Math Answers

A blow-moulded polymer container can be considered as a cylinder with flat ends. Its capacity is 2
litre and it has thin walls of uniform thickness.
(a) Produce expressions for its volume and surface area.
(b)Using differential calculus, find the dimensions of the cylinder which will result in the
(c)minimum amount of polymer being used for its manufacture.
To be provided

A transmission line with characteristic impedance 250 Ω is connected to a pure
resistive load of 1000 Ω through a λ/4 line segment. Calculate the characteristic
impedance of this segment to achieve perfect matching between the load and the
transmission line.

b) A surface wave with frequency ω = 3 × 10^(−5)
rad/ s
is propagating at a metal-free
space boundary. The plasma frequency of metal is 9 × 10^(15) rad and εr = 7. Estimate
k. If the amplitude of the wave is 10^(5)
Vm−1
at the surface, what will be its value at
the depth of 0.2 µm in the metal? Ignore the collision effect.

If y = e^{m tan^(-1)x},show that (1+x^2)y n+1 + ( 2nx-m)yn + n(n-1)=0

2 kg of water is heated from 0°C to 100°C and converted into steam at the same
temperature. Calculate the increase in entropy, given that specific heat of water is
4.18 × 103
J kg−1
K−1
and Latent heat of vaporisation is 2.27 × 107
J kg−1
.

If a random variable U has the I-distribution with n degrees of freedom, show tbat
Z = U' has the F-distribution with I and n degrees of freedom. (S)

Obtain the maximum likelihood estimate of 0 in

f(x,9) •(l +9)x1

0 < x < 1

Prove that

.A.,( m,n )-- .Jm.Jn
m+n

for all m > O,n > O.Hence evaluate 13(10,7) and P(�·iJ (6)

) Let X ,X , ••• ,X. be a random sample from a normal population, N(µ,cr ).0btain
1 2 2
a (I -a) level confidence interval for µ when cr

is unknown

) Let the p.d.f. ofa continuous random variable X be given by

f(x)=Pe P1, x�O

Find the maximum likelihood estimator of p based on a sample X" X2 , •••• , x..

) Let X and Y have the continuous joint distribution with p.d.f.

l 1- if x2 + y

f(x,y)= na''

2 :Sa'
0 , otherwise

where a> 0. Compute the marginal densities of X and Y. Check whether X and Y are independent.