First slide

Theory of expressions

Question

The value of $p$ for which both the roots of the equation $4 x^{2} - 20 p x + (25 p^{2} + 15 p - 66) = 0$ are less than 2, lies in:

Moderate

A

$(\frac{4}{5}, 2)$
B

$(2, \infty)$
C

$(- 1, \frac{- 4}{5})$
D

$(- \infty, - 1)$

Solution

The given equation is $4 x^{2} - 20 p x + (25 p^{2} + 15 p - 66) = 0$
Here Discriminant $\geq 0$
$\Rightarrow 400 p^{2} - 16 (25 p^{2} + 15 p - 66) \geq 0$ $(∵ b^{2} - 4 a c \geq 0)$
$\Rightarrow - 240 p + 1056 \geq 0$
$\Rightarrow p \leq \frac{1056}{240}$ $\Rightarrow$ $p \leq \frac{22}{5}$
Now roots are less than $2$
$∴ f (2) > 0$ and their sum $< 4$
$\Rightarrow p^{2} - p - 2 > 0$ and $p < \frac{4}{5}$
$\Rightarrow (P - 2) (p + 1) > 0$ and $p < \frac{4}{5}$
$\Rightarrow p > 2$ or $p < - 1$ and $p < \frac{4}{5}$
Combining, we get $p \in (- \infty, - 1)$ .

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App