Answer: Option C. -> 11111
$$\therefore \sqrt {123454321} = 11111$$

Answer: Option E. -> None of these
$$\eqalign{
& \sqrt {\sqrt {17956} + \sqrt {24025} } \cr
& = \sqrt {134 + 155} \cr
& = \sqrt {289} \cr
& = 17 \cr} $$

Answer: Option A. -> 3
Sum of prime numbers greater than 4 but less than 16
$$\eqalign{
& = \left( {5 + 7 + 11 + 13} \right) \cr
& = 36 \cr
& \therefore \frac{1}{4} \times 36 \cr
& = 9 \cr
& = {3^2} \cr} $$

Answer: Option C. -> 6
The number of digits of the square root of a perfect square number of n digits is
$$\eqalign{
& {\text{(i)}}\frac{n}{2}{\text{, if n is even}} \cr
& {\text{(ii)}}\frac{{n + 1}}{2}{\text{, if n is odd}} \cr
& {\text{Here, }}n = 12 \cr
& {\text{So, required number of digits}} \cr
& = \frac{n}{2} \cr
& = \frac{{12}}{2} \cr
& = 6{\text{ }} \cr} $$

Answer: Option A. -> 12
$$\eqalign{
& {\text{Let,}} \cr
& {\text{ }}\frac{x}{{\sqrt {128} }} = \frac{{\sqrt {162} }}{x} \cr
& {\text{Then,}} \cr
& \Leftrightarrow {x^2} = \sqrt {128 \times 162} \cr
& \Leftrightarrow {x^2} = \sqrt {64 \times 2 \times 18 \times 9} \cr
& \Leftrightarrow {x^2} = \sqrt {{8^2} \times {6^2} \times {3^2}} \cr
& \Leftrightarrow {x^2} = 8 \times 6 \times 3 \cr
& \Leftrightarrow {x^2} = 144 \cr
& \Leftrightarrow x = \sqrt {144} \cr
& \Leftrightarrow x = 12 \cr} $$

Answer: Option E. -> None of these
$$\eqalign{
& {\text{Let }}\frac{x}{{1776}} = \frac{{111}}{x} \cr
& {\text{Then, }} \cr
& \Leftrightarrow {x^2} = 111 \times 1776 \cr
& \Leftrightarrow {x^2} = 111 \times 111 \times 16 \cr
& \Leftrightarrow x = \sqrt {{{\left( {111} \right)}^2} \times {{\left( 4 \right)}^2}} \cr
& \Leftrightarrow x = 111 \times 4 \cr
& \Leftrightarrow x = 444 \cr} $$

Answer: Option D. -> None of these
$$\eqalign{
& {\text{Let,}} \cr
& {\text{ }}\frac{{4\frac{1}{2}}}{x} = \frac{x}{{32}} \cr
& {\text{Then,}} \cr
& \Leftrightarrow {x^2} = 32 \times \frac{9}{2} \cr
& \Leftrightarrow {x^2} = 144 \cr
& \Leftrightarrow x = \sqrt {144} \cr
& \Leftrightarrow x = 12 \cr} $$

Answer: Option D. -> 68
$$\eqalign{
& \Leftrightarrow \frac{{52}}{x}{\text{ = }}\sqrt {\frac{{169}}{{289}}} \cr
& \Leftrightarrow \frac{{52}}{x} = \frac{{13}}{{17}} \cr
& \Leftrightarrow x = \left( {\frac{{52 \times 17}}{{13}}} \right) \cr
& \Leftrightarrow x = 68 \cr} $$

Answer: Option D. -> 45
$$\eqalign{
& {\text{Method 1:}} \cr
& {\text{Let the missing number be }}x \cr
& {\text{Then, }} \cr
& \Leftrightarrow {x^2} = 15 \times 135 \cr
& \Leftrightarrow x = \sqrt {15 \times \left(15 \times 9 \right)} \cr
& \Leftrightarrow x = \sqrt {{{15}^2} \times {3^2}} \cr
& \Leftrightarrow x = 15 \times 3 \cr
& \Leftrightarrow x = 45 \cr
& \cr
& {\text{Method 2:}} \cr
& {\text{Let the missing number be }}x \cr
& {\text{Then, }} \cr
& \Leftrightarrow {x^2} = 15 \times 135 \cr
& \Leftrightarrow {x^2} = 2025 \cr
& \Leftrightarrow x = \sqrt {{2025} } \cr
& \Leftrightarrow x = 45 \cr} $$

Answer: Option B. -> 15
$$\eqalign{
& {\text{Given expression,}} \cr
& = \sqrt {176 + 49} \cr
& = \sqrt {225} \cr
& = 15 \cr} $$