Question

The number of integer value(s) of $k$ for which the expression $x^2-2(4k-1)x+15k^2$ $-2k-7>0$ 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 $x^2-2(4k-1)+x+15k^2-2k-7>0$
Its discriminant, $D=b^2-4ac$
$={\{-2(4k-1)\}}^2-4\times 1\times (15k^2-2k-7)$
$=4{(4k-1)}^2-4(15k^2-2k-7)$
$=4[{(4k-1)}^2-(15k^2-2k-7)]$
$=4[16k^2-8k+1-15k^2+2k+7]$
$=4[k^2-6k+8]$
$=4[k^2-4k-2k+8|=4|(k-4)(k-2)]$ 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.