Hall Ticket No Question Paper Code: AHS002
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
B.Tech I Semester End Examinations (Supplementary) February 2017
Regulation: IARE-R16
LINEAR ALGEBRA AND ORDINARY DIFFERENTIAL EQUATIONS
(Common to all branches)
Time: 3 Hours Max Marks: 70
Answer ONE Question from each Unit
All Questions Carry Equal Marks
All parts of the question must be answered in one place only
UNIT I
1. Find the rank of the matrix
2
6666664
1 1 1 6
1 1 2 5
3 1 1 8
2 2 3 7
3
7777775
by reducing it into normal form.
Find the inverse of the matrix
2
6664
2 3 1
1 2 3
3 1 2
3
7775
using elementary row operations.
2. Find the inverse of A
2
6664
1 2 3
2 4 5
3 5 6
3
7775
by Gauss Jordon method.
Find the rank of the matrix
2
6666664
1 3 1
1 2 3 1
1 0 1 1
0 1 1 1
3
7777775
by reducing it into echolon form.
UNIT II
3. Find the Eigen values and Eigen vectors of the matrix
2
6664
1 1
1 0 0
1
3
7775

Find a matrix P such that P-1AP is diagonal matrix, where A
2
6664
2 2 1
1 3 1
1 2 2
3
7775

Page 1 of 2
4. Show that
i. a square matrix A and its transpose AT have same eigen values
ii. product of two Unitary matrices is Unitary.
Find a matrix P which diagonalises the matrix A
2
6664
1 1
0 2
1 1 1
3
7775

UNIT III
5. Solve the differential equation x dy
dx y x2y2
In a murder investigation, a corpse was found by a detective at exactly 8 PM. Being alert, the
detective also measured the body temperature and found it to be 70o F. Two hours later, the
detective measured the body temperature again and found it be 60o F. If the room temperature
is 50o F and assuming that the body temperature of the person before death was 98.6o at
what time did the murder occur?
6. Find the orthogonal trajectories of the family of circles passing through the origin and the centres
on the x axis.
Solve the differential equation x

1 x2 dy/dx

2x2 1
y x3
UNIT IV
7. Solve the differential equation

D3 2D2 5D 6
y y y0 y
00
1
Solve the differential equation
h
1)2
D2 1
y ex
8. Solve the differential equation

D2 5D 6
y x cos x cos 2x
A circuit consists of an inductance of 2 Henrys, a resistance of 4 Ohms and capacitance of 0.05
Farads. If q i 0 at t 0. Find and when there is a constant electromagnetic field
of 100 V.
UNIT V
9. If U
y-x/xy z-x/xz
then find the value of x2Ux y2Uy z2Uz
Examine the function sin x sin y sin for extreme values.
10. Find the extreme values of the function x4 y4 2x2 4xy 2y2
If x increases at the rate of 2 cm/sec at the instant when x 3 cm, and y 1 cm, at what rate
must y be changing in order that 2xy 3x2y shall be neither increasing nor decreasing?