IIT-JEE 2003Question 406
If $f$ is an even function then prove that $$\int_0^{\pi/2}f(\cos2x)\cos x\,dx=\sqrt2\int_0^{\pi/4}f(\sin2x)\cos x\,dx.$$
Definite Integration
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IIT-JEE 2004Question 407
Prove by permutation or otherwise that $\dfrac{(n^{2})!}{(n!)^{n}}$ is an integer $(n\in I^{+})$.
Permutation and Combination
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IIT-JEE 2004Question 408
$A$ and $B$ are two independent events. $C$ is the event in which exactly one of $A$ or $B$ occurs. Prove that $P(C)\ge P(A\cup B)\,P\left(\overline{A}\cap\overline{B}\right)$.
Probability
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