Question

If $\alpha$ and $\beta$ are the roots of ${4x}^2-16x+c=0,$ c>0 such that $1<\alpha <2<\beta <3$, then the no.of integer values of c is

• A. $17$
• B. $14$
• C. $18$
• D. $15$

Answer 1

The correct option is D $15$

Given, $\alpha$ and $\beta$ are roots of $4x^2-16x+x=0$

where $a=4$

$b=-16$

$x>0$

now, $\alpha \beta =\frac{a}{a}$

$\Rightarrow \alpha \beta =\frac{c}{4}$

$\Rightarrow \beta =\frac{c}{4\alpha }$ ………(i)

we have $2<\beta <3$

$\Rightarrow 2<\frac{c}{4\alpha }<3$ [from $\left(1\right)$]

$\Rightarrow 8<\frac{c}{\alpha }<12$

$\Rightarrow 8\alpha <c<12\alpha$ ………(ii)

and $1<\alpha <2$

$\Rightarrow \alpha >1\Rightarrow 8\alpha >8$ ……….(iii)

and $\alpha <2$

$\Rightarrow 12\alpha <24$ ………..(iv)

Substituting (iii) and (iv) in (ii) we get

$8<8\alpha <c<12\alpha <24$ $[\alpha =9,10,11,12,13,14,15,16,13,18,19,20,21,22,23]$

$\Rightarrow 8<c<24$

Therefore number of integer values of c is $15$.