KVPY Sample Paper KVPY Stream-SX Model Paper-30

The number of integral values of 'n', for which \[{{n}^{4}}-6{{n}^{2}}+25\] is prime number, is

A) 2

B) 1

C) 8

D) 0

Correct Answer: A

Solution :

\[{{n}^{4}}-6{{n}^{2}}+25={{n}^{4}}+10{{n}^{2}}+25-16{{n}^{2}}\]

\[=({{n}^{2}}+5-4n)({{n}^{2}}+5+4n)\]

Since \[{{n}^{4}}-6{{n}^{2}}+25\]is prime,

\[\therefore \] either \[{{n}^{2}}+5-4n=1\] i.e. \[n=2\]

or \[{{n}^{2}}+5+4n=1\] i.e. \[n=-2\]

\[\therefore \] Number of integral solutions is 2.