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.

a₁₂ = ?

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?