(A) Let $y = 2\alpha + 1$,then $\alpha = \frac{y - 1}{2}$.Since $\alpha$ is a root of $x^3 + 2x - 5 = 0$,we substitute $\alpha$ into the equation:$(\f

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

(A) Let $y = 2\alpha + 1$,then $\alpha = \frac{y - 1}{2}$.Since $\alpha$ is a root of $x^3 + 2x - 5 = 0$,we substitute $\alpha$ into the equation:$(\frac{y - 1}{2})^3 + 2(\frac{y - 1}{2}) - 5 = 0$$\frac{y^3 - 3y^2 + 3y - 1}{8} + y - 1 - 5 = 0$$y^3 - 3y^2 + 3y - 1 + 8y - 48 = 0$$y^3 - 3y^2 + 11y - 49 = 0$Comparing this with $x^3 + bx^2 + cx + d = 0$,we get $b = -3, c = 11, d = -49$.Therefore,$|b + c + d| = |-3 + 11 - 49| = |-41| = 41$.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Relation between roots and coefficients

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 2x - 5 = 0$ and the equation $x^3 + bx^2 + cx + d = 0$ has roots $2\alpha + 1, 2\beta + 1, 2\gamma + 1$,then the value of $|b + c + d|$ is (where $b, c, d$ are constants):

A
$41$
B
$39$
C
$40$
D
$43$

Explore More

Browse All

Question Bank

All Subjects

Class 11 Questions

Class 11

Mathematics Questions

Chapter

4-2.Quadratic Equations and Inequations

Subtopic

Relation between roots and coefficients

The sum of the cubes of all the roots of the equation $x^{4}-3x^{3}-2x^{2}+3x+1=10$ is

DifficultJEE MAIN 2022

View Solution

The cubic equation whose roots are the squares of the roots of $x^3-2x^2+10x-8=0$ is

DifficultAP EAMCET 2014

View Solution

If $\alpha$ and $\beta$ are the roots of $x^2+7x+3=0$ and $\frac{2\alpha}{3-4\alpha}, \frac{2\beta}{3-4\beta}$ are the roots of $ax^2+bx+c=0$ and $GCD(a, b, c) = 1$,then $a+b+c=$

EasyAP EAMCET 2019

View Solution

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 2(m^2 + 1)x + m^4 + m^2 + 1 = 0$,then find the value of $\alpha^2 + \beta^2$.

Medium

View Solution

If $\tan A$ and $\tan B$ are the roots of the equation $5x^2 - 4x + 1 = 0$, then the value of $A + B$ is...

DifficultMHT CET 2026

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 $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + 2x - 5 = 0$ and the equation $x^3 + bx^2 + cx + d = 0$ has roots $2\alpha + 1, 2\beta + 1, 2\gamma + 1$,then the value of $|b + c + d|$ is (where $b, c, d$ are constants):

Similar Questions

The sum of the cubes of all the roots of the equation $x^{4}-3x^{3}-2x^{2}+3x+1=10$ is

The cubic equation whose roots are the squares of the roots of $x^3-2x^2+10x-8=0$ is

If $\alpha$ and $\beta$ are the roots of $x^2+7x+3=0$ and $\frac{2\alpha}{3-4\alpha}, \frac{2\beta}{3-4\beta}$ are the roots of $ax^2+bx+c=0$ and $GCD(a, b, c) = 1$,then $a+b+c=$

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 2(m^2 + 1)x + m^4 + m^2 + 1 = 0$,then find the value of $\alpha^2 + \beta^2$.

If $\tan A$ and $\tan B$ are the roots of the equation $5x^2 - 4x + 1 = 0$, then the value of $A + B$ is...

Vedclass Test Series

Exam Paper Generator

Online Exam Module