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
$S = 0$
$S = ax + (1 - a){x^2}{\rm{ }}\forall a \in (0,\infty )$
$S = ax + (1 - a){x^2}{\rm{ }}\forall a \in R$
$S = ax + (1 - a){x^2}{\rm{ }}\forall a \in (0,2)$
The number of positive integers $x$ satisfying the equation $\frac{1}{x}+\frac{1}{x+1}+\frac{1}{x+2}=\frac{13}{2}$ is.
If the sum of two of the roots of ${x^3} + p{x^2} + qx + r = 0$ is zero, then $pq =$
Number of natural solutions of the equation $xyz = 2^5 \times 3^2 \times 5^2$ is equal to
The number of real solutions of the equation $e ^{4 x }+4 e ^{3 x }-58 e ^{2 x }+4 e ^{ x }+1=0$ is..........
Let $a$ , $b$ , $c$ are roots of equation $x^3 + 8x + 1 = 0$ ,then the value of
$\frac{{bc}}{{(8b + 1)(8c + 1)}} + \frac{{ac}}{{(8a + 1)(8c + 1)}} + \frac{{ab}}{{(8a + 1)(8b + 1)}}$ is equal to