Let [x] represent the greatest integer less than or equal to `x` If [`sqrt(n^2+lambda)]=[n^2+1]+2` , where `lambda,n in N ,` then `lambda` can assume `(2n+4)d iffe r e n tv a l u s` `(2n+5)d iffe r e n tv a l u s` `(2n+3)d iffe r e n tv a l u s` `(2n+6)d iffe r e n tv a l u s`

Options

A. (2n+4) different values
B. (2n+3) different values
C. (2n+5) different values
D. (3n+6) different values

Answer

C

Explanation

We have
`[sqrt(n^(2)+1)]=n`
`:.[sqrt(n^(2)+lambda)]=[sqrt(n^(2)+1)]+2`
`implies[sqrt(n^(2)+lambda)]=n+2`
`implies n+2 lesqrt(n^(2)+lambda) lt n+3`
`implies (n+2)^(2)lelambda lt(n+3)^(2)`
`implies 4n+4lelambda6n+9`
`implies lambda=4n+4,4n+5,....,6n+8`.
Hence, `lambda` can assume (2n+5) distinct values.