The value of p for which both the roots of th | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let $f(x) = 4x^2 - 20px + (25p^2 + 15p - 66) = 0$ .....$(i)$The roots of $(i)$ are real if $D \ge 0$:$D = (-20p)^2 - 4(4)(25p^2 + 15p - 66) = 400p

Vedclass · Accepted Answer

$(-\infty, -1)$

Vedclass · Answer

(D) Let $f(x) = 4x^2 - 20px + (25p^2 + 15p - 66) = 0$ .....$(i)$The roots of $(i)$ are real if $D \ge 0$:$D = (-20p)^2 - 4(4)(25p^2 + 15p - 66) = 400p^2 - 16(25p^2 + 15p - 66) = 16(66 - 15p) \ge 0$$\Rightarrow 66 - 15p \ge 0$ $\Rightarrow p \le \frac{22}{5}$ .....$(ii)$For both roots to be less than $2$:$1)$ $D \ge 0 \Rightarrow p \le \frac{22}{5}$$2)$ $f(2) > 0 \Rightarrow 4(2)^2 - 20p(2) + 25p^2 + 15p - 66 > 0$$16 - 40p + 25p^2 + 15p - 66 > 0$ $\Rightarrow 25p^2 - 25p - 50 > 0$ $\Rightarrow p^2 - p - 2 > 0$$(p - 2)(p + 1) > 0 \Rightarrow p  2$ .....$(iii)$$3)$ Vertex $x_v Taking the intersection of $(ii), (iii),$ and $(iv)$:$p < -1$.

Column-$I$	Column-$II$
$A$. Imaginary roots	$P$. $a \in (-1, 4)$
$B$. One root less than $3$ and other greater than $3$	$Q$. $a \in (-\infty, -1)$
$C$. One root less than $1$ and other greater than $3$	$R$. $a \in (-\infty, -4/3)$

Quadratic expression	The minimum value
i) $x^2 + 4x + 6$	a) $1$
ii) $x^2 - 2x + 5$	b) $2$
iii) $x^2 + 6x + 18$	c) $4$
iv) $x^2 - 4x + 5$	d) $9$

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

Similar Questions

If $f(x) = x^2 - 2(4K - 1)x + g(K) > 0$ for all $x \in R$ and for $K \in (a, b)$. If $g(K) = 15K^2 - 2K - 7$,then:

Match the following quadratic expressions with their minimum values:
Quadratic expression The minimum value
i) $x^2 + 4x + 6$ a) $1$
ii) $x^2 - 2x + 5$ b) $2$
iii) $x^2 + 6x + 18$ c) $4$
iv) $x^2 - 4x + 5$ d) $9$

If $a$ and $b$ are the maximum and minimum values of the quadratic expressions $1-2x-5x^2$ and $x^2-2x+5$ respectively,then the set of all values of $x$ for which the expression $5ax^2+bx+7$ is positive,is

The maximum and minimum values of $\frac{x^2 + x + 1}{x^2 - x + 1}$ are respectively ...........

Vedclass Test Series

Exam Paper Generator

Online Exam Module