First slide

Introduction to Determinants

Question

If $|\begin{array}{ccc} x^{n} & x^{n + 2} & x^{2 n} \\ 1 & x^{a} & a \\ x^{n + 5} & x^{a + 6} & x^{2 n + 5} \end{array}| = 0, \forall x \in R$ , where $n \in N$ , then a =

Moderate

A

n
B

n - 1
C

n + 1
D

none of these

Solution

Taking x⁵ common from last row, we get
$x^{5} |\begin{array}{ccc} x^{n} & x^{n + 2} & x^{2 n} \\ 1 & x^{a} & a \\ x^{n} & x^{a + 1} & x^{2 n} \end{array}| = 0, \forall x \in R$
$\Rightarrow a + 1 = n + 2 or a = n + 1$ (as it will make first and third row is identical)

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