Mathematics MCQ on Mathematical Reasoning for JEE and Engineering Exam 2022
MCQ on Mathematical Reasoning: To be an expert in JEE Mathematics, it is absolutely necessary to practice and be familiar will all the concepts as well as the questions of different types. This is essential to gain mastery over the subject. We have also often heard the common saying, “Practice Makes a Man Perfect”, hence students have to practice, practice and practice till they master the subject.
In this post we are providing you MCQ on Mathematical Reasoning, which will be beneficial for you in upcoming JEE and Engineering Exams.
MCQ on Mathematical Reasoning
Q1. 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) p ⇒ p
When p is true and q is false, then p∨q is true, q ⇒ p is true and p ∧ (∼q) is true. (Therefore, both p and ∼q are true)
Here, p ⇒ q is not true as a true statement cannot imply a false statement.
Q2. 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
Since ∼r is true, therefore, r is false. Also, q ⇒ r is true, therefore, q is false. (Therefore, a true statement cannot imply a false one) Also, p ⇒ q is true, therefore, p must be false.
Q3. The statement ~(p ↔ ~q)is ________.
p | q | ∼q | p↔∼q | ∼(p↔∼q) | p↔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 |
Q4. 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 ⇒ q.
Therefore, its contrapositive is given by ∼ q ⇒ ∼ p
Now,
∼p: two triangles are not identical.
∼q: two triangles are not similar.
Therefore, ~ q ⇒ ~ p: If two triangles are not similar, then these are not identical.
Q5. Consider the following statement
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
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).
Q6. What is the negation of the compound proposition? If the examination is difficult, then I shall pass if I study hard.
If p: Examination is difficult
q: I shall pass
r: I study hard
Given result is P ⇒ (r ⇒ q)
Now, ∼ (r ⇒ q) = r ∧ ∼q ∼(p ⇒ (r ⇒ q)) = p ∧ (r ∧ ∼q)
The examination is difficult and I study hard but I shall not pass.
Q7. The Boolean Expression (p ∧ ~q) ∨ q ∨( ~p ∧ q) is equivalent to ___________.
a) p ∧ q
b) p = q
c) p ∨ q
d) None of these
[(p ∧ ∼q) ∨ q] ∨ (∼p ∧ q) = (p ∨ q) ∧ (∼q ∨ q) ∨ (∼p ∧ q)
= (p ∨ q) ∧ [t ∨ (∼p ∧ q)]
= (p ∨ q) ∧ t
= p ∨ q
Q8. The contrapositive of the inverse of p ⇒ ~q is ________.
The inverse of p ⇒ ∼q is ∼ p ⇒ q
The contrapositive of ∼ p ⇒ q is ∼ q ⇒ p. [Because the contrapositive of p ⇒ q is ∼ q ⇒ p.]
Q9. If p and q are two statements, then (p ⇒ q) ⇔ ( ~q ⇒ ~p) is a ______.
p⇒q | ∼p⇒∼q | p⇒q⇔∼q⇒∼p |
T | T | T |
F | F | T |
T | T | T |
T | T | T |
Therefore, it is a tautology. Hence, the given proposition is a tautology
Q10. The statement p → (p → q) is equivalent to ________.
p → (q → p) = −p (q → p) = ∼p ∨ (∼q ∨ p)
(Since p ∨∼p is always true) = ∼p ∨ p ∨ q = p → (p ∨ q)
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