# Anna University Question Papers – Mathematics – I SEM

ANNA UNIVERSITY:CHENNAI 600 025

MODEL QUESTION PAPER

B.E. / B.Tech. Degree Examinations

First Semester

MA 131 – Mathematics I

Time: 3 hours.                                                                                       Max.marks:100

PART- A (10 x 2 Marks = 20 Marks)

1. If  are the eigenvalues of a matrix  what are the eigenvalues of  and
2. If  write  in terms of  and  using Cayley-Hamilton theorem.
3. Find the radius of curvature at any point on the curve
4. Find the envelope of the family where  is a parameter.
5. If  evaluate
6. Find the direction cosines of the line drawn from the point (1,0,1) to (1,1,-1).
7. Find the equation of th sphere on the line joining (1,1,1) and (2,2,2) as diameter.
8. If  prove that
9. Find the particular integral of

10. Solve:

PART-B (5 x 16 Marks = 80 Marks)

Question No.11 has no choice; Questions 12 to 15 have one choice   (EITHER-OR TYPE) each.

11. Reduce the quadratic form  into a canonical form by means of an

orthogonal transformation.  Determine its nature. Find a set of non-zero values for  for which the above quadratic form is zero.

12. (a) (i)  Find the image of the point  in the plane        (6)

(ii) Show that the lines  and  are coplanar.  Find the coordinates of their point of intersection and the equation of the plane containing them.                        (10)

OR

(b)   (i) Find the equation of the sphere passing through the points  and having its centre on the line                                                                            (8)

(ii) Find the tangent planes to the sphere    that are parallel to the plane                                                           (8)

13. (a) Find the evolute of the cycloid:

OR

(b) (i) Find the evolute of the parabola  considering it as the envelope of its normals.                                                                                                            (8)

(ii) Find the equation of the circle of curvature of  at    (8)

14. (a) (i) Obtain terms up to the third degree in the Taylor series expansion of    around the point                                                                        (10)

(ii) By differentiating under the integral sign, show that

(6)

OR

(b) (i) If  and  find  without actual substitution.    (6)

(ii) Show that the points on the surface  nearest to the origin are at a distance  from it.                                                                                 (10)

15. (a) (i) Solve  given that     (8)

(ii) Solve by the method of variation of parameters:      (8)

OR

(b) (i) Solve:                                                 (8)

(ii) Solve by reducing the order  given that  is a solution.                                                                                 (8)

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