**Chapter 31: Mathematical Reasoning – Exercise 31.1**

**Mathematical Reasoning – Exercise – 31.1 – Q.1(i)**

**Ans.**

A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.

The sentence "Listen to me, Ravi!" is an exclamatory sentence, so, it is not a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(ii)**

**Ans.**

A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.

This sentence is always false, because there are sets which are not finite. Hence, it is a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(iii)**

**Ans.**

A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.

This sentence is always false, because there are non-empty sets whose intersection is empty. Hence, it is a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(iv)**

**Ans.**

Some cats are black and some not. So, the given sentence may or may not be true. Hence, it is not a statement

**Mathematical Reasoning – Exercise – 31.1 – Q.1(v)**

**Ans.**

The sentence "Are all circles round?" is an interrogative sentence. So, it is not a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(vi)**

**Ans.**

The All triangles have three sides." is a true declarative sentence. So, it is a true statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(vii)**

**Ans.**

This sentence is always false, because there are rhombuses that are not squares. Hence, it is a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(viii)**

**Ans.**

If x > 0,

x^{2} + 5 |x| + 6 = 0

⇒ x^{2} + 5x + 6 = 0

⇒ x = - 3 or x = - 2

But, since x > 0, the equation has no roots.

If x < 0,

x^{2} + 5 |x| + 6 = 0

⇒ x^{2 }- 5x + 6 = 0

Which has no real roots.

So, the sentence x^{2} + 5x + 6 = 0 is always true

Hence, it is a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(ix)**

**Ans.**

It is not a statement.

The sentence "This sentence is a statement: cannot be assigned a truth value of either true or false. Because either assignment contradicts the sense of the sentence.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(x)**

**Ans.**

The sentence “Is the earth round?” is an interrogative sentence. So, it is not a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(xi)**

**Ans.**

The sentence "Go!" is an exclamatory sentence. So, it is not a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(xii)**

**Ans.**

“The real number x is less than 2" is not a statement, because its truth or falsity cannot be confirmed without knowing the value of x.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(xiii)**

**Ans.**

'There are 35 days in a month” is a false declarative sentence. So, it is a false statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(xiv)**

**Ans.**

“Mathematics is difficult” is true for those who may not like mathematics. But, for others, it may not be true. So, the given sentence may or may not be true. Hence, it is not a statement

**Mathematical Reasoning – Exercise – 31.1 – Q.1(xv)**

**Ans.**

This sentence is always true. Hence, it is a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.1(xvi)**

**Ans.**

This sentence is always false, because (-1 x 8 - 8). Hence, it is a statement.

**Mathematical Reasoning – Exercise – 31.1 – Q.2**

**Ans.**

Example (1):

“Who lost this watch?" is an interrogative sentence.

Hence, it is not a statement.

Example (2):

The sentence: x + 2. 9 is an open sentence. Its truth value cannot be confirmed unless we are given the value of x. So, it is not a statement.

Example (3): The sentence "May god bless you” is an exclamatory sentence. So, it is not a statement.