Write the general term in the expansion of x 2 - 1 x 12 , x ≠ 0 .

Given, x 2 - 1 x 12 General term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ General term in the expansion of x 2 - 1 x 12 = C r 12 x 2 12 - r - 1 x r = ( - 1 ) r C r 12 x 24 - 2 r x - r = ( - 1 ) r C r 12 x 24 - 3 r .

Find the 3 r d term in the expansion of 3 x 2 - 2 x 8 .

Given, 3 x 2 - 2 x 8 ( r + 1 ) t h term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ 3 r d term in the expansion of 3 x 2 - 2 x 8 = C 2 8 3 x 2 8 - 2 - 2 x 2 = 8 ! 2 ! 6 ! · 3 6 x 12 - 2 2 x - 2 ∵ C r n = n ! r ! ( n - r ) ! = 8 × 7 2 × 1 · 729 x 10 4 ∴ 3 r d term in the expansion of 3 x 2 - 2 x 8 = 81648 x 10 .

Find the 10 t h term in the expansion of 2 x 2 + 1 x 12 .

Given, 2 x 2 + 1 x 12 ( r + 1 ) t h term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ 10 t h term in the expansion of 2 x 2 + 1 x 12 = C 9 12 2 x 2 12 - 9 1 x 9 = 12 ! 913 ! · 2 3 x 6 · x - 9 ∵ C r n = n ! r ! ( n - r ) ! = 12 × 11 × 10 3 × 2 × 1 · 8 x - 3 ∴ 4 t h term in the expansion of 2 x 2 + 1 x 12 = 1760 x - 3 .

Find the fourth term from the end in the expansion of x 3 2 - 2 x 2 9 .

Given, x 3 2 - 2 x 2 9 ( r + 1 ) t h term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ 4 t h term in the expansion of x 3 2 - 2 x 2 9 = C 3 9 x 3 2 3 - 2 x 2 9 - 3 = 9 ! 3 ! 6 ! · x 9 8 · 64 x 12 ∵ C r n = n ! r ! ( n - r ) ! = 9 × 8 × 7 3 × 2 × 1 · 8 x 3 ∴ 4 t h term from end in the expansion of x 3 2 - 2 x 2 9 = 672 x - 3 .

Find the middle term in the expansion of: x 2 - 1 x 6 .

Given, x 2 - 1 x 6 Middle term in the expansion of ( a + b ) n = n 2 + 1 t h term, when n is even. = n + 1 2 t h and n + 3 2 t h terms, when n is odd. ∴ Middle term in the expansion of x 2 - 1 x 6 is 6 2 + 1 t h term = 4 t h term ( r + 1 ) t h term in the expansion of ( a + b ) n is C r n a n - r b r . ∴ 4 t h term in the expansion of x 2 - 1 x 6 = C 3 6 x 2 3 - 1 x 3 = 6 ! 313 ! x 3 - 1 ∵ C r n = n ! r ! ( n - r ) ! = 6 × 5 × 4 3 × 2 × 1 - x 3 = - 20 x 3 ∴ Middle term in the expansion of x 2 - 1 x 6 = - 20 x 3 .