Compre+Part+B - Comprehensive Exam Question Paper PDF

Preview

Summary

Comprehensive Exam Question Paper...

Description

BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI, HYDERABAD CAMPUS FIRST SEMESTER 2013-2014 MATHEMATICS-I: MATH F111 Comprehensive Exam (Closed Book) PART-B Date: 09-12-2013 Day: Monday

Time: 10:30 – 12:00 Max Marks: 40

Answer all questions. Start answering each question from fresh page. 1. Find an equation for the circle of curvature of the curve

( π2 , 1)

r ( t ) =t ^i+sint ^j

.

at

[7M]

2. Find the values of the constants a, b and c so that the directional derivative of  axy 2  byz  cz 2 x 3 at (1, 2, -1) has a maximum magnitude 64 in a direction parallel to the z-axis.

[6M]

3. Using cylindrical coordinates and triple integral, find the volume of the solid that is bounded by the graph of y = x 2 +z 2∧2 y = x 2+z 2 +4. [6M] 4. Without using line integral, calculate the counter-clock wise circulation and outward flux for the field and curve given below:  F =( −2 y 2 ) ^i+(4 xy ) ^j C : The boundary of the region in the first quadrant determined by the graphs

of y =0, y = √ x∧ y =2−x .

[8M]

2 5. Find the area of the region inside the curve r 6 cos 2 and outside the curve

r  3 . Sketch the curves and region neatly.

[5M] 

 (  1)

n 1

 2 x  3 n

n

n3 6. Find the radius and interval of convergence for the given series n 1 . x Also find for what values of , does the given series converge (a) absolutely, (b) conditionally? [8M]

****All the Best****...