Correct Answer: 1 : 9

Solution : Given: The expression px ³ – 2x ² – qx + 18 is completely divisible by  (x ² – 9).
If (x ² – 9) = 0
⇒ (x – 3)(x + 3) = 0
⇒ x = 3, – 3
So the roots of the equation (x ² – 9) are 3 and – 3.
So, 3 and – 3 will also be the roots of the equation px ³ – 2x ² – qx + 18 = 0
We know that if $\alpha$ is the root of the equation $f(x)=0$, then  $f(\alpha)=0$
⇒ p(3) ³ – 2(3) ² – q(3) + 18 = 0 and p(–3) ³ – 2(–3) ² – q(–3) + 18 = 0
⇒ 27p – 3q – 18 + 18 = 0 and –27p – 18 + 3q + 18 = 0
⇒ 27p = 3q
⇒ $\frac{\text{p}}{\text{q}} = \frac{1}{9}$
⇒ p : q = 1 : 9
Hence, the correct answer is 1 : 9.