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 x3 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 x4 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 (x2 + 2) (x – 3)
3x2 + 2x = A(x2 + 2) + (Bx + C) (x – 3)
x = 3
⇒ 27 + 6 = A(9 + 2)
⇒ 33 = 11A
⇒ A = 3
Equating the coefficients of x2
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 xn 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 xn 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 x2 terms in (1)
0 = A + B
⇒ B = -A
⇒ B = -3