STUPIDSID
1(a)
State Dirichlet conditions for the expansion of f(x) as Fourier series. Examine whether f(x)=sin(1/x) can be expanded in Fourier series in [-?,?]
5 M
1(b)
Find Laplace transform of:
5 M
1(c)
Find Z {f(k)} where f(k) is given by:
5 M
1(d)
Express the function f(x) as a Fourier integral hence evaluate the integral that follows:
5 M
2(a)
Define linear dependence and independence of vectors. If the vectors (0,1,a),(1,a,1) and (a,1,0) are linearly dependent then find the value of 'a'
6 M
2(b)
Find Laplace transform of:
6 M
2(c)
Find {f(k)} if F(z) is as given below and if ROC of F(Z) is:
(i)|z|<2 (ii)2<|z|<3 (iii) |z|>3
(i)|z|<2 (ii)2<|z|<3 (iii) |z|>3
8 M
3(a)
Determine the value of a and b for which the system:
x + 2y + z = 6
x + 3y + 5z = 9
2x + 5y + az = b
has (i)no solution (ii)unique solution (iii)infinite solutions.
Find the solutions in case of (ii) and (iii)
x + 2y + z = 6
x + 3y + 5z = 9
2x + 5y + az = b
has (i)no solution (ii)unique solution (iii)infinite solutions.
Find the solutions in case of (ii) and (iii)
6 M
3(b)
Evaluate the following:
6 M
3(c)
Find the Fourier series for f(x) in (0,2?) -
f(x) = x ... (0 < x ? ?)
= 2? - x ... (? ? x < 2?)
Hence deduce that
f(x) = x ... (0 < x ? ?)
= 2? - x ... (? ? x < 2?)
Hence deduce that
8 M
4(a)
Find two non singular matrices P and Q such that PAQ is in the normal form where
6 M
4(b)
Find L(|cost|)
6 M
4(c)
(i) If A, B are Hermitian prove that AB-BA is skew Hermitian
(ii) Show that A is Hermitian and iA is skew Hermitian if:
(ii) Show that A is Hermitian and iA is skew Hermitian if:
8 M
5(a)
Solve y'' + 2y = r(t); y(0) = 0, y'(0) = 0
Using Laplace Transform where
r(t) = 1 ... (0 ? t ? 1)
= 0 ... (t > 1)
Using Laplace Transform where
r(t) = 1 ... (0 ? t ? 1)
= 0 ... (t > 1)
6 M
5(b)
Find the complex form of the Fourier series of the function
f(x) = x2 + x ... (-? < x < ?)
f(x) = x2 + x ... (-? < x < ?)
6 M
5(c)
Find z(an), z(cos n?), z(sin n?)
8 M
6(a)
Show that ex is equal to:
6 M
6(b)
Find the inverse Laplace Transform of
6 M
6(c)
Obtain the Fourier series for the function
f(x) = x ... (-? < x < 0)
= -x ... (0 < x < ?)
Hence deduce that:
f(x) = x ... (-? < x < 0)
= -x ... (0 < x < ?)
Hence deduce that:
8 M
7(a)
Find the Fourier Transform of:
6 M
7(b)
Using Laplace Transform evaluate
6 M
7(c)
Show that the system of equations
ax + by + cz = 0
bx + cy + az = 0
cx + ay + bz = 0
has a non Trivial solution if a+b+c=0 or if a=b=c.
Find the non Trivial solution when the condition is satisfied
ax + by + cz = 0
bx + cy + az = 0
cx + ay + bz = 0
has a non Trivial solution if a+b+c=0 or if a=b=c.
Find the non Trivial solution when the condition is satisfied
8 M
More question papers from Applied Mathematics 3