VTU 21MATCS41 Set-2 Solved Model Question Paper

Mathematical Foundations for Computing, Probability & Statistics Computer Science & Allied Engg. branches-21MATCS41 Set-2 Solved Model Question Paper

Module 1

1.A] Define tautology. Show that {(pVq)^(p→r)^(q→r)} → r is a tautology by constructing the truth table.

1.B] Prove the following using the laws of logic [¬p^(¬q ^ r)] v [(q^r) v (p ^ r] ⇔ r.

1.C] For any two odd integers m and n, show that i) m + n is even ii) mn is odd.

or

2.A] Define i) open statement ii) Quantifiers

2.B] Write the following argument in symbolic form and then establish the validity:
If A gets the Supervisor’s position and works hard, then he will get a raise.
If he gets a raise, then he will buy a car.
He has not purchased a car.
Therefore he did not get the Supervisor’s position or he did not work hard.

2.C] For the following statements, the universe comprises all non-zero integers. Determine the truth value of each statement.

Module 2

3.A] Let A = {1, 2, 3, 4} and B={1, 2, 3, 4, 5, 6}
i) How many functions are there from A to B? How many of these are one-to-one? How many are onto?
ii) How many functions are there from B to A? How many of these are onto? How many are one-to-one?

3.B] Let 𝐴 = {1, 2, 3, 4, 5} × {1, 2, 3, 4, 5} and define R on A by (x1,y1) R(x2,y2) if x1+y1=x2+y2

i) Verify that R is an equivalence relation on A
ii) Determine the equivalence classes [(1, 3)], [(2.4)] and [(1, 1)].

3.C] Define i) Simple graph ii) Complete graph iii) Sub graph iv) Spanning sub graph v) Induced subgraph vi) Complement of a graph vii) Euler Circuit. Give one example each.

or

4.A] Draw the Hasse diagram representing the positive divisors of 36.

4.B]

4.C] Define Graph isomorphism. Determine whether the following graphs are isomorphic or not.

Module 3

5.A] Calculate the coefficient of correlation and obtain the lines of regression for the following data:

x	1	2	3	4	5	6	7	8	9
y	9	8	10	12	11	13	14	16	15

5.B] Fit a curve by y=ax^b for the following data.

x	1	2	3	4	5
y	0.5	2	4.5	8	12.5

5.C] Fit a straight line in the least square sense for the following data

x	50	70	100	120
y	12	15	21	25

or

6.A] The following are the percentage of marks in Mathematics(x) and Statistics (y) of nine students. Calculate the rank correlation coefficient.

x	38	50	42	61	43	55	67	46	72
y	41	64	70	75	44	55	62	56	60

6.B] Fit a parabola y=ax² + bx + c for the data and hence estimate y at x = 6.

x	1	2	3	4	5
y	10	12	13	16	19

6.C] With usual notation, compute means, x̄,Ȳ, and correlation coefficient r from the following lines of regression: y=0.516x+33.73 and x=0.512y+32.52

Module 4

7.A] A random variable 𝑋 has the following probability function:

x	0	1	2	3	5	6	7
P(x)	0	k	2k	3k	k2	2k2	7k2 + k

Find k and evaluate: P(X<6), P(X>=6) and P(0<X<5).

7.B] Find the mean and standard deviation of the Poisson distribution.

7.C] The marks of 1000 students in an examination follow a normal distribution with a mean 70 and a standard deviation 5. Find the number of students whose marks will be (i) less than 65 (ii) more than 75 (iii)between 65 and 75

or

8.A]

8.B] The probability that a pen manufactured by a company will be defective is 1/10. If 12 such pens are manufactured, find the probability that (a) exactly two will be defective (b) at least two will be defective (c) none will be defective

8.C] If the probability of a bad reaction from a certain injection is 0.001, determine the chance that out of 2000 individuals more than two will get a bad reaction

Module 5

9.A] X and Y are independent random variables. X takes values 2, 5, and 7 with probability 1/2, 1/4, and 1/4 respectively. Y take values 3, 4, and 5 with the probability 1/3, 1/3 and 1/3.
a) Find the joint probability distribution of X and Y.
b) Show that the covariance of X and Y is equal to zero

9.B] A coin was tossed 400 times and head turned up 216 times. Test the hypothesis that the coin is unbiased at 5% level of significance.

9.C] In experiments on pea breeding, the following frequencies of seeds were obtained

Round & Yellow	Wrinkled & Yellow	Round & green	Wrinkled & green	Total
315	101	108	32	556

Theory predicts that the frequencies should be in proportions 9:3:3:1. Examine the correspondence between theory and experiment.

or

10.A] Explain the terms: (i) Null hypothesis (ii) Confidence intervals (iii) Type-I and Type-II errors.

10.B] A sample of 100 students is taken from a large population. The mean height of the students in this sample is 160 cm. Can it be reasonably regarded that in the population the mean height is 165 cm and the standard deviation is 10 cm at 5% level of significance?

10.C] The nine items of a sample have the following values: 45, 47, 50, 52, 48, 47, 49, 53,51. Does the mean of these differ significantly from the assumed mean of these differ significantly from the assumed mean of 47.5?(t_0.05=2.31 for 8 degree of freedom)