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Standard X

Mathematics

Nature of Roots

The value of ...

Question

The value of $k$ for which the equation $x^{2} + 2 (k - 1) x + k + 5 = 0$ passes atleast one positive root are

[4, \infty)

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(- \infty, - 1) \cup (4, \infty)

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- 1,

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[- \infty, - 1]

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Solution

The correct option is C $- 1,$
By quadratic formula, larger root is
$\begin{matrix} \frac{- 2 (k - 1) + \sqrt{4 {(k - 1)}^{2} - 4 (k + 5)}}{2} \Rightarrow - k + 1 + \sqrt{k^{2} - 3 k - 4} \to (i) \end{matrix}$
The value of D ( part under root) must be $\geq 0$
So, k must be $\geq 4$
Now, if $\geq 4$ , then equation 1 is >0 and we require $k^{2} - 2 k - 4 > {(k - 1)}^{2}$ which is contradiction and if $k ⩽ - 1$ then, equation 1 is positive as required.
So, required value of $k ⩽ - 1$

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The value of k for which the equation x2+2(k−1)x+k+5=0 passes atleast one positive root are

The value of $k$ for which the equation $x^{2} + 2 (k - 1) x + k + 5 = 0$ passes atleast one positive root are