Electronics Engineering (Semester 4)
TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1 (a) Let X be a continuous random variable with probability distribution:
Find K and P(1 ≤ X ≤ 3)(5 marks)
1 (b) A relation R is the set of integers is defined by xRy if and only if x<y+1. Examine whether R is:
i) Reflective
ii) Symmetric
iii) Transitive(5 marks)
1 (c) Find the eigen values and eigen vectors corresponding to following matrix:
(5 marks)
1 (d) Find Laurent's series for -
(5 marks)
2 (a) Seven dice are thrown 729 times. How many times do you expect at least four dice to show three or five?(7 marks)
2 (b) Evaluate the following:
(7 marks)
2 (c) Show that the set of matrices
a, b ∈ z form an integral domain. Is it a field?(6 marks)
3 (a) Evaluate ∮c tan z dz where C is:
(i) is the circle |z|=2
(ii) is the circle |z|=1
(7 marks)
3 (b) "Is the following function injective, surjective?
f : R → R, f( x ) = 2x2 + 5x - 3"
(7 marks)
3 (c) Fit a binomial distribution to the following data:
|
X |
0 |
1 |
2 |
3 |
4 |
|
Frequency |
12 |
66 |
109 |
59 |
10 |
(6 marks)
4 (a) "If X is a normal variate with mean 10 and standard deviation 4, find:
i) P( | X - 14 | < 1)
ii) P( 5 ≤ X ≤ 18 )
iii) P( X ≤ 12)"
(7 marks)
4 (b) Let (G,*) be a group. Prove that G is an Abelian group if and only if (a * b)2 = a2 * b2.Where a2 stands for a * a.(7 marks)
4 (c) Using Poisson distribution find the approximate value of: 300C2(0.02)2(0.98)298 + 300C3(0.02)3(0.98)297.(6 marks)
5 (a) Show that the matrix
$A=\begin{bmatrix}
1 &-6 &-4 \\\\0
&4 &2 \\\\0
&-6 &-3
\end{bmatrix}$
is similar to a diagonal matrix. Also find the transforming matrix and the diagonal matrix
(7 marks)
5 (b) A die was thrown 132 times and the following frequencies were observed:
| No. obtained : | 1 | 2 | 3 | 4 | 5 | 6 | Total |
| Frequency : | 15 | 20 | 25 | 15 | 29 | 28 | 132 |
(7 marks)
5 (c) If C is a circle |z| = 1, using the integral
where K is real, show that
(6 marks)
6 (a) Let A = {1, 2, 3, 5, 6, 10, 15, 30} and R be the relation 'is divisible by'. Obtain the relation matrix and draw the Hasse diagram.(7 marks)
6 (b) A certain injection administered to 12 patients resulted in the following changes of blood pressure:
5, 2, 8, -1, 3, 0, 6, -2, 1, 5, 0, 4.
Can it be concluded that the injection will be in general accompanied by an increase in blood pressure?(7 marks)
6 (c) If X1 has mean 5 and variance 5, X2 has mean -2 and variance 3. If X1 and X2 are independant random variables, find -
i) E(X1 + X2), V(X1 + X2)
ii) E(2X1 + 3X2 - 5), V(2X1 + 3X2 - 5).(6 marks)
7 (a) A random variable X has the following probability distribution:
| X: | -2 | 3 | 1 |
| P (X = x): | 1/3 | 1/2 | 1/6 |
Find:
(i) Moment Generating function
(ii) First two raw moments
(iii) First two central moments
(7 marks)
7 (b) Verify Cayley Hamilton Theorem for matrix A and hence find A-1 where
(7 marks)
7 (c) A random sample of 50 items gives the mean 6.2 and standard deviation 10.24. Can it be regarded as drawn from a normal population with mean 5.4 at 5% level of significance?
(6 marks)
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