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A1100 Pages: 3

Module II
Answer any two questions, each carries 5 marks
OW OW OW
If W = f(x-y,y-z,z-x) , Show that ‏سے ہس ہس‎ = 0
ox @ az

If f(x,y) = ysinx+e™ cosy, find ‏کر‎

೫೫ '

State the second partial test for local(relative)extreme values.

An aquarium with rectangular sides and bottom (no top) is to hold 32 liters of
water. Find its dimension so that least quantity of material is required for its

construction.
Module III
Answer any two questions, each carries 5 marks

Find the unit tangent 70) and unit normal N(t) to the curve
F(t) = ९। costi +e! 511 ( + e'k at t=0.

⋅⋅ ↰ ⋅⋅−−↶⋅↥⋀⊳
⊱⊔≸⊃⋢⊃∘⋮⊜⋔⊟⊑≻∘⋦⋯∘⋂⋁⊖∁⇇∘↾∘⋮∂⋯∘⋁⋯≣⊑⊃∂⋔∁∣⊜↥⊊⊺∶↾≳∣ ⊹∃⊣∍∫≖ find the

displacement and distance traveled over the time interval 2 5154.
Find the equation of tangent plane to the surface x°+y?+z°=2S5at
P(3,0,4).Also find the parametric equation for the normal line to the surface at P.

Module IV
Answer any two questions, each carries 5 marks

Evaluate 1 ச dxdy , where the region R is given by 2yR

Evaluate [क्र 00/00 over the cardioid r = a(1—cos@) above the initial line.

Evaluate the triple integral fff --sin(z)aV over the rectangular box

O02 6

Module V
Answer any three questions, each carries 5 marks

2 ies. ‏پ2‎ നന്ന
Prove that V? f(r) =f" (r)+— /'(1) where ‏قرول دم‎ +y? 427.
Evaluate [றன்‌ + சஸ்‌ + ‏تند‎ along the twisted cubic x=f,y=f°,z=¢° from

(0,0,0) to (1,1,1)

Find the potential function for the vector field

F= (sinz + ycosx)i +(sinx + 2 605 y)j +(sin y+xcos 21

Find the work done in moving a particle in the force field

F =(x+y)i-x°j+(y+z)k along the curve defined by x? =4y,z =x,0
Show that foz —Ddxt+(z+xz4+2°)dy+(y+xyt+2yz)dz is independent of the
£

path of integration. Find the scalar potential and the value of the integral from
(1,2,2) to (2,3,4).

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