Question

Assertion (A)
If −5 is a root of $2x^2+2px-15=0\mathrm{and}p(x^2+x)+k=0$ 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 $(b^2-4ac)=0.$

(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.

Answer 1

(a) Both Assertion (A) and Reason (R) are true and Reason (R) is a correct explanation of Assertion (A).

$$\begin{gathered} \text{Reason}\left(\text{R}\right)\text{is true}\text{.} \\ \mathrm{It}\mathrm{is}\mathrm{given}\mathrm{that}-5\text{is a root of 2}{x}^2+2px-15=0;\mathrm{therefore},\text{we have:} \\ \text{2}\times {(-5)}^2+2p\times (-5)-15=0 \\ \Rightarrow 50-10p-15=0 \\ \Rightarrow 10p=35 \\ \Rightarrow p=\frac{35}{10}=\frac{7}{2} \\ \text{Now,} \\ \frac{7}{2}(x^2+x)+k=0 \\ \Rightarrow 7x^2+7x+2k=0 \\ \text{For equal roots, we have:} \\ D=0 \\ \Rightarrow 49-4\times 7\times 2k=0 \\ \Rightarrow 56k\text{= 49} \\ \Rightarrow k=\frac{49}{56}=\frac{7}{8} \\ \text{Thus, Assertion}\left(\text{A}\right)\text{is true and Reason}\left(\text{R}\right)\text{is a correct explanation of Assertion}\left(\text{A}\right). \end{gathered}$$