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1

### IIT-JEE 1990

Subjective
Prove that for any positive integer $$k$$,
$${{\sin 2kx} \over {\sin x}} = 2\left[ {\cos x + \cos 3x + ......... + \cos \left( {2k - 1} \right)x} \right]$$
Hence prove that $$\int\limits_0^{\pi /2} {\sin 2kx\,\cot \,x\,dx = {\pi \over 2}}$$

Solve it.
2

### IIT-JEE 1989

Subjective
If $$f$$ and $$g$$ are continuous function on $$\left[ {0,a} \right]$$ satisfying
$$f\left( x \right) = f\left( {a - x} \right)$$ and $$g\left( x \right) + g\left( {a - x} \right) = 2,$$
then show that $$\int\limits_0^a {f\left( x \right)g\left( x \right)dx = \int\limits_0^a {f\left( x \right)dx} }$$

Solve it.
3

### IIT-JEE 1988

Subjective
Evaluate $$\int\limits_0^1 {\log \left[ {\sqrt {1 - x} + \sqrt {1 + x} } \right]dx}$$

$${1 \over 2}\left[ {\log 2 + {\pi \over 2} - 1} \right]$$
4

### IIT-JEE 1988

Subjective
Find the area of the region bounded by the curve $$C:y=$$
$$\tan x,$$ tangent drawn to $$C$$ at $$x = {\pi \over 4}$$ and the $$x$$-axis.

$${1 \over 2}\left[ {\log 2 - {1 \over 2}} \right]$$ sq. units

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