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Question

# Assertion (A) The polynomial p(x) = x3 + x has one real zero. Reason (R) A polynomial of nth degree has at most n zeros. (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

## $\left(\mathrm{b}\right)\mathrm{Both}\mathrm{Assertion}\left(\mathrm{A}\right)\mathrm{and}\mathrm{Reason}\left(\mathrm{R}\right)\mathrm{are}\mathrm{true}\mathrm{but}\mathrm{Reason}\left(\mathrm{R}\right)\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{correct}\mathrm{explanation}\mathrm{of}\mathrm{Assertion}\left(\mathrm{A}\right).\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}p\left(x\right)=0\phantom{\rule{0ex}{0ex}}=>\left({x}^{3}+x\right)=0\phantom{\rule{0ex}{0ex}}=>x\left({x}^{2}+1\right)=0\phantom{\rule{0ex}{0ex}}=>x=0\text{or}{x}^{2}+1=0\phantom{\rule{0ex}{0ex}}\text{But}{x}^{2}+1\ne 0,\text{for any real value of}x.\left[\because {x}^{2}+1>0\right]\phantom{\rule{0ex}{0ex}}\therefore p\left(x\right)\text{has one real zero, i}\text{.e}\text{., 0}\text{.}\phantom{\rule{0ex}{0ex}}\therefore \text{Asseration}\left(\text{A}\right)\text{is true and reason}\left(\text{R}\right)\text{does not clearly explain Assertion}\left(\text{A}\right).\phantom{\rule{0ex}{0ex}}$

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