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1

IIT-JEE 1992

Subjective
Determine a positive integer $$n \le 5,$$ such that $$$\int\limits_0^1 {{e^x}{{\left( {x - 1} \right)}^n}dx = 16 - 6e} $$$

Answer

$$n = 3$$
2

IIT-JEE 1992

Subjective
Sketch the region bounded by the curves $$y = {x^2}$$ and
$$y = {2 \over {1 + {x^2}}}.$$ Find the area.

Answer

$$\left( {\pi - {2 \over 3}} \right)$$ sq. units
3

IIT-JEE 1991

Subjective
If $$'f$$ is a continuous function with $$\int\limits_0^x {f\left( t \right)dt \to \infty } $$ as $$\left| x \right| \to \infty ,$$ then show that every line $$y=mx$$
intersects the curve $${y^2} + \int\limits_0^x {f\left( t \right)dt = 2!} $$

Answer

Solve it.
4

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

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

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