## Engineering Mathematics, Question Paper of MCA Semester 1, Download Question Paper 1

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Roll No…..
Total No. of Questions: 9

B. Tech (Sem.-1st)
ENGINEERING MATHEMATICS-I
SUBJECT CODE: BTAM-101
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.

SECTION-A
Q1.
(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 SECTION-B
Q2.
(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.

Q3.

(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.

Q4.
(a) If then find the value of n that will make (b) State Euler’s theorem and use it to prove that x Q5.
(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.

SECTION-C
Q6.
(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.

Q7. (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.
Q8.
(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+cos and outside the circle r=2.
Q9.
(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.