JEE Main Mathematical Previous Year Questions With Solutions on Reasoning

Mathematical reasoning works to decide the reality upsides of the given assertions. This subject is covered under the JEE Main just and not in JEE Advanced. It conveys four checks and is probably the most straightforward idea in the prospectus. Numerical Logic is a subject that manages the standards of thinking.

Numerical thinking is additionally called a study of evidence. In this article, JEE competitors can get a bunch of inquiries posed in earlier year tests on Mathematical prevailing upon definite arrangements.

 

What is Mathematical Reasoning and what are its types?

Mathematical Reasoning is one of the subjects in arithmetic where the legitimacy of numerically acknowledged articulations is resolved utilizing intelligence and Maths abilities.

There are two major kinds of reasoning in Maths:

• Inductive reasoning
• Deductive reasoning

Question 1: The contrapositive of the inverse of p ⇒ ~q is

Solution:

The contrapositive if the inverse is ∼ q ⇒ 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 houses are identical.

q: Two houses are similar.

Clearly, the symbolized form can be written as p ⇒ q.

Therefore, its contrapositive is given by ∼ q ⇒ ∼ p

Now,

∼p: two houses are not identical.

∼q: two houses are not similar.

Therefore, ~ q ⇒ ~ p: If two houses are not similar, then these are not identical.

The correct answer is (A).

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

A) p ∨ q

B) p ⇒ q

C) p ∧ ( ~q)

D) q ⇒ p

Solution:

When p is true and q is false, we know, p∨q is true, q ⇒ p is true, and p ∧ (∼q) is true.

Here, p ⇒ q is not a true statement among all the options.

The correct answer is (B).

Question 4: ~( p ∨ q) ∨ (~p ∧ q) is equivalent to ____________.

Solution:

∼ (p ∨ q) ∨ (∼p ∧ q) = (∼p ∧ ∼q) ∨ (∼p ∧ q).

(or)

∼ (p ∨ q) ∨ (∼p ∧ q) =∼q ∧ ∼(∼ (p ∧ q)).

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

A) p ∧ q

B) p ∧ ~q

C) ~p ∧ q

D) ~p ∧ ~q

Solution:

By constructing the truth table for the respective function, it can be noted from the table that ∼ (∼p ⇒ q) is the same as ∼p ∧ ∼q.

The correct answer is (D).

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

Solution: By constructing the truth table for the respective function, it can be noted from the table,

(p ∧ ∼r) ⇒ (∼p ∨ q) is F Then, p = T, q = F, r = F.

Question 7: If p and q are two statements, then (p ⇒ q) ⇔ ( ~q ⇒ ~p) is a ___________.

Solution:

The given proposition is a tautology.

A Tautology is a compound assertion/function in Math which always gives Truth value as resultant. It doesn’t make any difference what the singular part comprises of; the outcome in repetition is true 100% of the time. The opposite of tautology is a contradiction.

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

A) p is false

B) p is true

C) q is true

D) None of these

Solution: (A)

The correct solution is ‘p’ must be false to satisfy the above true statements.

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

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

Solution:

If x: Examination is difficult

y: I shall pass

z: I study hard

Given result is x ⇒ (z⇒ y)

Now, ∼ (z ⇒ y) = z ∧ ∼y ∼(x ⇒ (z ⇒ y)) = x ∧ (z ∧ ∼y)

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

Question 10: The statement p → (p → q) is equivalent to __________.

Solution:

p → (q → p)

=> −p (q → p)

=> ∼p ∨ (∼q ∨ p).

(we know that p ∨∼p is always true, Hence) = ∼p ∨ p ∨ q = p → (p ∨ q)

Question 11: The Boolean Expression (p ∧ ~q) ∨ q ∨( ~p ∧ q) is equivalent to ___________.

Solution:

Given,

[(p ∧ ∼q) ∨ q] ∨ (∼p ∧ q)

= (p ∨ q) ∧ (∼q ∨ q) ∨ (∼p ∧ q)

= (p ∨ q) ∧ [t ∨ (∼p ∧ q)] [Since, ~p ∨ p = t]

= (p ∨ q) ∧ t [Since, t ∨ p = t ]

= p ∨ q [Since, t ∧ p = p]

=>p ∨ q.

Question 12:

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 P∧∼R.

Suman is brilliant and dishonest if and only if Suman is rich is Q ↔ (P ∧ ∼R)

Negative of the statement is expressed as ∼ (Q ↔ (P ∧ ∼ R).

FAQs

What is inductive reasoning?

In the Inductive technique for numerical thinking, the legitimacy of the assertion is checked by a specific arrangement of rules, and afterwards it is summed up. The standard of numerical acceptance utilizes the idea of inductive thinking. As inductive thinking is summed up, it isn’t considered in mathematical verifications. Here is a model which will assist with understanding inductive thinking in math better.

What do you mean by deductive reasoning?

The head of deductive reasoning is something contrary to the guideline of acceptance. As opposed to inductive thinking, in insightful thinking, we apply the standards of an overall case to a given assertion and make it valid for specific explanations. The guideline of numerical acceptance utilizes the idea of deductive reasoning (as opposed to its name). The beneath given model will assist with understanding the idea of insightful thinking in math.

Why is Mathematical Reasoning Important?

Numerical thinking is significant as it assists with creating decisive reasoning and getting Maths in a more significant manner. The ideas of thinking not just assist the understudies with having a more profound comprehension of the subject yet, in addition, helps in having a more extensive point of view to consistent proclamations.