. (a)
(b)
BJKE-B-MTH
SECTION A
Prove that a subgroup of a cyclic group is cyclic. Let G be a cyclic group
with generator a. If the order of G is infinite, then prove that G is
isomorphic to (Z, +).
Find the relative extrema of the function
fix, y) = 4y? + x2 — 12y?— 36y + 2.
Prove that in the interval 0 < x < 1, the function f(x) = x2 is uniformly
continuous while f(x) = R is not uniformly continuous.
x
Prove that x; = 2, xg = 1,xg3=0 isa feasible solution to the following
set of equations :
2x1 —Xq + 3xg=3
— 6x1 + 3x9 + 7xg=-9
Is the solution basic ? Justify your answer. If the solution is not basic,
reduce it to a basic feasible one.
Find a bilinear transformation which maps the points z = 0, —i, —1 into
w =i, 1, 0 respectively.
@ Prove that every group is isomorphic to a group of permutations.
Gi) Let A = {1, 2, 3} and et S3 denote the symmetric group on
3 elements. Then is Sg an abelian or non-abelian group ?
(i) Find the volume of the region above the xy-plane bounded by the
paraboloid z= x? + y? and the cylinder x2 + y2 = a2.
M
Gi) Provethat lim } & ==.
Moco x'+4 8
2
Let 12) = In (1 + 2). Expand f(z) in Taylor series about z = 0.
Determine the region of convergence of the series.
Gi) Find Laurent series about the indicated singularity for the
function,
?
ay Ao
3. (a 0) Prove that if u,(x), n = 1, 2, 3, ...... are continuous in [a, b] and if
>; u,(x) converges uniformly to the sum S(x) in [ bl, then S(x) is
continuous in [a, 0].
Gi) Prove that an absolutely convergent series is convergent. Show
thatl--—+—=- 3 is conditionally convergent.
(3) 9 If 15 a normal subgroup of a group G and if H is any subgroup of
G, then prove that
Hv N=HN=NH
where H v N denotes the join of H and N.
(ii) | State the Second Isomorphism Theorem of groups and apply it to
the case G=ZxZxZ, = 2 × £ × {0} and N= {0} & 2.
(c) ` Consider the LPP :
Minimize
2 = 1031 + 2x9
subject to
21 + 2824 233 > 1
21 - 2±3 1
- ® + 3x3 > 3,
x; > 0, fori= 1, 2, 3.
Solve the dual of the above LPP and find the minimum value of z.
BJKE-B-MTH 3
15
. (a) (1 State and prove Cauchy’s integral formula. Thus evaluate
COS 2
02
70
where C is the circle | > - 1 | = 3.
(1) 8080 the Residue Theorem and apply it to evaluate
| € dz
(2-1) (@ + 8(2
0
where C is given by |z| =
(b) Prove that the integral domain 2 is a Unique Factorization Domain and
a Euclidean Domain.
() Five workers perform five jobs and the operating cost is given below, but
there is a restriction that the worker C cannot perform the third job and
B cannot perform the fifth job. Find the optimal assignment and the
optimal assignment cost.
1 1 1
A | 24 29 18 32 19
3 | 17 26 34 22 -
01.99 10: 25
1 22 ` 18 28 190 24
2 28; 10 31 24 2?
BJKE-B-MTH 4
Q5. (a)
(b)
(c)
(d)
(e)
Q6. (a)
(b)
BJKE-B-MTH
SECTION B
Consider a particle of mass m moving in a plane under attractive force
k
r2
directed towards the origin, where k > 0. Using the polar coordinates
(r, 6) write the corresponding Lagrangian and obtain the equations of
motion. Also show that the angular momentum is conserved. 8
A fanction f, defined on [0, 1], is such that f{0) = 0, (5) =-1,4p=9.
Find the quadratic polynomial p(x) which agrees with f(x) for x = 0, > as
3
[ES
re
Draw the logic circuit which realises the Boolean function
L=(A+B).(A+C)+C(A+B.C) and simplify it. Draw the simplified
circuit also. 8
<1 for 0<x<1, show that |f{x)-p(x)| < a forO<x<1. 8
In a 2-dimensional flow there are sources at (a, 0), (— a, 0) and sinks at
(0, a), (0, —a), all are of equal strength. Determine the stream function
and show that the circle through these four points is a streamline. 8
Solve
10 ‘
Yar + Uny + yy = — Sin (x + y) 8
Find the solution of
u,—uuy +u= 0
for the initial values x (s) = 0, Yo(s) = 8, up(s) = — 2s.
Does the solution break down for any finite x ? Is the solution unique ? 15
Find a root of the equation sin x + cos x = 1, lying in (0, 2), by
Regula-Falsi method, correct up to four significant digits. 10
(c)
Q7. (a)
(b)
(ec)
BJKE-B-MTH
.
For a dynamical system having two degrees of freedom, the Lagrangian
is given by L = . (a2 ay - a3) = & (a2 + a; ), where q, and qo are
generalized coordinates.
Find the corresponding Hamiltonian and derive the Hamiltonian
equations of motion.
Show further that the generalized momentum corresponding to q, is
constant.
Show that the system exhibits a simple harmonic motion with respect to
the generalized coordinate qo. 15
Solve : pe (2,
Uy — Uy = 0, O<x<z, t>0
u(0, t) = u(2, t) = 0,
= ons
u(x, 0) = sin 5?
u(x, 0) = 0.
Write down the flow-chart of Runge-Kutta method of 4 order to find
y(0'8) for & oxy, 70) = % taking = 0-2.
Also solve the above IVP to find y(0-4) by Runge-Kutta method
(4th order). 15
Consider 2-dimensional Navier-Stokes equations of a steady fluid flow.
Show that there exists a stream function ‘P(x, y) for such a flow.
Find the equation satisfied by (x, y). 15
6
8. (3)
(b)
BJKE-B-MTH
Show that
fix, , 2, , = x2p? + y2q?- 4 = 0
and g(x, y, 2, , q)=qy-a=0,
where a is a constant, are compatible and hence solve , y, 2, p, 0) = 0.
Is it complete integral ? 15
State the sufficient condition for convergence of the Gauss-Seidel
iteration method and solve the following system of equations by using
this method : 15
6-72] + 1.17 + 2-2 905
2-12] - 1522 + 8-425 = 28:8
:12] + 9:452 ¬ 1-5xg = 229
(correct up to 3-significant digits)
There is a doublet at (c, 0) in a 2-dimensional flow. A cylinder of radius
a (a < €) with z-axis as axis of the cylinder was introduced into the flow.
Find the complex potential and image system for the flow. 10
