Let f(x)=x^4+a x^3+b x^2+c be a polynomial with real coefficients such that f(1)=-9 . Suppose that i 3 is a root of the equation 4 x^3+3 a x^2+2 b x=0 , where i= -1 . If α_1, α_2, α_3 , and α_4 are all the roots of the

Correct answer: 0 f(1)=1+a+b+c=-9 a+b+c=-10.....(1) 4 x^3+3 a x^2+2 b x=0 roots are 3 i,- 3 i, 0 4 x^2+3 a x+2 b=0 < ^ 3i _ - 3i a=0 ~\&~ 2 ~b 4 =( 3 i)(- 3 i) b=6 use a, b in (1) c=-16 array ll & f(x)=x^4+6 x^2-16=0 & (x^2+8)(x^2-2)=0 & x= ± 8 i, ± 2 array |α_1|^2+|α_2|^2+|α_3|^2+|α_4|^2=20

JEE Advanced 2024Paper 1

Exam: JEE Advanced
Year: 2024
Paper: Paper 1
Subject: Mathematics
Chapter: Quadratic Equation

JEE Advanced 2024 · Paper 1 · Mathematics · Quadratic Equation

Let f(x)=x^4+a x^3+b x^2+c be a polynomial with real coefficients...

Let

f(x)=x^4+a x^3+b x^2+c be a polynomial with real coefficients such that

f(1)=-9

. Suppose that

i \sqrt{3}

is a root of the equation

4 x^3+3 a x^2+2 b x=0

, where

i=\sqrt{-1}

. If

\alpha_1, \alpha_2, \alpha_3

, and

\alpha_4

are all the roots of the equation

f(x)=0

, then \left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2

is equal to ________.

Correct answer

Step-by-step solution

f(1)=1+a+b+c=-9 \quad \Rightarrow \quad a+b+c=-10.....(1)

4 x^3+3 a x^2+2 b x=0

roots are

\sqrt{3} i,-\sqrt{3} i, 0

\Rightarrow4 x^2+3 a x+2 b=0 < ^{\sqrt{3i}}_{-{\sqrt{3i}}}

\Rightarrow \quad \mathrm{a}=0 ~\&~ \frac{2 \mathrm{~b}}{4}=(\sqrt{3} \mathrm{i})(-\sqrt{3} \mathrm{i})

\mathrm{b}=6

use

\mathrm{a}, \mathrm{b}

in (1)

\Rightarrow \mathrm{c}=-16

\begin{array}{ll}\Rightarrow & \mathrm{f}(\mathrm{x})=\mathrm{x}^4+6 \mathrm{x}^2-16=0 \\ & \left(\mathrm{x}^2+8\right)\left(\mathrm{x}^2-2\right)=0 \\ \Rightarrow \quad & \mathrm{x}= \pm \sqrt{8} \mathrm{i}, \pm \sqrt{2}\end{array}

\Rightarrow \quad\left|\alpha_1\right|^2+\left|\alpha_2\right|^2+\left|\alpha_3\right|^2+\left|\alpha_4\right|^2=20

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