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

12th Standard

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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)

(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 $\sim (pV(qVr)\equiv \left( \sim p \right) \wedge \left( \sim q\wedge \sim r \right)$