##
**
****MATHEMATICS REASONING QUIZ-9**Dear Readers,

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.

**MATHEMATICS REASONING QUIZ-9**

**Q1.**If p= He is intelligent q=He is strong Then, symbolic form of statement "It is wrong that he is intelligent or strong," is

Solution

The symbolic form of given statement is ∼(p∨q)

The symbolic form of given statement is ∼(p∨q)

**Q2.**Which of the following is false?

Solution

Clearly, p∨∼pis always true. So, it is a tautology We have, ∼(∼p)↔p≅p↔p So, ∼(∼p)↔p is always true. So, it is a tautology We know that p→q≅∼p∨q ∴p∧(p→q)≅p∧(∼p∨q)≅(p∧∼p)∨(p∧q) ≅c∨(p∧q)≅p∧q ∴p∧(p→q)→p≅p∧q→p which is a tautology So, option (c) is false

Clearly, p∨∼pis always true. So, it is a tautology We have, ∼(∼p)↔p≅p↔p So, ∼(∼p)↔p is always true. So, it is a tautology We know that p→q≅∼p∨q ∴p∧(p→q)≅p∧(∼p∨q)≅(p∧∼p)∨(p∧q) ≅c∨(p∧q)≅p∧q ∴p∧(p→q)→p≅p∧q→p which is a tautology So, option (c) is false

**Q4.**Which of the following is logically equivalent to ∼(p↔q)?

Solution

We have, p↔q≅(p→q)∧(q→p) ≅(∼p∨q)∧(∼q∨p) ∴∼(p↔q)≅∼(∼p∨q)∨∼(∼q∨p) ≅(p∧∼q)∨(q∧∼p)

We have, p↔q≅(p→q)∧(q→p) ≅(∼p∨q)∧(∼q∨p) ∴∼(p↔q)≅∼(∼p∨q)∨∼(∼q∨p) ≅(p∧∼q)∨(q∧∼p)

**Q5.**If p→(q∨r) is false, then the truth values of p,q,r are respectively

**Q6.**The logically equivalent proposition of p→q is

**Q7.**Which of the following is logically equivalent to p∧q?

Solution

We have, p→q≅~p∨q ∴p→∼q≅∼p∨∼q≅∼(p∧q) So, option (a) is not correct ∼p∨∼q=∼(p∧q) So, option (b) is not correct ∼(p→∼q)=∼(∼p∨∼q)=p∧q So, option (c) is incorrect

We have, p→q≅~p∨q ∴p→∼q≅∼p∨∼q≅∼(p∧q) So, option (a) is not correct ∼p∨∼q=∼(p∧q) So, option (b) is not correct ∼(p→∼q)=∼(∼p∨∼q)=p∧q So, option (c) is incorrect

**Q8.**Dual of (x

^{'}∧y

^{'})

^{'}=x∨y is

ax+by+bz=0

bx+ay+bz=0

bx+by+az=0

Will have a non-trivial solution, is