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Indian Forest Service (Main)
Examination, 2024 JBNV-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.
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.
JBNV-B-MATH 1
Ql. (a)
(b)
(c)
(d)
(e)
Q2. (a)
JBNV-B-MATH
SECTION A
Let R be the ring of n x n matrices over reals. Show that R has only two
ideals namely {0} and R.
1 1
Show that the series — + =
(1+a)P (2+a)P (3+a)?
..,a>Ois
(i) absolutely convergent if p > 1.
(ii) conditionally convergent if 0 < p< 1.
If f(x) = (x — a)?" (x — b)?™+1, where m, n are positive integers, show
that f has neither a maximum nor a minimum at a and f has a local
minimum at b.
Let f(z) = u(r, 6) + iv(r, 6) be an analytic function. If u = — r® sin 30, then
construct the corresponding analytic function f(z) in terms of z.
Find all optimal solutions of the following linear programming problems
graphically :
(i) Maximize
Z = 3x1 + 6x9
subject to
Xp + xg<S8
Xj —xgs4
2x1 —Xg24
Xj, Xg20
(ii) The LPP in part (i) with the first constraint x, + x9 < 8 changed to
Xi + 2x9 <12.
(i) If G be a group of even order, then show that there exists an
element ‘a’ other than the identity element, such that a? = e.
(ii) Prove that an ideal S of the ring Z of all integers is a maximal
ideal, if S is generated by some prime integer.
(b)
(c)
Q3. (a)
(b)
(c)
Q4. (a)
JBNV-B-MATH
co
Examine the convergence of the improper integral = dx and
0
hence evaluate it. 15
Determine the poles of the function f(z) = CE and the residue
z—-1)" (z-
at each pole and hence evaluate fe (z)dz where C is the circle |z| = 2-5. 15
Cc
2
Jem at
Evaluate lim —__.. 10
x0 x?
Prove that in a Unique Factorization Domain R, an element is prime if
and only if it is irreducible. 15
Solve the LPP :
Maximize
Z = 2x1 + Xo + X3
subject to
Xj + 2x9-—xg <3
X1 — 2X9 — 5x3 2-9
X1, Xq, X3 20
by the simplex method. Write its dual problem and from the optimal
table of the given problem, obtain the optimal solution of the dual
problem. 15
Let G and H be finite groups, such that
ged (|G
Hj)=1
>
Show that the trivial homomorphism is the only homomorphism from G
into H. 10
00) (i) Find the image of |z—3i|= 3 under the mapping w = 8
2
(11) 1 the value of the integral
1+i
| (x-y +1±2) 02
0
along the straight line from 2 = 0 0 2 = 1 + 1. 7
() Find the initial basic feasible solution of the following minimum cost
transportation problem by the Vogel’s Approximation Method (VAM).
Using it, find the optimal solution and the minimum transportation cost.
Is the optimal solution unique ? If not, find an alternative optimal
solution. 15
Destination
12] Dg Dg 04 Availability
113 5 8 2 50
Origin 02 | 5 7 2 9 | 40
03 7 1 3 4130
Demand 40 35 25 20
JBNV-B-MATH 4
SECTION B
Q5. (a) Obtain the partial differential equation by eliminating the arbitrary
function f from the equation f(x + y + z, x2 + y2 +22) =0. 8
(b) Obtain the following approximate quadrature formula :
3
i} f(x) dx = 3 [f{0) + 3f(1) + 32) + £3). 8
0
(c) (i) Convert (523.0234375),, into an equivalent octal number and then
convert it to its binary form.
Gi) Ifx = (1D2.2),¢ and y = (52E.02),¢, then find the value of x + y in
decimal system. 4+4
(d) The velocity components in an unsteady three dimensional flow are
given by u = ap v= oF w= 7 . Describe the streamlines and
pathlines. 8
(e) Is a system of two particles which are connected by a rod of constant
length holonomic ? Justify your answer. 8
Q6. (a) Using Charpit’s method, find the complete’ integral of
ya + 3xp = 2 (z—y?p?), where p = andq= = . 10
(b) Write down the algorithm for solving the differential equation
Ms = f(x, y), y(xg) = yp, numerically by Euler’s method with step length
h up tox =x, = Xq + nh.
Solve the following differential equation for x=1 with step length
h = 0-2 by using Euler’s method :
x =x24y, y(0)=1. 649
(c) Derive the Hamilton equations for holonomic systems and use them to
discuss the motion of a simple pendulum. 15
JBNV-B-MATH 5
Q7. (a)
(b)
(c)
8. (a)
(b)
JBNV-B-MATH
Show that the iteration formula for the Newton-Raphson method for
finding the 800 root of a positive real number a 18 :
ਗਤ , where K > 0. + [ 5 = ہیلا
n
Use this formula to find 3/13 , correct up to three decimal places. 3+7
Find the general solution of the partial differential equation
[72 - (D’)? - 81) + 8012 = (1 _ ) (1 _ 7) + eX + 27,
where D = and D’= -6_ 15
Ox oy
Consider an inviscid incompressible fluid flow with velocity
” 148 246
gravitational constant. Find the pressure at a point (x,y,z) if
-- A
= [x 3 ) under the body force F= —gzk, where g is the
Consider a source and a sink of equal strength at points ( within
a fixed circular boundary + = 2. Determine the equation of
streamlines. 10
Find the solution of the heat equation
2
20 0 < ± < £ > 0
ot
under the boundary conditions 00, £) = 0 = , £) and the initial
condition
x, O<x<—
u(x, 0) = 2 15
7- ®, < ± < #
2
6
(c) Use Gauss-Jordan elimination method to solve the following system of
equations :
921 + × + ±3 = 7
22] + 22 + 533 = 13
2] + 4x9 + x3 = 9-4
correct up to 2-significant figures.
JBNV-B-MATH 7
15
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