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)

one

Answer 2

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.