Let $${H_1},{H_2},....,{H_n}$$ be mutually exclusive and exhaustive events with $$P\left( {{H_1}} \right) > 0,i = 1,2,.....,n.$$ Let $$E$$ be any other event with $$0 < P\left( E \right) < 1.$$
STATEMENT-1:
$$P\left( {{H_1}|E} \right) > P\left( {E|{H_1}} \right).P\left( {{H_1}} \right)$$ for $$i=1,2,....,n$$ because

STATEMENT-2: $$\sum\limits_{i = 1}^n {P\left( {{H_i}} \right)} = 1.$$

The minimum of distinct real values of $$\lambda ,$$ for which the vectors $$ - {\lambda ^2}\widehat i + \widehat j + \widehat k,$$ $$\widehat i - {\lambda ^2}\widehat j + \widehat k$$ and $$\widehat i + \widehat j - {\lambda ^2}\widehat k$$ are coplanar, is

Let $$\overrightarrow a \,,\,\overrightarrow b ,\overrightarrow c $$ be unit vectors such that $${\overrightarrow a + \overrightarrow b + \overrightarrow c = \overrightarrow 0 .}$$ Which one of the following is correct ?

Consider the following linear equations $$ax+by+cz=0;$$ $$\,\,\,$$ $$bx+cy+az=0;$$ $$\,\,\,$$ $$cx+ay+bz=0$$

Match the conditions/expressions in Column $$I$$ with statements in Column $$II$$ and indicate your answer by darkening the appropriate bubbles in the $$4 \times 4$$ matrix given in the $$ORS.$$

$$\,\,\,$$ Column $$I$$
(A)$$\,\,a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$
(B)$$\,\,$$ $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$
(C)$$\,\,a + b + c \ne 0$$ and $${a^2} + {b^2} + {c^2} \ne ab + bc + ca$$
(D)$$\,\,$$ $$a + b + c = 0$$ and $${a^2} + {b^2} + {c^2} = ab + bc + ca$$

$$\,\,\,$$ Column $$II$$
(p)$$\,\,\,$$ the equations represents planes meeting only at asingle point
(q)$$\,\,\,$$ the equations represents the line $$x=y=z.$$
(r)$$\,\,\,$$ the equations represent identical planes.
(s) $$\,\,\,$$ the equations represents the whole of the three dimensional space.