JEE Main Mathematical Reasoning Previous Year Questions With Solutions

Mathematical reasoning deal to determine the truth values of the given statements. This topic is covered under the JEE Main only and not in JEE Advanced. It carries four marks and is one of the easiest concepts in the entire syllabus. Mathematical Logic is a subject which deals with the principles of reasoning. Mathematical reasoning is also called a science of proof. In this article, JEE aspirants can get a set of questions asked in previous year exams on Mathematical reasoning with detailed solutions.

JEE Main Mathematical Reasoning Previous Year Questions With Solutions

Question 1: The contrapositive of the inverse of p â‡’ ~q is

Solution:

The inverse of p â‡’ âˆ¼q is âˆ¼ p â‡’ q

The contrapositive of âˆ¼ p â‡’ q is âˆ¼ q â‡’ p. [Because the contrapositive of p â‡’ q is âˆ¼ q â‡’ p.]

Question 2: Which of the following is the contrapositive of if two triangles are identical, then these are similar??

A) if two triangles are not similar, they are not identical

B) If two triangles are not identical, then these are not similar

C) If two triangles are not identical, then these are similar

D) If two triangles are not similar, then these are identical

Solution:

Consider the following statements

p: Two triangles are identical.

q: Two triangles are similar.

Clearly, the given statement in symbolic form is p â‡’ q.

Therefore, its contrapositive is given by âˆ¼ q â‡’ âˆ¼ p

Now,

âˆ¼p: two triangles are not identical.

âˆ¼q: two triangles are not similar.

Therefore, ~ q â‡’ ~ p: If two triangles are not similar, then these are not identical.

Question 3: If p is true and q is false, then which of the following statements is not true?

A) p âˆ¨ q

B) p â‡’ q

C) p âˆ§ ( ~q)

D) p â‡’ p

Solution:

When p is true and q is false, then pâˆ¨q is true, q â‡’ p is true and p âˆ§ (âˆ¼q) is true. (Therefore, both p and âˆ¼q are true)

Here, p â‡’ q is not true as a true statement cannot imply a false statement.

Question 4: ~( p âˆ¨ q) âˆ¨ (~p âˆ§ q) is equivalent to ____________.

Solution:

âˆ¼ (p âˆ¨ q) âˆ¨ (âˆ¼p âˆ§ q) = (âˆ¼p âˆ§ âˆ¼q) âˆ¨ (âˆ¼p âˆ§ q) =âˆ¼q âˆ§ âˆ¼(âˆ¼ (p âˆ§ q))

Question 5: Which of the following is logically equivalent to ~( ~ p â‡’ q)?

A) p âˆ§ q

B) p âˆ§ ~q

C) ~p âˆ§ q

D) ~p âˆ§ ~q

Solution:

It is clear from the table that âˆ¼ (âˆ¼p â‡’ q) is equivalent to âˆ¼p âˆ§ âˆ¼q.

Question 6: If (p âˆ§ ~r) âˆ§ ( ~p / q) is false, then write the truth values of p, q and r.

Solution:

Since,

(p âˆ§ âˆ¼r) â‡’ (âˆ¼p âˆ¨ q) is F Then, p = T, q = F, r = F.

Question 7: If p and q are two statements, then (p â‡’ q) â‡” ( ~q â‡’ ~p) is a ___________.

Solution:

 pâ‡’q âˆ¼pâ‡’âˆ¼q pâ‡’qâ‡”âˆ¼qâ‡’âˆ¼p T T T F F T T T T T T T

Therefore, it is a tautology. Hence, the given proposition is a tautology.

Question 8: If each of the following statements is true, then P â‡’ ~q, q â‡’ r, ~r

A) p is false

B) p is true

C) q is true

D) None of these

Solution:

Since âˆ¼r is true, therefore, r is false. Also, q â‡’ r is true, therefore, q is false. (Therefore, a true statement cannot imply a false one) Also, p â‡’ q is true, therefore, p must be false.

Question 9: What is the negation of the compound proposition?

If the examination is difficult, then I shall pass if I study hard.

Solution:

If p: Examination is difficult

q: I shall pass

r: I study hard

Given result is P â‡’ (r â‡’ q)

Now, âˆ¼ (r â‡’ q) = r âˆ§ âˆ¼q âˆ¼(p â‡’ (r â‡’ q)) = p âˆ§ (r âˆ§ âˆ¼q)

The examination is difficult and I study hard but I shall not pass.

Question 10: The statement p â†’ (p â†’ q) is equivalent to __________.

Solution:

p â†’ (q â†’ p) = âˆ’p (q â†’ p) = âˆ¼p âˆ¨ (âˆ¼q âˆ¨ p)

(Since p âˆ¨âˆ¼p is always true) = âˆ¼p âˆ¨ p âˆ¨ q = p â†’ (p âˆ¨ q)

Question 11: The statement ~(p â†” ~q)is ________.

Solution:

 p q âˆ¼q pâ†”âˆ¼q âˆ¼(pâ†”âˆ¼q) pâ†”q T T F F T T T F T T F F F T F T F F F F T F T T

Question 12: The Boolean Expression (p âˆ§ ~q) âˆ¨ q âˆ¨( ~p âˆ§ q) is equivalent to ___________.

Solution:

[(p âˆ§ âˆ¼q) âˆ¨ q] âˆ¨ (âˆ¼p âˆ§ q) = (p âˆ¨ q) âˆ§ (âˆ¼q âˆ¨ q) âˆ¨ (âˆ¼p âˆ§ q)

= (p âˆ¨ q) âˆ§ [t âˆ¨ (âˆ¼p âˆ§ q)]

= (p âˆ¨ q) âˆ§ t

= p âˆ¨ q

Question 13:

 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 expressed as

Solution:

Suman is brilliant and dishonest is Pâˆ§âˆ¼R.

Suman is brilliant and dishonest if and only if Suman is rich is Q â†” (P âˆ§ âˆ¼R)

Negative of the statement is expressed as âˆ¼(Q â†” (P âˆ§ âˆ¼R).