# NCERT Class 11-Math՚s: Chapter – 14 Mathematical Reasoning Part 7 (For CBSE, ICSE, IAS, NET, NRA 2022)

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**Question 14**:

Show that the following statement is true.

For any real numbers if , then when .

**Answer**:

We prove the statement ‘*p*’ is true by contrapositive method and by Direct Method.

Direct Method for any real number given

for some .

Contrapositive Method The contrapositive statement of ‘’ is ″ For any real numbers if , where , then .

Given

**Question 15**:

Check the validity of the statements

(i) is a multiple of and .

(ii) is a multiple of or .

**Answer**:

(i) Let

Where is a multiple of ″ is true

is a multiple of ″ is true

Hence *p* is true.

(ii) Let , where

is a multiple of ″ , is true.

is a multiple of ″ , is true.

Hence *q* is true.

## Objective Type Questions

### Choose the Correct Answer Out of the Four Options Given Against Each of the Examples 16 to 18 (M. C. Q.)

**Question 16**:

Which of the following is a statement?

(A) Roses are black.

(B) Mind your own business.

(C) Be punctual.

(D) Do not tell lies

**Answer**:

(A) is the correct answer as the sentences in (B) , (C) and (D) are neither true nor false. Infect all these sentences are advices.

**Question 17**:

The negation of the statement

“It is raining and weather is cold.” is

(A) It is not raining and weather is cold.

(B) It is raining or weather is not cold.

(C) It is not raining or weather is not cold.

(D) It is not raining and weather is not cold.

**Answer**:

(C) is the correct answer as it satisfies

The options (A) , (B) and (D) do not satisfy .

**Question 18**:

Which of the following is the converse of the statement?

“If Billu secure good marks, then he will get a bicycle.”

(A) If Billu will not get bicycle, then he will not secure good marks.

(B) If Billu will get a bicycle, then he will secure good marks.

(C) If Billu will get a bicycle, then he will not secure good marks.

(D) If Billu will not get a bicycle, then he will secure good marks.

**Answer**:

(B) is the correct answer since the statement is the converse of the statement .

## 14.3 EXERCISE

### Short Answer Type

**Question 1**:

Which of the following sentences are statements? Justify

(i) A triangle has three sides

(ii) is a complex number

(iii) Sky is red

(iv) Every set is an infinite set

(v)

(vi)

(vii) Where is your bag?

(viii) Every square is a rectangle

(ix) Sum of opposite angles of a cyclic quadrilateral is

(x)

**Answer**:

(i) to (v) and (viii) to (x) are statements.