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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.

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