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1

IIT-JEE 1994

Subjective
Show that $$\int\limits_0^{n\pi + v} {\left| {\sin x} \right|dx = 2n + 1 - \cos \,v} $$ where $$n$$ is a positive integer and $$\,0 \le v < \pi .$$

Answer

$$2n + 1 - \cos \gamma $$
2

IIT-JEE 1993

Subjective
Evaluate $$\int_2^3 {{{2{x^5} + {x^4} - 2{x^3} + 2{x^2} + 1} \over {\left( {{x^2} + 1} \right)\left( {{x^4} - 1} \right)}}} dx.$$

Answer

$${1 \over 6}\log 6 - {1 \over {10}}$$
3

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

$$n = 3$$
4

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

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Class 12