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JEE Main 201219 MayOnline
Exam
JEE Main
Year
2012
Date
19 May
Mode
Online
Subject
Mathematics
Chapter
Quadratic Equation

JEE Main 2012 · 19 May · Online · Mathematics · Quadratic Equation

Let p, q, r ∈ R and r>p>0 . If the quadratic equation p x^2+q x+r=0...

Let p,q,r∈Rp, q, r \in R and r>p>0r>p>0. If the quadratic equation px2+qx+r=0p x^2+q x+r=0 has two complex roots α\alpha and β\beta, then ∣α∣+∣β∣|\alpha|+|\beta| is

Options

  1. Aequal to 1
  2. Bless than 2 but not equal to 1
  3. Cgreater than 2
  4. Dequal to 2

Correct answer

C. greater than 2

Step-by-step solution

Given quadratic equation is px2+qx+r=0D=q2−4pr\begin{aligned} & p x^2+q x+r=0 \\ & D=q^2-4 p r \end{aligned} Since α\alpha and β\beta are two complex root ∴β=αˉ⇒∣β∣=∣αˉ∣⇒∣β∣=∣α∣(∵∣αˉ∣=∣α∣) Consider ∣α∣+∣β∣=∣α∣+∣α∣(∵∣β∣=∣α∣)=2∣α∣>2.1=2(∵∣α∣>1)\begin{array}{ll} \therefore \beta=\bar{\alpha} \Rightarrow|\beta|=|\bar{\alpha}| \Rightarrow|\beta|=|\alpha| \\ (\because|\bar{\alpha}|=|\alpha|) \\ \text{ Consider } \\ |\alpha|+|\beta|=|\alpha|+|\alpha| & (\because|\beta|=|\alpha|) \\ =2|\alpha|>2.1=2 & (\because|\alpha|>1) \end{array} Hence, ∣α∣+∣β∣|\alpha|+|\beta| is greater than 2 .
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