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 ___________.

Your input ____
2
IIT-JEE 2010 Paper 2 Offline
MCQ (Single Correct Answer)
+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
Numerical
+4
-0

Let $k$ be a positive real number and let

$$ \begin{aligned} A & =\left[\begin{array}{ccc} 2 k-1 & 2 \sqrt{k} & 2 \sqrt{k} \\ 2 \sqrt{k} & 1 & -2 k \\ -2 \sqrt{k} & 2 k & -1 \end{array}\right] \text { and } \\\\ \mathbf{B} & =\left[\begin{array}{ccc} 0 & 2 k-1 & \sqrt{k} \\ 1-2 k & 0 & 2 \sqrt{k} \\ -\sqrt{k} & -2 \sqrt{k} & 0 \end{array}\right] . \end{aligned} $$

If $\operatorname{det}(\operatorname{adj} A)+\operatorname{det}(\operatorname{adj} B)=10^6$, then $[k]$

is equal to _________.

[ Note : adj M denotes the adjoint of a square matrix M and $[k]$ denotes the largest integer less than or equal to $k$ ].

Your input ____
4
IIT-JEE 2010 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1
Let $S=\{1,2,3,4\}$. The total number of unordered pairs of disjoint subsets of $S$ is equal to :
A
25
B
34
C
42
D
41
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