Question

The integer $k$, for which the inequality $x^2-2(3k-1)x+8k^2-7>0$ is valid for every $x$ in $R$ is :

• A. $3$
• B. $2$
• C. $4$
• D. $0$

Answer 1

The correct option is A

$3$

Explanation for the correct answer:

Compute the value of $k$.

An inequality $x^2-2(3k-1)x+8k^2-7>0$ is given and that means the quadratic equation $x^2-2(3k-1)x+8k^2-7=0$ has imaginary roots.

Therefore, for this quadratic equation $D<0$.

$$\begin{gathered} 4{(3k-1)}^2-4(8k^2-7)<0 \\ \Rightarrow 9k^2+1-6k-8k^2+7<0 \\ \Rightarrow k^2-6k+8<0 \\ \Rightarrow (k-2)(k-4)<0 \end{gathered}$$

So, the value of $k$ lies between $2$ and $4$.

Since, $k$ is an integer. therefore, the value of $k$ is $3$.

Hence, option $A$ is the correct answer.