Question

Assertion (A): If –5 is a root of $$2x^{2}+2px-15=0$$ and the equation $$p(x^{2}+x)+k=0$$ has equal roots then $$k=\frac{7}{8}$$.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has equal roots, if $$(b^{2}-4ac)=0$$.

Options

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

• Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).

• Assertion (A) is true and Reason (R) is false.

• Assertion (A) is false and Reason (R) is true.

Answer 1

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

Explanation:

Step 1 – Assertion: A is true. Substituting $$x=-5$$ into the first equation gives $$p=\frac{7}{2}.$$ The second equation is $$px^{2}+px+k=0.$$ For equal roots, $$p^{2}-4pk=0.$$ Since $$p=\frac{7}{2}\neq 0,$$ this gives $$k=\frac{p}{4}=\frac{7}{8}.$$

Step 2 – Reason: R is true. A quadratic equation has equal roots when its discriminant is zero.

Step 3 – Link: Applying this discriminant condition to the second equation gives the asserted value of $$k.$$