GUJARAT UNIVERSITY B.E. SEM – 3 (CE/IT) Question Bank Advance Maths Each question is of equal Marks (10 Marks) 1 Q.1 Find the Fourier Series for () x fx e in the interval 0 2 x . Q.2 Expand () sin fx x x as a Fourier series in the interval 0 2 x . Q.3 Find the Fourier series of 2 () 2 fx x x in the interval (0,3). Hence deduce that 2 2 2 2 1 1 1 ... 1 2 3 12 . Q.4 Find the Fourier series of the function 2 2 0 () 0 x x fx x x . Q.5 Find the Fourier series of the function 0 1 () 0 1 ( 2) 1 2 x x fx x x x . Hence show that 1 1 1 1 ... 1 3 5 7 4 . Q.6 Find the Fourier series of 2 () fx x in the interval 0 x a , ( ) () fx a fx . Q.7 If () cos fx x , expand () fx as a Fourier series in the interval ( , ) , ( 2) () fx fx . Q.8 For the function () fx defined by () fx x , in the interval ( , ) . Obtain the Fourier series. Deduce that 2 2 2 2 1 1 1 ... 1 3 5 8 . Q.9 Given 1 0 () 1 0 x x fx x x . Is the function even of odd ? Find the Fourier series for () fx and deduce the value of 2 2 2 1 1 1 ... 1 3 5 . Q.10 Find the Fourier series of the periodic function () fx ; () fx k when 0 x and () fx k when 0 x , and ( 2) () fx fx . Q.11 Half range sine and cosine series of () ( ) fx x x in (0, ) Q.12 Find the Fourier series for the function ,0 1 () 2 ,1 2 x x fx x x Q.13 Find the Fourier series for f(x) defined by f(x) = 2 4 x x when -< x <and f(x + 2) = f(x) and hence show that 12 ....... 4 1 3 1 2 1 1 1 2 2 2 2 2
12
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GUJARAT UNIVERSITY B.E. SEM – 3 (CE/IT)
Question Bank Advance Maths
Each question is of equal Marks (10 Marks)
1
Q.1 Find the Fourier Series for ( ) xf x e in the interval 0 2x .
Q.2 Expand ( ) sinf x x x as a Fourier series in the interval 0 2x .
Q.3 Find the Fourier series of 2( ) 2f x x x in the interval (0,3). Hence deduce that 2
2 2 2
1 1 1...
1 2 3 12
.
Q.4 Find the Fourier series of the function
2
2
0( )
0
x xf x
x x
.
Q.5
Find the Fourier series of the function
0 1
( ) 0 1
( 2) 1 2
x x
f x x
x x
. Hence show that
1 1 1 1...
1 3 5 7 4
.
Q.6 Find the Fourier series of 2( )f x x in the interval 0 x a , ( ) ( )f x a f x .
Q.7 If ( ) cosf x x , expand ( )f x as a Fourier series in the interval ( , ) ,
( 2 ) ( )f x f x .
Q.8 For the function ( )f x defined by ( )f x x , in the interval ( , ) . Obtain the
Fourier series. Deduce that 2
2 2 2
1 1 1...
1 3 5 8
.
Q.9 Given
1 0( )
1 0
x xf x
x x
. Is the function even of odd ? Find the Fourier
series for ( )f x and deduce the value of 2 2 2
1 1 1...
1 3 5 .
Q.10 Find the Fourier series of the periodic function ( )f x ; ( )f x k when 0x
and ( )f x k when 0 x , and ( 2 ) ( )f x f x .
Q.11 Half range sine and cosine series of ( ) ( )f x x x in (0, )
Q.12 Find the Fourier series for the function
,0 1( )
2 ,1 2
x xf x
x x
Q.13 Find the Fourier series for f(x) defined by f(x) =
2
4
xx when -< x < and
f(x + 2) = f(x) and hence show that 12
.......4
1
3
1
2
1
1
1 2
2222
GUJARAT UNIVERSITY B.E. SEM – 3 (CE/IT)
Question Bank Advance Maths
Each question is of equal Marks (10 Marks)
2
Q.14 Find the Fourier series for the function
;0 1( )
0;1 2
x xf x
x
.
Q.15 If f(x) = x in 0 < x <
2
= - x in 2
< x <
2
3
= x - 2 in 2
3< x < 2
Prove that f(x) =
222 5
5sin
3
3sin
1
sin4 xxx
Q.16 If f(x) =
l
x when 0 < x < l
= l
xl 2 when l < x < 2l
Prove that f(x)
.......
5cos
5
13cos
3
1cos
14
2
12222 l
x
l
x
l
x
I
Q.17 When x lies between and p is not an integer, prove that
sin px =
.........
3
3sin3
2
2sin2
1
sinsin
2222222 p
x
p
x
p
xp
Q.18 Find the Fourier series for the function ( ) axf x e in ( , )l l
Q.19 Half range sine and cosine series of ( ) 2 1f x x in (0,1)
Q.20 Half range sine and cosine series of 2x in (0, )
Q.21 Find Half range sine and cosine series for 2( 1)f x x in 0,1
Q.22
Attempt the following.
GUJARAT UNIVERSITY B.E. SEM – 3 (CE/IT)
Question Bank Advance Maths
Each question is of equal Marks (10 Marks)
3
If
12–0
031–
2–21
A find A3 and A–1 using Cayley Hamilton Theorem.
Show that the matrix 1 11
1 13
i
i
is unitary.
Q.23
Attempt the following.
Using Cayley-Hamilton theorem, find the inverse of5 3
3 2
.
If A = 3 1
1 2
show that 2 5 7 0A A I , Where I is a unit matrix of second
order.
Q.24
Attempt the following.
Define Hermitian matrix. If 2 3 1 3
5 4 2
i iA
i i
show that *AA is a Hermitian
matrix.
Using Gauss –Jordan Method , find the inverse of
2 1 2
2 2 1
1 2 2
.
Q.25
Find the eigenvalues & eigenvectors of the following matrix
221
131
122
A .
GUJARAT UNIVERSITY B.E. SEM – 3 (CE/IT)
Question Bank Advance Maths
Each question is of equal Marks (10 Marks)
4
Q.26
Find the eigenvalues & eigenvectors of the following matrix
8 –6 2
–6 7 –4
2 –4 3
A
.
Q.27
Attempt the following.
Find A 1 by Gauss Jordan Method, where A =
3 3 4
2 3 4
0 1 1
.
Find characteristic equation ,eigen value and eigen vectors of matrix A ifA=
2 0 1
0 2 0
1 0 2
.
Q.28
If
1 0 3
2 1 1
1 1 1
A
find A–1 using Cayley Hamilton Theorem.
Q.29
If
1 1 2
0 2 0
0 0 3
A
find A–1 using Cayley Hamilton Theorem.
Q.30
Verify cayley-Hamilton theorem for the matrix A, where
2 1 1
1 2 1
1 1 2
A
GUJARAT UNIVERSITY B.E. SEM – 3 (CE/IT)
Question Bank Advance Maths
Each question is of equal Marks (10 Marks)
5
Q.31
Verify cayley-Hamilton theorem for the matrix A, where
7 2 2
6 1 2
6 2 1
A
Q.32
Verify cayley-Hamilton theorem for the matrix A, where
3 2 4
4 3 2
2 4 3
A
Q.33
Find the eigen values and eigen vectors of the matrix
8 6 2
6 7 4
2 4 3
A
Q.34
Find the eigen values and eigen vectors of the matrix
2 2 3
2 1 6
1 2 0
A
Q.35
Find the eigen values and eigen vectors of the matrix
2 1 1
1 1 2
1 2 1
A
Q.36
Attempt the following.
Prove that the matrix
1 1(1 ) ( 1 )
2 2
1 1(1 ) (1 )
2 2
i i
A
i i
ia unitary and find 1A .
Show that
3 7 4 2 5
7 4 2 3
2 5 3 4
i i
A i i
i i
is a Hermitian matrix.
GUJARAT UNIVERSITY B.E. SEM – 3 (CE/IT)
Question Bank Advance Maths
Each question is of equal Marks (10 Marks)
6
Q.37
Attempt the following.
Show that the matrix i i
Ai i
is unitary matrix, if
2 2 2 2 1 .
Q.38
Show that every square matrix can be uniquely expressed as P iQ , where P
and Q are Hermitian matrices.
Q.39 Solve the following equations :
( a ) (D - 2)2 y = 8(e2x + sin2x + x2) (b) ( D2 + D) y = x2 + 2x + 4
Q.40 Solve the following equations :
(a) (D2 + 1) y = x2 cosx (b) 2 2 3( 1) cosh 2xD y e x x
Q.41 Solve the following equations :
(a) 4 2 2 2( 2 1) cosD D y x x (b) (D2 + 2) y = e-2x + cos3x + x2
Q.42 Solve the following equations :
(a) ( D2 + 2D+1) y = x ex sinx (b) (D2 - 9) y = e3x cos2x