For an ellipse with eccentricity $e = \frac{1}{2}$,the centre is at the origin. If one directrix is $x = 4$,then the equation of the ellipse is:

In the circuit shown,the reading of the Ammeter is doubled after the switch is closed. Each resistor has a resistance $1\,\Omega$ and the ideal cell has an $e.m.f.$ $10\,V$. Then,the Ammeter has a coil resistance equal to ............ $\Omega$

Assertion: Addition of silver ions to a mixture of aqueous sodium chloride and sodium bromide solution will first precipitate $AgBr$ rather than $AgCl$.
Reason: $K_{sp}$ of $AgCl > K_{sp}$ of $AgBr$.

Two identical metal plates show the photoelectric effect when light of wavelength ${\lambda _A}$ falls on plate $A$ and ${\lambda _B}$ falls on plate $B$,where ${\lambda _A} = 2{\lambda _B}$. The maximum kinetic energy is:

If $x > 0$ and $x^y = e^{x-y}$,then $\frac{dy}{dx}$ is equal to

An observer is standing $500 ~m$ away from a vertical hill. Starting between the observer and the hill,a police van sounding a siren of frequency $1000 ~Hz$ moves towards the hill with a uniform speed. If the frequency of the sound heard directly from the siren is $970 ~Hz$,the frequency of the sound heard after reflection from the hill (in $Hz$) is about,(velocity of sound $= 330 ~m/s$):