Question

Make the correct alternative in following question:

If xⁿ $-$ 1 is divisible by x $-$ $\lambda$, then the least positive integral value of $\lambda$ is

(a) 1 (b) 2 (c) 3 (d) 4

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{P}\left(n\right):x^n-1\mathrm{is}\mathrm{divisible}\mathrm{by}\left(x-\lambda \right). \\ \mathrm{As},\mathrm{for}n=1, \\ \mathrm{P}\left(1\right)=x^1-1=x-1 \\ \mathrm{As},\mathrm{P}\left(1\right)\mathrm{must}\mathrm{be}\mathrm{divisible}\mathrm{by}\left(x-\lambda \right). \\ \Rightarrow \left(x-1\right)\mathrm{must}\mathrm{be}\mathrm{divisible}\mathrm{by}\left(x-\lambda \right). \\ \mathrm{So},\lambda =1 \end{gathered}$$

Hence, the correct alternative is option (a).