Question

what is the a answer of this MCQ ​

Answer 1

Answer:

Yes, 2 is a prime number.

According to the definition of prime numbers, any whole number which has only 2 factors is known as a prime number. Now, the factors of 2 are 1 and 2. Since there are exactly two factors of 2, it is a prime number.

Answer 2

✪ $\sqrt{2}$ is a/an number:-

$\star$ Even

$\star$ Prime

$\star$ Rational

$\star$ Irrational $\huge\bold\orange{✓}$

$\:$

$\huge\bold\pink{Step\: by\:Step:-}$

Given:-

$\sqrt{2}$

To prove:-

$\sqrt{2}$ is an irrational number.

Proof:-

Let us assume that $\sqrt{2}$ is a rational number.

So it can be expressed in the form $\dfrac{p}{q}$ where p, q are co-prime integers and q≠0

$\sqrt{2}$ = $\dfrac{p}{q}$

Here p and q are coprime numbers and q ≠ 0

Solving:-

$\sqrt{2}$ = $\dfrac{p}{q}$

On squaring both the side we get,

$\implies$ 2 = $(\dfrac{p}{q}) {}^{2}$

$\implies$2q² = p²……………………………..(1)

$\dfrac{{p}^{2} }{2}$ = q²

So 2 divides p and p is a multiple of 2.

⇒ p = 2m

⇒ p² = 4m² ………………………………..(2)

From equations (1) and (2), we get,

2q² = 4m²

⇒ q² = 2m²

⇒ q² is a multiple of 2

⇒ q is a multiple of 2

Hence, p, q have a common factor 2. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

$\therefore$ $\sqrt{2}$ is an irrational number.