Question

Assertion (A): If the equation $$x^{2}-kx+1=0$$ has no real roots then $$-2<k<2$$.

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

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 but Reason (R) is not a correct explanation of Assertion (A).

Explanation:

Step 1 – Assertion: A is true. Having no real roots requires a negative discriminant: $$(-k)^{2}-4(1)(1)<0.$$ Thus $$k^{2}<4,$$ which gives $$-2<k<2.$$

Step 2 – Reason: R is true. A quadratic equation has real roots when its discriminant is non-negative.

Step 3 – Link: R states the condition for real roots, but does not directly state or derive the negative-discriminant condition needed for no real roots.