Find the coefficient of x - 6 in 3 x - 4 x 10 .

Given, 3 x - 4 x 10 r + 1 th term in the expansion of a + b n is given by T r + 1 = C r n a n - r b r . The general term in 3 x - 4 x 10 is T r + 1 = - 1 r C r 10 3 x 10 - r 4 x r ⇒ T r + 1 = - 1 r · C r 10 3 10 - r ( 4 ) r x 10 - 2 r ...(i) For coefficient of x - 6 , put 10 - 2 r = - 6 ⇒ 2 r = 10 + 6 ⇒ 2 r = 16 ⇒ r = 8 Put r = 8 in (i), we get T 8 + 1 = ( - 1 ) r · 10 C 8 3 10 - 8 ( 4 ) 8 x 10 - 16 = C 8 10 3 2 4 8 x - 6 ∴ Coefficient of x - 6 in 3 x - 4 x 10 is C 8 10 3 2 4 8 = C 2 10 3 2 4 8 = 10 × 9 1 × 2 × 9 × 4 8 = 405 × 4 8

Find the coefficient of x 11 in 2 x 2 + 3 x 3 13 .

Given, 2 x 2 + 3 x 3 13 The general term in 2 x 2 + 3 x 3 13 is T r + 1 = C r 13 2 x 2 13 - r 3 x 3 r = C r 13 ( 2 ) 13 - r 3 r x 26 - 2 r x - 3 r ⇒ T r + 1 = C r 13 ( 2 ) 13 - r ( 3 ) r x 26 - 5 r ...(i) For coefficient of x 11 , put 26 - 5 r = 11 ⇒ 5 r = 15 ⇒ r = 3 Put r = 3 in (i), we get T 3 + 1 = C 3 13 ( 2 ) 10 ( 3 ) 3 x 26 - 15 ⇒ T 4 = 13 × 12 × 11 1 × 2 × 3 · 2 10 · 3 3 · x 11 ∴ Coefficient of x 11 in 2 x 2 + 3 x 3 13 is ( 286 ) 2 10 3 3

Find the coefficient of x 2 in 7 x 3 - 2 x 2 9 .

Given, 7 x 3 - 2 x 2 9 The general term in 7 x 3 - 2 x 2 9 is T r + 1 = - 1 r · C r 9 7 x 3 9 - r 2 x 2 r = - 1 r · C r 9 7 9 - r 2 r x 27 - 3 r x - 2 r ⇒ T r + 1 = - 1 r · C r 9 7 9 - r 2 r x 27 - 5 r ...(i) For coefficient x 2 , put 27 - 5 r = 2 ⇒ 5 r = 25 ⇒ r = 5 Put r = 5 in (i), we get T 5 + 1 = - 1 5 · C 5 9 7 4 2 5 x 27 - 25 = - 9 × 8 × 7 × 6 × 5 5 × 4 × 3 × 2 × 1 7 4 2 5 x 2 ∴ Coefficient of x - 7 in 2 x 2 3 - 5 4 x 5 7 is - 126 × 7 4 × 2 5

Which term is independent of x in the expansion of x 3 - 4 x 2 10 and find the value of term.

Given, x 3 - 4 x 2 10 The general term in x 1 2 3 - 4 x 2 10 is T r + 1 = - 1 r · C r 10 x 1 2 3 10 - r 4 x 2 r = - 1 r · C r 10 1 3 10 - r 4 r · x 5 - r 2 · x - 2 r = - 1 r · C r 10 1 3 10 - r 4 r · x 5 - r 2 - 2 r T r + 1 = - 1 r · C r 10 1 3 10 - r 4 r · x 10 - 5 r 2 ...(i) For the term independent of x , put 10 - 5 r 2 = 0 ⇒ 5 r = 10 ⇒ r = 2 Put r = 2 in (i), we get T 2 + 1 = - 1 2 · C 2 10 1 3 8 4 2 · x 0 ∴ Term independent of x is T 3 = 80 729 .

Find the term independent of x in the expansion of 3 x 3 + 5 x 25 .

Given, 3 x 3 + 5 x 25 The general term in 3 x 3 + 5 x 25 is T r + 1 = C r 25 3 x 3 25 - r 5 x r = C r 25 3 25 - r 5 r · x - 1 3 ( 25 - r ) x r 2 = C r 25 3 25 - r 5 r · x - 25 3 + r 3 + r 2 ⇒ T r + 1 = C r 25 3 25 - r 5 r · x - 50 + 2 r + 3 r 6 ...(i) For term independent of x , put - 50 + 5 r 6 = 0 ⇒ 5 r = 50 ⇒ r = 10 Put r = 10 in (i), we get T 10 + 1 = C 10 25 ( 3 ) 15 ( 5 ) 10 x 0 ∴ Term independent of x is T 11 = C 10 25 ( 3 ) 15 ( 5 ) 10 .