See the electrical circuit shown in the figure. Which of the following equations is a correct equation for it?

At a certain place,the angle of dip is $60^{\circ}$ and the horizontal component of the earth's magnetic field $(B_H)$ is $0.8 \times 10^{-4} ~T$. The earth's overall magnetic field is

Consider the following statements about manganate and permanganate ions. Identify the correct statements:
$(A)$ The geometry of both manganate and permanganate ions is tetrahedral.
$(B)$ The oxidation states of $Mn$ in manganate and permanganate are $+7$ and $+6$, respectively.
$(C)$ Oxidation of $Mn(II)$ salt by peroxodisulphate gives manganate ion as the final product.
$(D)$ Manganate ion is paramagnetic and permanganate ion is diamagnetic.
$(E)$ Acidified permanganate ion reduces oxalate, nitrite and iodide ions.
Choose the correct answer from the options given below:

In a $\triangle ABC$,if $a \cos^2 \frac{C}{2} + c \cos^2 \frac{A}{2} = \frac{3b}{2}$,then the sides of the triangle are in

Suppose $A$ and $B$ are the points at which the line $x+y-\lambda=0$ meets the pair of straight lines $x^2+y^2-2x-4y+2=0$. If $\angle AOB=90^{\circ}$,then a value of $\lambda$ is

The specific conductance of a $0.1 \ N \ KCl$ solution at $23 \ ^oC$ is $0.012 \ \Omega^{-1} \ cm^{-1}$. The resistance of the cell containing the solution at the same temperature was found to be $55 \ \Omega$. The cell constant will be ............. $cm^{-1}$.