Question

The number of integer value(s) of $k$ for which the expression $ {{x}^{2}}-2(4k-1)x+15{{k}^{2}} $ $ -2k-7>0 $ for every real number $x$ is/are

• A. None
• B. one
• C. finitely many, but greater than 1
• D. infinitely many

Answer 1

Answer: (B)

Given, expression is $ {{x}^{2}}-2(4k-1)+x+15{{k}^{2}}-2k-7>0 $
Its discriminant, $ D={{b}^{2}}-4ac $
$={{\{-2(4k-1)\}}^{2}}-4\times 1\times (15{{k}^{2}}-2k-7) $
$=4{{(4k-1)}^{2}}-4(15{{k}^{2}}-2k-7) $
$=4[{{(4k-1)}^{2}}-(15{{k}^{2}}-2k-7)] $
$=4[16{{k}^{2}}-8k+1-15{{k}^{2}}+2k+7] $
$=4[{{k}^{2}}-6k+8] $
$=4[{{k}^{2}}-4k-2k+8|=4|(k-4)(k-2)] $ Now, for real values of x, $ D<0 $
$ \Rightarrow $ $ (k-4)(k-2)<0 $
$ \Rightarrow $ $ k<4 $ or $ k>2 $
$ \therefore $ Integer value of k is 3. Hence, number of integer value of k is noe.