Question

The set of values of $k$ for which roots of the equation $x^2-3x+k=0$ lie in the interval $(0,2)$ is

• A. $(2,\infty )$
• B. $(0,\infty )$
• C. $(-\infty ,\frac{9}{4})$
• D. $(2,\frac{9}{4}]$

Answer 1

The correct option is D $(2,\frac{9}{4}]$

Both the roots will lie in the given interval $(k_1,k_2)$ if the following conditions are satisfied:

$D\ge 0$ .....(1)

$k_1<-\frac{b}{2a}$ .....(2)

$af\left(k_1\right)>0$ .......(3)

$af\left(k_2\right)>0$ ......(4)

Here, $k_1=0$ and $k_2=2$

From $\left(1\right)$ we get $k\le \frac{9}{4}$

From $\left(2\right)$ we get $0<\frac{3}{2}<2$

From $\left(3\right)$ we get $k>0$

From $\left(4\right)$ we get $k>2$

From $\left(1\right),\left(2\right),\left(3\right)$ and $\left(4\right)$ we get the set of values for $k$ to be $(2,\frac{9}{4}]$.