Engineering Tutorials

B.E. Ist Semester Examination,

May – 2009

Mathematics – 1

Paper – Math – 1

Note: Attempt five questions in all, selecting two questions from each part.

1. (a) Test the convergence or divergence of the series : 6

Σ |

(b) Discuss the convergence of the series 7

x +

(c) State, with reasons, the values of x for which the series.

X – converges

2. (a) Compute to four decimal places, the value of cos 32^o, by use to Taylor’s series. 6

(b) If p the radius of curvature at any point P on the parabola y² = 4ax and S be its focus, then show that p2 varies as (SP)³. 8

(c) Show that the asymptotes of the curve x²y² = a² (x² + y²) form a square of side 2a. 6

3. (a) If u = x² tan^-1 (y/x) – y² tan^-1 (x/y), evaluate 10

x2

(b) If f (x, y) = tan^-1 (xy), compute f (0.9,-1.2) approximately. 10

4. Find the maximum and minimum distances from the origin to the curve 5x² + 6xy + 5y² – 8 = 0.

(b) Evaluate dx 10

Part – B

5. (a) Find the volume of solid formed by revolving a loop of the lemniscate r² = a² cos 2θ about

the initial line 10

(b) Evaluate dy dx by changing the order of integration. 10

6. (a) Evaluate ∫∫∫ (x + y + z ) dx dx dz over the tetrahedron bounded by the planes x = 0, y = 0, z =

0 and x + y + z = 1. 10

(b) Show that . 10

7.(a) Find the constants a and b so that the surface ax² – byz = (a + 2) x is orthogonal to the surface

4x²y + z3 = 4 at the point (1, – 1 , 2) 10

(b) If v₁ and v₂ be the vectors joining the fixed points (x₁,y₁,z₁) and (x₂,y₂,z₂) respectively to a

variable point (x, y, z), prove that 10

8. (a) verify stoke’s theorem for F = (x² + y²) i – 2xy j taken around the rectangle bounded by the

lines x = ± a, y = 0, y = b 10

(b) Using divergence theorem, evaluate R. N ds where S is the surface of the

sphere x² + y² + z² = 9 10

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