## MATHEMATICS REASONING QUIZ-1

As per analysis for previous years, it has been observed that students preparing for JEE MAINS find Mathematics out of all the sections to be complex to handle and the majority of them are not able to comprehend the reason behind it. This problem arises especially because these aspirants appearing for the examination are more inclined to have a keen interest in Mathematics due to their ENGINEERING background.

Furthermore, sections such as Mathematics are dominantly based on theories, laws, numerical in comparison to a section of Engineering which is more of fact-based, Physics, and includes substantial explanations. By using the table given below, you easily and directly access to the topics and respective links of MCQs. Moreover, to make learning smooth and efficient, all the questions come with their supportive solutions to make utilization of time even more productive. Students will be covered for all their studies as the topics are available from basics to even the most advanced.

Q1. The contrapositive of "If two triangles are identical, then these are similar" is
•  If two triangles are not similar, then these are not identical
•  If two triangles are not identical, then these are not similar
•  If two triangles are not identical, then these are similar
•  if two triangles are not similar, then these are identical
Solution
Let p: Two triangles are identical q: Two triangles are similar Clearly, the given statement in symbolic form is p→q. ∴ Its contrapositive is given by ~q→~p. ie, If two triangles are not similar, then these are not identical.

Q2.The proposition p→∼(p∧∼q) is :
•  The proposition p→∼(p∧∼q) is
•  A tautology
•  Either a tautology or a contradiction
•  Neither a tautology nor a contradiction
Solution
p→∼(p∧∼q) ≅∼p∨∼(p∧∼q)≅∼p∨(∼p∨q)≅∼p∨q Clearly, it is neither a tautology nor a contradiction

Q3.  Which of the following is a proposition?
•   I am a lion
•  A half open door is half closed
•  A triangle is a circle and 10 is a prime number
•  Logic is an interesting subject
Solution
Clearly, statement in option (c) is false. So, it has definite truth value. Hence, it is a proposition

Q4. Consider the following statements : p∶I shall pass,q∶I study The symbolic representation of the proposition “I shall pass iff I study” is
•  p→q
•  q→p
•  p→~q
•  p↔q
Solution

Q5.If p:4 an even prime number q:6 is a divisor of 12 and r:the HCF of 4 and 6 is 2, then which one of the following is true?
•  (p∨q)
•  (p∨q)∧~r
•
d~(q∧r)∨p
•  ~p∨(q∧r)
Solution
∴p:4is an even prime number. q:6 is a divisor of 12 And r: the HCF of 4 and 6 is 2 ∴ ∼p∨(q∧r)is true.

Q6. Which of the following is logically equivalent to ~(~p→q)?
•  p∧q
•  p∧∼q
•  ∼p∧q
•  ∼p∧∼q
Solution
We have, p→q≅∼p∨q ∴∼(∼p→q)≅∼(p∨q)≅∼p∧∼q

Q7.The negation of the compound proposition p∨(∼p∨q) is
•  (p∧∼q)∧∼p
•  (p∧∼q)∨∼p
•  (p∧∼q)∨∼p
•  None of these
Solution
We have, ∼{p∨(∼p∨q)} ≅∼{(p∨∼p)∨q}≅∼(t∨q)≅∼t≅c Also, (p∧∼q)∧∼p≅(p∧∼p∧∼q)≅c∧∼q≅c So, option (a) is correct

Q8.A compound sentence formed by two simple statements p and q using connective 'or' is called
•  Conjunction
•  Disjunction
•  Implication
•  None of these
Solution
A compound sentence formed by two simple statements p and q using connective 'or' is called disjunction

Q9.The logically equivalent proposition of p⇔q is
•  (p∧q)∨(p∧q)
•  (p⇒q)∧(q⇒p)
•  (p∧q)∨(q⇒p)
•  (p∧q)⇒(p∨q)
Solution
(p⇒q)∧(q⇒p) means p⇔q

Q10. If truth value of p∨q is true, then truth value of ∼p∧q is
•  False if p is true
•  True if p is true
•  False if q is true
•  True if q is true
Solution

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