Let $f(x) = x^3 + bx^2 + cx + d$ where $0 < b^2 < c$. Then $f(x)$:

If the distance of the point $P(1, -2, 1)$ from the plane $x + 2y - 2z = \alpha$,where $\alpha > 0$,is $5$,then the foot of the perpendicular from $P$ to the plane is:

In a competition,$A, B$,and $C$ are participating. The probability that $A$ wins is twice that of $B$,and the probability that $B$ wins is twice that of $C$. What is the probability that $A$ loses?

The following reaction occurs in the Blast Furnace where iron ore is reduced to iron metal:
$Fe_2O_{3(s)} + 3CO_{(g)} \rightleftharpoons 2Fe_{(l)} + 3CO_{2(g)}$
Using the Le Chatelier's principle,predict which one of the following will not disturb the equilibrium?

If $A$ and $B$ are not disjoint sets,then $n(A \cup B) =$

Which one of the following ions has the same number of unpaired electrons as those present in $V^{3+}$ ion?