C.S.E. (MAIN)
MATHEMATICS 2006
PAPER-I
Time allowed: 3 hours Maximum Marks 300
INSTRUCTIONS
Each question is printed both in Hindi and in English.
Answers must be written in the medium specified in the Admis'sion
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 Questions 1 and 5 which are compulsory,
and any three of the remaining questions selecting at
least one question from each Section.
Assume suitable data if considered necessmy and indicate the
same clearly.
The number of marks carried by each question is indicated at
the end of the question.
SECTION'A'
Q. 1. Attempt any five ofthe following:
Let V be the vector space of all 2 x 2 matrices over the field F. Prove that V has dimension 4 by exhibiting a basis for V.
State Caylay-Hamilton theorem and using it, find the inverse of
<img src='./qimages/1145-1b.jpg'>

Find a and b so that exists,where

<img src='./qimages/1145-1c.jpg'>

Express <img src='./qimages/1145-1d.jpg'> in terms of Gamma function and hence evaluate the integral
<img src='./qimages/1145-1d1.jpg'>

pair of tangents to the conic ax2 by2 1 intercepts a constant distance 2k on the y-axis. Prove that the locus of their point of intersection is the conic. ax2(ax2+by2-1)= bk2(ax2-1)2

Show that the length of the shortest distance between the line z x tan alpha, y 0 and any tangent to the ellipse x2 sin2 alpha y2= a2, z=0 is constant.

Q.2.(a)If T IR2->IR2 is defined by compute the matrix of T relative to the basis beta=
Using elementary row operations, find the rank of the matrix
<img src='./qimages/1145-2b.jpg'>

(c)Investigate for what values of lamda and mew the equations

x+2y+3z=10
lamda Z =mew
have-
no solution;
a unique solution;
infinitely many solutions.

Find the quadratic form corresponding to the symmetric matrix <img src='./qimages/1145-2d.jpg'>
Is this quadratic form positive definite ?Justify your answer.
Q.3.(a) Find the values of a and b such that
<img src='./qimages/1145-3a.jpg'>


<img src='./qimages/1145-3b1.jpg'>
show that
<img src='./qimages/1145-3b2.jpg'>

(c)Change the order of integration in
<img src='./qimages/1145-3c.jpg'>
and hence evaluate it.

Find the volume of the uniform ellipsoid
<img src='./qimages/1145-3d.jpg'>


Q.4.(a) If PSP' and QSQ' are the two perpendicular focal chords of a conic 1/r 1 ecos theta, prove that 1/SP·SP'+1/SQ·SQ' is constant.
Find the equation of the sphere which touches the plane at the point and cuts orthogonally the sphere x2 y2+ z2 4x 6y 4 0
Show that the plane ax+ by+ cz 0 cuts the cone xy+yz+zx 0 in perpendicular lines, if
If the plane lx my nz p passes through the extremities of three conjugate semidiameters of the ellipsoid x2/a2+y2/b2+z2/c2=1 prove that a2 l2 b2 m2 c2 n2 3 p2
SECTION'B'
Q. 5.Attempt any five of the following

Find the family of curves whose tangents form an angle pie/4 with the hyperbolas xy c,c>0.
Solve the differential equation
<img src='./qimages/1145-5b.jpg'>

A particle is free to move on a smooth vertical circular wire of radius a. It is projected horizontally from the lowest point with velocity 2 root ga. Show that the reaction between the particle and the wire is zero after is time
<img src='./qimages/1145-5c.jpg'>
The middle points of opposite sides of a jointed quadrilateral are connected by light rods of lengths If be the tensions in these rods, prove that
<img src='./qimages/1145-5d.jpg'>

(e)Find the depth of the centre of pressure of a triangular lamina with a vertex in the surface of the liquid and other two vertices at depths b and c from the surface.

Find the values of constants b and c so that the directional derivative of the function. axy2 byz cz2 x3 at the point has maximum magnitude 64 in the direction parallel to z-axis.
Q.6.(a) Solve:
<img src='./qimages/1145-6a.jpg'>

Solve the equation <img src='./qimages/1145-6b.jpg'> using the substitution y u and xy v and find its singular solution, where p=dy/dx

Solve the differential equation
<img src='./qimages/1145-6c.jpg'>

(d)Solve the differential equation
<img src='./qimages/1145-6d.jpg'>by the method of
variation of parameters.

Q.7.(a) A particle, whose mass is is acted upon by a force towards the origin. If it starts from rest at a distance show that it will arrive at origin in time pie/4.

If u and V are the velocity of projection and the terminal velocity respectively of a particle rising vertically against a resistance varying as the square of the velocity, prove that the time taken by the particle to reach the highest point is
<img src='./qimages/1145-7b.jpg'>

Show that the length of an endless chain, which will hang over a circular pulley of radius c so as to be in contact with two third of the circumference of the pulley is
<img src='./qimages/1145-7c.jpg'>

A uniform rod of length 2a, can turn freely about one end, which is fixed at a height above the surface of the liquid. If the densities of the rod and liquid be row and sigma,show that the rod can rest either in a vertical position or inclined at an angle theta to the vertical such that
<img src='./qimages/1145-7d.jpg'>

Q.8.(a)If <img src='./qimages/1145-8a.jpg'> determine a vector R bar satisfying the vector equations.
<img src='./qimages/1145-8a1.jpg'>
Prove that rn r bar is an irrotational vector for any value of but is solenoidal only if n+3 =0.
If the unit tangent vector t bar and binormal b bar make angles theta and pie respectively with a constant unit vector a bar,prove that
<img src='./qimages/1145-8c.jpg'>

Verify Stokes' theorem for the function
<img src='./qimages/1145-8d.jpg'>
integrated round the square in the plane z 0 and bounded by the lines x x and a>0.