IIT-JEE 2007 · Paper 2 · Mathematics Q63

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MathematicsSingle correct+4 / -1Question 63 of 66

Passage

Paragraph for Question Nos. 61 to 63

If a continuous function $f$ defined on the real line $\mathbf R$ , assumes positive and negative values in $\mathbf R$ then the equation $f(x)=0$ has a root in $\mathbf R$ . For example, if it is known that a continuous function $f$ on $\mathbf R$ is positive at some point and its minimum value is negative then the equation $f(x)=0$ has a root in $\mathbf R$ .

Consider $f(x)=ke^x-x$ for all real $x$ where $k$ is a real constant.

For $k>0$ , the set of all values of $k$ for which $ke^x-x=0$ has two distinct roots is

\left(0,\frac1e\right)

\left(\frac1e,1\right)

\left(\frac1e,\infty\right)

(0,1)

Answer and worked solution

Official answer: A

Two roots. With $k>0$ , $f\to+\infty$ as $x\to\pm\infty$ , so there are two roots exactly when the minimum value $1+\ln k$ is negative:
$\ln k<-1\iff0<k<\frac1e.$

(A)

Topics:Maxima and minima

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IIT-JEE 2007 · Paper 2, Mathematics question 63