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

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Total No. of Questions: 9                              

B. Tech (Sem.-1^st)

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+cosand 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.