1
IIT-JEE 2007
MCQ (Single Correct Answer)
+4
-1
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.

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

A
$${1 \over e}$$
B
$$1$$
C
$$e$$
D
$${\log _e}2$$
2
IIT-JEE 2007
MCQ (Single Correct Answer)
+4
-1
If a continuous function $$f$$ defined on the real line $$R$$, assumes positive and negative values in $$R$$ then the equation $$f(x)=0$$ has a root in $$R$$. For example, if it is known that a continuous function $$f$$ on $$R$$ is positive at some point and its minimum value is negative then the equation $$f(x)=0$$ has a root in $$R$$.
Consider $$f\left( x \right) = k{e^x} - x$$ for all real $$x$$ where $$k$$ is real constant.

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

A
$$\left( {0,{1 \over e}} \right)$$
B
$$\left( {{1 \over e},1} \right)$$
C
$$\left( {{1 \over e},\infty } \right)$$
D
$$\left( {0,1} \right)$$
3
IIT-JEE 2005 Screening
MCQ (Single Correct Answer)
+2
-0.5
If $$P(x)$$ is a polynomial of degree less than or equal to $$2$$ and $$S$$ is the set of all such polynomials so that $$P(0)=0$$, $$P(1)=1$$ and $$P'\left( x \right) > 0\,\,\forall x \in \left[ {0,1} \right],$$ then
A
$$S = \phi $$
B
$$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,2} \right)$$
C
$$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,\infty } \right)$$
D
$$S = ax + \left( {1 - a} \right){x^2}\,\,\forall \,a \in \left( {0,1} \right)$$
4
IIT-JEE 2004 Screening
MCQ (Single Correct Answer)
+2
-0.5
If $$f\left( x \right) = {x^3} + b{x^2} + cx + d$$ and $$0 < {b^2} < c,$$ then in $$\left( { - \infty ,\infty } \right)$$
A
$$f\left( x \right)$$ is a strictly increasing function
B
$$f\left( x \right)$$ has a local maxima
C
$$f\left( x \right)$$ is a strictly decreasing function
D
$$f\left( x \right)$$ is bounded
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