Answer : Option A

Explanation :

LCM of 21, 36, 66 = 2772
ie, all multiples of 2772 are divisible by 21, 36 and 66
Prime factorization of 2772 is,
2772 = 2 × 2 × 3 × 3 × 7 × 11
ie, to make it a perfect square, we have to multiply it by 7 and 11
Hence, required number = 2772 × 7 × 11 = 213444

Answer : Option D

Explanation :

$MF#%\begin{align}&\dfrac{\sqrt{7}}{2} - \dfrac{10}{\sqrt{7}} + \sqrt{175}\\\\
&= \dfrac{\sqrt{7}}{2} - \dfrac{10}{\sqrt{7}} + \sqrt{7 \times 25}\\\\
&= \dfrac{\sqrt{7}}{2} - \dfrac{10}{\sqrt{7}} + 5\sqrt{7}\\\\
&=\dfrac{\left(\sqrt{7}\right)^2 - (2 \times 10) + (5\sqrt{7} \times 2\sqrt{7}) }{2\sqrt{7}}\\\\
&=\dfrac{7 - 20 + 70 }{2\sqrt{7}} = \dfrac{57}{2\sqrt{7}} \\\\&= \dfrac{28.5}{\sqrt{7}}
= \dfrac{28.5}{2.645} = \dfrac{28500}{2645} = 10.77\end{align}$MF#%

$MF#%\text{Please note that }\dfrac{57}{2\sqrt{7}}\text{ can be solved further in the below lines as well}
$MF#%

$MF#%\dfrac{57}{2\sqrt{7}} = \dfrac{57 \times \sqrt{7}}{2\sqrt{7} \times \sqrt{7}} = \dfrac{57\sqrt{7}}{14} \\\\
= \dfrac{57 \times 2.645}{14} = \dfrac{150.765}{14} = 10.77$MF#%

Given Expression = 25 x 14 x 11 = 5. 11 5 14

Answer : Option D

Explanation :

$MF#%\begin{align}&3\sqrt{5} + \sqrt{125}= 17.88\\\\
&\Rightarrow 3\sqrt{5} + \sqrt{25 \times 5}= 17.88\\\\
&\Rightarrow 3\sqrt{5} + 5\sqrt{5}= 17.88\\\\
&\Rightarrow 8\sqrt{5}= 17.88\\\\
&\Rightarrow \sqrt{5}= \dfrac{17.88}{8} = 2.235\\\\\\\\
&\sqrt{80} + 16\sqrt{5}= \sqrt{16 \times 5} + 16\sqrt{5} \\\\&= 4\sqrt{5} + 16\sqrt{5}= 20\sqrt{5} = 20 \times 2.235 = 44.7\end{align}$MF#%

Given exp. = √[(12.1 + 8.1)(12.1 - 8.1)/(0.25)(0.25 + 19.95)]
=√[(20.2 * 4) /( 0.25 * 20.2)]
=√4 / 0.25
= √400 / 25
=√16 = 4.

A number ending in 8 can never be a perfect square.

Answer : Option D

Explanation :

$MF#%\sqrt{5.4756} = 2.34$MF#%

L.C.M. of 21, 36, 66 = 2772.

Now, 2772 = 2 x 2 x 3 x 3 x 7 x 11

To make it a perfect square, it must be multiplied by 7 x 11.

So, required number = 2² x 3² x 7² x 11² = 213444

x = (3 + 1) x (3 + 1) = (3 + 1)² = 3 + 1 + 23 = 2 + 3. (3 - 1) (3 + 1) (3 - 1) 2

y = (3 - 1) x (3 - 1) = (3 - 1)² = 3 + 1 - 23 = 2 - 3. (3 + 1) (3 - 1) (3 - 1) 2

x² + y² = (2 + 3)² + (2 - 3)²

   = 2(4 + 3)

   = 14

Answer : Option D

Explanation :

$MF#%\sqrt{1\dfrac{9}{16}} = \sqrt{\dfrac{25}{16}} = \dfrac{\sqrt{25}}{\sqrt{16}} = \dfrac{5}{4} = 1\dfrac{1}{4}$MF#%