For natural numbers m- n if 1-ym1-yn-1-a1 y-a2 y2- and a1-a2-10- then m- n is A- 45-35B- 35-45C- 20-45D- 35-20

The correct option is B ( 1 y )^m ( 1 + y )^n= 1 + a_1y+... +a_2y^2+... ⇒ (1 ^mC_1y+^mC_2y^2 ⋯)(1+^nC_1y+^nC 2y^2+⋯)=1+a_1y+a_2y^2+⋯ ⇒ a_1 = ^nC_1 ^mC_1, ...

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Byju's Answer

Standard XII

Mathematics

Middle Terms

For natural n...

Question

For natural numbers m, n if ${(1 - y)}^{m} {(1 + y)}^{n} = 1 + a_{1} y + a_{2} y^{2} + . . . a n d a_{1} = a_{2} = 10, t h e n (m, n)$ is

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Solution

The correct option is B
${(1 - y)}^{m} {(1 + y)}^{n} = 1 + a_{1} y + . . . + a_{2} y^{2} + . . .$
$\Rightarrow (1 -^{m} C_{1} y +^{m} C_{2} y^{2} - \dots) (1 +^{n} C_{1} y +^{n} C - 2 y^{2} + \dots) = 1 + a_{1} y + a_{2} y^{2} + \dots$
$\Rightarrow a_{1} =^{n} C_{1} -^{m} C_{1}, a_{2} =^{m} C_{1}^{n} C_{1} +^{n} C_{2} \Rightarrow n - m = 10, m^{2} + n^{2} - m - n - 2 m n = 20$
$\Rightarrow n - m = 10, {(m - n)}^{2} - (m + n) = 20$
$\Rightarrow n - m = 10, n + m = 10, n + m = 80 \Rightarrow m = 35, n = 45.$

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For natural numbers m, n if (1−y)m(1+y)n=1+a1y+a2y2+...and a1=a2=10,then(m,n) is

The correct option is B (1−y)m(1+y)n=1+a1y+...+a2y2+... ⇒(1−mC1y+mC2y2−⋯)(1+nC1y+nC−2y2+⋯)=1+a1y+a2y2+⋯ ⇒a1=nC1−mC1,a2=mCn1C1+nC2⇒n−m=10,m2+n2−m−n−2mn=20 ⇒n−m=10,(m−n)2−(m+n)=20 ⇒n−m=10,n+m=10,n+m=80⇒m=35,n=45.

For natural numbers m, n if ${(1 - y)}^{m} {(1 + y)}^{n} = 1 + a_{1} y + a_{2} y^{2} + . . . a n d a_{1} = a_{2} = 10, t h e n (m, n)$ is