In a certain culture of bacteria,the rate of increase is proportional to the number present. It is found that the number doubles in $4$ hours. Then the number of times the bacteria are increased in $12$ hours is

The number of solutions of $\frac{dy}{dx} = \frac{y+1}{x-1}$,when $y(1) = 2$ is

Let $f:[0,2] \rightarrow R$ be a function which is continuous on $[0,2]$ and is differentiable on $(0,2)$ with $f(0)=1$. Let $F(x)=\int_0^{x^2} f(\sqrt{t}) dt$ for $x \in [0,2]$. If $F'(x)=f'(x)$ for all $x \in (0,2)$,then $F(2)$ equals

The equation of the family of curves for which the length of the normal is equal to the radius vector is

Let a curve $y=f(x), x \in(0, \infty)$ pass through the points $P\left(1, \frac{3}{2}\right)$ and $Q\left(a, \frac{1}{2}\right)$. If the tangent at any point $R(b, f(b))$ to the given curve cuts the $y$-axis at the point $S(0, c)$ such that $bc=3$,then $(PQ)^2$ is equal to $.........$.

The rate of growth of bacteria in a culture is proportional to the number of bacteria present and the bacteria count is $1000$ at initial time $t = 0$. The number of bacteria is increased by $20\%$ in $2$ hours. If the population of bacteria is $2000$ after $\frac{k}{\log_{e}\left(\frac{6}{5}\right)}$ hours,then $\left(\frac{k}{\log_{e} 2}\right)^{2}$ is equal to