First slide

Harmonic Progression

Question

$If a x^{3} + b x^{2} + c x + d is divisible by a x^{2} + c, then a, b, c, d \underline{are} in$

Moderate

A

AP
B

GP
C

HP
D

None of these

Solution

$Since a x^{3} + b x^{2} + c x + d is divisible by a x^{2} + c, therefore, when a x^{3} + b x^{2} + c x + d$
$is divided by a x^{2} + c the remainder should be zero, Now when$
$a x^{3} + b x^{2} + c x + d is divided by a x^{2} + c, then the remainder is \frac{b c}{a} - d$
$\begin{array}{l} ∴ \frac{b c}{a} - d = 0 \\ \Rightarrow b c = a d \\ \Rightarrow \frac{b}{a} = \frac{d}{c} \\ Hence, from this a, b, c, d are not necessarily in G.P. \end{array}$

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