MTH401 Current final term Spring 2013 complete paper with MCQ’s
There are all the answers marked about I am confirmed that they are
correct, you can find unmarked answers in handouts. And almost 75%
MCQs were from this file, and I also add some new questions which
were new but easy.
The annihilator operator of the function
( D  6)2
y  e6x
is
( D  6)3
D6
D6
Correct answer
After converting the given differential equation
the function f(x) is
(Sec3x)/4
Correct answer
(Sec3x)/64
None of them
Secx
Wronskian of the function
0
1
4 y //  64 y  sec3x
yc  c1  c2 cos x  c3 sin x
into standard form,
is
Correct answer
2
3
In the infinite series of (x-a) which can be written as

 c ( x  a)
n 0
n
n
 c0  c1 ( x  a )  c2 ( x  a ) 2  ...
the number a is called the
Radius of power series
Centre of power series
Base of power series
None of them
Correct answer

 a ( x  a)
n 1
n
n
Suppose that a power series
is represented by a function “f”whose domain
is the interval of the convergence of the power series. That function “f” is continuous,
differentiable and integrable on
(a + R, a - R)
(R -a, R + a)
(a - R, a + R)
None of them
Correct answer
The interval of convergence for the function secx is
(  ,   )
(
 
, )
2 2
correct answer

( , )
2
None of them
dy
 2 xy  0
dx
The solution of the linear first order differential equation
2
y  ex
Correct answer
is

y
n 0
x2n
n!

y  ex
y
n 0
2
x2n
n!
Both
&
None of them
X  L 
The quantity
Reactance of circuit
Impedance of circuit
Quasi of circuit
None of them
1
C
is called
Correct answer
d 2x
dx
 2
 2x  0
2
dt
dt
For the equation of free damped motion
the roots are
2
2
2
2
m1      
m1      
 2   2  0 then system is said to be
&
If
Under damped
Over damped
Critically damped
Correct answer
None of them
x(t )  Ae t sin[  2   2   ]
The time interval between two successive maxima of
called
Phase period
Correct answer
Quasi-period
Both the period
None of them
is
d 2x
dx
 5  4x  0
2
dt
dt
The given differential equation
Over damped
Critically damped
Under damped
None of them
is
The standard unit for measurement of inductance is
Volt
Ohms
Henry
Correct answer
None of them
Which of the rule in matrices under multiplication does not hold true?
Commutative law
Associative law
Identity law
None of them
x y z
1 2 3
A 
& B   p q r

5 6 7 
l m n
If
2 4
23
3 3
None of them
a
b 
o 
then the order of
Correct answer
matirx A  B
is
d  x   3  7  x   4 
 
     sin t
dt  y  1 1  y   8 
The given system without the use of matrices
dx
dy
 3 x  7 y  4sin 2t ;
 x  y  8cos 2t
dt
dt
is
dx
dy
 3x  7 y  4sin t ;
 x  y  8cos t
dt
dt
dx
dy
 3 x  7 y  4sin t;
 x  y  8sin t
dt
dt
Correct answer
None of them
Suppose that {X1, X2, X3,…,Xn} is a set of n solutions vectors on an interval I, of a
homogeneous system X/=AX. The set is said to be a fundamental set of solutions of the
system on the interval I if the solution vectors are
Linearly dependent
Linearly independent
Correct answer
Homogeneous
None of them
The coefficient matrix of the following homogeneous system of differential equation
dx
dy
 3x  2 y ,
 x  2 y is
dt
dt
3 2 
2 2 


3 1 
 2 2


3 2 
1 2 


Correct answer
None of them
3  18
A

2  9 
The matrix
has an eigen value of multiplicity
1
2
3
4
1
A   2
 2
The matrix
Single root of A
triple root of A
double root of A
None of them
2
1
2
 2
 2 
1 
has eigen values
  1,  1,5 where   1
is a
4
1
0
0
4
0
0
1  0 gives
4
 = 4 of multiplicity of 1
 = 4 of multiplicity of 2
 = 4 of multiplicity of 3
None of them
dy
dx
 2x ,  3y
dt
dt
For the system of differential equations
(are)
the independent variable(s) is
x, t
y, t
x, y
t
dy
dx
 2x ,  3y
dt
dt
For the system of differential equations
(are)
x, t
y, t
x, y
t
the dependent variable(s) is
If L denote the linear differential operators with constant coefficients, then
represents the
L1
L2
L4
L3
L1L4  L2 L3
L1
L3
L4
L2
L1
L2
L3
L4
Correct answer
None of them
Wronskian of x,x2 is
x
x2
x3
0
dy x  y

dx
x
The general solution of differential equation
y
e x  cx
y
e x  cy
x
y
e  cx

e
x
y
 cx
The form of the exact solution to
dy
2
 3 y  e  x , y 0  5
dx
is
Ae 1.5 x  Bxe  x
.is given by
Ae1.5 x  Be  x
Ae1.5 x  Bxe  x
Ae 1.5 x  Be  x
If m and n are non negative integers and
Pn ( x)
is a Legendre’s polynomial then
1
 P ( x)P ( x)dx  0
m
n
for m  n
1
Correct answer
1
 P ( x)P ( x)dx  0
m
n
for m  n
1
1
 P ( x)P ( x)dx  0
m
n
for m  0
1
1
 P ( x)P ( x)dx  0
m
n
for n  0
1
If A is a square matrix and its determinant is zero, then
A is singular matrix. Correct answer
A is non singular matrix.
A is scalar matrix.
A is diagonal matrix.
An electronic component of an electronic circuit that has the ability to store charge and
opposes any change of voltage in the circuit is called
Inductor
Resistor
Capacitor
correct answer
None of them
Operator method is the method of the solution of a system of linear homogeneous or
linear non-homogeneous differential equations which is based on the process of
systematic elimination of the
Dependent variables
correct answer
Independent variable
Choice variable
None of them
Any linear differential equation of the form
dny
d n1 y
dy
an x n n  an 1 x n 1 n 1   a1 x  a0 y  g ( x) where a0 , a1 , a2 ..., an are constants.
dx
dx
dx
is
called
Homogeneous equation
Polar equation
Equi-dimensioanl equation or Cauchy eular
None of them
For eigen values   5,5 of a matirx
correct answer
3 4 
A

 1 7 
,there exists ......... eigen vectors.
infinite
one
two
three
( x 2  64)( x 2  36) y  xy  y  0
Ordinary points of
0,1
are
8,-8
6,-6
None of others.
x  x0
A singular point
of the given equation
a regular singular point if
( x  x0 ) P( x)
is analytic at x0
a2 ( x) y //  a1 ( x) y /  a0 ( x) y  0
( x  x0 )Q( x)
is analytic at x0
( x  x0 ) P( x) & ( x  x0 ) 2 Q( x)
are analytic at x0.
Correct answer
None of them
Singular points of the equation
x= -2, 2
None of them
x =2
x=-2
1
A
0
( x 2  4)2 y //  ( x  2) y /  y  0
are
0
1
The matrix
has ..............
Real and unequal value
Repeated & real eigen value
Complex eigen value
None of them
Let  be an eigen value of a non zero square matrix A. Then the equation
det( A   I )  0
is called
is said to be
Trivial equation
Characteristics equation
Non-trivial equation
None of them
correct answer
If y1=x2 is solution of the differential equation, then formula for finding Second solution
x 2 y //  2 y  0
of
is
e 2 x
dx
2
 x

y2  x 2 
e 2 x
dx
2
 x

y2  x 
y2 

x 2 



y2  x 2 

For
9
-9
10
-10
2
ex
dx
x4
2
dx
x4
y sin 2 x  y 2 cos x  c where y(0)  3; the valueof cis
------.
M  x, y  dx  N  x, y  dy  0
A differential equation
M N

x
y
M N

y
x
is said to be an exact if -----.
M N Correct answer

y
x
M N

y
y
Logistics equation is an Example of
Linear
Non linear
Bernoullis
One more name was there I forgot it.
dy
 (Cotx) y  cos 2 ( x)
dx
The integrating factor of the differential equation
is ------------
  ln | Sinx |
  ln | Cosx |
  Sinx
  Cosx
Give two examples of Bessel’s Differential Equation? Marks 3
What is wronskian? Marks 3
Give principle of superposition to find out any homogeneous equation marks
Define general linear equation of order n? 5 marks
Marks 5
d3y
d2y
dy

4
 5  4x
3
2
dx
dx
dx
Write the differential equation
in the form L(y) = g(x) where
L is a differential operator with constant coefficients
Marks 5
Deduce the Special Case of Logistic Equation “Epidemic spread”?
Write down the system of differential equations
dx
dy
 6x  y ,
 x  3 y  9t  9
dt
dt
in form of
X '  AX  F (t )