Question

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

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A)
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A)
• C. (A) is true but (R) is false
• D. (A) is false but (R) is true

Answer 1

Answer: (D)

$ \because t=x+\frac{1}{x}$
$\therefore x^{4}-3 x^{3}-2 x^{2}-3 x+1=0$
$\Rightarrow x^{2}-3 x-\frac{3}{x}+\frac{1}{x^{2}}-2=0$
$\Rightarrow \left(x^{2}+\frac{1}{x^{2}}\right)-3\left(x+\frac{1}{x}\right)-2=0$
$\Rightarrow t^{2}-3 t-4=0$ is a quadratic in variable $t,$
so Reason (R) is true
$\therefore t=-1,4$
$\Rightarrow x=\omega, \omega^{2}, 2 \pm \sqrt{3}$
$\therefore $ Assertion $(A)$ is false as roots does not lie in [0,3]