I C.S.E-Mnins 2002
I MATHEMATICS

PAPER-I
SECTION-A
1.Attempt any of the following:

Show that the mapping where is linear
and non singular.(12)
A square matrix A is non singular If and only if the constant h!nu in its characteristic polynomial
iS different from zero.(12)
show that <img src='./qimages/1102-1c.jpg'>
Show that <img src='./qimages/1102-1d.jpg'>
Show that the equation
9x2-16y2-l8x-32y-151=0
represents a hyperbola. Obtain its eccentricity and foci.(12)
(f)Find the co-ordinates of the centre of the sphere inscribed in the tetrahedron formed by the
planes
and
2.(a)Let be a linear mapping given by
Obtain bases for its null space and range space.(15)

(b)Let A be a real 3*3 symmetric matrix with eigen values 0,0 and 5. If the corresponding
eigen vectors are and then find the matrix A
Solve the following system of Linear equations:
X1-2x2-3x3+4x4
-x1-3x2 5x4-2x 0
2x1+x-2x3+3x4-4x5=17

Use Cayley- Hamilton theorem to find the inverse of the following matrix:
<img src='./qimages/1102-2d.jpg'>

3.(a)let sin1/x
0 x=0
Obtain condition on p such that f iS continous at x=0
and f is differentiable at x=0.

Consider the set of triangles having given base and a given vertex angle. Show that the
triangle having the maximum area will be isosceles.

if the roots of the equation
<img src='./qimages/1102-3c.jpg'>


(d)Find the centre of gravity of the region bottndcd by the and both
axes in the first quadrant, the density benign p kxy. where k is constant

Tangents are drawn from any point on the ellipse x2/a2+y2/b2=1 to the circle x2-y2=r2,
Show that the chords of contact are tangents to the ellipse a2x2+b2y2 r4.

Consider a rectangular parallelepiped with edges obtain the shortest distance between
one of its diagonals and an edge which does not intersect this diagonal
Show that the feed of the six normals drawn from any point (alpha,beta,gaama) ellipsoid
x2/a2+y2/b2= 1
lie on the cone
<img src='./qimages/1102-4c.jpg'>

variable plane parallel to the plane
0 meet the coordinate axes of A,Band C. Show that the circle ABC lies on the
conic
<img src='./qimages/1102-4d.jpg'>

SECnON-B
5. Attempt any five of the following:
Solve:x dy/dx+3y=x3y2

Find the value of lamda for which all solution of
x2 d2y/dx2+3y dy/dx-lamda y=0
tend to zero as x-->infinity

A particle of mass m is acted upon by a force towards the origin.
If it starts from rest at a distance a from the origin, show that the time taken by it to reach the origin is pie/4.( 12)

Obtain the equation of the curve in which a string hangs under gravity from two fixed points
(not lying in a vertical line), when line mass density at each of its points varies as the radius
of curvature oft11c curve.(12)

half the ellipSe is vertically immersed in water with minor axis just in the surface. Find the
position of centre of pressure.

let R bar be the unit Vector along the vector r bar(t).Show that <img src='./qimages/.jpg'>

6.(a)Find the value of constant lamda such that the following differential equation becomes exact
<img src='./qimages/1102-6a.jpg'>
Further,for this value of lamda, solve the equation.
Solve:
(c)using the method of variation of parameter. find the solution of <img src='./qimages/.jpg'>
with and

where D=d/dx.

A heavy particle of mass m slides on a smooth are of a cycloid in a medium whose resistance
in mv2/2C, v being the velocity of the particle and c being the distance of the starting point
from the vertex.If the axis is Vertical and vertex upwards, find the velocity of the particle at
the cusp.

particle describes a curve with constant velocity and its angular velocity about a given
point O varies inversely as its distance from(). Show that the curve is an equiangular spiral.(15)

Five weightless rods of equal lengths are jointed together so as to form a rhombus ABCD
with a diagonal BD. lf a weight W be attached to C and the system be suspended from a point
A show that the trust BD is equal to W/root3.

solid cylinder floats in a liquid with its axis vertical. Let sigma be the ratio of the specific
gravity of the cylinder to that of the liquid
prove that the equilibrium is stable if the ratio of
the radius of the base to the height is greater than root 2sigma(1-sigma)

Find the curvature for the space curve:
x=acos theta,y=asin theta,z=a theta tan alpha

<img src='./qimages/1102-8b.jpg'>

let D be a closed and bounded region having boundary S. further, let f be a scalar function
having second order partial derivatives defines on it. Show that
<img src='./qimages/1102-8c.jpg'>

Find the vaule of constants a,b and c such that the maximum value of directional derivative
off= axy2 byz +cx2x2 at( is in the direction parallel to y-axis and has magnitude 6. 15)