Question

The value of $p$, for which both the roots of the equation $4x^2-20px+25p^2+15p-66=0$ are less than $2$, is

• A. $\left(\frac{4}{5},2\right)$
• B. $\left(-1,\frac{-4}{5}\right)$
• C. $\left(2,\infty \right)$
• D. $\left(-\infty ,-1\right)$

Answer 1

The correct option is D

$\left(-\infty ,-1\right)$

Explanation for the correct answer:

Step 1: Write the discriminant of the equation and apply condition for real roots

Let $f\left(x\right)=4x^2-20px+25p^2+15p-66$ $...\left(i\right)$

The equation has real roots when

$∆=b^2-4ac\ge 0$

$\Rightarrow$ ${\left(-20p\right)}^2-4\left(4\right)\left(25p^2+15p-66\right)\ge 0$

$\Rightarrow$ $16\left(66-15p\right)\ge 0$

$\Rightarrow$ $p\le \frac{22}{5}$ $...\left(ii\right)$

Both roots of equation $\left(i\right)$ must be less than $2$ according to the given condition

Step 2: Apply condition for roots of equation to be less than a given value

Hence $f\left(2\right)>0$

$\Rightarrow$ $4{\left(2\right)}^2-20p\left(2\right)+25p^2+15p-66>0$

$\Rightarrow$ $25p^2-25p-50>0$

$\Rightarrow$ $p^2-p-2>0$

$\Rightarrow$ $\left(p-2\right)\left(p+1\right)>0$

$\Rightarrow$ $p\in \left(-\infty ,-1\right)\cup \left(2,\infty \right)$ $...\left(iii\right)$

As each root is less than two , sum of roots should be less than $4$

Step 3: Write the condition for sum of roots to be less than a given value

$\Rightarrow$ $\frac{-\left(-20p\right)}{4}<4$

$\Rightarrow$ $p<\frac{16}{20}$

$\Rightarrow$ $p<\frac{4}{5}$ $...\left(iv\right)$

From $\left(ii\right),\left(iii\right),\left(iv\right)$ we get

$p\in \left(-\infty ,-1\right)$

Hence, option $\left(D\right)$ is the correct answer.