# NCERT Class 11 Mathematics Solutions: Chapter 14 – Mathematical Reasoning Miscellaneous Exercise Part 3 (For CBSE, ICSE, IAS, NET, NRA 2022)

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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) .

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.

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.

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