Question

If $\alpha ,\beta$ are the roots of the quadratic equation $4x^2-16x+k=0,k\in R$ such that $1<\alpha <2$ and $2<\beta <3$, then the number of integral values of $k$ is

• A. $5$
• B. $6$
• C. $3$
• D. $2$

Answer 1

The correct option is C $3$

Given,
$f\left(x\right)=4x^2-16x+k=0,k\in R$
and $1<\alpha <2$ and $2<\beta <3$
As $a=4>0$,
(i) $f\left(1\right)>0$
$\Rightarrow 4-16+k>0$
$k>12$
(ii) $f\left(3\right)>0$
$\Rightarrow 36-48+k>0$
$k>12$
(iii) Discriminant, $D>0$
$\Rightarrow 256-16k>0$
$k<16$
(iv) $f\left(2\right)<0$
$16-32+k<0$
$\Rightarrow k<16$
The integral value of $k=13,14,15$.
The no. of integral values are $3$.
The correct answer is option C.