QP Code 1 6 0 1 0 1 Register Number
VELALAR COLLEGE OF ENGINEERING AND TECHNOLOGY
(An Autonomous Institution, Affiliated to Anna University, Chennai)
Semester Examinations Apr May 2017 Regulations-2016
Programme: B.E/B.Tech Semester: 1 Max. Marks: 100 Duration 3 Hrs
Course Code Title: 16MAT11 MATHEMATICS-I
Knowledge
Levels
K1 Remembering K3 Applying K5 Evaluating
K2 Understanding K4 Analyzing K6 Creating
Part A Answer ALL Questions. 10 x 2 20 Marks
No. Question KL
1. If 3 are the eigen values of then find the eigen values of A2, and A-1. K4
2.
Show that the matrix P




sin
cos





cos
sin
is orthogonal.
K3
3.
Show that the series ...
5
1
4
1
3
1
2
1
1 2 2 2 2 is absolutely convergent.
K3
4. State the D'Alembert's ratio test for Convergence of a positive term series. K1
5. Find the radius of curvature for at the point where it cuts the y axis. K3
6.
Find the envelope of the family of straight lines
m
a
y mx where m is the parameter.
K5
7.
If
x
z
z
y
y
x
u find
z
u
z
y
u
y
x
u
x








K5
8.
If u f y z x prove that 0








z
u
y
u
x
u
K3
9. Find the area of a circle of radius with using double integration. K5
10.
Evaluate
1
0
2
1
x(x y)dydx
K5
Part B Answer ALL Questions. 5 x 16 80 Marks
No
Question
Marks KL
11. a Reduce the following quadratic forms into a canonical form by an orthogonal
reduction and find the rank, signature, index and the nature of the quadratic form.
x y z 4xy 2yz 4xz
2
3
2
3
2
6
16 K3
OR
b
Diagonalise the matrix A














2 1 3
2 3 1
6 2 2
using orthogonal
transformation.
16 K3
12. a i.
Test the convergence of the series






1
2 1
cos
n n

.
8 K6
ii. Test for conditional convergence of the following series
2 3 .......
5
1
2
4
1
1 2
3
1
2
1
3 3 3 3
8 K4
OR
b i. Use Integral test to discuss the nature of convergence of the series
.......
3.4
1
2.3
1
1.2
1

8 K4
ii. Test the convergence of the series
.......0 1
1 1 1 1 4
4
3
3
2
2








x
x
x
x
x
x
x
x
x
8 K4
13. a i.
Find the radius of curvature at the point



2
3

2
3a a
on the curve
3 . 3 3 x y axy
8 K4
ii.
Find the evolute of the ellipse 1 2
2
2
2

b
y
a
x
8 K4
OR
b i. Find the equation of circle of curvature at on xy=c2
8 K3
ii.
Find the envelope of the family of straight lines
b
y
a
x
where a
and b are connected by the relations
8 K3
14. a i.
Find the Taylor's series expansion of exsiny near the point




4


upto
the third degree terms.
8 K3
ii. A rectangular box open at the top, is to have volume of 32 cc. Find the
dimensions of the box, that requires the least material for its construction
8 K3
OR
b i.
If








x y
x y
u
3 3
1 tan Prove that u
y
u
x
u
x sin 2





.
8 K3
ii. .Find the maximum value of the largest rectangular parallelopiped that can
be inscribed in the ellipsoid 1 2
2
2
2
2
2

c
z
b
y
a
x
8 K3
15. a i.
Change the order of integration and then evaluate
a
a
x xydydx
0
2
2 .
8 K3,K
6
ii. Find the area between the parabola 2 y x and the line y x . 8 K3
OR
b i. Transform into polar coordinates and evaluate



0 0
2 2
e dydx x y
8 K3,K
6
ii. Find the area of the cardioid r 8 K3