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1

IIT-JEE 2008

MCQ (Single Correct Answer)
Consider the two curves $${C_1}:{y^2} = 4x,\,{C_2}:{x^2} + {y^2} - 6x + 1 = 0$$. Then,
A
$${C_1}$$ and $${C_2}$$ touch each other only at one point.
B
$${C_1}$$ and $${C_2}$$ touch each other exactly at two points
C
$${C_1}$$ and $${C_2}$$ intersect (but do not touch ) at exactly two points
D
$${C_1}$$ and $${C_2}$$ neither intersect nor touch each other
2

IIT-JEE 2007

MCQ (Single Correct Answer)
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 2007

MCQ (Single Correct Answer)
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
$${\log _e}2$$
4

IIT-JEE 2007

MCQ (Single Correct Answer)
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 line $$y=x$$ meets $$y = k{e^x}$$ for $$k \le 0$$ at

A
no point
B
one point
C
two points
D
more than two points

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