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 =

A point charge $$+Q$$ is placed just outside an imaginary hemispherical surface of radius $$R$$ as shown in the figure. Which of the following statements is/are correct?

Two coherent monochromatic point sources $${S_1}$$ and $${S_2}$$ of wavelength $$\lambda = 600\,nm$$ are placed symmetrically on either side of the center of the circle as shown. The sources are separated by a distance $$d=1.8$$ $$mm.$$ This arrangement produces interference fringes visible as alternate bright and dark spots on the circumference of the circle. The angular separation between two consecutive bright spots is $$\Delta \theta .$$ Which of the following options is/are correct?

The instantaneous voltages at three terminals marked $$X,Y$$ and $$Z$$ are given by

$${V_x} = {V_0}\,\sin \,\omega t,$$

$${V_Y} = {V_0}\,\sin $$ $$\left( {\omega t + {{2\pi } \over 3}} \right)$$

and $$Vz = {V_0}\sin \left( {\omega t + {{4\pi } \over 3}} \right)$$

An ideal voltmeter is configured to read $$rms$$ value of the potential difference between its terminals. It is connected between points $$X$$ and $$Y$$ and then between $$Y$$ and $$Z.$$ The reading(s) of the voltmeter will be