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A1801 Pages: 2
Module 111
Answer any two questions, each carries 5 marks.
Write the parametric equations of the tangent line to the graph of r(t) = Inti +
6711 + ttkatt = 2
^ particle moves along the curve r = (६ - 4t)i+ (८2 +4t)j +
(8 t? - 3t3 )k where t denotes time. Find 1
(i) the scalar tangential and normal components of acceleration at time t = 2
(ii) the vector tangential and normal components of acceleration at time t = 2
Find the equation to the tangent plane and parametric equations of the normal line
to the ellipsoidx? + y? + 422 = 12 atthe point (2, 2, 1)
Module IV
Answer any two questions, each carries 5 marks.

1 |
⋅ ⋅ x
Reverse the order of integration and evaluate [1
Ox ४.

If R is the region bounded by the parabolas y = x?and y* = x in the first

quadrant, evaluate 7 ( + 4

Use triple integral to find the volume of the solid bounded by the surfacey = x?

andthe planes y + 2 = 4,z = 0.

Module V
Answer any three questions, each carries 5 marks.
Ifr = xi+ yj + zkandr = |[r||, show that Vlog r =

Examine whether F = (x? - yz)i + (y2-2x)j + 60 - xy)kis a conservative
field. If so, find the potential function

Show that7?f (r) = 2 me + f(r), wherer = xityjtzk, 7 = || ۶
Compute the line integral ford - x*dy) along the triangle whose vertices are
(1,0), (0,1)and ( -1, 0)

Show that the line integral [( ysin xdx - 605 xdy) is independent of the path and

hence evaluate it from (0, 1) and (a, - 1)
Module VI
Answer any three questions, each carries 5 marks.
Using Green’s theorem, find the work done by the force field ரீட்‌ 1) =
(e*—y?)i + (cosy + 22) ona particle that travels once around the unit circle
x? + y* = 1 inthe counter clockwise direction.

Using Green’s theorem evaluate [൭൦ y’)dx+x’dy, where c is the boundary of
the area common to the curve y = x*andy = x

Evaluate the surface integral (3205 ൧൩൦൭ 5 is the part of the plane

x + y + 2 = 1 that lies in the first octant

Using divergence theorem, evaluate [[.) ds where

F = (x? + 9) ६ + z*j +(e” - 2) kand 5 is the surface of the rectangular solid
bounded by the co ordinate planes and the planes x = 3, y= 1, 2-3

Apply Stokes’s theorem to evaluate [Far , whereF = (x? — y?)i + 2xyjand ௦ is

the rectangle in the xy plane bounded by the lines x = 0,y =0,x =aandy=b
‏بد‎ KK

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