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# 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.

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Solution

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

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