NIMCET 2020 Question Paper with solutions

let ${\mathit{X}}_{\mathit{n}}=4^{\mathit{n}}-3\mathit{n}-1,\mathit{n}∊\mathit{N}.$ then ${\mathit{x}}_1=4-3-1=0$
And, for $\mathit{n}\ge 2,$ we have [using Binomial Theorem]
Thus, ${\mathit{X}}_{\mathit{n}}$ is some positive integral multiple of 9 for $\mathit{n}\ge 2$
Hence, X consists of all positive integral multiplies of 9 of the form
together with 0
Now, $\mathit{y}=\left\{9\left(\mathit{n}-1\right):\mathit{n}∊\mathit{N}\right\}.$ this shows that Y consists of all integral multiples of 9 together with 0.
In fact, we have
$\mathit{X}=\left\{0,9,54,243,.....\right\}$ and $\mathit{y}=\left\{0,9,18,278,36,54,....\right\}$
Thus X ⊂ Y