Let `lambda` be the greatest integer for which `5p^(2)-16,2plambda,lambda^(2)` are jdistinct consecutive terms of an AP, where `p in R`. If the common difference of the Ap is `((m)/(n)),n in N` and m ,n are relative prime, the value of `m+n` is

Options

A. 133
B. 138
C. 143
D. 148

Answer

C

Explanation

` :.5p^(2)-16,2plambda,lambda^(2)` are in AP, then
`4plambda =5p^(2)-16+lambda^(2)`
` implies 5p^(2)-4plambda+lambda^(2)-16=0 " " ".....(i)"`
`B-4AC ge0 " " [:.p in R] `
` implies 16lambda^(2)-4*5*(lambda^(2)-16)ge0`
` implies lambda ^(2) +80ge 0` or `lambda^(2)ge 80`
` implies -sqrt(80)le lambda le sqrt(80)`
`:. lambda=8 " "[" greatest integer "]`
From Eq. (i), `5p^(2)-32p+48=0`
`implies (p-4)(5p-12)=0`
`:. p=4, p=(12)/(5)`
` implies p=(12)/(5), p ne 4` [for p=4 all terms are equal]
Now, common difference `=lambda^(2)-2plambda`
`64-16xx(12)/(5)=64(1-(3)/(5))=(128)/(5)=(m)/(n) " "[ " given "]`
`:.m=128` and `n=5`
Hence, `m+n=143`