The Lucas sequence $$L_n$$ is defined by the recurrence relation:

$${L_n} = {L_{n - 1}} + {L_{n - 2}}$$, for $$n \ge 3$$,

with $${L_1} = 1$$ and $${L_2} = 3$$.

Which one of the options given is TRUE?

Let $$A = \left[ {\matrix{ 1 & 2 & 3 & 4 \cr 4 & 1 & 2 & 3 \cr 3 & 4 & 1 & 2 \cr 2 & 3 & 4 & 1 \cr } } \right]$$ and $$B = \left[ {\matrix{ 3 & 4 & 1 & 2 \cr 4 & 1 & 2 & 3 \cr 1 & 2 & 3 & 4 \cr 2 & 3 & 4 & 1 \cr } } \right]$$.

Let $$\mathrm{det}(A)$$ and $$\mathrm{det}(B)$$ denote the determinates of the matrices A and B, respectively.

Which one of the options given below is TRUE?

Geetha has a conjecture about integers, which is of the form

$$\forall x\left( {P(x) \Rightarrow \exists yQ(x,y)} \right)$$,

where P is a statement about integers, and Q is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha's conjecture?

Let $$f(x) = {x^3} + 15{x^2} - 33x - 36$$ be a real-valued function. Which of the following statements is/are TRUE?