If $f(x)$ is a polynomial of degree $n$ with rational coefficients and $1+2i, 2-\sqrt{3}$ and $5$ are three roots of $f(x)=0$,then the least value of $n$ is

$\frac{1}{e^{3x}}(e^x + e^{5x}) = a_0 + a_1x + a_2x^2 + \ldots$
$\Rightarrow 2a_1 + 2^3a_3 + 2^5a_5 + \ldots$ is equal to

Two cylinders $A$ and $B$ fitted with pistons contain equal amounts of an ideal diatomic gas at $300\,K$. The piston of $A$ is free to move,while that of $B$ is held fixed. The same amount of heat is given to the gas in each cylinder. If the rise in temperature of the gas in $A$ is $30\,K$,then the rise in temperature of the gas in $B$ is .... $K$.

The percentage of gold in $14$ carat gold is: (in $\%$)

If $(1, a)$ and $(b, 2)$ are conjugate points with respect to the circle $x^2+y^2=25$,then $4a+2b$ is equal to

The total number of $\sigma$ and $\pi$-bonds in pyrophosphoric acid are respectively