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JEE Advanced 2007Paper 1
Exam
JEE Advanced
Year
2007
Paper
Paper 1
Subject
Mathematics
Chapter
Quadratic Equation

JEE Advanced 2007 · Paper 1 · Mathematics · Quadratic Equation

Let α, β be the roots of the equation x^2-p x+r=0 and (α)/(2), 2 β...

Let α,β\alpha, \beta be the roots of the equation x2−px+r=0x^2-p x+r=0 and α2,2β\frac{\alpha}{2}, 2 \beta be the roots of the equation x2−qx+r=0x^2-q x+r=0. Then, the value of rr is

Options

  1. A29(p−q)(2q−p)\frac{2}{9}(p-q)(2 q-p)
  2. B29(q−p)(2p−q)\frac{2}{9}(q-p)(2 p-q)
  3. C29(q−2p)(2q−p)\frac{2}{9}(q-2 p)(2 q-p)
  4. D29(2p−q)(2q−p)\frac{2}{9}(2 p-q)(2 q-p)

Correct answer

D. 29(2p−q)(2q−p)\frac{2}{9}(2 p-q)(2 q-p)

Step-by-step solution

The equation x2−px+r=0x^2-p x+r=0 has roots (α,β)(\alpha, \beta) and the equation x2−qx+r=0x^2-q x+r=0 has roots (α2,2β).⇒r=αβ and α+β=p and α2+2β=q⇒β=2q−p3 and α=2(2p−q)3⇒αβ=r=29(2q−p)(2p−q)\begin{array}{ll} \left(\frac{\alpha}{2}, 2 \beta\right) . & \\ \Rightarrow & r=\alpha \beta \text{ and } \alpha+\beta=p \text{ and } \frac{\alpha}{2}+2 \beta=q \\ \Rightarrow & \beta=\frac{2 q-p}{3} \text{ and } \alpha=\frac{2(2 p-q)}{3} \\ \Rightarrow & \alpha \beta=r=\frac{2}{9}(2 q-p)(2 p-q) \end{array}
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