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Download Tamil Nadu 12th Standard Maths question papers, model tests, one-mark questions, important questions, and public exam papers in PDF format. Free study materials and answer keys for TN State Board students.

Questions:

Generic 2

5 x 2 = 10

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

Write the converse, inverse, and contrapositive of each of the following implication.
If x and y are numbers such that x = y, then x² = y²

Let A = {a +\(\sqrt5\) b : a,b∈Z}. Check whether the usual multiplication is a binary operation on A.

On Z, define \(⊗ \mathrm{by}(m * n)\) = mn + nm: ∀m, n∈Z. Is ⊗ binary on Z?

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.

Generic 5

3 x 5 = 15

Show that -(p ∧ q) ≡ (~p) v (~q).

Prove p⟶(q⟶r) ☰ (p ∧ q)⟶r without using truth table.

Establish the equivalence property connecting the bi-conditional with conditional: p ↔️ q ≡ (p ➝ q) ∧ (q⟶ p)

Multiple Choice Question

15 x 1 = 15

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

(a)

(b)

-7

(c)

(d)

-49

10.

'+' is not a binary operation on ___________

(a)

(b)

(c)

(d)

Q- {0}

11.

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

12.

If a * b = a²b² - ab then 3 * (1 * 1)

(a)

(b)

(c)

(d)

13.

A binary operation * is defined on the set of positive rational numbers Q⁺ by a*b = \(\frac { ab }{ 4 } \). Then 3 * \(\left( \frac { 1 }{ 5 } *\frac { 1 }{ 2 } \right) \) is _____________

(a)

\(\frac { 3 }{ 160 } \)

(b)

\(\frac { 5 }{ 160 } \)

(c)

\(\frac { 3 }{ 10 } \)

(d)

\(\frac { 3 }{ 40 } \)

14.

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

(a)

(b)

(c)

(d)

-1

15.

The identity element in the group {R - {1},x} where a * b = a + b - ab is __________

(a)

(b)

(c)

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

(d)

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

16.

Which of the following is a tautology?

(a)

p ν q

(b)

p ∧ q

(c)

q v ~ q

(d)

q ∧ ~ q

17.

The identity element of \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\} \) |x \(\in \) R, x ≠ 0} under matrix multiplication is __________

(a)

\(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \)

(b)

\(\left( \begin{matrix} \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \\ \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \end{matrix} \right) \)

(c)

\(\left( \begin{matrix} \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \\ \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \end{matrix} \right) \)

(d)

\(\left( \begin{matrix} \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \\ \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \end{matrix} \right) \)

18.

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

19.

The dual of ᄀ(p V q) V [p V (p ∧ ᄀr)] is

(a)

ᄀ(p ∧ q) ∧ [p V (p ∧ ᄀr)]

(b)

(p ∧ q) ∧ [p ∧ (p V ᄀr)]

(c)

ᄀ(p ∧ q) ∧ [p ∧ (p ∧ r)]

(d)

ᄀ(p ∧ q) ∧ [p ∧ (pV ᄀr)]

20.

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

(a)

(b)

(c)

(d)

21.

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)

22.

Which one is the inverse of the statement (pVq)➝(pΛq)?

(a)

(p∧q)➝(pVq)

(b)

ᄀ(pvq)➝(p∧q)

(c)

(ᄀpvᄀq)➝(ᄀp∧ᄀq)

(d)

(ᄀp∧ᄀq)➝(ᄀpVᄀq)

23.

Which one of the following statements has truth value F?

(a)

Chennai is in India or \(\sqrt 2\) is an integer

(b)

Chennai is in India or \(\sqrt 2\) is an irrational number

(c)

Chennai is in China or \(\sqrt 2\) is an integer

(d)

Chennai is in China or \(\sqrt 2\) is an irrational number

Generic 3

3 x 3 = 9

24.

In (z, *) where * is defined as a * b = a + b + 2. Verify the commutative and associative axiom.

25.

Let G = {1, i, -1, -i} under the binary operation multiplication. Find the inverse of all the elements.

26.

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

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Answers Key:

Generic 2

5 x 2 = 10

p	~p	p v (~q)
T	F	T
F	T	T

The last column contains only T
The given statement is a tautology

If x and y are numbers such that x = y, then x²= y²
Converse statement :
If x and y are numbers such that x²= y² then x = y
Inverse statement :
If x and y are numbers such that x ≠ y then x²≠ y²
Contrapositive statement :
If x and y are numbers such that x²≠ y² then x ≠ y

A = {a+\(\sqrt5\) b:a,b ∈ z}
Let C = a+\(\sqrt5\) b
B = c+\(\sqrt5\)d∈A
where a, b, c, d ∈ Z
[∵ ac + 5bd∈Z and ad+bc ∈Z]
∴ B = (a+\(\sqrt5\)b).(c+\(\sqrt5\)d)
= ac+\(\sqrt5\)ad+cb\(\sqrt5\) + 5bd
= (ac+5bd)+\(\sqrt5\)(ad+bc)∈A
∴ C.B ∈A∀ a, b, c, d∈Z
∴ Usual multiplicaition is a binary operation on.

Given m*n = mⁿ + n^m ∀m, n∈Z
Let us take - 2, 2∈Z
Consider m = -3, n = 2
(m * n) = (-2 * 2) = (-2)² + (2)^-2
\(=4+\frac { 1 }{ 4 } =\frac { 17 }{ 4 } \notin Z\)
∴ * is not a binary operation on Z

Then the four types of conditional statements corresponding to p and q are respectively listed below.
(i) p →q : (conditional statement) “If the number of primes is infinite then Ooty is in Kerala”.
(ii) q → p : (converse statement) “If Ooty is in Kerala then the number of primes is infinite”
(iii) ¬p → ¬q (inverse statement) “If the number of primes is not infinite then Ooty is not in Kerala”.
(iv) ¬q → ¬p (contrapositive statement) “If Ooty is not in Kerala then the number of primes is not infinite”.

Generic 5

3 x 5 = 15

p	q	~p	~q	(p∧q)	~(p∧q)	(~p)v(~q)
T	T	F	F	T	F	F
T	F	F	T	F	T	T
F	T	T	F	F	T	T
F	F	T	T	F	T	T

The entries in column (6) and column (7) are equal.
∴ ~(P∧q) ≡ (~p) v (~q)

Prove that p⟶(q⟶r) = (p ∧ q)⟶r without using truth table. From example we know that p⟶ q = ~p V q
Consider LHS = p⟶(q⟶r)
= p ⟶ (~q V r) [Implication Law]
= ~p V (~q V r) [Implication Law]
= (~p V ~q) V r [associative property]
= ~(p ∧ q) V r [using Demorgan's law]
= (p ∧ q) ⟶ r [Implication Law]
Hence proved.

p	q	p ➝ q	q⟶ p	p ↔️ q	(p ➝ q) ∧ (q⟶ p)
T	T	T	T	T	T
T	F	F	T	F	F
F	T	T	F	F	F
F	F	T	T	T	T

The entries in the columns corresponding to p ↔ q and ( p ⟶ q) ∧ (q ⟶ p) are identical and hence they are equivalent

Multiple Choice Question

15 x 1 = 15

(a)
7

10.

(d)
Q- {0}

11.

(d)
p v q is true

12.

(a)
0

13.

(d)
\(\frac { 3 }{ 40 } \)

14.

(d)
-1

15.

(a)
0

16.

17.

(c)
\(\left( \begin{matrix} \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \\ \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \end{matrix} \right) \)

18.

(b)
a*b = b*a ∀ a, b \(\in \) S

19.

(d)
ᄀ(p ∧ q) ∧ [p ∧ (pV ᄀr)]

20.

21.

(b)
ㄱr⟶(p∨q)

22.

23.

Generic 3

3 x 3 = 9

24.

(a) Let a, b \(\in \) 2.
The a * b = a + b + z = b + a + z = b * a
(z, *) has commutative property.
(b) Let a, b, c \(\in \) G
(a * b) * c = (a+b+2) * c = a + b + 2 + c + 2
= a + b + c + 4
∴ z is associate under *.

25.

Clearly 1 is the identity element of (G₁)
Inverse of 1 is 1 [∴ (1)(1) = 1]
Inverse of i is -i [∴ (i) (-i) = -i² = 1]
Inverse of -1 is -1 [∴ (-1)(-1) = 1]
Inverse of is i [∴ (-i)(i) = -i² = 1]

26.

Let a = 2, b = -5
∴ a * b = a^b
⇒ 2 * -5 = 2^-5 = \(\frac{1}{2^5}\) ∉ z
* is not a binary operation on z.