(D) Let $f(x) = ax^2 + bx + c = a(x - \alpha)(x - \beta)$.Given that $\alpha < k < \beta$,the factors $(k - \alpha)$ and $(k - \beta)$ have opposite s

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

(D) Let $f(x) = ax^2 + bx + c = a(x - \alpha)(x - \beta)$.Given that $\alpha Specifically,$(k - \alpha) > 0$ and $(k - \beta) For the expression $f(k) = a(k - \alpha)(k - \beta)$ to be negative,we must have $a(k - \alpha)(k - \beta) Multiplying by $a$,we get $a^2(k - \alpha)(k - \beta) < 0$,which is equivalent to $a(ak^2 + bk + c) < 0$ or $a^2k^2 + abk + ac < 0$.

Class 11 Mathematics 2 Complex Numbers and Quadratic Equations: Quadratic Equations Quadratic expressions and Position of roots

Easy

If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2 + bx + c = 0$ and $k$ is a real number,then the condition for $\alpha < k < \beta$ is given by:

A
$ac > 0$
B
$ak^2 + bk + c = 0$
C
$ac < 0$
D
$a(ak^2 + bk + c) < 0$

Explore More

Browse All

Question Bank

All Subjects

Class 11 Questions

Class 11

Mathematics Questions

Chapter

2 Complex Numbers and Quadratic Equations: Quadratic Equations

Subtopic

Quadratic expressions and Position of roots

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:

Difficult

View Solution

If $a, b, c$ are real numbers and $a > 0$,what is the minimum value of the quadratic expression $ax^2 + bx + c$ for real $x$?

Medium

View Solution

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

DifficultJEE MAIN 2013

View Solution

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

Difficult

View Solution

The number of integral values of $m$ for which the quadratic expression $(1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m)$ is always positive for all $x \in R$ is:

DifficultJEE MAIN 2019

View Solution

Vedclass Products

For Students

Vedclass Test Series

Mock tests in real JEE/NEET style with performance analysis. 5-day free trial.

Start Free Trial

For Teachers

Exam Paper Generator

Generate Set A/B/C/D exam papers from 7.5L+ questions in 2 minutes. 3 chapters free.

Try Free

For Institutes

Online Exam Module

Live online exams with unlimited students, 360° analytics & white-label branding.

See Demo

If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^2 + bx + c = 0$ and $k$ is a real number,then the condition for $\alpha < k < \beta$ is given by:

Similar Questions

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:

If $a, b, c$ are real numbers and $a > 0$,what is the minimum value of the quadratic expression $ax^2 + bx + c$ for real $x$?

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

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

The number of integral values of $m$ for which the quadratic expression $(1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m)$ is always positive for all $x \in R$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module