The value of p such that both the roots of th | vedclass

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

Question

(A) Let $f(x) = (p - 5)x^2 - 2px + (p - 4)$.For the roots to be real and satisfy the given conditions,we analyze the sign of $f(x)$ at specific points

Vedclass · Accepted Answer

$\left( \frac{49}{4}, 24 \right)$

Vedclass · Answer

(A) Let $f(x) = (p - 5)x^2 - 2px + (p - 4)$.For the roots to be real and satisfy the given conditions,we analyze the sign of $f(x)$ at specific points.Case $1$: If $p - 5 > 0$ (i.e.,$p > 5$),the parabola opens upward.For one root to be in $(0, 2)$ and the other in $(2, 3)$,we must have $f(0) > 0$,$f(2)  0$.$f(0) = p - 4 > 0 \implies p > 4$.$f(2) = (p - 5)(4) - 2p(2) + (p - 4) = 4p - 20 - 4p + p - 4 = p - 24 $f(3) = (p - 5)(9) - 2p(3) + (p - 4) = 9p - 45 - 6p + p - 4 = 4p - 49 > 0 \implies p > \frac{49}{4}$.Combining $p > 5$,$p  \frac{49}{4}$,we get $p \in \left( \frac{49}{4}, 24 \right)$.Case $2$: If $p - 5 For one root to be in $(0, 2)$ and the other in $(2, 3)$,we must have $f(0)  0$,and $f(3) $f(0) = p - 4 $f(2) = p - 24 > 0 \implies p > 24$. This is impossible since $p Thus,the interval is $\left( \frac{49}{4}, 24 \right)$.

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 $3$	$R$. $a \in (-\infty, -4/3)$

The value of $p$ such that both the roots of the equation $(p - 5)x^2 - 2px + (p - 4) = 0$ are positive,one is less than $2$ and the other lies between $2$ and $3$,lies in the interval:

Similar Questions

The values of $a$ for which one root of the equation $x^2 - (a + 1)x + a^2 + a - 8 = 0$ exceeds $2$ and the other is lesser than $2$,are given by

Consider the quadratic equation $(c - 5)x^2 - 2cx + (c - 4) = 0$,where $c \ne 5$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $(0, 2)$ and the other root lies in the interval $(2, 3)$. Then the number of elements in $S$ is

Let $f(x) = (1 + b^2)x^2 + 2bx + 1$ and $m(b)$ be the minimum value of $f(x)$. If $b$ can take any real value,what is the range of $m(b)$?

The value of $k$ for which the equation $x^2 - 3x + k = 0$ has at least one real root in $[0, 1]$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module