These questions on the Mathematics paper of the Foundation Scholarship Examination for Engineering 2001 were set by R. Timoney.
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(a) State and prove the formula expressing the Laplace transform L[f (n) ](s) (of the nth
derivative of a function f (t) (t > 0)) in terms of L[f ](s).
(b) Compute (from the definition) the Laplace transform of the step function
0, t < a
Ha (t) =
1, t ≥ a
where a > 0.
(c) Use Laplace transform methods and complex numbers to solve the differential equation
d2 x
dx
1, 0 ≤ t < c
+ 2 + 2x =
0, t ≥ c
dt2
dt
with initial conditions x(0) =
11
dx
(0) = 0.
dt
(a) Define the Z transform of a sequence (xk )∞
k=0 . Define the convolution of two sequences
∞
(xk )∞
and
(y
)
and
show
that
the
Z
transform
of the convolution is the product of
k k=0
k=0
the Z transforms.
(b) Find the Z transform of
k
X
(k − j)π
j sin
3
j=1
!∞
k=0
(c) Which of the following discrete linear systems are stable (vk represents the input, xk the
output at step k).
i.
2yk+2 + 3yk+1 + yk = vk
ii.
3yk+2 + 9yk+1 + 2yk = vk
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