The $pK_a$ of a weak acid is $4.8$. What should be the ratio of $\frac{[\text{acid}]}{[\text{salt}]}$,if a buffer of $pH=5.8$ is required?

The coefficient of $x^{24}$ in the expansion of $(1+x^2)^{12}(1+x^{12})(1+x^{24})$ is

$(x^2+y^2) dy = xy dx$. If $y(x_0) = e$ and $y(1) = 1$,then the value of $x_0$ is equal to

The random variable $X$ takes the values $1, 2, 3, \ldots, m$. If $P(X=n) = \frac{1}{m}$ for each $n$,then the variance of $X$ is

The pressure and density of a diatomic gas $\left(\gamma = \frac{7}{5}\right)$ change adiabatically from $(P, d)$ to $(P^{\prime}, d^{\prime})$. If $\frac{d^{\prime}}{d} = 32$,then $\frac{P^{\prime}}{P}$ is

In which of the following is incomplete double circulation observed?