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12th Standard Maths English Medium Discrete Mathematics Reduced Syllabus Important Questions With Answer Key 2021

12th Standard

    Reg.No. :
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Maths

Time : 01:00:00 Hrs
Total Marks : 100

      Multiple Choice Questions


    15 x 1 = 15
  1. The operation * defined by a*b =\(\frac{ab}{7}\) is not a binary operation on

    (a)

    Q+

    (b)

    Z

    (c)

    R

    (d)

    C

  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 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

  4. In the last column of the truth table for ¬( p ∨ ¬q) the number of final outcomes of the truth value 'F' are

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    4

  5. The proposition p ∧ (¬p ∨ q) is

    (a)

    a tautology

    (b)

    a contradiction

    (c)

    logically equivalent to p ∧ q

    (d)

    logically equivalent to p ∨ q

  6. If * is defined by a * b = a2 + b2 + ab + 1, then (2 * 3) * 2 is

    (a)

    20

    (b)

    40

    (c)

    400

    (d)

    445

  7. Which of the following is a contradiction?

    (a)

    p v q

    (b)

    p ∧ q

    (c)

    q v ~ q

    (d)

    q ∧ ~ q

  8. 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)

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

  9. Define * on Z by a*b = a+b+1 ∀ a,b \(\in \) Z. Then the identity element of z is

    (a)

    1

    (b)

    0

    (c)

    1

    (d)

    -1

  10. If p is true and q is unknown, then _________

    (a)

    ~ p is true

    (b)

    p v (~p) is false

    (c)

    p ∧ (~p) is true

    (d)

    p v q is true

  11. '+' is not a binary operation on

    (a)

    ~

    (b)

    z

    (c)

    c

    (d)

    Q- {0}

  12. '-' is a binary operation on

    (a)

    ~

    (b)

    Q-{0}

    (c)

    R-{0}

    (d)

    Z

  13. In (N, *), x * y = max(x, y), x, y \(\in \) N then 7 * (-7)

    (a)

    7

    (b)

    -7

    (c)

    0

    (d)

    -49

  14. In (S, *), is defined by x * y = x where x, y \(\in \) S, then

    (a)

    associative

    (b)

    Commutative

    (c)

    associative and commutative

    (d)

    neither associative nor commutative

  15. The number of commutative binary operations which can be defined on a set containing n elements is

    (a)

    n\(\frac { n(n+1) }{ 2 } \)

    (b)

    \({ n }^{ { n }^{ 2 } }\)

    (c)

    \(n^{ \frac { n }{ 2 } }\)

    (d)

    n2

    1. 2 Marks


    10 x 2 = 20
  16. Write the statements in words corresponding to ¬p, p ∧ q , p ∨ q and q ∨ ¬p, where p is ‘It is cold’ and q is ‘It is raining.’

  17. 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

  18. Fill in the following table so that the binary operation ∗ on A = {a,b,c} is commutative.

    * a b c
    a b    
    b c b a
    c a   c
  19. Write the converse, inverse, and contrapositive of each of the following implication.
    (i) If x and y are numbers such that x = y, then x2 = y2
    (ii) If a quadrilateral is a square then it is a rectangle.

  20. Construct the truth table for the following statements.
    ( p V q) ∧ ¬q

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

  22. Check whether dot product is defined on the set of vectors. Explain?

  23. Is it possible to define a binary operation * on any non empty set S such that a * \(\\ \\ b=\cfrac { a+b }{ a-b } \) ,given that a and b are integers.

  24. Form the truth table of (~P)➝(~q).

  25. p:N is divisible buy 4 and q:N is an even number. Whether p➝q is true.

    1. 3 Marks


    10 x 3 = 30
  26. How many rows are needed for following statement formulae?
    p ∨ ¬ t ( p ∨ ¬s)

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

  28. 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 AVB

  29. 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 AΛB

  30. 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

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

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

  33. If in a pair (S,*) where S is a nonemty set and * is a binary operation defined on S as a2=e for all a દ s, then that * is commutative given that e is the identify element.

  34. Show that (Z3-[0],X3) Satisfies closure, identiy and inverse properties.

  35. If in (S,*) satisfying closure, associative, identity and inverse axiom,\(a\ast { b }^{ 2 }={ a }^{ 2 }\ast b\) for some a,b∊S, then prove that a=b.

    1. 5 Marks


    7 x 5 = 35
  36. Verify
    (i) closure property,
    (ii) commutative property,
    (iii) associative property,
    (iv) existence of identity, and
    (v) existence of inverse for following operation on the given set
    m*n=m+n-mn; m,n ∈Z

  37. Verify
    (i) closure property,
    (ii) commutative property,
    (iii) associative property,
    (iv) existence of identity, and
    (v) existence of inverse for the operation +5 on Z5 using table corresponding to addition modulo 5.

  38. Verify
    (i) closure property,
    (ii) commutative property,
    (iii) associative property,
    (iv) existence of identity, and
    (v) existence of inverse for the operation ×11 on a subset A = {1,3,4,5,9}
    of the set of remainders {0,1,2,3,4,5,6,7,8,9,10}

  39. Let A be Q\{1}. Define ∗ on A by x*y = x + y − xy . Is ∗ binary on A? If so, examine the commutative and associative properties satisfied by ∗ on A.

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

  41. Let Q, be the set of all nonzero rational numbers and k is a nonzero fixed rational number and * be a binary operation defined as a*b=kab. Show that (Q,*) satisfies closure, associative, inverse and commutative properties.

  42. Prove by using truth table \(\sim (pV(qVr)\equiv \left( \sim p \right) \wedge \left( \sim q\wedge \sim r \right) \) 

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