Which of the following sentences are statements? In case of a statement, mention whether it is true or false.

(i) Paris is in France.

(ii) Each prime number has exactly two factors.

(iii) The equation x² + 5|x| + 6 = 0 has no real roots.

(iv) (2 + √3) is a complex number.

(v) Is 6 a positive integer?

(vi) The product of -3 and -2 is -6.

(vii) The angles opposite the equal sides of an isosceles triangle are equal.

(viii) Oh! It is too hot.

(ix) Monika is a beautiful girl.

(x) Every quadratic equation has at least one real root.

(i) The sentence ‘Paris is in France’ is a statement. Paris is located in France, so the sentence given is true, so it is a statement. The statement is true.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(ii) The sentence ‘Each prime number has exactly two factors’ is a statement. It is a mathematically proven fact that each prime number has exactly two factors, so the given sentence is true. Hence it is a statement. The statement is true.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(iii) The sentence ‘The equation x² + 5|x| + 6 = 0 has no real roots.’ Is a statement. x² + 5|x| + 6 = 0 do not have real roots.

Case 1: (x ≥ 0)

|x| = x: (x ≥ 0)

x² + 5|x| + 6 = 0

x² + 5x + 6 = 0

(x + 2) (x + 3) = 0

x = -2 and x = -3

But we assumed x ≥ 0. So it is a contradiction.

Case 2: (x <0)

|x| = x: (x <0)

x² + 5|x| + 6 = 0

x² - 5x + 6 = 0

(x - 2) (x - 3) = 0

x = 2 and x = 3

But we assumed x < 0. So it is a contradiction.

So, there are no real roots for the equation x² + 5|x| + 6 = 0

So, the given sentence is true, and it is a statement.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(iv) The sentence ‘(2 + √3) is a complex number’ is a statement.

A number which can be expressed in the form ‘a+ib’ is a complex number, (2 + √3) cannot be expressed in ‘a+ib’ form, so 2 + √3 is not a complex number. So the given sentence is a statement, and it is false.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(v) The sentence ‘Is 6 a positive integer?’ is a question, so it is not a statement.

Note: A sentence which is in the form of an order, exclamation and question is not a statement.

(vi) The sentence ‘The product of -3 and -2 is -6’ is a statement.

Because, the product of -3 and -2 is 6 not -6, the given sentence is false. Hence the given sentence is a statement. This statement is false.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(vii) The sentence given is a statement. It is mathematically proven that the angles opposite to the equal sides of an isosceles triangle are equal. So the given sentence is true, and it is a statement.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(viii) The sentence ‘Oh! It is too hot’ is not a statement. It is an exclamation, and hot is subjective, it is not a fact, and it is an opinion. So, the given sentence is not a statement.

Note: A sentence which is in the form of an order, exclamation and question is not a statement.

(ix) The sentence ‘Monica is a beautiful girl’ is not a statement. The given sentence is an opinion; this can be true for some cases, false for some other case. So, the given sentence is not a statement.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.

(x) The given sentence is a statement.

Because not every quadratic equation will have a real root. So the given sentence is false. It is a statement. This statement is false.

Note: A sentence is called a mathematically acceptable statement if it is either true or false but not both.