Practicing the Intermediate 2nd Year Maths 2A Textbook Solutions Inter 2nd Year Maths 2A Partial Fractions Solutions Exercise 7(d) will help students to clear their doubts quickly.

## Intermediate 2nd Year Maths 2A Partial Fractions Solutions Exercise 7(d)

Question 1.

Find the coefficient of x^{3} in the power series expansion of \(\frac{5 x+6}{(x+2)(1-x)}\) specifying the region in which the expansion is valid.

Solution:

Question 2.

Find is the coefficient of x^{4} in the power series expansion of \(\frac{3 x^2+2 x}{\left(x^2+2\right)(x-3)}\) specifying the interval in which the expansion is valid.

Solution:

Let \(\frac{3 x^2+2 x}{\left(x^2+2\right)(x-3)}=\frac{A}{x-3}+\frac{B x+C}{x^2+2}\)

Multiplying with (x^{2} + 2) (x – 3)

3x^{2} + 2x = A(x^{2} + 2) + (Bx + C) (x – 3)

x = 3

⇒ 27 + 6 = A(9 + 2)

⇒ 33 = 11A

⇒ A = 3

Equating the coefficients of x^{2}

3 = A + B

⇒ B = 3 – A = 3 – 3 = 0

Equating the constants,

2A – 3C = 0

⇒ 3C = 2A = 6

⇒ C = 2

Question 3.

Find the coefficient of x^{n} in the power series expansion of \(\frac{x-4}{x^2-5 x+6}\) specifying the region in which the expansion is valid.

Solution:

Let \(\frac{x-4}{x^2-5 x+6}=\frac{A}{x-2}+\frac{B}{x-3}\)

Multiplying with (x – 2) (x – 3)

x – 4 = A(x – 3) + B(x – 2)

x = 2

⇒ -2 = A(2 – 3) = -A

⇒ A = 2

x = 3

⇒ -1 = B(3 – 2) = B

⇒ B = -1

Question 4.

Find the coefficient of x^{n} in the power series expansion of \(\frac{3 x}{(x-1)(x-2)^2}\)

Solution:

∴ 3x = A(x – 2)^{2} + B(x – 1) (x – 2) + C(x – 1) ……..(1)

putting x = 1,

3 = A(1 – 2)^{2}

⇒ A = 3

putting x = 2,

6 = C(2 – 1)

⇒ C = 6

Now equating the co-efficient of x^{2} terms in (1)

0 = A + B

⇒ B = -A

⇒ B = -3