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1

IIT-JEE 1995

Subjective
Let $${I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.$$ Use mathematical induction to prove that $${I_m} = m\,\pi ,m = 0,1,2,........$$

Answer

Solve it.
2

IIT-JEE 1994

Subjective
In what ratio does the $$x$$-axis divide the area of the region
bounded by the parabolas $$y = 4x - {x^2}$$ and $$y = {x^2} - x?$$

Answer

$$121:4$$
3

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

$$2n + 1 - \cos \gamma $$
4

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

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

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Class 12