Question

Assertion :If equation $a{x}^2+bx+c=0$ and $x^2-3x+4=0$ have exactly one root common, then at least one of $a,b,c$ is imaginary. Reason: If $a,b,c$ are not all real, then equation $a{x}^2+bx+c=0$ can have one real root and one one root imaginary.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$a{x}^2+bx+c=0$ has two complex conjugate roots only if all the coefficients are real. If all the coefficients are not real then it is not necessary that both the roots are imaginary. Hence, statement 2 is true.

Now, equation $x^2-3x+4=0$ has two complex conjugate roots. If $a{x}^2+bx+c=0$ has all coefficients real, then there will be two common roots. But if there is only one root common, then at least one of $a,b,c$ must be non-real.

Thus, both the statements are true and statement 2 is correct explanation of statement 1