UPSC IFS 2017 Question Paper for Mathematics Paper-I

I.F.Su EXAM-(M)2017 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. 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/pan 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

Q1. (a) Let A be a square matrix of order 3 such that each of its diagonal
elements is 'a' and each of its off-diagonal elements is 1. If B = bA is
orthogonal, determine the values of a and b. 8

(b) Let V be the vector space of all 2 x 2 matrices over the field R. Show that
W is not a subspace of V, where
(i) W contains all 2 x 2 matrices with zero determinant.
W consists of all 2 x 2 matrices A such that A2 = A. 8

(c) Using the Mean Value Theorem, show that
f(x) is constant in [a, b], if f'(x) = 0 in [a, b].
f(x) is a decreasing function in (a, b), if f'(x) exists and is < 0
everywhere in (a, b). 8

(d) Let u(x, y) = ax2 + 2hxy + by2 and v(x, y) = Ax2 + 2Hxy + By2. Find the
acu v)
Jacobian J = ' , and hence show that u, v are independent unless
a(x, y)
a b h
8
ABH =

(e) Find the equations of the planes parallel to the plane
3x - 2y + 6z + 8 = 0 and at a distance 2 from it. 8

Q2. (a) State the Cayley-Hamilton theorem. Verify this theorem for the matrix

1 0 2-

A= 0 -1 1 . Hence find A-1. I

1 0

(b) Show that
n/2 F(13 +1c1
)q
1 2 2
sinP 0 cosq 0 de =
2 Frp+q+2)
2 )

FSI-P-MTH 2

Hence evaluate the following integrals:
n/2
(i) sin4 x cos5 x dx
0
1
x3(i _ x2)512 dx

1
x4(1— x)3 dx 10

Find the maxima and minima for the function
f(x, y) = x3 + y3 - 3x - 12y + 20.
Also find the saddle points (if any) for the function. 10

Show that the angles between the planes given by the equation

2x2 - y2 + 3z2 - xy + 7zx + 2yz = 0 is tan-1 . /0
4
Q3. (a) Reduce the following matrix to a row-reduced echelon form and hence
find its rank : 10
-1 2 -1
2 4 4 2
A=
0 0 1 5
1 6 3 2
Given that the set fu, v, wl is linearly independent, examine the sets
(i) + v, v + w, w + ul
tu + v, u - v, u - 2v + 2w)

for linear independence. /0

2)
Evaluate the integral e-(x2 + Y dx dy, by changing to polar
00

2 ,frc
coordinates. Hence show that .
I. e-x dx = /0
2.
0
Find the angle between the lines whose direction cosines are given by
the relations / + m + n = 0 and 2/m + 2/n - mn O. /0
FSI-P-MTH 3

Q4. (a) Find the eigenvalues and the corresponding eigenvectors for the matrix
[ 0 —2
A= . Examine whether the matrix A is diagonalizable. Obtain
1 3
a matrix D (if it is diagonalizable) such that D = P-1 A P. /0

A function f(x, y) is defined as follows:
22
x
if (0, 0)
x 2 +y2 '
0, if (x, y) = (0, 0)

Show that fxy(0, 0) = fyx(0, 0). /0

Find the equation of the right circular cone with vertex at the origin and
whose axis makes equal angles with the coordinate axes and the
generator is the line passing through the origin with direction ratios
(1,— 2, 2). /0

Find the shortest distance and the equation of the line of the shortest
x—3 y—8 z—3
distance between the lines . — and

x+3 y+7 z—6
/0
—3 2 4

FSI-P-MTH 4

SECTION B

Q5. Solve

(2D3 - 7D2 + 7D - 2) y = ewhere
-8x D= 8
dx •

Solve the differential equation

x2 d2Y 2x - 4y = x4 . 8
cbc2 dx

A particle is undergoing simple harmonic motion of period T about a
centre 0 and it passes through the position P (OP = b) with velocity v in
the direction OP. Prove that the time that elapses before it returns to P
.
T-
is-tan
vir 8
it K2mb

A heavy uniform cube balances on the highest point of a sphere whose
radius is r. If the sphere is rough enough to prevent sliding and if the
side of the cube be ±
711. , then prove that the total angle through which
2
the cube can swing without falling is 90°. 8

Prove that
v2rn = n (n + 1) ril-2
-> ->
and that nrr is irrotational, where r=lr I= Vx2 + y2 + z2 . 8

Q6. Solve the differential equation

dy2 + 2•
— dy
(dxJ
— y cot x = y2.
dx
15

A string of length a, forms the shorter diagonal of a rhombus formed of
four uniform rods, each of length b and weight W, which are hinged
together. If one of the rods is supported in a horizontal position, then
2W (2b2 -a2 )
prove that the tension of the stringis 10
b-j4b2 -a2

FSI-P-MTH 5

(c) Using Stokes' theorem, evaluate

f[(x + y) dx + (2x — z) dy + (y + z) dz],

where C is the boundary of the triangle with vertices at (2, 0, 0), (0, 3, 0)
and (0, 0, 6). 15

Q7. (a) Solve the differential equation
3
e3x(dy 1) ± (u.y) e2y 0.
10
dx dx

A planet is describing an ellipse about the Sun as a focus. Show that its
velocity away from the Sun is the greatest when the radius vector to the
planet is at a right angle to the major axis of path and that the velocity
. 2icae
then is , where 2a is the major axis, e is the eccentricity and T
TV1— e2

is the periodic time. 10

A semi-ellipse bounded by its minor axis is just immersed in a liquid, the
density of which varies as the depth. If the minor axis lies on the
surface, then find the eccentricity in order that the focus may be the
centre of pressure. 10

Evaluate

A
if
where S is the surface of the cone, z = 2 — ilx2 + y2 above xy-plane and
A A A
f = (x — z) i + (x3 + yz) j — 3xy2 k . 10

FSI-P-MTH 6

d 2y
Q8. (a) Solve + 4y = tan 2x by using the method of variation of parameter. /0
dx2
A particle moves in a straight line, its acceleration directed towards a
1
5 va
fixed point 0 in the line and is always equal to(a ut =., when it is at a
x` /
distance x from 0. If it starts from rest at a distance a from 0, then
6
8 11-
prove that it will arrive at 0 with a velocity a Veii, after time — /0
15 p.

Find the curvature and torsion of the circular helix

r = a (cos 0, sin 0, 0 cot 13),

is the constant angle at which it cuts its generators. /0

If the tangent to a curve makes a constant angle a, with a fixed line,
then prove that lc cos a ± sin a = 0.
K
Conversely, if — is constant, then show that the tangent makes a

constant angle with a fixed direction. /0

7

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