Question

Assertion (A)
If one zero of the polynomial p(x) = (k² + 4) x² + 9x + 4k is the reciprocal of the other zero, then k = 2.

Reason (R)
If (x − α) is a factor of the polynomial p(x), then α is a zero of p(x).

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).
(b) Both assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Answer 1

$$\begin{gathered} \left(\text{b}\right)\text{The Reason}\left(\text{R}\right)\text{is the factor theorem}\text{.} \\ \text{Let}\mathrm{\alpha }\text{and}\frac{1}{\mathrm{\alpha }}\text{be the zeroes of}p\left(x\right). \\ \text{Then product of the zeroes=}\mathrm{\alpha }\times \frac{1}{\mathrm{\alpha }}=\frac{\text{constant term}}{\text{co-efficient of}{x}^2}=\frac{4k}{k^2+4} \\ =>\frac{4k}{k^2+4}=1 \\ =>k^2-4k+4=0 \\ =>{(k-2)}^2=0 \\ =>k=2 \\ \therefore \text{Assertion}\left(\text{A}\right)\text{is true, but reason}\left(\text{R}\right)\text{is not its correct explanation.} \end{gathered}$$