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**ANNA UNIVERSITY:CHENNAI 600 025**

**MODEL QUESTION PAPER**

**B.E. / B.Tech. Degree Examinations**

**First Semester**

**MA 131 – Mathematics I**

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Time: 3 hours. Max.marks:100

**Answer All Questions **

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

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

10. Solve:

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**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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