Information Technology (Semester 4)
Total marks: 80
Total time: 3 Hours
INSTRUCTIONS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Draw neat diagrams wherever necessary.
1.a.
A random discrete variable x has the probability mass function given
x |
-2 |
-1 |
0 |
1 |
2 |
3 |
P(x) |
0.2 |
k |
0.1 |
2k |
0.1 |
2k |
Find (i) k (ii) E(X) (iii) V(X).
(5 marks)
00
1.b.
Find smallest positive integer modulo 5, to which $3^2$.$3^3$.$3^4$.$3^{10}$ is congruent.
(5 marks)
00
1.c.
Given two lines of regression line $6y = 5x+90$, $15x = 8y+130$
Find (i) $\bar x$, $\bar y$ (ii) correlation coefficient of r
(5 marks)
00
1.d.
Show that G={1,-1,i,-i} is a group under usual multiplication of complex number.
(5 marks)
00
2.a.
Show that $111^{333}+333^{111}$ is divisible by 7.
(6 marks)
00
2.b.
The following table gives the number of accidents in a city during a week. Find whether the accidents are uniformly distributed over a week.
Day |
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |
Total |
No of accidents |
13 |
15 |
9 |
11 |
12 |
10 |
14 |
84 |
(6 marks)
00
2.c.
i. Write the following permutation as the product of disjoint cycles
f = (1 3 2 5) (1 4 5 ) (2 5 1)
ii. Simply as sum of product (A+B) (A+B') (A'+B) (A'+B').
(8 marks)
00
3.a.
Find gcd (2378, 1769) using Euclidean Algorithm. Also find x and y such that $2378x+1769y = gcd(2378,1769)$
(6 marks)
00
3.b.
Give an example of a graph which has
(i) Eulerian circuit but not a Hamiltonian circuit
(ii) Hamiltonian circuit but not an Eulerian circuit
(iii) Both Hamiltonian circui and Eulerian circuit
(6 marks)
00
3.c.
Show that $D_{10},\leq$ is a lattice. Draw its Hase diagram.
(8 marks)
00
4.a.
Calculate the coefficient of correlation between x and y from the following data
x |
23 |
27 |
28 |
29 |
30 |
31 |
33 |
35 |
36 |
39 |
y |
18 |
22 |
23 |
24 |
25 |
26 |
28 |
29 |
30 |
32 |
(6 marks)
00
4.b.
Let G be a group of all permutations of degree 3 on 3 symbol 1,2 & 3. Let H = {I, (1 2)}
be a subgroup of G. Find all distinct left cosets of H in G and hence index of H.
(6 marks)
00
4.c.
i. The average marks scored by 32 boys is 72 with standard deviation of 8 while that for 36 girls is 70 with standard deviation 6. Test at 5% LOS whether the boys perform better than the girls.
ii. A random sample of 15 items gives the mean 6.2 and variance 10.24.
Can it be regarded as drawn from a normal population with mean 5.4 at 5% LOS?
(8 marks)
00
5.a
Derive mgf of binomial distribution and hence find its mean and variance.
(6 marks)
00
5.b.
It was found that the burning life of electric bulbs of a particular brand was normally distributed with the mean 1200 hrs and the S.D. of 90 hours,Estimate the number of bulbs in a lot of 2500 bulbs having the burning life : (i) more than 1300 hours (ii) between 1050 and 1400 hours.
(6 marks)
00
5.c.
(i) Find inverse of $8^{-1}$(mod 77) using Euler's theorem.
(ii) Find the Jacobi's symbol of $(\frac{32}{15})$.
(8 marks)
00
6.a
Solve $x \equiv 1(mod3) $, $x \equiv 2(mod5) $, $x \equiv 3(mod7)$
(6 marks)
00
6.b.
Given L = {1,2,4,5,10,20} with divisibility relation. Verify that (L,$\leq$) is a distributive but not complimented Lattice.
(6 marks)
00
6.c.
(i) Draw a complete graph of 5 vertices.
(ii) Given an example of tree. (Sketch the tree).
(8 marks)
00