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

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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).

Answer: (i)

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.

This is a contradiction.

So, our assumption is wrong.

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

The given statement is true.

Answer: (ii)

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.

Answer:

The given statement can be written in five different ways as follows.

(i) A triangle is equiangular implies that it is an obtuse-angled triangle.

(ii) A triangle is equiangular only if it is an obtuse-angled triangle.

(iii) For a triangle to be equiangular, it is necessary that the triangle is an obtuse-angled triangle.

(iv) For a triangle to be an obtuse-angled triangle, it is sufficient that the triangle is equiangular.

(v) If a triangle is not an obtuse-angled triangle, then the triangle is not equiangular.