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Discrete Mathematics Model Question Paper

12th Standard EM

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Maths

Time : 00:45:00 Hrs
Total Marks : 40
    5 x 1 = 5
  1. A binary operation on a set S is a function from

    (a)

    S ⟶ S

    (b)

    (SxS) ⟶ S

    (c)

    S⟶ (SxS)

    (d)

    (SxS) ⟶ (SxS)

  2. In the set Q define a⊙b= a+b+ab. For what value of y, 3⊙(y⊙5)=7?

    (a)

    y=\(\frac{2}{3}\)

    (b)

    y=\(\frac{-2}{3}\)

    (c)

    y=\(\frac{-3}{2}\)

    (d)

    y=4

  3. Which one is the contrapositive of the statement (pVq)⟶r?

    (a)

    ㄱr➝(ㄱp∧ㄱq)

    (b)

    ㄱr⟶(p∨q)

    (c)

    r⟶(p∧q)

    (d)

    p⟶(q∨r)

  4. Which of the following is a tautology?

    (a)

    p ν q

    (b)

    p ៱ q

    (c)

    q v ~ q

    (d)

    q ៱ ~ q

  5. 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)

    \(\frac { a }{ a-1 } \)

  6. 5 x 2 = 10
  7. 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.

  8. Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give
    verbal sentence describing each of the following statements.
    (i) ¬p
    (ii) p ∧ ¬q
    (iii) ¬p ∨ q
    (iv) p➝ ¬q
    (v) p↔q

  9. In the set of integers under the operation * defined by a * b = a + b - 1. Find the identity element.

  10. Let S be the set of positive rational numbers and is defined by a * b =\(\frac{ab}{2}\). Then find the identity element and the inverse of 2.

  11. Let G = {1, w, w2) where w is a complex cube root of unity. Then find the universe of w2. Under usual multiplication.

  12. 5 x 3 = 15
  13. 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
    Ze = the set of all even integers

  14. Determine whether ∗ is a binary operation on the sets given below.
    (A*v)=a√b is binary on R

  15. Let \(A=\left( \begin{matrix} 1 & 0 \\ 0 & 1 \\ 1 & 0 \end{matrix}\begin{matrix} 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{matrix} \right) ,B=\left( \begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 0 & 1 \end{matrix} \right) ,C=\left( \begin{matrix} 1 & 1 \\ 0 & 1 \\ 1 & 1 \end{matrix}\begin{matrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix} \right) \)be any three boolean matrices of the same type.
    Find (A∨B)∧C 

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

  17. Construct the truth table for (-p) v (q ∧ r)

  18. 2 x 5 = 10
  19. How many rows are needed for following statement formulae?
    p ∨ ¬ t ( p ∨ ¬s)

  20. How many rows are needed for following statement formulae?
    (( p ∧ q) ∨ (¬r ∨¬s)) ∧ (¬ t ∧ v))

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