Find the value of \(\alpha \) for which the coefficients of the middle terms in the expansions of \({ \left( 1+\alpha x \right) }^{ 4 }\)and \({ \left( 1-\alpha x \right) }^{ 6 }\)are equal.

Find the coefficient of x⁷ in \({ \left( { ax }^{ 2 }+\frac { 1 }{ bx } \right) }^{ 11 }\)and x^-7 in \({ \left( { ax }-\frac { 1 }{ b{ x }^{ 2 } } \right) }^{ 11 }\)and find the relation between a and b so that these coefficients are equal.

Prove that there is no term involving x⁸ in the expansion of\({ \left( { 2x }^{ 2 }-\frac { 3 }{ x } \right) }^{ 11 }\).

Prove that (1+y+y²+...)² = (1+2y+3y²+...)

Show that the ratio of the coefficient of x¹⁰ in (1-x²)¹⁰ and the term independent of x in \(\left( x-\frac { 2 }{ x } \right) ^{ 10 }\) is 1:32.

Find the expansion of (3x² - 2ax + 3a²)³ using binomial theorem.

Find the middle terms in the expansion of \(\left( 2x-\frac { { x }^{ 2 } }{ 6 } \right) ^{ 9 }\)

In the expansion of (x + a)ⁿ, sums of odd and even terms are P and Q respectively, Prove that
(i) 2 (P² + Q²) = (x + a)²ⁿ + (x - a)²ⁿ
(ii) p² - Q² = (x² - a²)ⁿ

If the fourth term in the expansion of \(\left( ax+\frac { 1 }{ x } \right) ^{ n }is \frac { 20 }{ 27 } \) then find the value of a and n.

The coefficients of three consecutive terms in the expansion of (1 + x)ⁿ are in the ratio 1 : 6 : 30. Find n.

The coefficient of (m + 1) ^th term in the expansion of (1 +x)²ⁿ is equal to the coefficient of (m + 3)^th term. Show that m + 1 = n.