Class 11 Mathematics Mathematical Reasoning MCQs Set 03

Let p be the statement ‘x is an irrational number’, q be the statement ‘y is a transcendental number’ and r be the statement ‘x is a rational number iff y is a transcendental number’.
Statement I r is equivalent to either q or p.
Statement II r is equivalent to ~ ( p↔~ p).

Question: The statement p →(q → p) is equivalent to
a) p → (p ↔ q)
b) p → (p → q)
c) p → (p ∨ q)
d) p → (p ∧ q)
Answer: d

Question: The false statement in the following is
a) p ∧ (~ p) is a contradiction
b) (p ⇒ q) ⇔ (~ q ⇒ ~ p) is a contradiction
c) ~ (~ p) ⇔ p is a tautology
d) p ∨ (~ p) is a tautology
Answer: d

Question: Consider the statement∼ 'For an integer n, if n3 – 1 is even, then n is odd.' The contrapositive statement of this statement is∼
a) For an integer n, if n is even, then n3 – 1 is odd.
b) For an intetger n, if n3 – 1 is not even, then n is not odd.
c) For an integer n, if n is even, then n3 – 1 is even.
d) For an integer n, if n is odd, then n3 – 1 is even.
Answer: a

Question: ( p ∧ ~ q) ∧ (~ p ∧ q) is
a) a tautology
b) a contradiction
c) both a tautology and a contradiction
d) neither a tautology nor a contradiction
Answer: b

Question: The proposition S : ( p⇒ q) ⇔(~ p ∨ q) is
a) a tautology
b) a contradiction
c) either a tautology or a contradiction
d) neither a tautology nor a contradiction
Answer: a

Question: Let p and q be two statements. Then,(~ p ∨ q) ∧ (~ p ∧~ q) is a
a) tautology
b) contradiction
c) neither tautology nor contradiction
d) both tautology and contradiction
Answer: c

Question: The contrapositive of the statement 'If I reach the station in time, then I will catch the train' is ∼
a) If I do not reach the station in time, then I will catch the train.
b) If I do not reach the station in time, then I will not catch the train.
c) If I will catch the train, then I reach the station in time.
d) If I will not catch the train, then I do not reach the station in time.
Answer: d

Question: The statement ( p→(q→ p))→( p→( p ∨ q)) is ∼
a) equivalent to ( p ∧ q)∨ (~ q)
b) a contradiction
c) equivalent to ( p ∨ q) ∧ (~ p)
d) a tautology
Answer: d

Question: If p and q are two statements, then ( p⇒ q) ⇔(~ q ⇒ ~ p) is a
a) contradiction
b) tautology
c) neither contradiction nor tautology
d) None of the options
Answer: b

Question: If p and q are simple proposition, then (~ p ∧ q) ∨ (~ q ∧ p) is false when p and q are respectively
a) T, T
b) T, F
c) F, F
d) F, T
Answer(a,b,c)

Question: Contrapositive of the statement ∼ ‘If a function f is differentiable at a, then it is also continuous at a’, is ∼
a) If a function f is continuous at a, then it is not differentiable at a.
b) If a function f is not continuous at a, then it is not differentiable at a.
c) If a function f is not continuous at a, then it is differentiable at a
d) If a function f is continuous at a, then it is differentiable at a.
Answer: b

Question: If p and q are two statements, then ~( p ∧ q) ∨~ (q⇔ p) is
a) tautology
b) contradiction
c) neither tautology nor contradiction
d) either tautology or contradiction
Answer: c

Question: The contrapositive of the statement 'If you are born in India, then you are a citi en of India', is :
a) If you are not a citi en of India, then you are not born in India.
b) If you are a citi en of India, then you are born in India.
c) If you are born in India, then you are not a citi en of India.
d) If you are not born in India, then you are not a citi en of India.
Answer: a

Question: Contrapositive of the statement ‘If two numbers are not equal, then their squares are not equal’, is :
a) If the squares of two numbers are equal, then the numbers are equal.
b) If the squares of two numbers are equal, then the numbers are not equal.
c) If the squares of two numbers are not equal, then the numbers are not equal.
d) If the squares of two numbers are not equal, then the numbers are equal.
Answer: a

Question: The propositions ( p⇒~ p) ∧ (~ p⇒ p) is
a) Tautology and contradiction
b) Neither tautology nor contradiction
c) Contradiction
d) Tautology
Answer: c

Question: The contrapositive of the following statement, 'If the side of a square doubles, then its area increases four times', is :
a) If the area of a square increases four times, then its side is not doubled.
b) If the area of a square increases four times, then its side is doubled.
c) If the area of a square does not increases four times, then its side is not doubled.
d) If the side of a square is not doubled, then its area does not increase four times.
Answer: c

Question: The contrapositive of the statement 'If it is raining, then I will not come', is :
a) If I will not come, then it is raining.
b) If I will not come, then it is not raining.
c) If I will come, then it is raining.
d) If I will come, then it is not raining.
Answer: d

Question: The contrapositive of the statement 'if I am not feeling well, then I will go to the doctor' is
a) If I am feeling well, then I will not go to the doctor
b) If I will go to the doctor, then I am feeling well
c) If I will not go to the doctor, then I am feeling well
d) If I will go to the doctor, then I am not feeling well.
Answer: c

Question: If p and q are two statements, then statement p⇒ q ∧ ~ q is
a) tautology
b) contradiction
c) neither tautology nor contradiction
d) None of the options
Answer: c

Question: Let p and q be two statements, then ( p ∨ q) ∨ ~ p is
a) tautology
b) contradiction
c) Both tautology and contradiction
d) None of the options
Answer: a

Question: Contrapositive of the statement 'If two numbers are not equal, then their squares are not equal'. is :
a) If the squares of two numbers are not equal, then the numbers are equal.
b) If the squares of two numbers are equal, then the numbers are not equal.
c) If the squares of two numbers are equal, then the numbers are equal.
d) If the squares of two numbers are not equal, then the numbers are not equal.
Answer: c

Question: The statement p ∨~ pis
a) tautology
b) contradiction
c) neither a tautology nor a contradiction
d) None of the options
Answer: a

Question: The statement ( p⇒ q) ⇔(~ p ∧ q) is a
a) tautology
b) contradiction
c) neither tautology nor contradiction
d) None of the options
Answer: c

Question: The proposition ~ ( p⇒ q) ⇒(~ p ∨ ~ q) is
a) a Tautology
b) a Contradiction
c) either a Tautology or a Contradiction
d) neither a Tautology nor a Contradiction
Answer: a

Question: The negation of the compound proposition is p ∨ (~ p ∨ q)
a) (p ∧ ~ q) ∧ ~ p
b) (p ∨ ~ q) ∨ ~ p
c) (p ∧ ~ q) ∨ ~ p
d) None of the options
Answer: a

Question: The proposition ( p⇒~ p) ∧ (~ p ⇒ p) is
a) contigency
b) neither Tautology nor Contradiction
c) contradiction
d) tautology
Answer: c

Question: Let p and q denote the following statements
p : The sun is shining
q: I shall play tennis in the afternoon
The negation of the statement 'If the sun is shining then I shall play tennis in the afternoon', is
a) q ⇒: p
b) q ∧ : p
c) p ∧ : q
d) : q ⇒: p
Answer: c

Question: Consider the following statements :
P : Suman is brilliant
Q : Suman is rich.
R : Suman is honest
the negation of the statement 'Suman is brilliant and dishonest if and only if suman is rich' can be equivalently expressed as :
a) ~ Q « ~ P ∨ R
b) ~ Q « ~ P ∧ R
c) ~ Q « P ∨ ~ R
d) ~ Q « P ∧ ~ R
Answer: d

Question: Consider the following two statements :
P : If 7 is an odd number, then 7 is divisible by 2.
Q : If 7 is a prime number, then 7 is an odd number.
If V1 is the truth value of the contrapositive of P and V2 is the truth value of contrapositive of Q, then the ordered pair (V1, V2) equals:
a) (F, F)
b) (F, T)
c) (T, F)
d) (T, T)
Answer: a

Directions : Each of these questions contans two statements : statement I (Assertion) and Statement II (Reason). Each of these questions also has four alternatite choice, only one of which is the correct answer.You has to select one of the codes a), b), c), d) given below.
a) Statement I is true, Statement II is true; Statement II is a correct explanation for Statement I.
b) Statement I is true, Statement II is true; Statement II is not a correct explanation for Statement I.
c) Statement I true, Statement II is false.
d) Statement I is false, Statement II is true.

Question: Statement I The statement [ p ∧ ( p→ q)]→ q is a tautology.
Statement II If all truth values of a statement is true, then the statement is a tautology.
Answer: a

Question: Let p: Ice is cold and q : blood is green be two statements, then
Statement I p ∨ q : Ice is cold or blood is green.
Statement II p ∧ q : Ice is not cold or blood is green.
Answer: c

Question: Let S :~ ( p⇔ q) ∨ ~ (q⇔ p)
Statement I The statement S is logically equivalent to ( p⇔q)
Statement II The logically equivalent proposition of pÛ q is ( p⇔ q) ∧ (q⇔p)
Answer: a

Question: If p→ q be any conditional statement,
Statement I The converse of p→ q is the statement q→ p.
Statement II The inverse of p→ q is the statement ~ q→~ p.
Answer: c

Question: Suppose p, q and r be any three statements.
Statement I The statement p→(q→ r) is a tautology.
Statement II ( p ∧ q) → r and p→(q→ r) are identical.
Answer: d

Question: Let p be the statement, 'Mr A passed the examination', qbe the statement, 'Mr A is sad' and r be the statement 'It is not true that Mr A passed therefore he is sad.'
Statement I r ≡ p⇔ q
Statement II The logical equivalent of p⇔ qis~ p ∨ q.
Answer: d

Question: Statement I ~ ( p ↔ ~ q) is equivalent to p ↔ q
Statement II ~ ( p ↔ ~ q) is a tautology
Answer: b

Question: Statement I ~ (A ⇔ ~ b) is equivalent to A ⇔ B.
Statement II A ∨ (~ (A ∧ ~ B)) a tautology.
Answer: b

Question: Consider
Statement I ( p∧~ q) ∧(~ p∧ q) is a fallacy.
Statement II ( p→ q) ↔(~ q→~ p) is a tautology.
Answer: b