VBQs Mathematical Reasoning Class 11 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 11 Mathematics with solutions. The following Mathematical Reasoning Class 11 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 11 examinations.

**Mathematical Reasoning VBQs Class 11 Mathematics**

**Question. For integers m and n, both greater than 1, consider the following three statements :****P :** m divides n**Q :** m divides n^{2}**R :** m is prime,

then

(a) Q ∧ R→ P

(b) P ∧ Q→ R

(c) Q→ R

(d) Q→ P

## Answer

A

**Question. The statement p → (q → p) is equivalent to :**

(a) p→q

(b) p→( p ∨ q)

(c) p→( p→q)

(d) p→( p ∧ q)

## Answer

B

**Question. The following statement (p → q) → [(~p → q) → q] is :**

(a) a fallacy

(b) a tautology

(c) equivalent to ~ p → q

(d) equivalent to p → ~q

## Answer

B

**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. The negation of ~ s ∨ (~ r ∧ s) is equivalent to :**

(a) s ∨ (r ∨ ~ s)

(b) s ∧ r

(c) s ∧ ~ r

(d) s ∧ (r ∧ ~ s)

## Answer

B

**Question. Let p and q be any two logical statements and r ~ p → (~ p ∨ q) . If r has a truth value F, then the truth values of p and q are respectively :**

(a) F, F

(b) T, T

(c) T, F

(d) F, T

## Answer

C

**Question. The Boolean expression ~ (p ∨ q) ∨ (~ p ∧ q) is equivalent to :**

(a) p

(b) q

(c) ~q

(d) ~p

## Answer

D

**Question. If p → (~ p ∨ ~ q) is false, then the truth values of p and q are respectively.**

(a) T, F

(b) F, F

(c) F, T

(d) T, T

## Answer

D

**Question.** **Statement-1:** The statement A→(B → A) is equivalent to A→(A ∨ B).**Statement-2:** The statement ~ [(A ∧ B) → ( ~ A ∨ B)] is a Tautology.

(a) Statement-1 is false; Statement-2 is true.

(b) Statement-1 is true; Statement-2 is true; Statement- 2 is not correct explanation for Statement-1.

(c) Statement-1 is true; Statement-2 is false.

(d) Statement-1 is true; Statement-2 is true; Statement- 2 is the correct explanation for Statement-1.

## 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. 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 V _{1} is the truth value of the contrapositive of P and V_{2} is the truth value of contrapositive of Q, then the ordered pair (V_{1}, V_{2}) equals:**

(a) (F, F)

(b) (F, T)

(c) (T, F)

(d) (T, T)

## Answer

C

**Question. Let p and q be two Statements. Amongst the following, the Statement that is equivalent to p → q is**

(a) p ∧ ~ q

(b) ~ p ∨ q

(c) ~ p ∧ q

(d) p ∨ ~ q

## Answer

B

**Question. The negation of the statement “If I become a teacher, then I will open a school”, is :**

(a) I will become a teacher and I will not open a school.

(b) Either I will not become a teacher or I will not open a school.

(c) Neither I will become a teacher nor I will open a school.

(d) I will not become a teacher or I will open a school.

## Answer

A

**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. The logically equivalent preposition of p ⇔ q is**

(a) ( p⇒q) ∧ (q⇒ p)

(b) p ∧ q

(c) ( p ∧ q) ∨ (q⇒ p)

(d) ( p ∧ q)⇒(q ∨ p)

## Answer

A

**Question. The expression ∼ (∼ p → q) is logically equivalent to :**

(a) ∼ p ∧ ∼ q

(b) p ∧ ∼ q

(c) ∼ p ∧ q

(d) p ∧ q

## Answer

A

**Question. The logical statement (p ⇒ q) ∧ (q ⇒∼ p) : is equivalent to:**

(a) p

(b) q

(c) ~p

(d) ~q

## Answer

C

**Question. Let p, q, r be three statements such that the truth value of ( p ∧ q) → (~ q ∨ r) is F. Then the truth values of p, q, r are respectively :**

(a) T, F, T

(b) T, T, T

(c) F, T, F

(d) T, T, F

## Answer

D

**Question. If the Boolean expression (p ⊕ q) ∧ (: p ⊙ q) is equivalent to p ∧ q , where ⊕,⊙ ∈ {∧ , ∨} then the ordered pair (⊕, ⊙) is:**

(a) (∨ , ∧)

(b) (∨ , ∨)

(c) (∧ , ∨)

(d) (∧ , ∧)

## Answer

C

**Question. The Boolean expression ~ ( p⇒(~q)) is equivalent to :**

(a) p ∧ q

(b) ⇒ ~ p

(c) p ∨ q

(d) (~ p)⇒ q

## Answer

A

**Question. The negation of the Boolean expression p ∨ (~ p ∧ q) is equivalent to :**

(a) p∧ ~ q

(b) ~ p∧ ~ q

(c) ~ p∨ ~ q

(d) ~ p ∨ q

## Answer

B

**Question. Which one of the following is a tautology?**

(a) (p ∧ (p → q)) → q

(b) q → (p ∧ (p → q))

(c) p ∧ (p ∨ q)

(d) p ∨ (p ∧ q)

## Answer

A

**Question. The proposition p→~ ( p ∧ ~ q) is equivalent to :**

(a) q

(b) (~ p) ∨ q

(c) (~ p) ∧ q

(d) (~ p) ∨ (~ q)

## Answer

B

**Question. If the truth value of the statement p → (~q ∨ r) is false (F), then the truth values of the statements p, q, r are respectively.**

(a) T, T, F

(b) T, F, F

(c) T, F, T

(d) F, T, T

## Answer

A

**Question. Given the following two statements :****(S _{1}) :** (q ∨ p) → ( p ↔ ~ q) is a tautology.

**(S**~ q ∧ (~ p ↔ q) is a fallacy. Then :

_{2}) :(a) both (S

_{1}) and (S

_{2}) are correct

(b) only (S

_{1}) is correct

(c) only (S

_{2}) is correct

(d) both (S

_{1}) and (S

_{2}) are not correct

## Answer

D

**Question. Which of the following statements is a tautology?**

(a) p ∨ (~q) → p ∧ q

(b) ~(p ∧ ~ q) → p ∨ q

(c) ~(p ∨ ~ q) → p ∧ q

(d) ~(p ∨ ~ q) → p ∨ q

## Answer

D

**Question. Which one of the following Boolean expressions is a tautology ?**

(a) (p ∧ q) ∨ (p ∧ ~ q)

(b) (p ∨ q) ∨ (p ∨ ~ q)

(c) (p ∨ q) ∧ (p ∨ ~ q)

(d) (p ∨ q) ∧ (~p ∨ ~ q)

## Answer

B

**Question. The logical statement [~ (~ p ∨ q) ∨ (p ∧ r)] ∧ (~ p ∧ r) is equivalent to:**

(a) (~ p ∧ ~ q) ∧ r

(b) ~ p ∨ r

(c) (p ∧ r) ∧ ~ q

(d) (p ∧ ~ q) ∨ r

## Answer

C

**Question. Which one of the following statements is not a tautology?**

(a) (p ∨ q) → (p ∨ (~ q))

(b) (p ∧ q) → (~ p) ∨ q

(c) p → (p ∨ q)

(d) (p ∧ q) → p

## Answer

A

**Question. Consider the following three statements:****P :** 5 is a prime number.**Q :** 7 is a factor of 192.**R :** L.C.M. of 5 and 7 is 35.**Then the truth value of which one of the following statements is true?**

(a) (~ P) ∨ (Q ∧ R)

(b) (P ∧ Q) ∨ (~ R)

(c) (~ P) ∧ (~ Q ∧ R)

(d) P ∨ (~ Q ∧ R)

## Answer

D

**Question. If p → (p ∧ ~q) is false, then the truth values of p and q are respectively:**

(a) F, F

(b) T, F

(c) T, T

(d) F, T

## Answer

C

**Question. The Boolean expression ((p ∧ q) ∨ (p∨ ∼ q)) ∧ (∼ p∧ ∼ q) is equivalent to :**

(a) p ∧ q

(b) p ∧(∼ q)

(c ) (∼ p) ∧ (∼ q)

(d) p∨ (∼ q)

## Answer

C

**Question. The negation of the Boolean expression x↔~ y is equivalent to:**

(a) (x ∧ y) ∨ (~ x∧ ~ y)

(b) (x ∧ y) ∧ (~ x∨ ~ y)

(c) (x∧ ~ y) ∨ (~ x ∧ y)

(d) (~ x ∧ y) ∨ (~ x∧ ~ y)

## Answer

A

**Question. If q is false and p ∧ q ↔ r is true, then which one of the following statements is a tautology?**

(a) (p ∨ r )→(p ∧ r )

(b) (p ∧ r )→(p ∨ r )

(c) p ∧ r

(d) p ∨ r

## Answer

B

**Question. If p ⇒(q ∨ r) is false, then the truth values of p, q, r are respectively:**

(a) F, T, T

(b) T, F, F

(c) T, T, F

(d) F, F, F

## Answer

A

**Question. Let A, B, C and D be four non-empty sets. The contrapositive statement of “If A ⊆ B and B ⊆ D, then A ⊆ C ” is:**

(a) If A ⊄ C, then A ⊆ B and B ⊆ D

(b) If A ⊆ C, then B ⊂ A or D ⊂ B

(c) If A ⊄ C, then A ⊄ B and B ⊆ D

(d) If A ⊄ C, then A ⊄ B or B ⊄ D

## Answer

D

**Question. The negation of the Boolean expression ~ s ∨ (~ r ∧ s) is equivalent to :**

(a) ~ s ∧ ~ r

(b) r

(c) s ∨ r

(d) s ∧ r

## Answer

D

**Question. The only statement among the following that is a tautology is**

(a) A ∧ (A ∨ B)

(b) A ∨ (A ∧ B)

(c) [A ∧ (A → B)]→ B

(d) B→[A ∧ (A →B)]

## Answer

C

**Question. If (p ∧ ~ q) ∧ (p ∧ r) → ~ p ∨ q is false, then the truth values of p, q and r are respectively**

(a) F, T, F

(b) T, F, T

(c) F, F, F

(d) T, T, T

## Answer

B

**Question. Which of the following is a tautology?**

(a) (~ p) ∧ (p ∨ q)→q

(b) (q →p) ∨ ~ (p→q)

(c) (~ q) ∨ (p ∧ q)→q

(d) (p→q) ∧ (q →p)

## Answer

A

**Question.Statement-1 :** ~ ( p ↔~ q) is equivalent to p ↔q .**Statement-2 :** ~ ( p ↔~ q) is a tantology

(a) Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(b) Statement-1 is true, Statement-2 is false.

(c) Statement-1 is false, Statement-2 is true.

(d) Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement -1

## Answer

B

**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

B

**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. Consider the following two statements.****Statement p:**

The value of sin 120° can be divided by taking q = 240° in the equation 2 sin θ/2 = √(1 + sinθ) − √(1 − sinθ).**Statement q:**

The angles A, B, C and D of any quadrilateral ABCD satisfy the equation cos( ½(A+C) ) + cos( ½(B+D) ) = 0

Then the truth values of p and q are respectively.

(a) F, T

(b) T, T

(c) F, F

(d) T, F

## Answer

A

**Question. 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-1 :** r is equivalent to either q or p**Statement-2 :** r is equivalent to ~(p↔~q).

(a) Statement -1 is false, Statement-2 is true

(b) Statement -1 is true, Statement-2 is true; Statement -2 is a correct explanation for Statement-1

(c) Statement -1 is true, Statement-2 is true; Statement -2 is not a correct explanation for Statement-1

(d) Statement -1 is true, Statement-2 is false

## Answer

None

**Question. The statement : ( p↔~q) is: **

(a) a tautology

(b) a fallacy

(c) eqivalent to p ↔ q

(d) equivalent to ~ p ↔ q

## Answer

C

**Question. Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement p⇒(q ∨ r ) is:**

(a) ( p ∨ q)⇒r

(b) ( p⇒q) ∨ ( p⇒r )

(c) ( p⇒~ q) ∧ ( p⇒r )

(d) ( p⇒q) ∧ ( p⇒~ r )

## Answer

B

**Question. Consider the statement: “For an integer n, if n ^{3} – 1 is even, then n is odd.” The contrapositive statement of this statement is:**

(a) For an integer n, if n is even, then n

^{3}– 1 is odd.

(b) For an intetger n, if n

^{3}– 1 is not even, then n is not odd.

(c) For an integer n, if n is even, then n

^{3}– 1 is even.

(d) For an integer n, if n is odd, then n

^{3}– 1 is even.

## Answer

A

**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. Negation of the statement:**

√5 is an integer of 5 is irrational is:

(a) √5 is not an integer or 5 is not irrational

(b) √5 is not an integer and 5 is not irrational

(c) √5 is irrational or 5 is an integer.

(d) √5 is an integer and 5 is irrational

## Answer

B

**Question. For any two statements p and q, the negation of the expression p ∨ (~ p ∧ q) is:**

(a) ~ p ∧ ~ q

(b) p ∧ q

(c) p ↔ q

(d) ~ p ∨ ~ q

## Answer

D

**Question. The contrapositive of the statement “If you are born in India, then you are a citizen of India”, is :**

(a) If you are not a citizen of India, then you are not born in India.

(b) If you are a citizen of India, then you are born in India.

(c) If you are born in India, then you are not a citizen of India.

(d) If you are not born in India, then you are not a citizen 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 Statement that is TRUE among the following is**

(a) The contrapositive of 3x + 2 = 8 ⇒ x = 2 is x ≠ 2 ⇒ 3x + 2 ≠ 8.

(b) The converse of tanx = 0 ⇒ x = 0 is x ≠ 0 ⇒ tan x = 0.

(c) p ⇒ q is equivalent to p ∨ ~ q.

(d) p ∨ q and p ∧ q have the same truth table.

## Answer

A

**Question. The proposition (~p) ∨ (p ∧ ~ q)**

(a) p→~ q

(b) p ∧ (~ q)

(c) q→ p

(d) p ∨ (~ q)

## Answer

B

**Question. The Boolean Expression (p ∧ ~ q) ∨ q ∨ (~ p∧q) is equivalent to:**

(a) p ∨ q

(b) p ∨ ~ q

(c) ~ p ∧ q

(d) p ∧ q

## Answer

A

**Question. Let S be a non-empty subset of R. Consider the following statement :****P :** There is a rational number x ∈ S such that x > 0.**Which of the following statements is the negation of the statement P ?**

(a) There is no rational number x ∈ S such than x ≤ 0.

(b) Every rational number x ∈ S satisfies x ≤ 0.

(c) x ∈ S and x ≤ 0 ⇒ x is not rational.

(d) There is a rational number x ∈ S such that x ≤ 0.

## Answer

B

**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. The contrapositive of the statement “I go to school if it does not rain” is**

(a) If it rains, I do not go to school.

(b) If I do not go to school, it rains.

(c) If it rains, I go to school.

(d) If I go to school, it rains.

## Answer

B

**Question. Contrapositive of the statement :**

‘If a function ƒ is differentiable at a, then it is also continuous at a’, is :

(a) If a function ƒ is continuous at a, then it is not differentiable at a.

(b) If a function ƒ is not continuous at a, then it is not differentiable at a.

(c) If a function ƒ is not continuous at a, then it is differentiable at a

(d) If a function ƒ is continuous at a, then it is differentiable at a.

## Answer

B

**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 proposition ~ ( p ∨ ~ q) ∨ ~ ( p ∨ q) is logically equivalent to:**

(a) p

(b) q

(c) ~ p

(d) ~ q

## Answer

C

**Question. Consider****Statement-1 :** (p ∧ ~ q) ∧ (~ p ∧ q) is a fallacy.**Statement-2 :** (p → q) ↔ (~ q → ~ p) is a tautology.

(a) Statement-1 is true; Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

(b) Statement-1 is true; Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.

(c) Statement-1 is true; Statement-2 is false.

(d) Statement-1 is false; Statement-2 is true.

## Answer

B