# NCERT Class 11-Math՚s: Chapter – 14 Mathematical Reasoning Part 11 (For CBSE, ICSE, IAS, NET, NRA 2023)

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**Question 12**:

Prove by direct method that for any integer , is always even.

[Hint: Two cases is even, is odd.]

**Answer**:

We have given,

Let us Assume, is even

Let , where k is natural number

Therefore, is even.

Now, Let us Assume n is odd

Let , where k is natural number

Therefore, is even.

Hence, is always even

**Question 13**:

Check the validity of the following statement.

(i) is divisible by and .

(ii) is a multiple of or .

**Answer**:

(i) p: 125 is divisible by 5 and 7

Let,

is divisible by .

is divisible .

Here, q is true and r is false.

Therefore, is False

Hence, is not valid.

(ii) is a multiple of or

Let,

is a multiple of .

is a multiple of .

Here, is false and is False

Therefore, is False

Hence, is not valid

**Question 14**:

Prove the following statement by contradiction method.

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

**Answer**:

Let p is false, as the sum of an irrational number and a rational number is irrational.

Let is irrational and n is rational number

But, we know that is irrational where as is rational which is contradiction.

Here, Our Assumption is False

Hence, P is true.

**Question 15**:

Prove by direct method that for any real numbers , if , then .

**Answer**:

We have Given for any real number if

To Find:

Explanation: Let us Assume

where x and y are real number

On squaring both sides we get

(Assumption)

Therefore,

Hence, Proved