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

12th Standard

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Maths

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

      Multiple Choice Questions


    15 x 1 = 15
  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. Which one of the following is a binary operation on N?

    (a)

    Subtraction

    (b)

    Multiplication

    (c)

    Division

    (d)

    All the above

  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. If a compound statement involves 3 simple statements, then the number of rows in the truth table is

    (a)

    9

    (b)

    8

    (c)

    6

    (d)

    3

  5. The truth table for (p ∧ q) ∨ ¬q is given below

    p q (p ∧ q) ∨ (¬q)
    T T (a)
    T F (b)
    F T (c)
    F F (d)

    Which one of the following is true?

    (a)
    (a) (b) (c) (d)
    T T T T
    (b)
    (a) (b) (c) (d)
    T F T T
    (c)
    (a) (b) (c) (d)
    T T F F
    (d)
    (a) (b) (c) (d)
    T F F F
  6. 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

  7. The proposition p ∧ (¬p ∨ q) is

    (a)

    a tautology

    (b)

    a contradiction

    (c)

    logically equivalent to p ∧ q

    (d)

    logically equivalent to p ∨ q

  8. Determine the truth value of each of the following statements:
    (a) 4+2=5 and 6+3=9
    (b) 3+2=5 and 6+1=7
    (c) 4+5=9 and1+2= 4
    (d) 3+2=5 and 4+7=11

    (a)
    (a) (b) (c) (d)
    F T T T
    (b)
    (a) (b) (c) (d)
    T F T F
    (c)
    (a) (b) (c) (d)
    T T F F
    (d)
    (a) (b) (c) (d)
    F F T T
  9. 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.

  10. The binary operation * defined on a set s is said to be commutative if

    (a)

    a*b \(\in \) S ∀ a, b \(\in \) S

    (b)

    a*b = b*a ∀ a, b \(\in \) S

    (c)

    (a*b) * c = a*(b*c) ∀ a, b \(\in \) S

    (d)

    a*b = e ∀ a, b \(\in \) S

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

    (a)

    32

    (b)

    33

    (c)

    39

    (d)

    31

  12. 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)

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

  13. The number whose multiplication universe does not exist in C.

    (a)

    0

    (b)

    1

    (c)

    0

    (d)

    1

  14. '-' is a binary operation on

    (a)

    ~

    (b)

    Q-{0}

    (c)

    R-{0}

    (d)

    Z

  15. 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

    1. 2 Marks


    10 x 2 = 20
  16. Examine the binary operation (closure property) of the following operations on the respective sets (if it is not, make it binary)
    a*b = a + 3ab − 5b2;∀a,b∈Z

  17. 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.

  18. Determine whether ∗ is a binary operation on the sets given below.
    a*b=min (a,b) on A={1,2,3,4,5)

  19. Determine the truth value of each of the following statements
    (i) If 6 + 2 = 5 , then the milk is white.
    (ii) China is in Europe or \(\sqrt3\) is an integer
    (iii) It is not true that 5 + 5 = 9 or Earth is a planet
    (iv) 11 is a prime number and all the sides of a rectangle are equal

  20. Show that p v (~p) is a tautology.

  21. 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.

  22. Is cross product commutative on the set of vectors? Justify your answer.

  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. If a☰b(mod n) and C☰d(mod n),check whether a+c☰(b+d) (mod n).

  25. Write the equivalent forms of p➝q and (~p)➝q.

    1. 3 Marks


    10 x 3 = 30
  26. 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

  27. Write down the
    (i) conditional statement
    (ii) converse statement
    (iii) inverse statement, and
    (iv) contrapositive statement for the two statements p and q given below.
    p: The number of primes is infinite.
    q: Ooty is in Kerala.

  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 (A∨B)∧C 

  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 (A∧B)∨C

  30. Show that ¬(p↔️q) ≡ p↔️¬q

  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 on the set Q of rational numbers, a binary operation * is defined as a*b=λ(a+b) were λ is a nonzero fixed number and its given that * is associative, then the value of λ and what can we say about the operation*?

  34. If a≡b(mod n) show that am≡bm (mod n).whwre m is anatural number.

  35. In in (S,*) satisfying closure, associative, identity and inverse axioms and\((a\ast b)^{ -1 }={ a }^{ -1 }\ast { b }^{ -1 }\) ∀a,b∈S, then prove that * is commutative. 

    1. 5 Marks


    7 x 5 = 35
  36. Establish the equivalence property connecting the bi-conditional with conditional: p ↔️ q ≡ (p ➝ q) ∧ (q⟶ p)

  37. 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.

  38. Prove that p➝(¬q V r) ≡ ¬pV(¬qVr) using truth table.

  39. Let S be a non-empty set and 0 be a binary operation on s defined by x 0 y = x; x, Y \(\in \) s. Determine whether 0 is commutative and association.

  40. Show that (2018)2017+(2020)2017≡0(mod2019).

  41. Examine whether there is a nonempty subset S of the set of real numbers such that it satisfies closure, associative, identity and inverse properties under a binary operation * defined as a*b=k where k is a fixed real number.

  42. Prove that (2019)10+(2020)10≡1025(mod 2018)

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