$\frac{(1+i)^{2011}}{(1-i)^{2009}}$ is equal to

$A$ coil of radius '$r$' is placed on another coil (whose radius is '$R$' and current through it is changing) so that their centers coincide. $(R \gg r)$. If both are coplanar,then the mutual inductance between them is proportional to

$A$ cylindrical vessel open at the top is $20 \ cm$ high and $10 \ cm$ in diameter. $A$ circular hole whose cross-sectional area is $1 \ cm^2$ is cut at the centre of the bottom of the vessel. Water flows from a tube above it into the vessel at the rate of $100 \ cm^3 s^{-1}$. The height of water in the vessel under steady state is ....... $cm$ (Take $g = 1000 \ cm s^{-2}$)

Solar radiation emitted by the sun resembles that emitted by a black body at a temperature of $6000\, K$. Maximum intensity is emitted at a wavelength of about $4800\, \mathring{A}$. If the sun was cooled down from $6000\, K$ to $3000\, K$,then the peak intensity would occur at a wavelength of ......... $\mathring{A}$.

The ratio of the mass percentages of $C$ and $H$,and $C$ and $O$ of a saturated acyclic organic compound $X$ are $4:1$ and $3:4$ respectively. Then,the moles of oxygen gas required for the complete combustion of two moles of organic compound $X$ is:

For the parabola $y^2+6y-2x+5=0$:
$(I)$ The vertex is $(-2,-3)$
$(II)$ The directrix is $y+3=0$
Which of the following is correct?