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Question

# Assertion (A) If −5 is a root of $2{x}^{2}+2px-15=0\mathrm{and}p\left({x}^{2}+x\right)+k=0$ has equal roots, then $k=\frac{7}{8}.$ Reason (R) The equation $a{x}^{2}+bx+c=0,a\ne 0$ has equal roots, if $\left({b}^{2}-4ac\right)=0.$ (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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## (a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A). $\text{Reason}\left(\text{R}\right)\text{is true}\text{.}\phantom{\rule{0ex}{0ex}}\mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}-5\text{is a root of 2}{x}^{2}+2px-15=0;\mathrm{therefore},\text{we have:}\phantom{\rule{0ex}{0ex}}\text{2}×{\left(-5\right)}^{2}+2p×\left(-5\right)-15=0\phantom{\rule{0ex}{0ex}}⇒50-10p-15=0\phantom{\rule{0ex}{0ex}}⇒10p=35\phantom{\rule{0ex}{0ex}}⇒p=\frac{35}{10}=\frac{7}{2}\phantom{\rule{0ex}{0ex}}\text{Now,}\phantom{\rule{0ex}{0ex}}\frac{7}{2}\left({x}^{2}+x\right)+k=0\phantom{\rule{0ex}{0ex}}⇒7{x}^{2}+7x+2k=0\phantom{\rule{0ex}{0ex}}\text{For equal roots, we have:}\phantom{\rule{0ex}{0ex}}D=0\phantom{\rule{0ex}{0ex}}⇒49-4×7×2k=0\phantom{\rule{0ex}{0ex}}⇒56k\text{= 49}\phantom{\rule{0ex}{0ex}}⇒k=\frac{49}{56}=\frac{7}{8}\phantom{\rule{0ex}{0ex}}\text{Thus, Assertion}\left(\text{A}\right)\text{is true and Reason}\left(\text{R}\right)\text{is a correct explanation of Assertion}\left(\text{A}\right).\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

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