Question : If the expression px3 – 2x2 – qx + 18 is completely divisible by (x2 – 9), then what is the ratio of p to q?
Option 1: 1 : 9
Option 2: 1 : 3
Option 3: 3 : 1
Option 4: 9 : 1
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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.
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