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 the interval

Moderate

A

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

$(- \infty, - 1)$
C

$(2, \infty)$
D

none of these

Solution

The given equation is $4 x^{2} - 20 p x + (25 p^{2} + 15 p - 66) = 0$ …..(1)
As (1) has real roots,

therefore, ${(- 20 p)}^{2} - 4.4 (25 p^{2} + 15 p - 66) \geq 0$ $(∵ b^{2} - 4 a c \geq 0)$

$\Rightarrow - 240 p + 1056 \geq 0 \Rightarrow p \leq \frac{22}{5}$

Since, both the roots of the equation (1) are Less than 2,

therefore, sum of the roots $< 4$

$\Rightarrow \frac{20 p}{4} < 4 \Rightarrow p < \frac{4}{5}$

Now, if $f (x) = 4 x^{2} - 20 p x + 25 p^{2} + 15 p - 66,$

Then graph of $y = f (x)$ is an upward parabola meeting $X -$ axis at points which lie on left of $x = 2$ and hence $f (2) > 0$

$\Rightarrow 16 - 40 p + 25 p^{2} + 15 p - 66 > 0$

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

$\Rightarrow p^{2} - p - 2 > 0 \Rightarrow (p - 2) (p + 1) > 0$

$\Rightarrow p < - 1 o r p > 2$

But $p < \frac{4}{5},$ therefore, $p < - 1$

i.e. $p \in (- \infty, - 1) .$

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