**Roll No…..**

**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 r

^{2}=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 xy

^{2}=4a^{2}(2a-x) and it asymptote.
(b)
The curve y

^{2}(a+x)=x^{2}(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 x^{2}+y^{2}+z^{2}=1/
b)
Expand x

^{2}y+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 x

^{2}+y^{2}=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 y

^{2}+yz^{3}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=(4x

^{3}i-x^{2}yj+x^{2}zk and S is the surface of the cylinder x2+y2=a^{2}bounded by the planes z=0, and z=b.
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