JEE MAIN 2020 NEWS
NATIONAL LEVEL ONLINE TEST
Mathematical Reasoning has less weightage in JEE Main Mathematics section as compared to chapters like Coordinate Geometry and Integral Calculus. However, the questions asked from this chapter require lesser logical thinking and are very easy to score. Read JEE Main Mathematics Syllabus
Around 1-2 questions are generally asked from this chapter which carry a total of 4 marks. Therefore, the total weightage of Mathematical Reasoning in JEE Main is 1-2%. Some important concepts tested under this chapter are Tautology, Fallacy, and various Mathematical Reasoning Laws.
Read the article to find notes on Mathematical Reasoning for JEE Main Preparation.
The truth table given below displays the truth values of p ∧ q ( p and q) and q ∧ p ( q and p) -
|Truth Table (p ∨ q, q ∨ p)|
|p||q||p ∧ q||q ∧ p|
|Rule: p ∧ q is true if and when p and q are true.|
The truth table of p ∨ q, q ∨ p is given below -
|Truth Table (p v q, q v p)|
|p||q||p ∨ q||q ∨ p|
|Rule: p ∨ q is false if and when both p and q are false.|
The truth table of ~p is given below -
|Truth Table (~p)|
|Rule: ~ is true if and when p is false.|
The truth table for p → q is given below -
|Truth Table (p → q, q → p)|
|p||q||p → q||q → p|
|Rule: p → q is false if and when p is true and q is false.|
The truth table for p ↔ q or q ↔ p is given below -
|Truth Table (p ↔ q, q ↔ p)|
|p||q||p ↔ q||q ↔ p|
|Rule: p ↔ q is true if and when both p and q have the same truth value.|
The truth table for p → q is given below -
|Truth Table (p → q)|
|p||q||p → q||~q → ~p (Contrapositive)||q ↔ p (Converse)|
The truth table for a tautology ‘p ∨ ~p’, p being a logical statement is given below -
|Truth Table (p ∨ ~p)|
|p||~p||p ∨ ~p|
The truth table for ‘p ∧ ~ p” which is a fallacy, p being a logical statement is given below -
|Truth Table (p ∧ ~p)|
|p||~p||p ∧ ~p|
Given below are some laws that are satisfied by statements -
Question 1: What is the contrapositive of the inverse of p ⇒ ~q?
Solution: The inverse of p ⇒ ∼q is ∼ p ⇒ q.
As the contrapositive of p ⇒ q is ∼ q ⇒ p,
Therefore, the contrapositive of ∼ p ⇒ q is ∼ q ⇒ p.
Question 2: Which of the given statements 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, they these are not similar
C) If two triangles are not identical, they are similar
D) If two triangles are not similar, they are identical
Solution: Look at the given statements -
p: Two triangles are identical.
q: Two triangles are similar.
Evidently, the given statement in representative form is p ⇒ q.
Thus, its contrapositive is expressed as ∼ q ⇒ ∼ p
∼p: two triangles are not identical.
∼q: two triangles are not similar.
Therefore, ~ q ⇒ ~ p: If two triangles are not similar, they are not identical.
Question 3: Suppose p is true and q is false, then which of the statements given below is not true?
A) p ∨ q
B) p ⇒ q
C) p ∧ ( ~q)
D) p ⇒ p
Solution: If p is true and q is false, then p∨q is true, q ⇒ p is true and p ∧ (∼q) is true.
Thus, both p and ∼q are true.
Here, p ⇒ q is not true as a true statement.
Question 4: What is ~( p ∨ q) ∨ (~p ∧ q) is equivalent to?
Solution: ∼ (p ∨ q) ∨ (∼p ∧ q) = (∼p ∧ ∼q) ∨ (∼p ∧ q) =∼q ∧ ∼(∼ (p ∧ q))
Question 5: Which of the options given below is logically equivalent to ~( ~ p ⇒ q)?
A) p ∧ q
B) p ∧ ~q
C) ~p ∧ q
D) ~p ∧ ~q
Solution: From the table given below, it is clear that ∼ (∼p ⇒ q) is equivalent to ∼p ∧ ∼q.
Question 6: Suppose (p ∧ ~r) ∧ ( ~p / q) is false, write the truth values of p, q and r.
Since (p ∧ ∼r) ⇒ (∼p ∨ q) is F.
Then, p = T, q = F, r = F.
Question 7: Suppose p and q are two statements, then what is (p ⇒ q) ⇔ ( ~q ⇒ ~p)?
Therefore, the (p ⇒ q) ⇔ ( ~q ⇒ ~p) is a tautology.
Question 8: Suppose each of the given 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: As ∼r is true, r is false. Also, q ⇒ r is true. Thus, q is false.
Hence, a true statement cannot indicate a false one.
p ⇒ q is also true.
Therefore, p is false.
Question 9: Find the negation of the compound proposition given below.
Suppose the examination is difficult, then I will pass if I study hard.
Solution: Let p: Examination is difficult
q: I shall pass
r: I study hard
Given result, P ⇒ (r ⇒ q)
Now, ∼ (r ⇒ q) = r ∧ ∼q ∼(p ⇒ (r ⇒ q)) = p ∧ (r ∧ ∼q)
The examination is difficult and I study hard but I will not pass.
Question 10: What is the statement p → (p → q) equivalent to?
Solution: p → (q → p) = −p (q → p) = ∼p ∨ (∼q ∨ p)
Since p ∨∼p is always true,
∼p ∨ p ∨ q = p → (p ∨ q)
Pro Tip - Use the elimination method whenever you feel stuck in a Mathematical reasoning problem.
Question 1: Take a look at the following statements -
P: Suman is brilliant
Q: Suman is rich
R: Suman is honest
How can the negation of the statement: Suman is brilliant and dishonest if and only if Suman is rich be expressed as?
Trick to solve this question: Express “Suman is brilliant and dishonest” first. You will get P∧∼R. Then, express “Suman is brilliant and dishonest if and only if Suman is rich” using Q. You will get Q ↔ (P ∧ ∼R). Write the expression of negation from this.
Solution: ∼(Q ↔ (P ∧ ∼R)
Question 2: The statement ~(p ↔ ~q)is ________.
Trick to solve this question: Use the truth table for easily identifying the answer. Find the nature of each element - p, q, and ~q. Then, find p ↔ ~q, ∼(p↔∼q), and p↔q.
Question 3: What is the Boolean Expression (p ∧ ~q) ∨ q ∨( ~p ∧ q) equivalent to?
Trick to solve this question: Write [(p ∧ ∼q) ∨ q] ∨ (∼p ∧ q) and find its equivalent. Continue shortening the equivalent until you arrive at the correct solution.
Solution: p ∨ q
1. Make your own formula book that you can refer to as often as possible.
2. Solve a minimum of 20 questions from Mathematics everyday.
3. Make use of logical thinking to solve JEE Main Mathematics questions.
4. Use NCERT, R. D Sharma, and Arihant books for learning and practice.
5. Solve as many mock tests and previous years’ questions as you can.
The article above comprises all important notes for JEE Main Mathematical Reasoning. Adoption of study tips will help in acing the topic.