Between two fractions $\frac{1}{2}$ and $\frac{1}{8},$ how many fractions are there in all?

The value of $\frac{(81)^{3.6} \times (9)^{2.7}}{(81)^{4.2} \times 3}$ is

What is the value of $9 \frac{1}{3} + 19 \frac{2}{3} + 20 \frac{3}{4} + 19 \frac{1}{4}$?

Choose the incorrect relation$(s)$ from the following:
$(i)$ $\sqrt{6}+\sqrt{2}=\sqrt{5}+\sqrt{3}$
$(ii)$ $\sqrt{6}+\sqrt{2}<\sqrt{5}+\sqrt{3}$
$(iii)$ $\sqrt{6}+\sqrt{2}>\sqrt{5}+\sqrt{3}$

If the sum of squares of two real numbers is $41$ and their sum is $9,$ then the sum of cubes of these two numbers is

The value of $\left(2-\frac{1}{3}\right)\left(2-\frac{3}{5}\right)\left(2-\frac{5}{7}\right) \ldots \left(2-\frac{999}{1001}\right)$ is