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Indian Forest Services (Main)
Examination-2025 DJSM-B-MATH
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.
Assume suitable data, if necessary, and indicate the same clearly.
Unless otherwise mentioned, symbols and notations have their usual standard meanings.
Answers must be written in ENGLISH only.
DJSM-B-MATH/55 1 [ P.T.O.
(b)
(c)
(d@)
(e)
SECTION—A
If H and K are finite subgroups of a group and HK = {hk|he H, ke K}, prove
that |HK |= LAN
|HnK|
Let S be a non-empty subset of R, bounded below, and T = {—x: x € S}. Prove
that the set T is bounded above and supT =-inf S.
Prove that the series Lbs bb tpt converges to > 10g2.
25 7
Find the image of |z| <1 under the bilinear transformation which maps z=], 4,
-1 onto w=i 0, -i respectively.
The forest department aims to afforest up to 100 hectares with teak and pine to
maximize CO, absorption, planting at least 10 hectares of each, and no more
than 60 hectares of teak. The table below provides the CO, absorption and
resource requirements :
Tree CO, Absorption Labour Water
(tons/ha/year) (hours/ha/week) (litres/ha/week)
Teak 20 40 200
Pine 15 20 150
2. (a)
(b)
()
The available resources are 3200 labour hours and 16000 litres of water per
week. Formulate this as a linear programming problem.
If pis an odd prime, prove that there is no group that has exactly p elements of
order p.
Let f,, (x) =nx(l- x)", xe [0, 1], ne N. Show that (i) the sequence {f,,} converges
to a function f on (0, 1] and (i) f is integrable on [0, 1] and lim ii Sr =f f, but
noo
10
still the convergence of the sequence {f,} is not uniform. 5+10
Evaluate the integral fp sin?) dx using the method of contour integration.
Evaluate
pee AE ay
+b?x? +a7y?
xe?
the field of integration E being the positive quadrant of the ellipse ~ tl ¥ a},
a?
b2
DJSM-B-MATH/55 2
15
10
(0)
(6)
4. (a)
(b)
(0)
5. (a)
Show that in the ring 2 «2, (i) the ideal S = (0 0): € 2) is a prime ideal but
not maximal and (1) the ideal T= [7 70 € ZxZ:3|n} is a maximal ideal.
10+5
Solve the following linear programming problem by simplex method :
Maximize 2 =3x, + 2×2 +3x3
subject to
2 72 723 54
220 +×2 +3x3 10
20 5222 + ×3 56
1, 22, 23 20
Is the solution unique? If not, find all the optimal solutions. 15
Let f(x) = ×” +a,_,x""}+...+a9 € Z[x], and suppose that is prime such
that p{a,, [ ..., 12100 and 2 100. Prove that f(x) is irreducible over 0. 10
Let /(2) = 2 Find the Laurent series expansion of f(z) in the following
(z-1)(z+2)
regions :
8 1<|2|<2
( |2|> 2 8+7
A company has 5 workers A, 8, C, D, E and 4 jobs J, 7, IV. The profit (in
rupees) that each worker earns from completing each job is given in the table
below :
! | ° | | ° | )
A 0 20 18 21
| | 19 16 17: 20
© 16 1⁄5 | 20 | 18
| 2 | 18 15 16
E 14 [195 | 13 15
Assign each job to exactly one worker in such a way that the profit is
maximized. 15
SECTION—B
Eliminate the arbitrary function F from the given equation , ) =0
and find the corresponding partial differential equation. 8
DJSM-B-MATH/55 [.7.
(0
(a)
(e)
(0
Find the interval in which the root of the equation xe* = 1 lies between 0 and 1,
obtained by using three iterations of bisection method. 8
Perform the operations (i) (+42) + (-13) and (ii) (-42) - (-13) in binary using signed
2’s complement representation for negative numbers in 8-bit system. Give the
final answer in decimal. 4+4
Derive the Lagrange’s equation for a particle of mass m that slides on a
frictionless wire hanging as a cycloid given by the equations x = 696-5176,
{ = 01 + ੦੦589). Further, if ०००५ = f (say), then transform the Lagrange’s
2
equation to the form + f = 0, where symbolizes time. 8
dt a
Prove that the equation of motion of a homogeneous inviscid liquid moving
under forces arising from a potential V may be written in the form
% 7 07301
× -- +V
a 3 (2 9
where 2 t, 2 , 2 respectively stand for velocity vector, time, vorticity vector,
pressure, density, and the gradient operator.
If the velocity 3, referred to cylindrical polar coordinates (7, 9 2), is given by
[० dor, ०] 0૨૬7૬૦)
7 2
= 2
0 lea” ] 7
2 7
where 0 is a constant, prove that the vorticity is given by
te [0 0, 0] Osrsa
[00] (
2 62
Find a complete integral of the equation “ + = 2, where p= Ox q= 10
Write down the algorithm for finding the integral J = f(x)dx using Simpson’s
4rd rule. Hence, find I = [)(4× - 3×2) dx taking 10 intervals. Also, find the
relative error. 7+8
A uniform rod of length 2a and mass m can rotate freely about a fixed end.
What is the least angular velocity required to start with, from the lowest
position so as to reach the top in order to make a complete revolution? Also,
find out the time consumed. 15
DJSM-B-MATH/55 4
7. (a)
In an examination, the number of candidates who secured marks between
certain limits was as follows :
Marks a 0-19 20-39 40-59 60-89 90-99
No. of candidates: 41 62 65 50 17
()
(c)
8. (a)
(b)
(c)
Estimate the number of candidates getting marks less than 50.
Find the general solution of the partial differential equation
(4D? -4DD’ + D’?)z = 8log(x +2y) +cos2xcosy
where fee 2 and Die.
x ay
A long infinite cylinder of radius a is placed in a uniform stream such that its
axis lies perpendicular to the stream. Besides, a circulation round the cylinder
is produced by a uniform line vortex through the origin. If the uniform stream
velocity is -Ui and the circulation is 2nk, then find out the complex velocity
potential. Show, by using the theorem of Blasius, that the cylinder experiences
an uplifting force.
A simple source, of strength m, is fixed at the origin O in a uniform stream of
incompressible fluid moving with velocity Ui. Show that the velocity potential »
—
at any point P of the stream is ue —Urcos®, where OP =r and 0 is the angle OP
r
makes with the direction i. Find the differential equation of the streamlines
and show that they lie on the surfaces Ur? sin? @-2mcos6 = constant.
Let u(x, t) be the solution of the wave equation
du da7u
3 a2 O<x<1, t>0
with the initial conditions u(x, 0) =sinx+sin2x+sin3x; SH, 0)=0, O<x<n
and the boundary condition u(0, t) = u(x, t) =0, t 20. Find the value of (3, n).
Using Runge-Kutta method of fourth order, find y at x=0-2, given that
a =3e* +2y, y(0) =0 and h=0-1, correct to four decimal places.
kkk
10
15
15
10
15
15
DJSM-B-MATH/55 5 26BS—348
