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

12th Standard

Reg.No. :
•
•
•
•
•
•

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)

(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)