Question

Assertion :The real roots of the equation $x^4-3x^3-2x^2-3x+1=0$ lies in $[0,3]$ . Reason: It is a quadratic equation in the variable $t=x+\frac{1}{x}$.

• 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. Assertion is incorrect and Reason is correct.

Answer 1

The correct option is D Assertion is incorrect and Reason is correct.

$\because t=x+\frac{1}{x}$

$\therefore x^4-3x^3-2x^2-3x+1=0$

$\Rightarrow x^2-3x-\frac{3}{x}+\frac{1}{x^2}-2=0$

$\Rightarrow (x^2+\frac{1}{x^2})-3(x+\frac{1}{x})-2=0$

$\Rightarrow t^2-3t-4=0$ is quadratic in variable $t$ so Reason $\left(R\right)$ is true

$\therefore t=-1,4$

$\Rightarrow x=w,w^2,2\pm \sqrt{3}$

$\therefore$ Assertion $\left(A\right)$ is false as roots does not lie in $[1,3]$.