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Indian Forest Services (Main)
Examination-2025 DJSM-U-MATH
MATHEMATICS
Paper - I
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.
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.
DJSM-U-MATH 1
SECTION A
Ql. (a) If a subspace W of R* is generated by the vectors (3, 8, —3, —5),
(1, - 2, 5, —3) and (2, 3, 1, —4), then find a basis and dimension of W.
Extend that basis to get a basis of R*.
(b) Find a row echelon matrix which is row equivalent to
00 -2 3 1
24 1 4
A= 3
12 -3 1 2
4 8 23 5
and find the rank of A.
(c) |Amreek has n number of children by his first wife. Shaina has (n + 1)
children by her first husband. They marry and have children of their
own also. The whole family now has 12 children. It is assumed that
children of Amreek from his first wife do not fight among themselves,
and likewise, children of Shaina by her first husband do not fight among
themselves. Find the maximum possible number of fights between
children that can take place.
(d) Prove that :
Bie
hiner YEP Gne) Gut 1)
2" r+)
(e) Find the equation of the plane passing through the points (2, 2, 1) and
(9, 3, 6) and perpendicular to the plane 2x + 6y + 6z = 9.
DJSM-U-MATH 2
Q2. (a) For the companion matrix C
0 1 0 ... 0
0 0 1 ... 0
0 0 0 ... 1
—ag -] -2 ... -81-]
of anth degree polynomial
(A) = AP + AM 1 + ... + 1 At ag,
prove that
(1)
(111)
00) )
(11)
(0) 0)
(ii)
DJSM-U-MATH
the characteristic polynomial is 0).
if A; is an eigenvalue of ©, then x; = [1 a, A}
associated eigenvector.
if Ay, ,... are distinct eigenvalues of C, then
V1CV = diag (Ay, Ag, படிக்
1 1 vee 1
2 2 we
where V = bs 2
0-1 -1 -1
7 72 Ay
5 is the
Find the expansion 08 (511 1 x) in terms of ascending powers of x.
Evaluate :
lim (xx? +1-x2)
Find the shortest distance between the straight lines
x-1 ১৩ Yee மக and > -5 এটাক iar ಕ:
2 3 4 4 4 5
Also show that the lines are coplanar.
Show that the enveloping cylinder of
the ` ellipsoid
x? y 2 22
—5 ++ 5+-5=1, whose generators are parallel to the line
2
, meets the plane z = 0 in circles.
0 4 Ja? - 9 _2
70
Q3. (a) Prove that 2—* < tanly-—tan-!x < ¥=* O<x<y.
1+ y2 1+x2’
. x 8 -1{4 x 1
Hence or otherwise, show that — +—— < tan-*}—| < = +=. 10
4 25 3 4 6
(b) (i) Diagonalize the quadratic form
5x} - 26x? + 10x3 + 4xXoXg + 14x9x) + 6x 1Xp.
Show that it is positive semi-definite and find a non-zero set of
values of x1, X9, X3 which makes the diagonalized form zero. 8
(ii) If W is a subspace of a finite dimensional vector space V(F), then
prove that W is finite dimensional and dim W < dim V. Also, prove
that dim W = dim V ifand only if W = V. 7
(c) Find the equation of the cone whose vertex is the point (1, 2, 3) and
guiding curve is : 15
x24 y2422=4, x+y+z=1.
Q4. (a) Find the equations of the spheres passing through the circle
x24 y2422-5=0, 2x+38y+z-3=0
and touching the plane 3x + 4z-—15=0. 10
(b) @ For the linear operator T : R® > R® defined by
T(x, y,z)=(x+y+z, 2y+z, 2y + 3z),
find the eigenvalues and the basis for eigenspace. 8
(ii) Prove that the necessary and sufficient condition for a linear
transformation y = Ax to preserve lengths is that the matrix A is
orthogonal. 7
(c) | Prove that the area included between the folium
x3 + y? = 8axy
and its asymptote is equal to the area of its loop. 15
DJSM-U-MATH 4
Q5. (a)
(b)
(c)
(d)
(e)
Q6. (a)
(b)
SECTION B
Solve the differential equation
dy y x+(1—x?)l/2
ax 23/2 ~ 22° 3
dx (1-x*) (1-x*)
Solve the differential equation
2
ad°y__ dy .
x* —+ —x —~ +4y =cos (log x) + x sin (log x). 8
af a g g
The displacement of a particle in a straight line is given by the equation
x = acosnt + bsinnt. Show that the particle describes a simple
i 5 : : PD) 3 itd Qa
harmonic motion whose amplitude is ya +b* and period is —. 8
n
A light ladder is supported on a rough floor and leans against a smooth
wall. How far up the ladder can a man climb without slipping taking
place ? 8
For vector fields v and Vv’ , prove that
> => => => =>
Vx(@ xV)=U(V.V)-V(V.0)4HV.V)Y-@.V)V. 8
Given that y=x+ is a solution of the differential equation
2
x? dy +X dy -y=0.
dx? dx
Find the other linearly independent solution and write down the general
solution of the given differential equation. 10
A projectile is launched with initial speed u making an angle (90 — 60)
with the vertical. An air resistance force — BY (B > 0) acts upon the
projectile, where V is the instantaneous velocity. Find the velocity Vv,
and show that the position vector rat any time t is
> mu A a Ste m- i m |A4
r =—(cos0j +sn6 k)}/1l-e ™ a t+—e mM-—|k,
B B B
where m is the mass of the projectile. 15
DJSM-U-MATH 5
(c) (i) Find the directional derivative of
F(x, , 2) = xy? - : + 22
at (1, - 1, 2) in the direction of 61 + 21 + 3k. Also find its
maximum value. 5
7 : ^ ^ 2
(ii) ` Verify Stokes’ theorem for f = (x + y) 1 + 22 ] + k, where
5 is the upper surface of the sphere + + 22 = 1 over 2 = 0 and
† is its boundary in the xy plane. 10
Q7. (a) framework ABCD consists of four equal, light rods smoothly jointed
together to form a square. It is suspended from a peg at A, and a weight
w is attached to C, the framework being kept in shape by a light rod
connecting B and D. Find the thrust in this rod. 10
(0) [086 the method of variation of parameters to show that the solution of
2
2 + ky = (x),
satisfying the conditions y(0) = 0 = x is given by
x=0
y(x) = |] o(t) sin k (x - £) dt. 15
0
(6) Determine a’ such that
at ^ dN ¢
T > > A ¬»
= , —=a 11, —= B
ds * ds * ds =
represents Frenet-Serret formulae. 5
(1) A curve in a space 15 represented by
A
=etcosti + £ j +etk.
Find the curvature and principal normal of this curve at t = 0. 10
DJSM-U-MATH 6
Qs. (a) Find the values of « and B such that the vectors
A A
f =(ax+y) + (y —3z) ; +(x+02) k and g” 7 oer + kk
x“+y
are solenoidal. 3
Gi) | Show that the vector
> A A A
f =(2x—yz) i +(2y—zx) j + (2z-xy) k
is irrotational and find a scalar function ¢ such that f= grad . 7
(b) Find the general and singular solutions of
Sxy = 2px? 2p2, p= WY. 15
dx
(c) A solid hemisphere is placed with its base inclined to the surface of a
liquid, in which it is completely immersed, at a given angle a. Show that
if the resultant thrust on the curved portion of the surface is equal to
twice the weight of the liquid displaced, then tan a = 2. 15
DJSM-U-MATH 7
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