If $S$ is a set of $P(x)$ is polynomial of degree $ \le 2$ such that $P(0) = 0,$$P(1) = 1$,$P'(x) > 0{\rm{ }}\forall x \in (0,\,1)$, then
$S = 0$
$S = ax + (1 - a){x^2}{\rm{ }}\forall a \in (0,\infty )$
$S = ax + (1 - a){x^2}{\rm{ }}\forall a \in R$
$S = ax + (1 - a){x^2}{\rm{ }}\forall a \in (0,2)$
If the expression $\left( {mx - 1 + \frac{1}{x}} \right)$ is always non-negative, then the minimum value of m must be
If $x$ be real, then the minimum value of ${x^2} - 8x + 17$ is
Let $\mathrm{a}=\max _{x \in R}\left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\}$ and $\beta=\min _{x \in R}\left\{8^{2 \sin 3 x} \cdot 4^{4 \cos 3 x}\right\}$
If $8 x^{2}+b x+c=0$ is a quadratic equation whose roots are $\alpha^{1 / 5}$ and $\beta^{1 / 5}$, then the value of $c-b$ is equal to:
If $\alpha ,\beta $are the roots of ${x^2} - ax + b = 0$ and if ${\alpha ^n} + {\beta ^n} = {V_n}$, then
The product of all real roots of the equation ${x^2} - |x| - \,6 = 0$ is