Resolve the fraction x 2 ( x - 1 ) ( x - 2 ) into partial fraction.

Given, x 2 ( x - 1 ) ( x - 2 ) Let, x 2 ( x - 1 ) ( x - 2 ) = 1 + A x - 1 + B x - 2 ⇒ x 2 = ( x - 1 ) ( x - 2 ) + A ( x - 2 ) + B ( x - 1 ) ...(i) Put x = 1 in (i), ⇒ 1 = A ( - 1 ) ⇒ A = - 1 Put x = 2 in (i), ⇒ 4 = B ( 1 ) ⇒ B = 4 ∴ x 2 ( x - 1 ) ( x - 2 ) = 1 - 1 x - 1 + 4 x - 2

Resolve the fraction x 3 ( x - 1 ) ( x + 2 ) into partial fraction.

Given, x 3 ( x - 1 ) ( x + 2 ) = x 3 x 2 + x - 2 = x x 2 + x - 2 - 1 x 2 + x - 2 + 3 x - 2 x 2 + x - 2 = x - 1 + 3 x - 2 x 2 + x - 2 = x - 1 + 3 x - 2 ( x - 1 ) ( x + 2 ) Let, 3 x - 2 ( x - 1 ) ( x + 2 ) = A x - 1 + B x + 2 ⇒ 3 x - 2 = A ( x + 2 ) + B ( x - 1 ) ...(i) Put x = 1 in (i), ⇒ 1 = A ( 3 ) ⇒ A = 1 3 Put x = - 2 in (i), ⇒ - 8 = B ( - 3 ) ⇒ B = 8 3 ∴ x 3 ( x - 1 ) ( x + 2 ) = x - 1 + 1 3 ( x - 1 ) + 8 3 ( x + 2 )

Resolve x 3 ( 2 x - 1 ) ( x - 1 ) 2 into partial fraction.

Given, x 3 ( 2 x - 1 ) ( x - 1 ) 2 Let, x 3 ( 2 x - 1 ) ( x - 1 ) 2 = 1 2 + A 2 x - 1 + B x - 1 + C ( x - 1 ) 2 ⇒ 2 x 3 = ( 2 x - 1 ) ( x - 1 ) 2 + 2 A ( x - 1 ) 2 + 2 B ( 2 x - 1 ) ( x - 1 ) + 2 C ( 2 x - 1 ) ...(i) Put x = 1 2 in (i), ⇒ 2 1 8 = 2 A 1 4 ⇒ A = 1 2 Put x = 1 in (i), ⇒ 2 ( 1 ) = 2 C ( 1 ) ⇒ C = 1 Put x = 0 in (i), ⇒ 0 = ( - 1 ) ( 1 ) + 2 A ( 1 ) + 2 B ( - 1 ) ( - 1 ) + 2 C ( - 1 ) ⇒ 2 A + 2 B - 2 C = 1 ⇒ 2 B = 1 + 2 C - 2 A ⇒ 2 B = 1 + 2 - 1 ⇒ B = 1 ∴ x 3 ( 2 x - 1 ) ( x - 1 ) 2 = 1 2 + 1 2 ( 2 x - 1 ) + 1 ( x - 1 ) + 1 ( x - 1 ) 2

Resolve x 3 ( x - a ) ( x - b ) ( x - c ) into partial fraction.

Given, x 3 ( x - a ) ( x - b ) ( x - c ) Let, x 3 ( x - a ) ( x - b ) ( x - c ) = 1 + A x - a + B x - b + C x - c ⇒ x 3 = ( x - a ) ( x - b ) ( x - c ) + A ( x - b ) ( x - c ) + B ( x - a ) ( x - c ) + C ( x - a ) ( x - b ) ...(i) Put x = a in (i), ⇒ a 3 = A ( a - b ) ( a - c ) ⇒ A = a 3 ( a - b ) ( a - c ) Put x = b in (i), ⇒ b 3 = B ( b - a ) ( b - c ) ⇒ B = b 3 ( b - a ) ( b - c ) Put x = c in (i), ⇒ c 3 = C ( c - a ) ( c - b ) ⇒ C = c 3 ( c - a ) ( c - b ) ∴ x 3 ( x - a ) ( x - b ) ( x - c ) = 1 + a 3 ( a - b ) ( a - c ) ( x - a ) + b 3 ( b - a ) ( b - c ) ( x - b ) + c 3 ( c - a ) ( c - b ) ( x - c )

E M B I B E

Intermediate Second Year Mathematics Paper 2A>Algebra>Chapter 7 - Partial Fractions>Exercise 7(c)

Telangana Board Solutions for Exercise 3: Exercise 7(c)

Author:Telangana Board

Telangana Board Mathematics Solutions for Exercise - Telangana Board Solutions for Exercise 3: Exercise 7(c)

Attempt the free practice questions from Exercise 3: Exercise 7(c) with hints and solutions to strengthen your understanding. Intermediate Second Year Mathematics Paper 2A solutions are prepared by Experienced Embibe Experts.