1

### IIT-JEE 2005 Screening

If $$f(x)$$ is a twice differentiable function and given that $$f\left( 1 \right) = 1;f\left( 2 \right) = 4,f\left( 3 \right) = 9$$, then
A
$$f''\left( x \right) = 2$$ for $$\forall x \in \left( {1,3} \right)$$
B
$$f''\left( x \right) = f'\left( x \right) = 5$$ for some $$x \in \left( {2,3} \right)$$
C
$$f''\left( x \right) = 3$$ for $$\forall x \in \left( {2,3} \right)$$
D
$$f''\left( x \right) = 2$$ for some $$x \in \left( {1,3} \right)$$
2

### IIT-JEE 2004 Screening

If $$y$$ is a function of $$x$$ and log $$(x+y)-2xy=0$$, then the value of $$y'(0)$$ is equal to
A
$$1$$
B
$$-1$$
C
$$2$$
D
$$0$$
3

### IIT-JEE 2001 Screening

Let $$f:\left( {0,\infty } \right) \to R$$ and $$F\left( x \right) = \int\limits_0^x {f\left( t \right)dt.}$$ If $$F\left( {{x^2}} \right) = {x^2}\left( {1 + x} \right)$$, then $$f(4)$$ equals
A
$$5/4$$
B
$$7$$
C
$$4$$
D
$$2$$
4

### IIT-JEE 2000

If $${x^2} + {y^2} = 1$$ then
A
$$yy'' - 2{\left( {y'} \right)^2} + 1 = 0$$
B
$$yy'' + {\left( {y'} \right)^2} + 1 = 0$$
C
$$yy'' + {\left( {y'} \right)^2} - 1 = 0$$
D
$$yy'' + 2{\left( {y'} \right)^2} + 1 = 0$$

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