If A denotes the sum of all the coefficients in the expansion of ( 1 - 3 x + 10 x 2 ) n and B denotes the sum of all the coefficients in the expansion of ( 1 + x 2 ) n , then [2024]

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Solution

Q.
If $A$ denotes the sum of all the coefficients in the expansion of ${(1 - 3 x + 10 x^{2})}^{n}$ and $B$ denotes the sum of all the coefficients in the expansion of ${(1 + x^{2})}^{n},$ then [2024]

1 $B = A^{3}$

2 $3 A = B$

3 $A = 3 B$

4 $A = B^{3}$

Ans.
(4)

Putting $x = 1$ in ${(1 - 3 x + 10 x^{2})}^{n},$ we get

$A = {(1 - 3 + 10)}^{n} = 8^{n}$ ...(i)

Putting $x = 1$ in ${(1 + x^{2})}^{n},$ we get

$B = {(1 + 1)}^{n} = 2^{n}$ ...(ii)

From (i) and (ii), we get $A = B^{3}$

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