UPSC IFS 2018 Question Paper for Mathematics Paper - I

1.F,S. EXAM-Q,112018 FSI-P-MTH
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. I 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.

FSI-P-MTH 1

SECTION A

Ql. Show that the maximum rectangle inscribed in a circle is a square. 8

2 2 0
Given that Adj A = 2 5 1 and det A = 2. Find the matrix A. 8
0 1 1

If f: [a, 131 -> R be continuous in [a, bl and derivable in (a, b), where
0 <a <b, show that for c e (a, b)
f(b) - f(a) = cf/(c) log (b/a). 8

Find the equations of the tangent planes to the ellipsoid
2x2 6y2 + = 27
3z2

which pass through the line
x-y-z=0 =x-y+2z-9. 8

Prove that the eigenvalues of a Hermitian matrix are all real. 8

Q2. Find the equation of the cylinder whose generators are parallel to the line
-
x= = z- and whose guiding curve is x2 + y2 = 4, z = 2. 10

Show that the matrices
1 1 -f 1. 0 3--
A= 1 2 1 and B = 0 2 2 are congruent. 10
-1 1 3 _3 2 0

If (I) and tif be two functions derivable in [a, b] and 4)(x) iiff(x) - w(x) 4)'(x)> 0
for any x in this interval, then show that between two consecutive roots of
l(x) = 0 in [a, bl, there lies exactly one root of w(x) = 0. 10

Show that the vectors al = (1, 0, - 1), a2 = (1, 2, 1), a3 = (0, - 3, 2) form a
basis for R3. Express each of the standard basis vectors as a linear
combination of al, a2, a3. 10

FSI-P-MTH 2

Q3. (a) Find the equation of the tangent plane that can be drawn to the sphere
x2 + y2 + z2 — 2x + 6y + 2z + 8 = 0,
through the straight line
3x — 4y — 8 = 0 = y — 3z + 2. 10

(b) If f = f(u, v), where u = ex cos y and v = ex sin y, show that
Of 82f 2 ( 52f of
ax2 + ay2 = + v2) ou2 +
ay2) 10

Let T : V2(R) —> V2(R) be a linear transformation defined by
T(a, b) = (a, a + b). Find the matrix of T, taking {e l, e2} as a basis for the
domain and {(1, 1), (1, —1)} as a basis for the range. 10

Evaluate ii(x2 + xy) dx dy over the region R bounded by xy = 1, y = 0,

y=x andx=2. 10

Q4. (a) Find the equations of the straight lines in which the plane 2x + y — z = 0
cuts the cone 4x2 — y2 + 3z2 = 0. Find the angle between the two straight
lines. /0

Show that the functions u = x + y + z, v = xy + yz + zx and
w = x3 + y3 + z3 — 3xyz are dependent and find the relation between
them. 10

Find the locus of the point of intersection of the perpendicular
x2 y2
generators of the hyperbolic paraboloid — = 2z. /0
a b2
If (n + 1) vectors al, a2, ..., an, a form a linearly dependent set, then
show that the vector a is a linear combination of al, a2, ah; provided
a2, ..., an form a linearly independent set. 10

FSI-P-MTH 3

SECTION B

Q5. Find the complementary function and particular integral for the equation
d 2y
y — xe x COS2 X
dx2

and hence the general solution of the equation. 8

d2y dy
Solve 2 —+y =xex log x (x >0) by the method of variation of
cbc2

parameters. 8

If the velocities in a simple harmonic motion at distances a, b and c from
a fixed point on the straight line which is not the centre of force, are u, v
and w respectively, show that the periodic time T is given by
u2 V2 W2
4 2
0 (b c) (c — a) (a — b) = a b c 8

1 1 1

From a semi-circle whose diameter is in the surface of a liquid, a circle is
cut out, whose diameter is the vertical radius of the semi-circle. Find the
depth of the centre of pressure of the remainder part. 8
—> A A A
If r =xi +yj + zk and f(r) is differentiable, show that
—>
div[f(r) r = rr(r) + 3f(r).
—>
r
Hence or otherwise show that div (— = 0. 8
r3

Q6. Solve the differential equation (y2 + 2x2y) clx + (2x3 — xy) dy = 0. 10

Let T1 and T2 be the periods of vertical oscillations of two different
weights suspended by an elastic string, and C1 and C2 are the statical
extensions due to these weights and g is the acceleration due to gravity.
4m2(C1 — C 2 )
Show that g 15
T2 _T 2 •
1 2
—> A
Show that F = (2xy + z3)I + x21 + 3xz2 k is a conservative force.
Hence, find the scalar potential. Also find the work done in moving a
particle of unit mass in the force field from (1, —2, 1) to (3, 1, 4). 15

FSI-P-MTH 4

Q7. (a) The end links of a uniform chain slide along a fixed rough horizontal rod.
Prove that the ratio of the maximum span to the length of the chain is

+ (1 +12)2
u log 1

where µ is the coefficient of friction. 10

Solve: 10
dy = 4x+6y+5
dx 3y+2x+4

A frame ABC consists of three light rods, of which AB, AC are each of
length a, BC of length a, freely jointed together. It rests with BC

horizontal, A below BC and the rods AB, AC over two smooth pegs E and
F, in the same horizontal line, at a distance 2b apart. A weight W is
suspended from A. Find the thrust in the rod BC. 10

Let a be a unit-speed curve in R3 with constant curvature and zero
torsion. Show that a is (part of) a circle. 10

Q8. (a) A solid hemisphere floating in a liquid is completely immersed with a
point of the rim joined to a fixed point by means of a string. Find the
inclination of the base to the vertical and tension of the string. 15

A snowball of radius r(t) melts at a uniform rate. If half of the mass of
the snowball melts in one hour, how much time will it take for the entire
mass of the snowball to melt, correct to two decimal places ? Conditions
remain unchanged for the entire process. 15

For a curve lying on a sphere of radius a and such that the torsion is
never 0, show that
+ Kr \ 2
= a2 . 10
1C) K2 t i

FSI-P-MTH 5

r