Question

What will be the answer of this question with solution ? please tell me ​

Answer 1

Answer:

(d) none of these

Step-by-step explanation:

Answer 2

$\Huge\underline{\text{Proper question}}$

An irrational number between 5 and 6 is

(A) $\dfrac{1}{2}(5+6)$

(B) $\sqrt{5+6}$

(C) $\sqrt{5\times6}$

(D) none of these

$\Huge\underline{\text{Idea}}$

$\red{\bigstar}\text{Rational numbers}$

We can write these numbers in a ratio of two numbers.

$\red{\bigstar}\text{Irrational numbers}$

Irrational numbers are real and not rational. So, they can't be written in $\dfrac{a}{b}$ form, where $b$ is nonzero, while $a$ and $b$ are integers.

If a square root of a perfect square is rational. Else, the square root is an irrational value.

First, let's see what numbers are irrational.

$\dfrac{1}{2}(5+6)$ is a rational number, because $\dfrac{11}{2}$ is rational. (Reject)

$\sqrt{5+6}$ is an irrational number, because $\sqrt{11}$ is irrational.

$\sqrt{5\times6}$ is an irrational number, because $\sqrt{30}$ is irrational.

Now let's see which number is between 5 and 6.

The main idea used here is that the squares of positive numbers are always increasing.

Let's take an example.

$1<2<3<4<\cdots$

$1^{2}<2^{2}<3^{2}<4^{2}<\cdots$

If we compare the radicand, we can compare the square roots, because they are always increasing.

Let's compare them.

$9<11<16$

$3<\sqrt{11}<4$

Hence, $\sqrt{11}$ is not between 5 and 6. (Reject)

$25<30<36$

$5<\sqrt{30}<6$

So, $\sqrt{30}$ is between 5 and 6.

$\Huge\underline{\text{Answer}}$

The right option is (c).