CS (MAIN) EXAM, 2010 MATHEMATICS Paper—I
TIII16 Allowed Three Hours i Maximum Marks 300
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 spea?ed in the Admission
Certi?cate.
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 otheriuise indicated.


1 Attempt any five of the following

If lambda1, lambda2, lambda3 are the eigenvalues of the matrix <img src='./qimages/909-1a.jpg'>

What is the null space of the differentiation transformation where Pn is the space of all polynonnals of degree n over the real numbers? What is the null space of the second derivative as a transforrnation of P ‘P What is the null space of the kth derivative?

A twice-differentiable function 1s such that O and O for a c b. Prove that there is at least one point Q for which O

Does the integral <img src='./qimages/909-1d.jpg'> <br><br>exist?If so, find its value.

Show that the plane x y 22 3 cuts the sphere x2 +y2 in a circle of radius l and find the equation of the sphere which has this circle as a great circle.

Show that the function is Riemann integrable in the interval where denotes the greatest integer less than or equal to oz. Can you give an example of a function that is not Riemann integrable on Compute integral 0 to 2 dx, where is as above.

<img src='./qimages/909-2a.jpg'> <br><br>find the unique linear transforination T:IR3—>]R3 so that M is the matrix of T with respect to the basis
beta v2 :v3
of IR3 and
beta' {W1 w2
of R2. Also find z). 20

Show that a box (rectangular parallelepiped) of maxirnurn volume V with prescribed surface area is a cube.

Show that the plane 3x 4y 2 O touches the paraboloid 3x2 4y2 lOz and find the point of contact.

Let A and B be n n matrices over reals. Show that I — BA is invertible if I AB is invertible. Deduce that AB and BA have the sarne eigenvalues. 20
Let D be the region determined by the inequalities x y z 8 and x2 y2. Compute <img src='./qimages/909-3b.jpg'>
Show that every sphere through the
circle
x2 +y2 —2a.x+r2 z=O
cuts orthogonally every sphere through the circle
x2 +22 y=0
In the n-space IRn, determine whether or not the set {e1 e2, e2 ,en 1 is linearly independent.
Let T be a linear transformation from a vector space V over reals into V such that T— T2 =I. Show that T is invertible .

f is a homogeneous function of degree n in x and and has continuous first— and second—order partial derivatives, then show that <img src='./qimages/909-4b1.jpg'> <img src='./qimages/909-4b2.jpg'>
Find the vertices of the skew quadrilateral formed by the four generators of the hyperboloid
x2/4+y2-z2=49
passing through and — 2). 20

Section—B

5. Attempt any five of the following
Consider the differential equation
y’ ax, x 0
where alfa is a constant. Show that— <img src='./qimages/909-5a.jpg'> 12
Show that the differential equation

admits an integrating factor which is a function of Hence solve the equation. 12

Find <img src='./qimages/909-5c.jpg'> 12
If vl,v2,v3 are the velocities at three points C of the path of a projectile, where the inclinations to the horizon are oz on —2B and if tl, t2 are the times of describing the arcs AB, BC respectively, prove that <img src='./qimages/909-5d.jpg'>
Find the directional derivative of y)=x2y3 +Xy at the point in the direction of a unit vector which makes an angle of ‘rt/3 with the x—axis. 12
Show that the vector ?eld de?ned by the vector function <img src='./qimages/909-5f.jpg'> is conservative. 12
6. Verify that
<img src='./qimages/909-6a.jpg'>
Hence show that——
if the differential equation M dx N dy O is homogeneous, then (Mx Ny} is an integrating factor unless Mx Ny
if the differential equation Mdx+Ndy is not exact but is of the form f1(xy)y dx y)x dy=0 then (Mx — is an integrating factor unless Mx — Ny =O. 20

A particle slides down the arc of a smooth cycloid whose axis is vertical and vertex lowest. Prove that the time occupied in falling down the first half of the vertical height is equal to the time of falling down the second half. 20
(c)Prove that
div f(div (grad f).v
where f is a scalar function. 20

Show that the set of ‘solutions of the homogeneous linear differential equation
y’ 0
on an interval forrns a vector subspace W of the real vector space of continuous functions on I. What is the dimension of

A particle moves with a central acceleration new[r5 being projected from an apse at a distance root 3 with velocity 3root(2new). Show that its path is the curve x4 +y4 =9.

Use the divergence theorem to evaluate <img src='./qimages/909-7c.jpg'> the boundary of the region bounded by the paraboloid z x2 y2 and the plane z 4y.

Use the method of undetermined coefficients to find the particular solution of
y"+y=sin
and hence find its general solution. 20

A solid hemisphere is supported by a string fixed to a point on its rim and to a point on a srnooth vertical wall with which the curved surface of the hemisphere is in contact. If 9 and ¢ are the inclinations of the string and the plane base of the hemisphere to the vertical, prove by using the principle of virtual work that
tan¢=3/8+tan teta 20

(c)Verify Green’s theorem for
e-x siny dx e-x cosy dy
the path of integration being the boundary of the square whose vertices are pie/2] and pie/2). 20