If the graph of $y = ax^3 + bx^2 + cx + d$ is symmetric about the line $x = k$ then
$k=c$
$k = -\frac{c}{b}$
$a + \frac{c}{{2b}} + k = 0$
none of these
If $\alpha , \beta , \gamma $ are roots of equation ${x^3} + a{x^2} + bx + c = 0$, then ${\alpha ^{ - 1}} + {\beta ^{ - 1}} + {\gamma ^{ - 1}} = $
If $|x - 2| + |x - 3| = 7$, then $x =$
Consider the equation ${x^2} + \alpha x + \beta = 0$ having roots $\alpha ,\beta $ such that $\alpha \ne \beta $ .Also consider the inequality $\left| {\left| {y - \beta } \right| - \alpha } \right| < \alpha $ ,then
Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?
$I$. It has at least one negative root.
$II$. It has at least one positive root.
$III$. Both its roots are real.
If ${x^2} + px + 1$ is a factor of the expression $a{x^3} + bx + c$, then