(D) Given $p(x) = f(x) - g(x) = (a - a_1)x^2 + (b - b_1)x + (c - c_1).$Since $p(x) = 0$ has only one root at $x = -1,$ the quadratic $p(x)$ must be a

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

(D) Given $p(x) = f(x) - g(x) = (a - a_1)x^2 + (b - b_1)x + (c - c_1).$Since $p(x) = 0$ has only one root at $x = -1,$ the quadratic $p(x)$ must be a perfect square of the form $p(x) = k(x + 1)^2$ for some constant $k = a - a_1 \ne 0.$Given $p(-2) = 2,$$k(-2 + 1)^2 = 2 \Rightarrow k(-1)^2 = 2 \Rightarrow k = 2.$Thus,$p(x) = 2(x + 1)^2.$Now,we need to find $p(2):$$p(2) = 2(2 + 1)^2 = 2(3)^2 = 2 \times 9 = 18.$

Class 11 Mathematics 2 Complex Numbers and Quadratic Equations: Quadratic Equations Solution of quadratic equations and Nature of roots

Difficult

Let $a \ne a_1 \ne 0,$ $f(x) = ax^2 + bx + c,$ $g(x) = a_1x^2 + b_1x + c_1,$ and $p(x) = f(x) - g(x).$ If $p(x) = 0$ only for $x = -1$ and $p(-2) = 2,$ then the value of $p(2)$ is

AIEEE 2011 All AIEEE

A
$9$
B
$6$
C
$3$
D
$18$

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

Solution of quadratic equations and Nature of roots

Previous Year

AIEEE 2011 Paper All AIEEE years →

The number of distinct quadratic equations $ax^2 + bx + c = 0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c$ such that $a \neq b \neq c$ from the set $\{0, 1, 2, 4\}$ is:

MediumAP EAMCET 2025

View Solution

The product and sum of the roots of the equation $|x|^2 - 5|x| - 24 = 0$ are respectively:

EasyAP EAMCET 2016

View Solution

The roots of the equation $(x-a)(x-a-1)+(x-a-1)(x-a-2)+(x-a)(x-a-2)=0$,where $a \in R$,are always:

DifficultAP EAMCET 2009

View Solution

If $3$ is a root of the equation $x^2 + kx - 24 = 0$,then $3$ is also a root of which of the following equations?

Medium

View Solution

The roots of the equation $x^3-3x-2=0$ are

EasyTS EAMCET 2005

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

Let $a \ne a_1 \ne 0,$ $f(x) = ax^2 + bx + c,$ $g(x) = a_1x^2 + b_1x + c_1,$ and $p(x) = f(x) - g(x).$ If $p(x) = 0$ only for $x = -1$ and $p(-2) = 2,$ then the value of $p(2)$ is

Similar Questions

The number of distinct quadratic equations $ax^2 + bx + c = 0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c$ such that $a \neq b \neq c$ from the set $\{0, 1, 2, 4\}$ is:

The product and sum of the roots of the equation $|x|^2 - 5|x| - 24 = 0$ are respectively:

The roots of the equation $(x-a)(x-a-1)+(x-a-1)(x-a-2)+(x-a)(x-a-2)=0$,where $a \in R$,are always:

If $3$ is a root of the equation $x^2 + kx - 24 = 0$,then $3$ is also a root of which of the following equations?

The roots of the equation $x^3-3x-2=0$ are

Vedclass Test Series

Exam Paper Generator

Online Exam Module