Anna University Coimbatore 2010 B.E Mathematics-I Model - Question 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 (20X2=40)
1. Find
the sum and product of the eigen values of the matrix A=
2. One
of the Eigen values of 
is -9.
Find the other two Eigen Values.
3. If A is a singular matrix of order three, 2 and 3 are the eigen values then find its third eigen value
4. State Cayley-Hamilton theorem.
5. If the sum of the Eigen Values of the matrix of the quadratic form equal to 0, then what will be the nature of the quadratic form?
6. Find the equation of the sphere whose diameter is the join of (2,-3,1) & (1,-2,1).
7. Test
whether the plane x = 3 touches the sphere 
.
8. Find
the centre and radius of the great circle on the sphere
.
9. Find
the radius of curvature at 
.
10. What is the curvature of 
at any point on it?
11. Define Evolute and Involute.
12. Find the envelope of the
family of lines 
, m being the
parameter.
13. If 
14. Find the stationary points
of
.
15. If 
16. A flat circular plate is
heated so that the temperature at any point (x ,y) is 
. Find the coldest point on
the plate.
17. Evaluate 
.
18. Evaluate 
.
19. Shade the region of
integration 
.
20. Why do we change the order of integration in multiple integrals? Justify your answer with an example.
PART-B Answer ANY FIVE questions (5 X 12 = 60)
21. a) Find the Eigenvalues and
Eigen vectors of the matix A=
.(6)
b) Find the Eigenvalues and Eigen vectors of the
matrix 
.(6)
22. Reduce the quadratic
form 
to canonical form through an
orthogonal transformation. (12)
23. a)Find the centre and
radius ofcircle
b)Find the equation of tangent
planes to the sphere 
which
passes through the line
.(6)
24. a)If 
, prove that
where
is the radius of curvature of
the curve.(6)
b) Find the equation of the circle
of curvature of the parabola 
at the
point (3,6) .(6)
25. a) A rectangular box open at the top is to have volume of 32 cc. Find the dimensions of the box requiring least material for its construction, by Lagranges method. (8)
b) Find the extreme values of the
function
.(4)
26. a)Change the order
of integration and then evaluate 
(8)
b) Evaluate by changing to
polars,the integral 
.(4)
27. a) Find the volume of the tetrahedron bounded by
the planes 
and the
Co-ordinate planes.(8)
b) Find the area of a circle of radius a by double integration in polar coordinates. (4)
28. a) Find the equation to the right circular cone whose
vertex is the origin, where the axis is the line 
and
which has a vertical angle of 300
(6)
b) Show that the plane 
touches
the sphere
and find
the point of contact (6)
| Earning: Approval pending. |
