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JEE Advanced 2009Paper 2
Exam
JEE Advanced
Year
2009
Paper
Paper 2
Subject
Mathematics
Chapter
Quadratic Equation

JEE Advanced 2009 · Paper 2 · Mathematics · Quadratic Equation

The smallest value of k , for which both the roots of the equation...

The smallest value of kk, for which both the roots of the equation x2−8kx+16(k2−k+1)=0x^2-8 k x+16\left(k^2-k+1\right)=0 are real, distinct and have values atleast 4 , is

Correct answer

2

Step-by-step solution

(i) Given, x2−8kx+16x^2-8 k x+16\left(k^2-k+1\right)=0Now,Now,\quad D=64\left\{k^2-\left(k^2-k+1\right)\right\} =64(k−1)>0∴k>116−32k+16(k2−k+1)≥0⇒k2−3k+2≥0⇒(k−2)(k−1)≥0⇒k≤1 or k≥2\begin{aligned} & =64(k-1)>0 \\ & \therefore \quad k>1 \\ & 16-32 k+16\left(k^2-k+1\right) \geq 0 \\ & \Rightarrow \quad k^2-3 k+2 \geq 0 \\ & \Rightarrow(k-2)(k-1) \geq 0 \Rightarrow k \leq 1 \text{ or } k \geq 2 \\ & \end{aligned} Hence,k=2
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