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1

### IIT-JEE 2003

Subjective
If $$f$$ is an even function then prove that
$$\int\limits_0^{\pi /2} {f\left( {\cos 2x} \right)\cos x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\sin 2x} \right)\cos x\,dx.}$$

Solve it.
2

### IIT-JEE 2002

Subjective
Find the area of the region bounded by the curves $$y = {x^2},y = \left| {2 - {x^2}} \right|$$ and $$y=2,$$ which lies to the right of the line $$x=1.$$

$$\left( {{{20} \over 3} - 4\sqrt 2 } \right)$$ sq. units.
3

### IIT-JEE 2001

Subjective
Let $$b \ne 0$$ and for $$j=0, 1, 2, ..., n,$$ let $${S_j}$$ be the area of
the region bounded by the $$y$$-axis and the curve $$x{e^{ay}} = \sin$$ by,
$${{jr} \over b} \le y \le {{\left( {j + 1} \right)\pi } \over b}.$$ Show that $${S_0},{S_1},{S_2},\,....,\,{S_n}$$ are in
geometric progression. Also, find their sum for $$a=-1$$ and $$b = \pi .$$

$${{\pi \left( {1 + e} \right)} \over {1 + {\pi ^2}}}\left( {{{{e^{n + 1}} - 1} \over {e - 1}}} \right)$$
4

### IIT-JEE 2000

Subjective
For $$x>0,$$ let $$f\left( x \right) = \int\limits_e^x {{{\ln t} \over {1 + t}}dt.}$$ Find the function
$$f\left( x \right) + f\left( {{1 \over x}} \right)$$ and show that $$f\left( e \right) + f\left( {{1 \over e}} \right) = {1 \over 2}.$$
Here, $$\ln t = {\log _e}t$$.

Solve it.

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