The solution set of the system of equations $x + y = \frac{2\pi}{3}$ and $\cos x + \cos y = \frac{3}{2}$,where $x$ and $y$ are real,is:

Leela and Madan pooled their music $CDs$ and sold them. They received as many rupees for each $CD$ as the total number of $CDs$ they sold. They shared the money as follows: Leela first takes $10$ rupees,then Madan takes $10$ rupees and they continue taking $10$ rupees alternately until Madan is left with less than $10$ rupees. Find the amount that is left for Madan at the end,with justification.

There are $100$ students in a class. In an examination,$50$ of them failed in Mathematics,$45$ failed in Physics,$40$ failed in Biology and $32$ failed in exactly two of the three subjects. Only one student passed in all the subjects. Then,the number of students failing in all the three subjects is:

There are $200$ individuals with a skin disorder. $120$ had been exposed to the chemical $C_{1}$,$50$ to chemical $C_{2}$,and $30$ to both the chemicals $C_{1}$ and $C_{2}$. Find the number of individuals exposed to chemical $C_{1}$ but not chemical $C_{2}$.

The number of elements in the set $\{n \in \{1, 2, 3, \ldots, 100\} \mid (11)^{n} > (10)^{n} + (9)^{n}\}$ is $.....$

The probability of failing in Physics is $20\%$,and the probability of failing in Mathematics is $10\%$. What is the probability of failing in at least one subject (in $\%$)?