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# If $\Delta=\begin{vmatrix}x^n&x^{n+2}&x^{2n}\\1&x^p&p\\x^{n+5}&x^{p+6}&x^{2n+5}\end{vmatrix}=0$ then $p$ is given by

$(a)\;x^n\qquad(b)\;(n+1)\qquad(c)\;either\;(a)\;or\;(b)\qquad(d)\;both\;(a)\;and\;(b)$

Consider the given options
Option (a) If $p=x^n$ as $\Delta=0$ Since $C_1$ and $C_3$ are same
and considering option (b) if $p=n+1$
$\Rightarrow \Delta =0$
Since $R_1$ and $R_3$ are same if $x^5$ is taken out in common from $R_2$
P can be given by both $p = x^n$ and p = n+1
Hence (d) is the correct answer.
edited Mar 13, 2014