Three lines $${L_1}:r = \lambda \widehat i$$, $$\lambda $$ $$ \in $$ R,

$${L_2}:r = \widehat k + \mu \widehat j$$, $$\mu $$ $$ \in $$ R and

$${L_3}:r = \widehat i + \widehat j + v\widehat k$$, v $$ \in $$ R are given.

For which point(s) Q on L₂ can we find a point P on L₁ and a point R on L₃ so that P, Q and R are collinear?

For $$a \in R,\,|a|\, > 1$$, let

$$\mathop {\lim }\limits_{n \to \infty } \left( {{{1 + \root 3 \of 2 + ...\root 3 \of n } \over {{n^{7/3}}\left( {{1 \over {{{(an + 1)}^2}}} + {1 \over {{{(an + 2)}^2}}} + ... + {1 \over {{{(an + n)}^2}}}} \right)}}} \right) = 54$$

Let f : R be a function. We say that f has

PROPERTY 1 if $$\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {\sqrt {|h|} }}$$ exists and is finite, and

PROPERTY 2 if $$\mathop {\lim }\limits_{h \to 0} {{f(h) - f(0)} \over {{h^2}}}$$ exists and is finite. Then which of the following options is/are correct?

Let x $$ \in $$ R and let $$P = \left[ {\matrix{ 1 & 1 & 1 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$, $$Q = \left[ {\matrix{ 2 & x & x \cr 0 & 4 & 0 \cr x & x & 6 \cr } } \right]$$ and R = PQP^$$-$$1, which of the following options is/are correct?