For natural numbers m, n, if $(1-\mathrm{y}{)}^{\mathrm{m}}(1+\mathrm{y}{)}^{\mathrm{n}}=1+{\mathrm{a}}_1\mathrm{y}+{\mathrm{a}}_2{\mathrm{y}}^2+\dots$$\mathrm{and} {\mathrm{a}}_1={\mathrm{a}}_2=10, \mathrm{then} (\mathrm{m},\mathrm{n}) \mathrm{is}$

detailed solution

Correct option is C

(1−y)m(1+y)n=1+a1y+a2y2+a3y3+…On differentiating w.r.t. y, we get −m(1−y)m−1(1+y)n+(1−y)mn(1+y)n−1=a1+2a2y+3a3y2+…  ----(i)On putting y =0 in Eq. (i), we get −m+n=a1=10      ∵a1=10, given …..(ii) On again differentiating Eq. (i), we get−m−(m−1)(1−y)m−2(1+y)n+(1−y)m−1n(1+y)n−1+n−m(1−y)m−1(1+y)n−1+(1−y)m(n−1)(1+y)n−2=2a2+6a3y+….........(iii)On putting Y= 0 in Eq. (iii), we get −m[−(m−1)+n]+n[−m+(n−1)]=2a2=20⇒ m(m−1)−mn−mn+n(n−1)=20⇒ m2+n2−m−n−2mn=20⇒ (m−n)2−(m+n)=20⇒ 100−(m+n)=20⇒ m+n=80            -------(iv)On solving Eqs. (ii) and (iv), we get m=35and n=45