1

### IIT-JEE 2010 Paper 2 Offline

Numerical
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 2009

Numerical
The maximum value of the function
$$f\left( x \right) = 2{x^3} - 15{x^2} + 36x - 48$$ on the set
$${\rm A} = \left\{ {x|{x^2} + 20 \le 9x} \right\}$$ is

3

### IIT-JEE 2009

Numerical
The maximum value of the function
$$f\left( x \right) = 2{x^3} - 15{x^2} + 36x - 48$$ on the set
$${\rm A} = \left\{ {x|{x^2} + 20 \le 9x} \right\}$$ is

4

### IIT-JEE 2009

Numerical
Let $$p(x)$$ be a polynomial of degree $$4$$ having extremum at
$$x = 1,2$$ and $$\mathop {\lim }\limits_{x \to 0} \left( {1 + {{p\left( x \right)} \over {{x^2}}}} \right) = 2$$.

Then the value of $$p (2)$$ is

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