## 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=ax^{2} + 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 | k^{2} | 2k^{2} | 7k^{2} + 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.

or

8.A]

**Module 5**

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 |

or

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