$A$ long wire carries a steady current. It is bent into a circle of one turn and the magnetic field at the centre of the coil is $B$. It is then bent into a circular loop of $n$ turns. The magnetic field at the centre of the coil will be

If the vectors $a \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b \hat{j}+\hat{k}$,and $\hat{i}+\hat{j}+c \hat{k}$ $(a \neq 1, b \neq 1, c \neq 1, a \neq b \neq c)$ are coplanar,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is:

Sunlight of intensity $1.3 ~kW ~m^{-2}$ is incident normally on a thin convex lens of focal length $20 ~cm$. Ignore the energy loss of light due to the lens and assume that the lens aperture size is much smaller than its focal length. The average intensity of light,in $kW m^{-2}$,at a distance $22 ~cm$ from the lens on the other side is.

What is absent in yeast?

$A$ metallic wire with tension $T$ and at temperature $30^{\circ} C$ vibrates with its fundamental frequency of $1 \ kHz$. The same wire with the same tension but at $10^{\circ} C$ temperature vibrates with a fundamental frequency of $1.001 \ kHz$. The coefficient of linear expansion of the wire is

If $A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$,then which one of the following holds for all $n \ge 1$ (by the principle of mathematical induction)?