BdMO National Higher Secondary :Problem Collection(2016)
Posted: Fri Feb 16, 2018 12:10 am
You will get question $1,2,3,4,5,6,7,8$ herehttp://matholympiad.org.bd/forum/viewto ... 878#p17476
Now the number 9 and The Final Question
9.
The integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$
(a)(3 POINTS:)Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$
Where $j$ is not a function of $x$,is $Z(j)=e^{j^{2}/4a} Z(0)$
(b)(10 POINTS):Show that,
$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n}$
Where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times...\times3\times 1$
(c)(7 POINTS):What is the number of ways to form $n$ pairs from $2n$ distinct objects?Interept the previous part of the problem in term of this answer.
[It was a 200 number exam,and this is one of the toughest problems.]
Now the number 9 and The Final Question
9.
The integral $Z(0)=\int^{\infty}_{-\infty} dx e^{-x^2}= \sqrt{\pi}$
(a)(3 POINTS:)Show that the integral $Z(j)=\int^{\infty}_{-\infty} dx e^{-x^{2}+jx}$
Where $j$ is not a function of $x$,is $Z(j)=e^{j^{2}/4a} Z(0)$
(b)(10 POINTS):Show that,
$\dfrac 1 {Z(0)}=\int x^{2n} e^{-x^2}= \dfrac {(2n-1)!!}{2^n}$
Where $(2n-1)!!$ is defined as $(2n-1)(2n-3)\times...\times3\times 1$
(c)(7 POINTS):What is the number of ways to form $n$ pairs from $2n$ distinct objects?Interept the previous part of the problem in term of this answer.
[It was a 200 number exam,and this is one of the toughest problems.]
