1
IIT-JEE 1997
Subjective
+5
-0
Let $$f(x)= Maximum$$ $$\,\left\{ {{x^2},{{\left( {1 - x} \right)}^2},2x\left( {1 - x} \right)} \right\},$$ where $$0 \le x \le 1.$$
Determine the area of the region bounded by the curves
$$y = f\left( x \right),$$ $$x$$-axes, $$x=0$$ and $$x=1.$$
2
IIT-JEE 1996
Subjective
+3
-0
Let $${A_n}$$ be the area bounded by the curve $$y = {\left( {\tan x} \right)^n}$$ and the
lines $$x=0,$$ $$y=0,$$ and $$x = {\pi \over 4}.$$ Prove that for $$n > 2,$$
$${A_n} + {A_{n - 2}} = {1 \over {n - 1}}$$ and deduce $${1 \over {2n + 2}} < {A_n} < {1 \over {2n - 2}}.$$
3
IIT-JEE 1995
Subjective
+5
-0
Let $${I_m} = \int\limits_0^\pi {{{1 - \cos mx} \over {1 - \cos x}}} dx.$$ Use mathematical induction to prove that $${I_m} = m\,\pi ,m = 0,1,2,........$$
4
IIT-JEE 1995
Subjective
+5
-0
Evaluate the definite integral : $$\int\limits_{ - 1/\sqrt 3 }^{1/\sqrt 3 } {\left( {{{{x^4}} \over {1 - {x^4}}}} \right){{\cos }^{ - 1}}\left( {{{2x} \over {1 + {x^2}}}} \right)} dx$$\$
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