Question

Assertion :The product of the real part of the roots of the equation $z^2+4z+4i=0$ and $2-2\sqrt{2}$ Reason: The roots of equation is given as $2(-1\pm \sqrt[4]{42}{e}^{-i\frac{\pi }{8}})$ and then take the product of real part.

• 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

Reason: Roots of equation $z^2+4z+4i=0$ is given by ${(z+2)}^2=4-4i=4(1-i)$

$\Rightarrow {\{z-(-2)\}}^2=4\sqrt{2}{e}^{-\frac{\pi }{4}i}$

$\Rightarrow z=-2\pm 2\left(2^{\frac{1}{4}}\right)e^{-\frac{\pi }{8}i}$ and product of real parts is $(2-2\sqrt{2})$ which shows that the reason is true.

Assertion: is true as the product of real parts of roots

$={(-2)}^2\{1-\sqrt{2}{\cos }^2\frac{\pi }{8}\}$

$=4\{1-\sqrt{2}{\left(\frac{\sqrt{2}+1}{2\sqrt{2}}\right)}^2\}=2-2\sqrt{2}$

So assertion is derived from reason.