10.

. The length, width and height of a rectangular box are measured with an error of atmost

14.
15.

20.

21.

22.

23.

B1A005 Pages: 3

Find the Taylor series expansion ೦8108 cos x about the point = (5)
Module II
5 3,34 07 ൮
{1५ = ‏وم(‎ (x + + + 2 —3xyz), Show that (= + D ‏د چٹ ) یئ‎ (5)

5%. Use a total differential to estimate the maximum percentage error that results if these

quantities are used to calculate the diagonal of the box. (5)
. Locate all relative extrema and saddle points of f(x,y) = ‏كير + كير‎ — 2x? + 4xy - 2y?
(5)
Module 111
. Find the angle between the surfaces ऋ + )2 +z? = 9 and z= x? + y? — 3at the point
(2,-1,2) (5)
Let ?=xit+ yj + 24 and r= ||, then prove that Vf(r) = ഥം (൭

Find an equation of the tangent plane to the ellipsoid 2x? + 397 + 27 = Yat the point
(2,1,1) and determine the acute angle that this plane makes with the XY plane. (5)

Module IV
, Change the order of integration and hence evaluate 1 ` xy dy dx (5)
. Evaluate i [४८ (५2 + y’) dx dy using polar co-ordinates (5)
. Find the volume of the paraboloid of revolution x? + y? =4z cut off by the 01೩70
2 = 4 (5)
Module V

Answer any 3 questions.

. Evaluate the line integral ‏مل‎ (xy + z*)ds वीणा (1,0,0) to (-1,0, z)along the helix 0

that is represented by the parametric equations x = cos ‏را‎ y = Sint, 2 =t (5)

Evaluate the line integral |. (क — x) dx + x’y dy along the curve C, y? = x? from (1, -

1) to (1,1) (5)
Find the work done by the force field ¢ = (८ + y)itxyj - 22% along the line segment
from (0,0,0) to(1,3,1) and then 10 (2, —1,5). (5)
Show that ¢ = (2xy + z3)i + ‏زمر‎ + 3xz?k is 8 conservative vector field. Also find its
scalar potential. (5)

Find the values of constants a,b,c so that F = (മ്മ + 8297 + (3x? - ‏زری‎ + (3x2? ~ yk
may be irrotational. For these values of a, b,c find the scalar potential of F (5)

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