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Thursday, November 26, 2015

Roll No…..
Total No. of Questions: 9                              

B. Tech (Sem.-1st)
Paper ID: [A1101] (2011 Batch)

Time: 3 Hrs.                                                                           Max. Marks: 60
Instruction to Candidate:
1. Section-A is Compulsory.
2. Attempt and Five questions from Section-B & C.
3. Select atleast Two questions from Section-B & C.

          (a) Identify the symmetries of the curve r2=cos
(b) Find the Cartesian co-ordinates of the point (5, tan-1(4/3) given in polar co-ordinates.
          (c) If u=F(x-y, y-z, z-x), the show that
(d) If u is a differentiable vector function of t of constant magnitude, then show that u.
(e) Change the Cartesian integral into an equivalent polar integral.
(f) For what values of a, b, c the vector function f=(x+2y+az) i-(bx-3y-z) j+(4x+cy+2z) k is irrotational.
(g) Give the physical interpretation of divergence of a vector point function.
          (h) What surface is represented by  
(i) If x= r cos and y= r sin, then find the value of  
          (j) Given that F (x, y, z) =0, then prove that
(a) Show that radius of curvature at any point (x,y) of the hypocycloid  is three times the perpendicular distance from the origin to the tangent at (x,y)
          (b) Trace the curve r=1+cos by giving all salient features in detail.


(a) Find the area included between the curve xy2=4a2(2a-x) and it asymptote.
(b) The curve y2(a+x)=x2(3a-x) is revolved about the axis of x. Find the volume generated by the loop.

(a) If  then find the value of n that will make
(b) State Euler’s theorem and use it to prove that x

(a) The temperature T at any point (x, y, z) in the space is T=400 x y z 2. Use lagrange’s multiplier method to find the highest temperature on the surface of the unit sphere x2+y2+z2=1/
b) Expand x2y+3y-2 in ascending powers of x-1 and y+2 by using  Taylor’s theorem.

(a) Evaluate  by changing the order of integration.
(b) Find the volume bounded by the cylinder x2+y2=4 and the planes y+z=4 and z=0.

(a) Prove that: grad div F=curl curl F+   2 F.
(b) Usethe stoke’s theorem to evaluate
Where C is the boundary of the triangle with vertices (2, 0, 0), (0, 3, 0), and (0, 0, 6) oriented in the anti-clock wise direction.
(a) Find the directional derivative of  f (x,y,z)= x y2+yz3 at (2,-1,1) in the direction of i+2j+2k.
(b) Find the area lying inside the cardiode r=2(1+cosand outside the circle r=2.
(a) State greens’ theorem in plane and use it to evaluate  where C is the triangle enclosed by y=0, x=
(b) State Divergence theorem use it to evaluate  where F=(4x3i-x2yj+x2zk and S is the surface of the cylinder x2+y2=a2 bounded by the planes z=0, and z=b.



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