If both the roots of the quadratic equation x | vedclass

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

Question

(A) Let $f(x) = x^2 - mx + 4$. For the roots $\alpha, \beta$ to be real,distinct,and lie in $[1, 5]$,the following conditions must be satisfied:$(1)$

Vedclass · Accepted Answer

$(4, 5)$

Vedclass · Answer

(A) Let $f(x) = x^2 - mx + 4$. For the roots $\alpha, \beta$ to be real,distinct,and lie in $[1, 5]$,the following conditions must be satisfied:$(1)$ Discriminant $D > 0$:$D = (-m)^2 - 4(1)(4) = m^2 - 16 > 0$$m^2 > 16 \Rightarrow m \in (-\infty, -4) \cup (4, \infty)$$(2)$ $f(1) > 0$ (since roots are distinct and lie in $[1, 5]$,$f(1)$ cannot be $0$ as $1$ is not a root):$f(1) = 1 - m + 4 = 5 - m > 0 \Rightarrow m $(3)$ $f(5) > 0$:$f(5) = 25 - 5m + 4 = 29 - 5m > 0 \Rightarrow m $(4)$ Vertex location $1 $1 Taking the intersection of all conditions:$m \in (4, \infty) \cap (-\infty, 5) \cap (-\infty, 5.8) \cap (2, 10) = (4, 5)$Thus,$m \in (4, 5)$. Option $A$ is correct.

If both the roots of the quadratic equation $x^2 - mx + 4 = 0$ are real and distinct and they lie in the interval $[1, 5]$,then $m$ lies in the interval.

Similar Questions

If the maximum value of $2x - 7 - ax^2$ cannot exceed $20$,then the minimum value of $a$ is

If the roots of the equation $x^2 - 2ax + a^2 + a - 3 = 0$ are real and less than $3$,then

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 - 3x + a = 0$,where $a \in R$ and $\alpha < 1 < \beta$,then which of the following is true?

If the minimum value of the quadratic expression $x^2+5x-2$ is $M$ and it occurs at $x=a$,then $\frac{M}{a}$ is equal to

Statement-$I$: If the roots $\alpha, \beta$ of the equation $x^2 + 2(a - 3)x + 9 = 0$,$a \in R$ satisfy $\alpha < 6 < \beta$,then $a < -3/4$.
Statement-$II$: If $f(x) = x^2 + 2(a - 3)x + 9$,then $f(6) < 0 \implies a < -3/4$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module