In order to eliminate the first degree terms from the equation $2x^2+4xy+5y^2-4x-22y+7=0$,the point to which the origin is to be shifted is

Using the Trapezoidal rule,find the approximate value of $\int_1^4 y \, dx$ based on the following table:

$x$	$1$	$2$	$3$	$4$
$y$	$0.7111$	$0.7222$	$0.7333$	$0.7444$

(in $.1833$)

The $Ca^{2+}$ and $F^{-}$ ions are located in a $CaF_{2}$ crystal,respectively,at face-centered cubic lattice points and in:

If the mass number of an atom is $A = 40$ and its electron configuration is $1s^2, 2s^2, 2p^6, 3s^2, 3p^6$,the number of neutrons and protons in its nucleus will be

The figure shows a rough sketch of a meter bridge. The galvanometer $(G)$ shows zero deflection at a length $\ell \, cm$. Now,$R_1$ and $R_2$ are interchanged,then the balancing length increases by $25 \, cm$. Find the ratio $R_1/R_2$.

When $100 \, V$ $DC$ is applied across a solenoid,a current of $1 \, A$ flows in it. When $100 \, V$ $AC$ is applied across the same coil,the current drops to $0.5 \, A$. If the frequency of the $AC$ source is $50 \, Hz$,the impedance and inductance of the solenoid are: