Anna University Coimbatore 2010 B.E Mathematics-I Model - exam 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 .
2. A is a singular matrix of order three, 2 and 3 are the eigen values. Find its third eigen value ?
3. If the sum of two Eigen values of a matrix is equal to its trace. Find its determinant value.
4. If the Eigen values of A are 1, 2, 3 then what are the eigen values of Adj A.
5. Find the equation of the plane through (1, 2, 3) parallel to the plane
6. Show that the spheres and intersect at right angles.
7. Find the equation of the cone whose vertex is the origin and base the circle
and show that the section of the cone by a plane parallel to the plane XOY is a hyperbola.
8. Define right circular cylinder.
9. Find the co-ordinates of the centre of curvature of the curve at the point (1,-1).
10. Find the radius of curvature on the curve at
11 Show that the radius of curvature at any point of the catenary
.
12. Find the envelope of
13..Find Taylors series expansion of near the point (-1,) up to the first degree terms .
14. Define Jacobian in two dimensions.
15. A flat circular plate is heated so that the temperature at any pointis
Find the coldest point on the plate.
16. Find when
17. Transform in to polar coordinates the integral
18. Express the area enclosed by and as double integral.
19. Sketch roughly the region of integration for the following double integral
20. Evaluate
PART-B (Answer ANY 5 questions) (5 X 12 = 60)
21.a) Using Cayley Hamiltons theorem find for the matix A=(8)
b) Find the eigen values of (4 )
22 a) Show that the plane touches the sphere also
find the point of contact.(6)
b) Find the equation of the right circular cylinder whose axis is and radius is 4.
Prove that the area of the section of this cylinder by the plane is.(6)
23 a) Find the circle of curvature at on (6)
b) Find the envelope of the straight line where a and b are parameters that are connected
by the relation. (6)
24. a) Find the radius of curvature of the parabola at t. (6)
b) Find the extreme values of the function.
25. Considering the Evolute of a curve as the envelope of its normal , find the Evolute of
.
26. a) Given the transformation and that is a function of u and v also of
x and y ,prove that (6)
b) Investigate for the maxima and minima if any, of the function (6)
27. a) The temperature T at any point (x, y, z) in space is T=, where c is a constant .Find the
highest temperature on the surface of the sphere . (6)
b) Expand in powers of x and y up to terms of third degree (6)
28. a) Find the volume of the ellipsoid by triple integration. (7)
b) Transform into polar coordinates and evaluate (5)
Earning: Approval pending. |