The bacteria increases at a rate proportional to the number of bacteria present. If the original number $N$ doubles in $4$ hours,then the number of bacteria in $12$ hours will be: (in $N$)

The equation of a curve whose normal at any point has a slope which is the same as the ordinate and which passes through $(1, -1)$ is $2x = k(3 - y^2)$. Then $k$ is:

At any point $(x, y)$ on a curve,if the length of the subnormal is $(x - 1)$ and the curve passes through $(1, 2)$,then the curve is a conic. $A$ vertex of the curve is:

Let $y = y(x)$ be the solution of the differential equation $x dy - y dx = \sqrt{x^2 - y^2} dx$,$x \geq 1$,with $y(1) = 0$. If the area bounded by the lines $x = 1$,$x = e^{\pi}$,$y = 0$ and the curve $y = y(x)$ is $\alpha e^{2\pi} + \beta$,then the value of $10(\alpha + \beta)$ is equal to ....... .

$A$ curve is such that the area of the region bounded by the coordinate axes,the curve,and the ordinate of any point on it is equal to the cube of that ordinate. The curve represents:

The principal increases continuously in a newly opened bank at the rate of $10 \%$ per year. An amount of Rs. $2000$ is deposited with this bank. How much will it become after $5$ years? $(e^{0.5} = 1.648)$