**How many strings of 5 digits have the property that the sum of their digits is 7 ?**

**(A) 66 (B) 330**

**(C) 495 (D) 99**

**Answer: B**

**EXPLANATION : **

**Let n=7 and r=5.**

^{n+r-1}C_{r-1}=330.

**Consider an experiment of tossing two fair dice, one black and one red. What is the probability that the number on the black die divides the number on red die ?**

**(A) 22 / 36 (B) 12 / 36**

**(C) 14 / 36 (D) 6 / 36**

**Answer: C**

**In how many ways can 15 indistinguishable fish be placed into 5 different ponds, so that each pond contains at least one fish ?**

**(A) 1001 (B) 3876**

**(C) 775 (D) 200**

**Answer: A**

**Explanation.**

^{15}C_{1}x^{14}C_{4}/15

**Consider the following statements:**

**(a) Depth – first search is used to traverse a rooted tree.**

**(b) Pre – order, Post-order and Inorder are used to list the vertices of an ordered rooted tree.**

**(c) Huffman’s algorithm is used to find an optimal binary tree with given weights.**

**(d) Topological sorting provides a labelling such that the parents have larger labels than their children.**

**Which of the above statements are true ?**

**(A) (a) and (b) (B) (c) and (d)**

**(C) (a) , (b) and (c) (D) (a), (b) , (c) and (d)**

**Answer: D**

**Consider a Hamiltonian Graph (G) with no loops and parallel edges. Which of the following is true with respect to this Graph (G) ?**

**(a) deg(v) ≥ n/2 for each vertex of G**

**(b) |E(G)| ≥ 1/2 (n-1)(n-2)+2 edges**

**(c) deg(v) + deg(w) ≥ n for every v and w not connected by an edge**

**(A) (a) and (b) (B) (b) and (c)**

**(C) (a) and (c) (D) (a), (b) and (c)**

**Answer: D**

**Consider the following statements :**

**(a) Boolean expressions and logic networks correspond to labelled acyclic digraphs.**

**(b) Optimal Boolean expressions may not correspond to simplest networks.**

**(c) Choosing essential blocks first in a Karnaugh map and then greedily choosing the largest remaining blocks to cover may not give an optimal expression.**

**Which of these statement(s) is/ are correct?**

**(A) (a) only (B) (b) only**

**(C) (a) and (b) (D) (a), (b) and (c)**

**Answer: D**

**Consider a full-adder with the following input values:**

**(a) x=1, y=0 and C _{i}(carry input) = 0**

**(b) x=0, y=1 and C _{i} = 1**

**Compute the values of S(sum) and C _{0} (carry output) for the above input values.**

**(A) S=1 , C _{0}= 0 and S=0 , C_{0}= 1 (B) S=0 , C_{0}= 0 and S=1 , C_{0}= 1**

**(C) S=1 , C _{0}= 1 and S=0 , C_{0}= 0 (D) S=0 , C_{0}= 1 and S=1 , C_{0}= 0**

**Answer: A**

Explanation.

Assume it, that it is written like (Carry,Sum.) in BINARY that is weighted.

X Y Carry(in) C Sum

0+ 0+0= 0 0 0.

0+ 0+1= 1 0 1.

0+ 1+0= 1 0 1.

0+ 1+1=(2) 1 0.

1+ 0+0=1= 0 1

1+ 0+1=(2) 1 0.

1+ 1+0=(2) 1 0.

1+ 1+1=(3) 1 1.

**“lf my computations are correct and I pay the electric bill, then I will run out of money. If I don’t pay the electric bill, the power will be turned off. Therefore, if I don’t run out of money and the power is still on, then my computations are incorrect.”**

**Convert this argument into logical notations using the variables c, b, r, p for propositions of computations, electric bills, out of money and the power respectively. (Where ¬ means NOT)**

**(A) if (c∧b) → r and ¬b → ¬p, then (¬r∧p)→¬c**

**(B) if (c∨b) → r and ¬b → ¬p, then (r∧p)→c**

**(C) if (c∧b) → r and ¬p → ¬b, then (¬r∨p)→¬c**

**(D) if (c∨b) → r and ¬b → ¬p, then (¬r∧p)→¬c**

**Answer: A**

**Match the following:**

**List – I List – II**

**(a) (p →q) ⇔ (¬q→¬p) (i) Contrapositive**

**(b) [(p∧q)→r]⇔[p→ (q→r)] (ii) Exportation law**

**(c) (p→q)⇔[(p∧¬q)→o] (iii) Reductio ad absurdum**

**(d) (p⇔q)⇔[(p→q)∧(q→p)] (iv) Equivalence**

**Codes:**

**(a) (b) (c) (d)**

**(A) (i) (ii) (iii) (iv)**

**(B) (ii) (iii) (i) (iv)**

**(C) (iii) (ii) (iv) (i)**

**(D) (iv) (ii) (iii) (i)**

**Answer: A**

**Consider a proposition given as:**

**“x≥6, if x ^{2}≥25 and its proof as:**

**If x≥6, then x ^{2}=x.x=6.6=36≥25**

**Which of the following is correct w.r.to the given proposition and its proof ?**

**(a) The proof shows the converse of what is to be proved.**

**(b) The proof starts by assuming what is to be shown.**

**(c) The proof is correct and there is nothing wrong.**

**(A) (a) only (B) (c) only**

**(C) (a) and (b) (D) (b) only**

**Answer: C**

Explanation.

Proof says any number greater than or equal to 6, its square is greater than or equal to 25.

6×6>25. 7×7>25——-.

Proposition 1 Fails because it does not prove that number (6,7,8—) square is not greater than 25.

Proposition 2 fails because there is no requirement of assumption as numbers are already given.

Proposition 3 is correct.

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