If the lines,joining the origin to the points of intersection of the curve $2x^2 - 2xy + 3y^2 + 2x - y - 1 = 0$ and the line $x + 2y = k$,are at right angles,then $k^2$ equals

$(a)$ Show that for every natural number $n$ relatively prime to $10$,there exists a natural number $m$ consisting only of digits $1$ such that $n$ divides $m$.
$(b)$ Hence or otherwise,show that every positive rational number can be expressed in the form $\frac{a}{10^b(10^c-1)}$ for some natural numbers $a, b, c$.

Which one of the following reactions does not occur?

In $YDSE$ setup,light of wavelength $640 \, nm$ is used with $d = 0.8 \, mm$ and $D = 1 \, m$. If intensity at central maximum is $I_0$ and its position is $y = 0$,then:

The maximum possible number of real roots of the equation $x^5 - 6x^2 - 4x + 5 = 0$ is

Two wires of the same material and length but diameters in the ratio $1:2$ are stretched by the same force. The elastic potential energy per unit volume for the wires, when stretched by the same force, will be in the ratio: (in $:1$)