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).

1.b. Find smallest positive integer modulo 5, to which $3^2$.$3^3$.$3^4$.$3^{10}$ is congruent.

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

1.d. Show that G={1,-1,i,-i} is a group under usual multiplication of complex number.

2.a. Show that $111^{333}+333^{111}$ is divisible by 7.

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

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').

3.a. Find gcd (2378, 1769) using Euclidean Algorithm. Also find x and y such that $2378x+1769y = gcd(2378,1769)$

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

3.c. Show that $D_{10},\leq$ is a lattice. Draw its Hase diagram.

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

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.

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?

5.a Derive mgf of binomial distribution and hence find its mean and variance.

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.

5.c. (i) Find inverse of $8^{-1}$(mod 77) using Euler's theorem.

(ii) Find the Jacobi's symbol of $(\frac{32}{15})$.

6.a Solve $x \equiv 1(mod3) $, $x \equiv 2(mod5) $, $x \equiv 3(mod7)$

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.c. (i) Draw a complete graph of 5 vertices.

(ii) Given an example of tree. (Sketch the tree).