$A$ point is moving on $y=4-2x^2$. The $x$-coordinate of the point is decreasing at the rate of $5 \text{ units/s}$. Then,the rate at which the $y$-coordinate of the point is changing when the point is at $(1,2)$ is

If the density of the earth is doubled keeping its radius constant,then acceleration due to gravity will be ........ $m/s^2$. $(g = 9.8\,m/s^2)$

If the mean of a Poisson distribution is $\frac{1}{2}$,then the ratio of $P(X=3)$ to $P(X=2)$ is

Two slabs $A$ and $B$ of different materials but of the same thickness are joined end to end to form a composite slab. The thermal conductivities of $A$ and $B$ are $k_1$ and $k_2$ respectively. $A$ steady temperature difference of $12^{\circ} C$ is maintained across the composite slab. If $k_1 = \frac{k_2}{2}$,the temperature difference across slab $A$ is (in $^{\circ} C$)

The radiation emitted by a star $A$ is $10,000$ times that of the sun. If the surface temperatures of the sun and the star $A$ are $6000 \ K$ and $2000 \ K$ respectively,the ratio of the radii of the star $A$ and the sun is (in $:1$)

$A$ block of $200 \ g$ mass moves with a uniform speed in a horizontal circular groove,with vertical side walls of radius $20 \ cm$. If the block takes $40 \ s$ to complete one round,the normal force by the side walls of the groove is: