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# (Download) UPSC IAS Mains Exam Paper - 2017 : Mathematics

## (Download) UPSC IAS Mains Exam Paper - 2017 : Mathematics

## MATHEMATICS

(Paper - 1)

**Time Allowed : Three Hours**

**Maximum Marks : 250**

## QUESTION PAPER SPECIFIC INSTRUCTIONS

**(Please read each of the following instructions carefully before
attempting questions)**

There are **EIGHT** questions divided in two Sections and printed both in
**HINDI** and in **ENGLISH.**

Candidate has to attempt **FIVE** questions in all.

Question Nos.** 1** and **5** are compulsory and out of the remaining,
**THREE** are to be attempted choosing at least **ONE** question from each
Section.

The number of marks carried by each question/part is indicated against it.

Answers must be written in the medium authorized in the Admission Certificate which must be stated clearly on the cover of this Question-cum-Answer (QCA) Booklet in the space provided. No marks will be given for answers written in a medium other than the authorized one.

Assume suitable data, if considered necessary, and indicate the same clearly.

Unless and otherwise indicated, symbols and notations carry their usual standard meanings.

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

## SECTION-A

**Q1.** (a) Let A = (2 2 / 1 3) find a non-singular matrix P such that P^{-1}
AP is a diagonal martix.

(b) Show that the similar matrices have the same characteristic polynominal.

(c) Intergrate the function F(x,y) = xy (x^{2} + y^{2}) over the
domain R: {-3<x^{2}-y^{2}<3, 1<xy<4}.

(d) Find the equation of the tangent pane at point (1,1,1) to the conicoid 3x^{2}-y^{2}
= 2z

(e) Find the shortest distance between the skew lines:

x-3/3 = 8-y/1 = z-3/1 and x+3/-3 = y+7/2 = z-6/4.

**Q2.** (a) Find the volume of the solid above the xy-plane and directly
below the portion of the elliptic paraboloid x^{2} + y^{2}/4 = z
which is cut off by the plane z=9.

(b) A palne passes through a fixed point (a,b,c) and cut the axis at the points
A, B, C respectively. Find the locus of the centre of the sphere which passes
through the origin O and A,B,C.

(c) Show that the plane 2x-2y+z+12 = 0 touches the sphere

x^{2}+y^{2}+z^{2}-2x-4y+2z-3 = 0 find the point of
contact.

(d) Suppose U and W are distint four dimensional subspaces of a vector space V,
where dim V = 6. Find the possible dimensions of subspace U^W.

**Q3. (a)**

(b) Prove that distinct non-zero eigenvectors of a matrix are linearly independent.

(c)

(d) Find the locus of the point of intersection of three mutually
perpendicular tangent planes to ax^{2} + by^{2} + cz^{2}
= 1.

**Q4.** (a) Reduce the following equation to the standard form and hence
determine the nature of the conicoid : x^{2}+y^{2}+z^{2}-yz-zx-xy-3x-6y-9z+21
= 0

(b) Consider the following system of equation in x,y,z:

x+2y+2z = 1

x+ay+3z = 3

x+11y+az = b.

(i) For which value of a does the system have a unique solution?

(ii) For which pair of value (a,b) does the system have more than one solution?

(c)

(d)

## SECTION-B

**Q5. **(a) Find the differential equation representing all the circles in
the x-y plane.

(b) Support that the streamlines, of the fluid flow are given by a family of
curves xy=c. Find the equipotential lines, that is, the orthogonal trajectories
of the family of curves representing the streamlines.

(c) A fixed wire is in the shape of the cardiod r = a (1 + cosq),
the initial line being the downward vertical. A small ring of mass m can slide
on the wire and is attached to the point r = 0 of the cardiod by an elastic
string of natural length a and modulus of elasticity 4 mg. The string is
released from rest when the string is horizontal. Show by using the laws of
conservation of energy that aq^{2}(1 + cosq)-g
cosq (1 - cosq) = 0, g
being the acceleration due to gravity.

(d) For what values of the constants a, b and c the vector V = (x +y+az) + (bx
+2-z) + (-3+ Cy+2z) k is irrotational. Find the divergence in cylindrical
coordinates of this vector with these values.

(e) The position vector of a moving point at time t is r = sinti+cos 2tj+ (t^{2}
+2y) k. Find the components of acceleration ā in the directions parallel to the
velocity vector ū and perpendicular to the plane of r and ū at time t= 0.

**Q6.** (a) (i) Solve the following simultaneous linear differential
equation:

(D+1)y = z+e^{x} and (D+1)z = y+e^{x} where y and z are
functions of independent variable x and D = d/dx.

(ii) If the growth rate of the population of bacteria at any time t is
proportional to the amount present at that time and population doubles in one
week, then how much bacterias can be expected after 4 weeks?

(b) (i) Consider the differential equation xy p^{2} - (x^{2}
+y^{2}-1) p +xy = 0 where p=d/dx Substituting u=x^{2} and v = y^{2}
reduce the equation to Clairaut's form in terms of u, v and p' = dv/du. Hence,
or otherwise solve the equation.

(ii) Solve the following initial value differential equations :

20y" +4y' +y = 0, y(0) = 3.2 and y' (0) = 0.

(c) A uniform solid hemisphere rests on a rough plane inclined to the horizon at an angle f with its curved surface touching the plane. Find the greatest admissible value of the inclination f for equilibrium. If f be less than this value, is the equilibrium stable ?

**Q7.** (a) Find the curvature vector and its magnitude at any point r = (q)
of the curve r= (a cosq, a sinq,
aq). Show that the locus of the feet of the
perpendicular from the origin to the tangent is a curve that completely lies on
the hyperboloid x^{2} + y^{2} - z^{2} = a^{2}.

(b) (i) Solve the differential equation:

(ii) Solve the Following differential equation using method of variation of
parameters:

(c) A particle is free to move on a smooth vertical circular wire of radius a. At time t = 0 it is projected along the circle from its lowest point A with velocity just sufficient to carry it to the highest point B. Find the time I at which the reaction between the particle and the wire is zero.

**Q8. (a)**

(b)

(c)

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## Printed Study Material for IAS Mains General Studies

## MATHEMATICS

(Paper - 2)

**Time Allowed : Three Hours**

**Maximum Marks : 250**

## QUESTION PAPER SPECIFIC INSTRUCTIONS

**(Please read each of the following instructions carefully before
attempting questions)**

There are **EIGHT** questions divided in two Sections and printed both in
**HINDI** and in **ENGLISH.**

Candidate has to attempt **FIVE** questions in all.

Question Nos.** 1** and **5** are compulsory and out of the remaining,
**THREE** are to be attempted choosing at least **ONE** question from each
Section.

The number of marks carried by each question/part is indicated against it.

Answers must be written in the medium authorized in the Admission Certificate which must be stated clearly on the cover of this Question-cum-Answer (QCA) Booklet in the space provided. No marks will be given for answers written in a medium other than the authorized one.

Assume suitable data, if considered necessary, and indicate the same clearly.

Unless and otherwise indicated, symbols and notations carry their usual standard meanings.

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

## SECTION-A

**Q1. **(a) Let x_{1} = 2 and x_{n+1} =
Öxn + 20, n = 1,2,3,...
Show that the sequence x_{1}, x_{2}, x_{3},.... is
convergent.

(b) Let G be a group of order n. Show that G is isomorphic to a subgroup of the
permutation group S_{n}:

(c) Find the supremum and the infimum of x/sin x on the interval (0,
p/2].

(d) Determine all entire functions f(z) such that 0 is a removable singularity
of f(1/z).

(e) Using graphical method, find the maximum value of

2x + y

subject to

4x + 3y < 12

4x + y <8

4x -y <8

x, y > 0.

**Q2.** (a)

where [x] denotes the largest integer less than
or equal to x.

(i) Determine all the real numbers t at which f is differentiable.

(ii) Determine all the real numbers t at which f is continuous but not
differentiable.

(b) Using contour integral method, prove that

(c) Let F be a field and F[X] denote the ring of polynomials over F in a single variable X. For f(X), g(X) e F[X] with g(X) = 0, show that there exist q(X), r(X) e F[X] such that degree (r(X)) < degree (g(X)) and f(X) = q(X) • g(X) + r(X).

**Q3.** (a) Show that the groups Z_{5}
x Z_{7} and Z_{35} are isomorphic.

(b) Let f = u + iv be an analytic function on the unit disc

(c) Solve the following linear programming
problem by simplex method :

Maximize

z = 3x_{1} + 5x_{2} + 4x_{3}subject to

2x_{1} + 3x_{2 }< 8

2x_{2} + 5x_{3} < 10

3x_{1} + 2x_{2} + 4x_{3} < 15

x_{1}, x_{2}, x_{3} > 0.

**Q4. **(a) For a function f: C → C and n >
1, let f^{(n)} denote the nth derivative off and f^{(0)} = f.
Let f be an entire function such that for some n > 1, f^{(n)} (1/k) = 0
for all k = 1, 2, 3, ..... Show that f is a polynomial.

(b) Find the initial basic feasible solution of the following transportation
problem using Vogel's approximation method and find the cost.

(c)

## SECTION-B

**Q5.** (a) Solve (D^{2} - 2DD' + D'^{2})
z = e^{x+2y} + x^{3} + sin2x,

Where

(b) Explain the main steps of the Gauss-Jordan
method and apply this method to find the inverse of the matrix

(c) Write the Boolean expression

z (y + z) (x + y + z)

in its simplest form using Boolean postulate rules. Mention the rules used
during simplification. Verify your result by constructing the truth table for
the given expression and for its simplest form.

(d) Let T be a closed curve in xy-plane and let
S denote the region bounded by the curve 1. Let

If f is prescribed at each point (x, y) of S and w is prescribed on the boundary
T of S, then prove that any solution w = w(x, y), satisfying these conditions,
is unique.

(e) Show that the moment of inertia of an
elliptic area of mass M and semi-axis a and b about a semi-diameter of length r
is 1/4Ma^{2}b^{2}/r^{2}. Further, prove that the moment
of inertia about a tangent is 5M/4p^{2}, where p is the perpendicular
distance from the centre of the ellipse to the tangent.

**Q6.** (a) Find a complete integral of the
partial differential equation

2 (pq + yp + qx) + x^{2} + y^{2 }= 0.

(b) For given equidistant values u__{1},
u_{0}, u_{1} and u_{2}, a value is interpolated by
Lagrange's formula. Show that it may be written in the form

Where x + y = 1

(c) Two uniform rods AB, AC, each of mass m and
length 2a, are smoothly hinged together at A and move on a horizontal plane. At
time t, the mass centre of the rods is at the point (€, n) referred to fixed
perpendicular axes Ox, Oy in the plane, and the rods make angles
q + $ with Ox. Prove
that the kinetic energy of the system is

Also derive Lagrange's equations of motion for the system if an external force
with components [X, Y] along the axes acts at A.

**Q7. **(a)

(b)

(c) A stream is rushing from a boiler through a conical pipe, the diameters
of the ends of which are D and d. If V and v be the corresponding velocities of
the stream and if the motion is assumed to be steady and diverging from the
vertex of the cone, then prove that

where K is the pressure divided by the density and is constant.

**Q8. **(a)

(b) Write an algorithm in the form of a flow chart for Newton-Raphson method.
Describe the cases of failure of this method.

(c) If the velocity of an incompressible fluid at the point (x, y, z) is given
by

then prove that the liquid motion is possible and that the velocity potential is
z/r^{3} Further, determine the streamlines.