1
IIT-JEE 2010 Paper 2 Offline
Numerical
+4
-0
Let $$f$$ be a function defined on $$R$$ (the set of all real numbers)
such that $$f'\left( x \right) = 2010\left( {x - 2009} \right){\left( {x - 2010} \right)^2}{\left( {x - 2011} \right)^3}{\left( {x - 2012} \right)^4}$$ for all $$x \in$$$$R$$

If $$g$$ is a function defined on $$R$$ with values in the interval $$\left( {0,\infty } \right)$$ such that $$f\left( x \right) = ln\,\left( {g\left( x \right)} \right),\,\,for\,\,all\,\,x \in R$$\$
then the number of points in $$R$$ at which $$g$$ has a local maximum is $$1$$.

2
IIT-JEE 2010 Paper 2 Offline
+4
-1
Let $$f$$ be a real-valued function defined on the interval $$(-1, 1)$$ such that
$${e^{ - x}}f\left( x \right) = 2 + \int\limits_0^x {\sqrt {{t^4} + 1} \,\,dt,}$$ for all $$x \in \left( { - 1,1} \right)$$,
and let $${f^{ - 1}}$$ be the inverse function of $$f$$. Then $$\left( {{f^{ - 1}}} \right)'\left( 2 \right)$$ is equal to
A
$$1$$
B
$${{1 \over 3}}$$
C
$${{1 \over 2}}$$
D
$${{1 \over e}}$$
3
IIT-JEE 2010 Paper 2 Offline
+4
-1

Consider the polynomial
$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$
Let $$s$$ be the sum of all distinct real roots of $$f(x)$$ and let $$t = \left| s \right|.$$

The real numbers lies in the interval

A
$$\left( { - {1 \over 4},0} \right)$$
B
$$\left( { - 11, - {3 \over 4}} \right)$$
C
$$\left( { - {3 \over 4}, - {1 \over 2}} \right)$$
D
$$\left( {0,{1 \over 4}} \right)$$
4
IIT-JEE 2010 Paper 2 Offline
+4
-1

Consider the polynomial
$$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$
Let $$s$$ be the sum of all distinct real roots of $$f(x)$$ and let $$t = \left| s \right|.$$

The area bounded by the curve $$y=f(x)$$ and the lines $$x=0,$$ $$y=0$$ and $$x=t,$$ lies in the interval

A
$$\left( {{3 \over 4},3} \right)$$
B
$$\left( {{{21} \over {64}},{{11} \over {16}}} \right)$$
C
$$\left( {9,10} \right)$$
D
$$\left( {0,{{21} \over {64}}} \right)$$
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