Question

Number of integral values of $k$ for which exactly one root of the equation $5x^2+(k+1)x+k=0$ lies in the interval $(1,3)$ is

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is D $8$

According to the question,

$$\begin{array}{c}if,f\left(x\right)=5x^2+(k+1)x+k=0\\ f\left(1\right)<0\\ \Rightarrow 5+k+1+k<0\\ \Rightarrow 6+2k<0\\ \therefore k<-3\\ andf\left(3\right)>0,\\ \Rightarrow 5(3{)}^2+(k+1)3+k>0\\ \Rightarrow 45+3k+3+k>0\\ \Rightarrow 48+4k>0\\ \therefore k>-12\\ Now,k=-12<k<-3\\ Valueofk=-11,-10,-9,-8,-7,-6,-5,-4\end{array}$$

So, that the No. of integral value of k=8.

and the correct option is D.