Question

Assertion (A): The quadratic equation $$x^{2}+5kx+16=0$$ has no real roots, if $$\frac{-8}{5}<k<\frac{8}{5}$$.

Reason (R): The equation $$ax^{2}+bx+c=0,\ a\neq 0$$ has no real root, if $$(b^{2}-4ac)<0$$.

विकल्प

• 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. The discriminant is $$ (5k)^{2}-4(1)(16)=25k^{2}-64.$$ It is negative when $$25k^{2}<64,$$ equivalently $$-\frac{8}{5}<k<\frac{8}{5}.$$ Thus, the equation has no real roots in the stated interval.

Step 2 – Reason: R is true. A quadratic equation has no real roots when its discriminant is negative.

Step 3 – Link: Applying the no-real-root condition in R gives exactly the interval stated in A.