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Which of the following sentences are statements?

Question:

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 $x^{2}+5|x|+6=0$ has no real roots.

(iv) $(2+\sqrt{3})$ 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.

 

 

Solution:

(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 $x^{2}+5|x|+6=0$ has no real roots.' Is a statement. $x^{2}+$ $5|x|+6=0$ do not have real roots.

Case 1: $(x \geq 0)$

$|x|=x:(x \geq 0)$

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

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

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

$x=-2$ and $x=-3$

But we assumed $x \geq 0$. So it is a contradiction.

Case $2:(x<0)$

$|x|=x:(x<0)$

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

$x^{2}-5 x+6=0$

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

$x=2$ and $x=3$

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

So, there are no real roots for the equation $x^{2}+5|x|+6=0$

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 ' $(2+\sqrt{3})$ is a complex number' is a statement.

A number which can be expressed in the form ' $a+i b$ ' is a complex number, $(2+\sqrt{3})$ cannot be expressed in ' $\mathrm{a}+\mathrm{ib}$ ' form, so $2+\sqrt{3}$ 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.

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Note: A sentence is called a mathematically acceptable statement if it is either true or ings to 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.

 

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