# Chapter 31: Mathematical Reasoning  – Exercise 31.2

## Mathematical Reasoning – Exercise – 31.2 – Q.1

The negation of the statement:

Banglore is the capital of Karnataka.

is

Banglore is not the capital of Karnataka.

The negation of the statement:

It rained on July 4,2005.

is

It did not rain on July 4, 2005.

The negation of the statement:

Ravish is honest.

is

Ravish is not honest.

The negation of the statement:

The earth is round.

is

The earth is not round.

The negation of the statement:

The sun is cold.

is

The sun is not cold.

## Mathematical Reasoning – Exercise – 31.2 – Q.2

The negation of the statements:

All birds sing.

is

Not all birds sing.

The negation of the statements:

Some even integers are prime.

is

No even integers is prime.

The negation of the statements:

There is a complex number which is not a real number.

is

All complex numbers are real numbers.

The negation of the statements:

I will not go to school.

is

I will go to school.

The negation of the statements:

Both the diagonals of a rectangle have the same length.

is

There is at least one rectangle whose both diagonals do not have the same length.

The negation of the statements:

All policemen are thieves.

is

No policemen is thief.

## Mathematical Reasoning – Exercise – 31.2 – Q.3

(i) The number x is not a rational number.

⇒ The number x is an irrational number.

∴ The statement “The number x is not an irrational number." is a negation of the first statement.

(ii) The number x is not a rational number.

⇒ The number x is an irrational number.

The statement "The number x is an irrational number” is not a negation of the first statement.

## Mathematical Reasoning – Exercise – 31.2 – Q.4

(i)

The negation of the statement:

p: For every positive real number x, the number (x -1) is also positive.

is

~p: There exists a positive real number x such that the number (x - 1) is not positive.

(ii)

The negation of the statement:

q: For every real number x, either x > 1 or x < 1.

is

~q: There exists a real number such that neither x > 1 or x <1.

(iii)

The negation of the statement:

r: There are exists a number x such that 0 < x <1.

is

~r: For every real number x, either x ≤ or x ≥ al.

## Mathematical Reasoning – Exercise – 31.2 – Q.5

The negation of the statement:

a + b = b + a is true for every real number a and b.

is

There exist real numbers a and b for which a + b ≠ b + a.

So, the given statement is not the negation of the first statement.

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