Let p, q be integers and let $$\alpha $$, $$\beta $$ be the roots of the equation, x² $$-$$ x $$-$$ 1 = 0 where $$\alpha $$ $$ \ne $$ $$\beta $$. For n = 0, 1, 2, ........, let a_n = p$$\alpha $$ⁿ + q$$\beta $$ⁿ.

FACT : If a and b are rational numbers and a + b$$\sqrt 5 $$ = 0, then a = 0 = b.

If a₄ = 28, then p + 2q =

Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density $$\rho $$ remains uniform throughout the volume. The rate of fractional change in density $$\left( {{1 \over \rho } {{d\rho } \over {dt}}} \right)$$ is constant. The velocity $$v$$ of any point on the surface of the expanding sphere is proportional to

A photoelectric material having work-function $${\phi _0}$$ is illuminated with light of wavelength $$\lambda \left( {\lambda < {{he} \over {{\phi _0}}}} \right).$$ The fastest photoelectron has a de-Broglic wavelength $${\lambda _d}.$$ A change in wavelength of the incident light by $$\Delta \lambda $$ result in a change $$\Delta {\lambda _d}$$ in $${\lambda _d}.$$ Then the ratio $$\Delta {\lambda _d}/\Delta \lambda $$ is proportional to

A symmetric star shaped conducting wire loop is carrying a steady state current $${\rm I}$$ as shown in the figure. The distance between the diametrically opposite vertices of the star is $$4a.$$ The magnitude of the magnetic field at the center of the loop is