Two ions having masses in the ratio $1:1$ and charges in the ratio $1:2$ are projected into a uniform magnetic field perpendicular to the field with speeds in the ratio $2:3$. The ratio of the radii of the circular paths along which the two particles move is

Find the equivalent capacitance across $A$ and $B$ in $\mu F$.

If $I_n = \int x^n \cdot e^{cx} \, dx$ for $n \geq 1$,then $c \cdot I_n + n \cdot I_{n-1}$ is equal to

$1+\frac{2}{4}+\frac{2 \cdot 5}{4 \cdot 8}+\frac{2 \cdot 5 \cdot 8}{4 \cdot 8 \cdot 12}+\frac{2 \cdot 5 \cdot 8 \cdot 11}{4 \cdot 8 \cdot 12 \cdot 16}+\ldots \ldots$ is equal to :

An electron revolves in a circle of radius $0.4 \text{ \AA}$ with a speed of $10^6 \text{ m/s}$ in a hydrogen atom. The magnetic field produced at the centre of the orbit due to the motion of the electron (in Tesla) is: $\left[\mu_0 = 4\pi \times 10^{-7} \text{ H/m}, e = 1.6 \times 10^{-19} \text{ C}\right]$

The length $x$ of a rectangle is decreasing at the rate of $3 \text{ cm/min}$ and the width $y$ is increasing at the rate of $2 \text{ cm/min}$. When $x = 10 \text{ cm}$ and $y = 6 \text{ cm}$,find the rate of change of the perimeter.