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 ___________.

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2
IIT-JEE 2010 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1
If the distance of the point $$P(1, -2, 1)$$ from the plane $$x+2y-2z$$$$\, = \alpha ,$$ where $$\alpha > 0,$$ is $$5,$$ then the foot of the perpendicular from $$P$$ to the planes is
A
$$\left( {{8 \over 3},{4 \over 3}, - {7 \over 3}} \right)$$
B
$$\left( {{4 \over 3},-{4 \over 3}, {1 \over 3}} \right)$$
C
$$\left( {{1 \over 3},{2 \over 3}, {10 \over 3}} \right)$$
D
$$\left( {{2 \over 3},-{1 \over 3}, {5 \over 3}} \right)$$
3
IIT-JEE 2010 Paper 2 Offline
Numerical
+4
-0
Two parallel chords of a circle of radius 2 are at a distance $$\sqrt 3 + 1$$ apart. If the chords subtend at the center , angles of $${\pi \over k}$$ and $${{2\pi } \over k},$$ where$$k > 0,$$ then the value of $$\left[ k \right]$$ is

[Note :[k] denotes the largest integer less than or equal to k ]

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4
IIT-JEE 2010 Paper 2 Offline
MCQ (Single Correct Answer)
+4
-1
For $$r = 0,\,1,....,$$ let $${A_r},\,{B_r}$$ and $${C_r}$$ denote, respectively, the coefficient of $${X^r}$$ in the expansions of $${\left( {1 + x} \right)^{10}},$$ $${\left( {1 + x} \right)^{20}}$$ and $${\left( {1 + x} \right)^{30}}.$$
Then $$\sum\limits_{r = 1}^{10} {{A_r}\left( {{B_{10}}{B_r} - {C_{10}}{A_r}} \right)} $$ is equal to
A
$$\left( {{B_{10}} - {C_{10}}} \right)$$
B
$${A_{10}}\left( {{B^2}_{10}{C_{10}}{A_{10}}} \right)$$
C
$$0$$
D
$${{C_{10}} - {B_{10}}}$$
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