C.S.E. (MAIN)
MATHEMATICS-2004
(PAPER-I)
Time allowed: 3 hours Max. 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 is.med 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 thantltat spec(fied in the Admission Certificate.
Candidates should attempt Questions I and 5 which are
compulsory, and any three of tile remaining questions selecting at
least one question from each Section.
Assume suitable data if considered necessmy and indicate the
same clearly.
All questions carry equal marks.
SECTION
Q.1.Attempt any five of the following:
(a)Let S be space generated by the vectors What is the dimension of the space S Find a basis for S.
Show that f:IR3->IR is a linear transformation, where 3x z. What is the dimension of the kernel? Find a basis for the kernel.

(c)Prove that the function f defined on by greatest integer x belongs to is integrable on and that <img src='./qimages/1156-1c.jpg'>

Show that src='./qimages/1156-1d.jpg'>

Prove that the locus of the foot of the perpendicular drawn from the vertex on a tangent to the parabola y2 4ax is y2 x3=0.
Find the equations of the tangent planes to the sphere x2 y2+ z2-4x 2y- 6z 5 which are parallel to the plane 2x+y-z=4.

Q.2.(a) Show that the linear transformation from IR3 to IR4 which is represented by the matrix <img src='./qimages/1156-2a.jpg'> is one-to-one. Find a basis for its image.
Verify whether the following system of equations is consistent
x+3z
-2x


(c)Find the characteristic polynomial of the matrix A=<img src='./qimages/1156-2c.jpg'> Hence find A and A6.

Define a positive definite quadratic form. Reduce the quadratic form x12+x32+2x1x2+2x2x3 to canonical form. Is this quadratic form positive definite
Q.3.(a) Let the roots of the equation in lamda.
(lamda-x)3 0
be w.Prove that
<img src='./qimages/1156-3a.jpg'>

(b)Prove that an equation of the form xn =alpha, where ne/N and alpha>0 is a real number, has a positive root.

(c)Prove that:<img src='./qimages/1156-3b.jpg'> when the integral is taken round the ellipse x2/a2+ y2/b2=1 and p is the length of three perpendicular from the centre to the tangent.
the function f is defined by <img src='./qimages/1156-3d.jpg'> then show that f possesses both the partial derivatives at but it is not continuous thereat.

Q.4.(a) Find the locus of the middle points of the chords of the rectangular hyperbola x2-y2 a2 which touch the parabola y2=4ax.

Prove that the locus of a line which meets the lines y ± mx, z c and the circle x2 y2 z=0 is c2 m2 mzx)2 c2 a2 m2(z2-c2)2.
Prove that the lines of intersection of pairs of tangent planes to ax2 by2 cz2 0 which touch along perpendicular generators lie on the cone a2(b c)x2 b2(c+a)y2 b)z2 0.

Tangent planes are drawn to the ellipsoid <img src='./qimages/1156-4d.jpg'> through the point (alpha, beta, gaama).prove that the perpendiculars to them through the origin generate the cone (alphax+ betay gaamaz)2 =a2x2 b2y2 +c2z2.
SECTION
Q.5.Attempt any five of the following
(a)Find the solution of the following differential equation
dy/dx+ycosx 1/2sin2x.
Solve y(xy 2x2y2) dx x2y2)dy 0.

A point moving with uniform acceleration describes distances s1 and s2 metres in successive intervals of time t1 and t2 seconds. Express the acceleration in terms of sl, s2 t1 and t2
A non uniform string hangs under gravity. Its cross-section at any point is inversely proportional to the tension at that point. Prove that the curve in which the string hangs is an arc of a parabola with its axis vertical.
A circular area of radius a is immersed with its plane vertical, and its centre at a depth c. Find the position of its centre of pressure.
Show that if A and B are irrotational, then A bar*B bar solenoidal.
Q.6.(a) Solve:<img src='./qimages/1156-6a.jpg'>
Reduce the equation (py 2p, where p=dy/dx to Clairaut's equation and hence solve it.
(c)Solve: <img src='./qimages/1156-6c.jpg'>
Solve the following differential equation:
<img src='./qimages/1156-6d.jpg'>

Q.7.(a) Prove that the velocity required to project a particle from a height h to fall at a horizontal distance a from a point of projection, is at least equal to <img src='./qimages/1156-7a.jpg'>

A car of mass 750 kg is running up a hill of l in 30 at a steady speed of 36 km/hr; the friction is equal to the weight of 40 kg. Find the work done in 1 second.

A uniform bar AB weights 12 N and rests with one part,AC of length 8m, on a horizontal table and the remaining part CB projecting over the edge of the table. If the bar is on the point of overbalancing when a weight of 5N is placed on it at a point 2m from A and a weight of7 N is hung from find the length of AB.

A cone, of given weight and volume, floats with its vertex downwards. Prove that the surface of the cone in contact with the liquid is least when its vertical angle is 2 tan-1( 1/root 2).
Q.8.(a) Show that the Frenet-Serret formulae can be written in the form
<img src='./qimages/1156-8a.jpg'>
Prove the identity
<img src='./qimages/1156-8b.jpg'>
Derive the identity
<img src='./qimages/1156-8c.jpg'>
where V is the volume bounded by the closed surface S.
(d)VerifY Stokes' theorem for
<img src='./qimages/1156-8c.jpg'>
where S is the upper half surface of the sphere x2 y2 z2 l and C is its boundary.