Complete solution set of the inequality $\left( {{{\sec }^{ - 1}}\,x - 4} \right)\left( {{{\sec }^{ 1}}\,x - 1} \right)\left( {{{\sec }^{ - 1}}\,x - 2} \right) \ge 0$ is
$\left[ {\sec 2\,,\,\sec \,1} \right]$
$\left[ {\sec 1\,,\,\sec \,2} \right]\, \cup \,\left[ {\sec \,4\,,\,\infty } \right)$
$\left( { - \infty \,,\,\sec \,2} \right]\, \cup \,\left[ {\sec \,1\,,\,\infty } \right)$
$\left( { - \infty \,,\,\sec \,4} \right]\, \cup \,\left[ {\sec \,2\,,\,\infty } \right)$
If two roots of the equation ${x^3} - 3x + 2 = 0$ are same, then the roots will be
Let $p_1(x)=x^3-2020 x^2+b_1 x+c_1$ and $p_2(x)=x^3-2021 x^2+b_2 x+c_2$ be polynomials having two common roots $\alpha$ and $\beta$. Suppose there exist polynomials $q_1(x)$ and $q_2(x)$ such that $p_1(x) q_1(x)+p_2(x) q_2(x)=x^2-3 x+2$. Then the correct identity is
If $\alpha , \beta , \gamma $ are roots of equation ${x^3} + a{x^2} + bx + c = 0$, then ${\alpha ^{ - 1}} + {\beta ^{ - 1}} + {\gamma ^{ - 1}} = $
Let $f(x)=x^4+a x^3+b x^2+c$ be a polynomial with real coefficients such that $f(1)=-9$. Suppose that $i \sqrt{3}$ is a root of the equation $4 x^3+3 a x^2+2 b x=0$, where $i=\sqrt{-1}$. If $\alpha_1, \alpha_2, \alpha_3$, and $\alpha_4$ are all the roots of the equation $f(x)=0$, then $\left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2$ is equal to. . . . . .