The smallest value of k , for which both the roots of the equation x 2 - 8 k x + 16 ( k 2 - k + 1 ) = 0 are real, distinct and have values at least 4, is [2009]

Question

The smallest value of $k$ , for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least 4, is [2009]

Smart Achievers · Accepted Answer

(2)

The given equation is $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$

$∵$ $Both the roots are real and distinct.$

$∴$ $D > 0 \Rightarrow {(8 k)}^{2} - 4 \times 16 (k^{2} - k + 1) > 0$

$\Rightarrow k > 1 ...(i)$

$∵$ $Both the roots are greater than or equal to 4$

$∴$ $α + β > 8 and f (4) \geq 0$

$\Rightarrow k > 1 ...(ii)$

$and 16 - 32 k + 16 (k^{2} - k + 1) \geq 0$

$\Rightarrow k^{2} - 3 k + 2 \geq 0 \Rightarrow (k - 1) (k - 2) \geq 0$

$\Rightarrow k \in (- \infty, 1] \cup [2, \infty) ...(iii)$

$Combining (i), (ii) and (iii), we get k \geq 2$

$∴$ $Smallest value of k = 2$

Solution

Q.
The smallest value of $k$ , for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least 4, is [2009]

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Solution

Q. The smallest value of k, for which both the roots of the equation x2-8kx+16(k2-k+1)=0 are real, distinct and have values at least 4, is [2009]

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Q.
The smallest value of $k$ , for which both the roots of the equation $x^{2} - 8 k x + 16 (k^{2} - k + 1) = 0$ are real, distinct and have values at least 4, is [2009]