The largest four digit number which is a perfect cube, is:

Answer: Option A

Answer: Option D

Explanation:

$$\begin{aligned} = \sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+\sqrt{225}}}}} \end{aligned}$$

$$\begin{aligned} =\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+15}}}} \end{aligned}$$

$$\begin{aligned} =\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{154+15}}}} \end{aligned}$$

$$\begin{aligned} =\sqrt{10+\sqrt{25+\sqrt{108+\sqrt{169}}}} \end{aligned}$$

$$\begin{aligned} =\sqrt{10+\sqrt{25+\sqrt{108+13}}} \end{aligned}$$

$$\begin{aligned} =\sqrt{10+\sqrt{25+\sqrt{121}}} \end{aligned}$$

$$\begin{aligned} =\sqrt{10+\sqrt{25+11}} \end{aligned}$$

$$\begin{aligned} =\sqrt{10+\sqrt{36}} \end{aligned}$$

$$\begin{aligned} =\sqrt{10+6} \end{aligned}$$

$$\begin{aligned} =\sqrt{16} = 4 \end{aligned}$$

Answer: Option B

Explanation:

Very obvious tip here is, after squre root the terms after decimal will be half (that is just a trick), works awesome at many questions like this.

Answer: Option D

Explanation:

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 = 2 x 2 x 3 x 3 x 7 x 7 x 11 x 11 = 213444

Answer: Option B