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. |