If $^nC_r$ denotes the number of combinations of $n$ things taken $r$ at a time,then the expression $^nC_{r+1} + ^nC_{r-1} + 2 \times ^nC_r$ equals

At $80\,^oC$,the vapour pressure of pure liquid $A$ is $520\,mmHg$ and that of pure liquid $B$ is $1000\,mmHg$. If a mixture of $A$ and $B$ boils at $80\,^oC$ and $1\,atm$ pressure,the amount of $A$ in the mixture is $........$ $mol\,\%$.

$\int_0^2 \frac{2x-2}{2x-x^2} dx$ is equal to

If $\int_{-1}^{4} f(x) \, dx = 4$ and $\int_{2}^{4} (3 - f(x)) \, dx = 7$,then the value of $\int_{2}^{-1} f(x) \, dx$ is

$A$ coin is tossed $n$ times. The probability of getting a head at least once is greater than $0.8$. Then,the least value of such $n$ is:

If the foot of the perpendicular from $(0,0,0)$ to a plane is $(1,2,3)$,then the equation of the plane is