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

(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

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

(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

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

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35,20

45,35

35,45

20,65

answer is C.

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Detailed Solution

$(1 - y)^{m} (1 + y)^{n} = 1 + a_{1} y + a_{2} y^{2} + a_{3} y^{3} + \dots$ On differentiating w.r.t. y, we get $- m (1 - y)^{m - 1} (1 + y)^{n} + (1 - y)^{m} n (1 + y)^{n - 1}$

$= a_{1} + 2 a_{2} y + 3 a_{3} y^{2} + \dots - - - - (i)$

On putting y =0 in Eq. (i), we get $- m + n = a_{1} = 10$ $[∵ a_{1} = 10, given] \dots ..(ii)$

On again differentiating Eq. (i), we get

$\begin{aligned} - m [- (m - 1) (1 - y)^{m - 2} (1 + y)^{n} + (1 - y)^{m - 1} n (1 + y)^{n - 1}] \\ + n [- m (1 - y)^{m - 1} (1 + y)^{n - 1} + (1 - y)^{m} (n - 1) (1 + y)^{n - 2}] \end{aligned}$

$= 2 a_{2} + 6 a_{3} y + \dots . . . . . . . . . (iii)$

On putting Y= 0 in Eq. (iii), we get

$\begin{array}{l} - m [- (m - 1) + n] + n [- m + (n - 1)] = 2 a_{2} = 20 \\ \Rightarrow m (m - 1) - mn - mn + n (n - 1) = 20 \\ \Rightarrow m^{2} + n^{2} - m - n - 2 mn = 20 \\ \Rightarrow (m - n)^{2} - (m + n) = 20 \\ \Rightarrow 100 - (m + n) = 20 \\ \Rightarrow m + n = 80 - - - - - - - (iv) \end{array}$