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)

$\because t=x+\frac{1}{x}$
$\therefore x^4-3x^3-2x^2-3x+1=0$
$\Rightarrow x^2-3x-\frac{3}{x}+\frac{1}{x^2}-2=0$
$\Rightarrow \left(x^2+\frac{1}{x^2}\right)-3\left(x+\frac{1}{x}\right)-2=0$
$\Rightarrow t^2-3t-4=0$ 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]