Anna University Coimbatore 2010 B.E Mathematics-I Model - university paper
DEPARTMENT OF MATHEMATICS
MODEL EXAMINATION
MATHEMATICS- I
(Common to all branches of I Year B.E./B.Tech.)
Max. Marks : 100 Time: 3hrs
PART-A Answer ALL Questions (20 X 2 = 40)
1. Find
the sum and product of the Eigen values of the matrix
.
2. Write
the matrix of the quadratic form
.
3. If A
is an orthogonal matrix, show that 
is also
orthogonal.
4. Verify
Cayley-Hamilton Theorem for
.
5. Find
the equation of the tangent plane at the point (1,-1, 2) to the sphere
.
6. Write down the equation of the whose diameter is the line joining (1,1,1) and
(-1,-1,-1).
7. Test
whether the plane 
touches the sphere
8. Find
the equation of the right circular cone whose vertex is at the origin,
whose axis is the line 
, and
which has semi-vertical angle of 
9. Find
the radius of curvature at any point (x, y) on
.
10. Find the radius of curvature
at 
on the curve 
11. Find the envelope of 
where 
is a parameter.
12. Show that the family of
straight lines 
has no envelope
where 
is the parameter.
13. Expand 
in the neighborhood of (
1,1).
14. Find the
Stationary points of
.
15. If 
and 
find
.
16. If 
Find 
17. Transform
in to polar coordinates the integral 
18. Express the
Volume bounded by 
and 
19. 
20. Shade the region of
integration in
.
PART-B Answer ANY FIVE Questions (5 X 12 = 60)
21. a) Using
Cayley Hamiltons theorem, find A4 for the matrix A=
.(6)
b) Reduce the
quadratic form 
to canonical form
through an
orthogonal transformation. (6)
22. a) Find the
equation to the tangent planes to the sphere 
which
are parallel to the
plane 
.(6)
b) Find the equation to the right circular cylinder of radius 2 and whose axis is the
line
.(6)
23. a) Find the
equation of the sphere passing through the circle 
and the centre of the
sphere
.(6)
b) Find the equation of the right circular cone whose vertex is at the origin and the guiding curve
is the
circle 
(6)
24. a) Find the radius of
curvature at 
on 
(6)
b) Find the
Evolute of the Parabola 
(6)
25. a) Find the circle of
curvature at 
.(6)
b) Find the equation of the sphere that passes through
the circle 
,
and
cuts the sphere
orthogonally.(6)
26.a) A rectangular box open at the top is to have volume 32 cc. Find the dimensions of the box ,
that requires the least material for its construction.(8)
b) Find the
minimum value of 
with the constant
.(4)
27.a) Find the volume of the
ellipsoid 
by triple integration.(8)
b) Evaluate 
over the area bounded
between the circles 
and 
.(4)
28.a) Change the order of
integration in 
.(8)
b) Find the area enclosed
by the curves 
and 
.(4)
| Earning: Approval pending. |
