SATHEE: Theory of Equations 5 Question 15

Theory of Equations 5 Question 15

15. The positive value of $k$ for which $k e^{x} - x = 0$ has only one root is

(a) $\frac{1}{e}$

(b) 1

(d) $\log_{e} 2$

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Answer:

Correct Answer: 15. (a)

Solution:

Let $f (x) = k e^{x} - x$

$\begin{aligned} f^{'} (x) & = k e^{x} - 1 = 0 \\ \Rightarrow & x & = - \ln k \end{aligned}$

$\begin{aligned} f^{''} (x) & = k e^{x} \\ ∴ {[f^{''} (x)]}_{x = - \ln k} & = 1 > 0 \end{aligned}$

Hence, $f (- \ln k) = 1 + \ln k$

For one root of given equation

$1 + \ln k = 0$

$\Rightarrow k = \frac{1}{e}$

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Theory of Equations 5 Question 15

15. The positive value of k for which kex−x=0 has only one root is

Answer:

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15. The positive value of $k$ for which $k e^{x} - x = 0$ has only one root is