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#### Discrete Mathematics Model Question Paper

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 35
4 x 1 = 4
1. Which one of the following statements has the truth value T?

(a)

sin x is an even function

(b)

Every square matrix is non-singular

(c)

The product of complex number and its conjugate is purely imaginary

(d)

$\sqrt 5$ is an irrational number

2. Which one of the following is not true?

(a)

Negation of a negation of a statement is the statement itself

(b)

If the last column of the truth table contains only T then it is a tautology.

(c)

If the last column of its truth table contains only F then it is a contradiction

(d)

If p and q are any two statements then p↔️q is a tautology.

3. The number of binary operations that can be defined on a set of 3 elements is

(a)

32

(b)

33

(c)

39

(d)

31

4. The identity element in the group {R - {1},x} where a * b = a + b - ab is

(a)

0

(b)

1

(c)

$\frac { 1 }{ a-1 }$

(d)

$\frac { a }{ a-1 }$

5. 2 x 2 = 4
6. If p is true and q is false, then which of the following is not true?
(1) p ⟶ q is F
(2) p v q is T
(3) p ∧ q is F
(4) p ⇔ q is F

7. Which of the following is not a contradiction?
(1) p v q
(2) p ∧ q
(3) p v ~q
(4) p ∧ ~p

8. 4 x 2 = 8
9. Let A =$\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix},B=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$be any two boolean matrices of the same type. Find AvB and A^B.

10. Write each of the following sentences in symbolic form using statement variables p and q.
(i) 19 is not a prime number and all the angles of a triangle are equal.
(ii) 19 is a prime number or all the angles of a triangle are not equal
(iii) 19 is a prime number and all the angles of a triangle are equal
(iv) 19 is not a prime number

11. Which one of the following sentences is a proposition?
(i) 4 + 7 =12
(ii) What are you doing?
(iii) 3n ≤ 8,1 n ∈ N
(iv) Peacock is our national bird
(v) How tall this mountain is!

12. Show that p v (q ∧ r) is a contingency.

13. 3 x 3 = 9
14. Verify the
(i) closure property,
(ii) commutative property,
(iii) associative property
(iv) existence of identity and
(v) existence of inverse for the arithmetic operation - on Z.

15. Construct the truth table for $(p\overset { \_ \_ }{ \vee } q)\wedge (p\overset { \_ \_ }{ \vee } \neg q)$

16. In (z, *) where * is defined by a * b = ab, prove that * is not a binary operation on z.

17. 2 x 5 = 10
18. Using the equivalence property, show that p ↔️ q ≡ ( p ∧ q) v (ㄱp ∧ ㄱq)

19. Verify (p ∧ -p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.