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1

IIT-JEE 1991

Subjective
Sketch the curves and identify the region bounded by
$$x = {1 \over 2},x = 2,y = \ln \,x$$ and $$y = {2^x}.$$ Find the area of this region.

Answer

$$\,{{4 - \sqrt 2 } \over {\log 2}} - {5 \over 2}\log 2 + {3 \over 2}$$
2

IIT-JEE 1991

Subjective
Evaluate $$\,\int\limits_0^\pi {{{x\,\sin \,2x\,\sin \left( {{\pi \over 2}\cos x} \right)} \over {2x - \pi }}dx} $$

Answer

$${8 \over {{\pi ^2}}}$$
3

IIT-JEE 1990

Subjective
Compute the area of the region bounded by the curves $$\,y = ex\,\ln x$$ and $$y = {{\ln x} \over {ex}}$$ where $$ln$$ $$e=1.$$

Answer

$${{{e^2} - 5} \over {4e}}$$
4

IIT-JEE 1990

Subjective
Show that $$\int\limits_0^{\pi /2} {f\left( {\sin 2x} \right)\sin x\,dx = \sqrt 2 } \int\limits_0^{\pi /4} {f\left( {\cos 2x} \right)\cos x\,dx} $$

Answer

Solve it.

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Class 12