Question

The value of $k$ for which the number $3$ lies between the roots of the equation $x^2+(1-2k)x+(k^2-k-2)=0$ is given by

• A. $2<k<5$
• B. $k<2$
• C. $2<k<3$
• D. $k>5$

Answer 1

The correct option is A $2<k<5$

Let $f\left(x\right)=x^2+(1-2k)x+k^2-k-2$ and $a=1>0$

The number $3$ lies between the roots of the given equation, if $f\left(3\right)<0$

Now, $f\left(3\right)=9+(1-2k)3+k^2-k-2=10-7k+k^2=k^2-7k+10$

Hence, $f\left(3\right)<0\Rightarrow k^2-7k+10<0$

$\Rightarrow (k-2)(k-5)<0\Rightarrow 2<k<5.$