C. S. (MAIN) EXAM 2009

MATHEMATICS

Paper-I

I Time Allowed Three Hours I I Maximum Marks 300 I

INSTRUCTIONS

Each question is printed both in Hindi and in English.

Answers must be written in the medium specified in the Admission Certificate issued to you, which must be stated clearly on the cover of the answer-book in the space provided for the purpose. No marks will be given for the answers written in a medium other than that specified in the Admission Certificate.

Candidates should attempt Question Nos. 1 and 5 which are compulsory, and any three of the remaining questions selecting at least one question from each Section.

The number of marks carried by each question is indicated at the end of the question.

Assume suitable data if considered necessary and indicate the same clearly. Symbols/ notations carry their usual meanings, unless otherwise indicated.

Section-A

1. Attempt any five of the following
Find a Hermitian and a skew-Hermitian
matrix each whose sum is the matrix
<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-1a.jpg">
Prove that the set V of the vectors x2 x3 x4 in R4 which satisfy the equations x1 x2 2x3 x4 0 and 2x1 3x2 x3 x4 is a subspace of R4 What is the dimension of this subspace? Find one of its bases. 12

Suppose that f" is continuous on and that f has three zeroes in the interval 2). Show that f has at least one zero in the interval 2). 12

If f is the derivative of some function defined on prove that there exists a number such that

<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-1d.jpg"> 12

A line is drawn through a variable point on the ellipse x2/a2+y2/b2 z 0 to meet two fixed lines y mx, z c and y z -c. Find the locus of the line. 12

Find the equation of the sphere having its centre on the plane 4x-5y-z and passing through the circle
x2+y2+z2-12x-3y+4z+8=0
3x+4y-5z+3 12

2. Let beta and beta dash be the two ordered bases of IR3 Then find a matrix representing the linear transformation lR3 which transforms beta into beta dash. Use this matrix representation to find where x 1).

If x 3 0 · 0 1 and y 4 0 · 0 with approximately what accuracy can you calculate the polar coordinates r and 9 of the point Express your estimates as percentage changes of the values that r and 8 have at the point 4). 20

Find a 2 x 2 real matrix A which is both orthogonal and skew-symmetric. Can there exist a 3 x 3 real matrix which is both orthogonal and skew-symmetric? Justify your answer. 20

3. Let 1R. 4 IR3 be a linear transformation defined by
1 x2 X3, (X3 X4 Xl X2, X3 X2, X4 X1)
Then find the rank and nullity of L. Also, determine null space and range space of L. 20

Let f :R2 1R be defined as
<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-3b.jpg">

Is f continuous at Compute partial derivatives off at any point if exist. 20

A space probe in the shape of the ellipsoid 4x2 y2 +4z2 16 enters the earth's atmosphere and its surface begins to heat. After one hour, the temperature at the point on the pro be surface is given by
8x2 +4yz-16z 600
Find the hottest point on the probe surface.

4. Prove that the set V of all 3 x 3 real symmetric matrices forms a linear subspace of the space of all 3 x 3 real matrices. What is the dimension of this subspace? Find at least one of the bases for V. 20

Evaluate
<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-4b.jpg">
where S is the outer side of the part of lhe sphere x2 y 2 z2 1 in the first octant.

Prove that the normals from the point x2 y2 to the paraboloid x2/a2+y2/b2=2z lie on the cone
<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-4c.jpg">

Section-B

5. Attempt any five of the following

A body 1s describing an ellipse of eccentricity e under the action of a central force directed towards a focus and when at the nearer apse, the centre of force is transferred to the other focus. Find the eccentricity of the new orbit in terms of the eccentricity of the original 20 orbit. 12

Find the Wronskian of the set of functions{3x2,[3x3]} on the interval and determine whether the set is linearly dependent on l]. 12

A uniform rod AB is movable about a hinge al A and rests with one end in contact with a smooth vertical wall. If the rod is incl ined at an angle of 30° with the horizontal, find the reaction at the hinge m. magnitude and direction. 12

A shot fired with a velocity V at an elevation a strikes a point P 1n a horizontal plane thi:ough the point of projection. If the point P 1s receding from the gun with velocity show that the elevation must be changed to where
sin 28 sin 2 a +-2v/vsin 8 12

Show that div (grad rn) n(n I)r n
where <img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-4e.jpg"> 12

Find the directional derivative of-
4xz3 3x2 y2 z2 at along z-axis;
x2yz 4xz2 at in the direction of
2i 2k. 6+6

6. Find the differential equation of the family of circles in the xy-plane passing through and 1). 20

Find the inverse Laplace transform of
20

Solve
20

7. One end of a light elastic string of natural length l and modulus of elasti city 2mg is attached to a fixed point 0 and the other end to a particle of mass m. The particle ini tially held at rest at O is let fall. Find the greatest extension of the string during the motion and show that the particle will reach O again after a time
<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-7a.jpg">

A particle is projected with velocity V from the cusp of a smooth inverted cycloid down the arc. S how that the time of reaching the vertex is

<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-7b.jpg">

where a is the radius of the generating circle.

On a rigid body, the forces l O(i 2k) j 2k) N and 6(2i k N are acting at points with position vectors i 2i 5k and 4i k respec tively. Reduce this system to a A A single force R acting at the point (4i 2 together with a couple G whose axis passes through this point. Does the A 20 point (4i 2j) lie on the central axis? 15

Find the length of an endless chain which will hang over a circular pulley of radius a so as to be in contact with three-fourth of the circumference of the pulley. 15

8. Find the work done in moving the particle once round the ellipse z 0 under the field of force given by

<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-8a.jpg"> 20

Using divergence theorem, evaluate
<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-8b.jpg">
where <img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-8b1.jpg"> and S is the surface of the sphere x2+y2+z2=a2 20

Find the value of
<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-8c.jpg">
taken over the upper portion of the surface x y 2ax az 0 and the bounding curve lies in the plane z when
<img src="E:sirishaCIVILS MAINS 2013CIVILS MAINS-20092009 mains optionals_sireeshaimages38-8c1.jpg">


Note English version of the Instructions is printed on the front cover of this question paper.