# NCERT Class 11 Mathematics Solutions: Chapter 14 –Mathematical Reasoning Miscellaneous Exercise Part 3

1. Check the validity of the statements given below by the method given against it.

(i) The sum of an irrational number and a rational number is irrational (by contradiction method).

(ii) If n is a real number with (by contradiction method).

The given statement is as follows.

the sum of an irrational number and a rational number is irrational.

Let us assume that the given statement, , is false.

That is, we assume that the sum of an irrational number and a rational number is rational.

So, where is irrational and are integers.

is a rational number and is an irrational number.

So, our assumption is wrong.

So, the sum of an irrational number and a rational number is rational.

The given statement is true.

The given statement, , is as follows.

If is a real number with , then .

Consider us assume that n is a real number with is not true.

That is,

Then, and n is a real number.

Squaring both the sides, we obtain

, which is a contradiction, since we have assumed that .

The given statement is true.

If is a real number with .

2. Write the following statement in five different ways, conveying the same meaning.

If triangle is equiangular, then it is an obtuse angled triangle.