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Indian Forest Service (Main)
Examination, 2024 JBNV-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.
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.
JBNV-U-MATH/47 t [PiF0?
(b)
(c)
(a)
(e)
(b)
(c)
SECTION—A
Let V=R*. Find a basis and dimension of the subspace
W={abhcod)jeV:a=bt+o c=b+d}
Describe explicitly a linear transformation from R° to R°, which has its range
spanned by (1, 0,-1) and (1, 2, 2).
Find the relation between the radii of a right circular cylinder and a cone if the
former with maximum possible curved surface area is inscribed in the latter.
1
Find the limit of (cotx — tan x)€e*, when x 30.
Show that if ax? +2hxy + by” +2gx+1=0 represents two straight lines, then
b<0 and bg? +h? =ab.
(0)
Let W, -{: ais Y¥ze ¢| and W, -{o Me ye ¢| be two subspaces of the
vector space of all 2x2 matrices over the complex field C. Show that
dim| M4 + Wa ) «inn
We WoW,
Evaluate the volume of the solid formed by rotating the curve r =a(l1+cos6)
about the initial line.
(i) Reduce the equation (c? +d?)(x? +y?) =(cc+dy+2f)? to its canonical
form and show that it represents a parabola. Find the latus rectum of the
parabola.
(i) A variable sphere passes through the points (0,0,+¢) and cuts the lines
y-xtan6=0=z2-c
y+xtan0=0=z+c
in the points P and Q. If |PQ|=2a (where a is a +ve number), then show
that the centre of all such spheres lies on the circle
x? +y? =(a? —c?)cosec?28, z=0.
JBNV-U-MATH/47 2
10
15
3.
4.
(a)
()
()
(a)
(b)
()
ae du du_1
If u=expisin™ wey , then show that x — + y— =—utan(log, wu).
>| 5 aa ax Yay 2 (loge u)
Reduce the matrix
22-1 6 4
A=|4 4 1 10 13
8 8 -1 26 23
to echelon form and then to row canonical form.
() Show that the equation of the plane containing the line reas =1,x=Oand
c
parallel to the line <= =1, y=0is *_Y¥_241.=0. Further show that if
ac a be
2d is the shortest distance between the given lines, then
1 a 1 a
a ene See
Gt bec? a4
(i) A variable plane is parallel to the plane = <0 and meets the
a c
coordinate axes in A, B, C respectively. Prove that the circle ABC lies on
the cone
ye(2 +2) +2x(£+2)+24(2+2) -0
eb ac ba
Find the equations of the generating lines of the hyperboloid
x2 y? 2?
4 9 16
passing through the point (2, 3, - 4).
Let T: R° -R® be defined by
T(x, y 2) =(6x-y+3z, -6x+4y-6z, -6x+2y-4z)
Find all the eigenvalues and corresponding eigenvectors.
() How many loops are generated of the curve r = asin3@? Find the sum of
the areas of all the loops.
(i) Deduce the asymptote of the curve rlog, 8 =a.
10
15
10
15
8
7
JBNV-U-MATH/47 3 [ P.T.O.
SECTION—B
5. (a) Solve the differential equation
2 2. 2
.
x x
x
2 dy
+ ¦ ® + # = 0, where p=—
12 | 2 ax 8
2
(b) Solve the differential equation 9.23 2y = 1) ×.
dx? ×
(c) A particle moves from rest at a distance a from a centre of force where
repulsion at distance x is 12 2. Show that its velocity at distance x is
and that the time it has taken is
15 1 2 ¬
—|vx* -ax+alo =+,/—-1
| a || 8
() Two uniform steel rods of equal size 1 hang from their junction and rest on a
symmetrically placed smooth vertical circular base of radius a. If each of the
rods subtends an angle 0 with the vertical line passing through the centre of
the circular base, show that, applying the principle of virtual work, the relation
obtained is 1 = 26010 5603. 8
(0) If is a solenoidal vector, then show that
રે _ ੦2੮2 کرون_
curl curl curl curl F = V“V*F = 8
6. (a) Solve the differential equation
2
+29y = xe +8in2x
(b) A particle moves in a path so that its acceleration is always directed to a fixed
... 9
(distance)?
distinguish between the three cases that arise. Further show that the square of
the periodic time varies as the cube of the major axis. 15
point and is equal to . Show that its path is a conic section and
>
© 0 Wr=xi+yj+zk, then find
Ir| 6
( Find the curvature and torsion of the curve x=acost, y=asint, z= bt. 9
JBNV-U-MATH/47 4
7. (a) Ifnear the surface of a celestial body having atmosphere, the gravity is almost
constant and the absolute temperature in its atmosphere is given by
2
z
T= To )t- ae
H being the height of the homogeneous atmosphere and n a constant quantity,
show that the pressure in the atmosphere will be given by
P= Po exp(sin™ 3]
nH
where pp is the pressure at z=0. 10
(b) Solve the differential equation
2
FY ~cotx WY - 1 -cotxiy = e* sin x 15
(c) (i) If satisfies V2 =0, then show that Vois both solenoidal and irrotational. 4
(i) Verify the divergence theorem for the function
F = (x? - ya\i +(y? - 2x)j +(2? -ay)k
taken over the parallelepiped 0< x <a, O<y<b,0<zsc. 11
8. (a) Verify Green’s theorem for
[ley +y?) ax + x?ayj
where C is bounded by the curves y= x and y=x?. 10
(b) Using the method of variation of parameters, solve the differential equation
2
2d*y, dy 2
x +x—~-y=x“logx, x>0
dx? "dx . . 15
(c) Establish a stability criterion if a rigid body is lying on another rigid body at a
point of contact, and also both have rough surfaces to prevent sliding and a
small area around the point of contact of both of them is circular.
A solid frustum of a paraboloid of revolution of height h and latus rectum 2a
rests with its vertex on that of a paraboloid of revolution of latus rectum 2b.
Find the stability condition. 15
kkk
JBNV-U-MATH/47 5 25SS—384
