Students get through Maths 2A Important Questions Inter 2nd Year Maths 2A Partial Fractions Important Questions which are most likely to be asked in the exam.

Intermediate 2nd Year Maths 2A Partial Fractions Important Questions

Question 1.
\(\frac{x+4}{\left(x^{2}-4\right)(x+1)}\) (Mar. 14)
Solution:
\(\frac{x+4}{\left(x^{2}-4\right)(x+1)}\) = \(\frac{A}{x+1}\) + \(\frac{B}{x+2}\) + \(\frac{c}{x-2}\)
Multiplying with (x2 – 4) (x + 1)
x + 4 = A(x2 – 4) + B(x + 1) (x – 2) + C (x + 1) (x + 2)
x = -1 ⇒ 3 = A(1 – 4) = -3A ⇒ A = -1
x = -2 ⇒ 2 = B(-2 + 1) (-2 – 2)
= 4B ⇒ B = + \(\frac{2}{4}\) = \(\frac{1}{2}\)
x = 2 ⇒ 6 = C(2 + 1)(2 + 2)
= 12C ⇒ C = \(\frac{1}{2}\)

Question 2.
\(\frac{x^{2}-x+1}{(x+1)(x-1)^{2}}\) (TS Mar. 15)
Solution:
Let \(\frac{x^{2}-x+1}{(x+1)(x-1)^{2}}\) = \(\frac{A}{x+1}\) + \(\frac{B}{x-1}\) + \(\frac{C}{(x-1)^{2}}\)
Multiplying with (x + 1) (x – 1)2
x2 – x + 1 = A(x – 1)2 + B(x + 1) (x – 1) + C(x + 1)
Put x = -1, 1 + 1 + 1 = A(4)
⇒ A = \(\frac{3}{4}\)
Put x = 1, 1 – 1 + 1 = C(2)
⇒ C = + \(\frac{1}{2}\)
Equating the coefficients of x2,
A + B = 1
⇒ B = 1 – A = 1 – \(\frac{3}{4}\) = \(\frac{1}{4}\)
Inter 2nd Year Maths 2A Partial Fractions Important Questions 22

Inter 2nd Year Maths 2A Partial Fractions Important Questions

Question 3.
Resolve the \(\frac{2 x^{2}+3 x+4}{(x-1)\left(x^{2}+2\right)}\) into partial fractions. (AP Mar. ’15, ’11; May ’11) (TS Mar. ’17)
Solution:
Let \(\frac{2 x^{2}+3 x+4}{(x-1)\left(x^{2}+2\right)}\) = \(\frac{A}{x-1}\) + \(\frac{\mathrm{Bx}+\mathrm{C}}{\mathrm{x}^{2}+2}\)
Multiplying with (x – 1) (x2 + 2)
2x2 + 3x + 4 = A(x2 + 2) + (Bx + C)(x – 1)
x = 1 ⇒ 2 + 3 + 4 = A(1 + 2)
9 = 3A ⇒ A = 3
Equating the coefficients of x2
2 = A + B ⇒ B = 2 – A = 2 – 3 = -1
Equating constants
4 = 2A – C ⇒ C = 2A – 4 = 6 – 4 = 2
\(\frac{2 x^{2}+3 x+4}{(x-1)\left(x^{2}+2\right)}\) = \(\frac{3}{x-1}\) + \(\frac{-x+2}{x^{2}+2}\)

Question 4.
Resolve \(\frac{x^{4}}{(x-1)(x-2)}\) into partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 25
Equating the coefficients of (x – 1) (x – 2),
15x – 14 = A(x – 2) + B(x – 1)
Put x = 1, 15 – 14 ⇒ A(-1) = A = -1
Put x = 2, 30 – 14 = B(1) ⇒ B = 16
∴ \(\frac{x^{4}}{(x-1)(x-2)}\) = x2 + 3x + 7 – \(\frac{1}{x-1}\) + \(\frac{16}{x-2}\)

Question 5.
Resolve \(\frac{x^{2}-3}{(x+2)\left(x^{2}+1\right)}\) into partial fractions. (AP Mar. ’17, ’16)
Solution:
Let \(\frac{x^{2}-3}{(x+2)\left(x^{2}+1\right)}\) = \(\frac{A}{x+2}\) + \(\frac{B x+C}{x^{2}+1}\)
Multiplying with (x + 2) (x2 + 1)
x2 – 3 = A(x2 + 1) + (Bx + C)(x + 2)
x = -2 ⇒ 4 – 3 = A(4 + 1)
1 = 5A ⇒ A = \(\frac{1}{5}\)
Equating the coefficients of x2
1 = A + B ⇒ B = 1 – A
= 1 – \(\frac{1}{5}\) = \(\frac{4}{5}\)
Equating the constants – 3 = A + 2C
2C = -3 – A = -3 – \(\frac{1}{5}\) = – \(\frac{16}{5}\)
C = –\(\frac{8}{5}\)
\(\frac{x^{2}-3}{(x+2)\left(x^{2}+1\right)}\) = \(\frac{1}{5(x+2)}\) + \(\frac{4 x-8}{5\left(x^{2}+1\right)}\)

Inter 2nd Year Maths 2A Partial Fractions Important Questions

Question 6.
Resolve \(\frac{1}{(x-1)^{2}(x-2)}\) into partial fractions. (May ’13)
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 23
Put x = 1 in (1)
1 = A(0) + B(1 – 2) + C(0)
⇒ -B = 1 ⇒ B = -1
Put x = 2 in (1)
⇒ 1 = A(0) + B(0) + C (2 – 1)2
⇒ C = 1
Equating the coefficients of x2 in (1)
0 = A + C ⇒ A = -C = -1
A = -1
∴ \(\frac{1}{(x-1)^{2}(x-2)}\) = \(\frac{-1}{x-1}\) – \(\frac{1}{(x-1)^{2}}\) + \(\frac{1}{x-2}\)

Question 7.
Resolve \(\frac{5 x+1}{(x+2)(x-1)}\) into Partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 1
⇒ 5x + 1 = A(x – 1) + B(x + 2) —– (1)
Put x = 1 in(1)
5(1) + 1,= A(0) + B(1 + 2)
⇒ 3B = 6 ⇒ B = 2
Put x = -2 in (1)
5(-2) + 1 = A (-2 – 1) + B(0)
⇒ -9 = -3A ⇒ A = 3
∴ \(\frac{5 x+1}{(x+2)(x-1)}\) = \(\frac{3}{x+2}\) + \(\frac{2}{x-1}\)

Inter 2nd Year Maths 2A Partial Fractions Important Questions

Question 8.
Resolve \(\frac{2 x+3}{5(x+2)(2 x+1)}\) into Partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 2
Inter 2nd Year Maths 2A Partial Fractions Important Questions 3

Question 9.
Resolve \(\frac{13 x+43}{2 x^{2}+17 x+30}\) into partial fractions.
Solution:
2x2 + 17x + 30 = 2x2 + 12x + 5x + 30
= 2x(x + 6) + 5(x + 6)
= (x + 6) (2x + 5)
Let \(\frac{13 x+43}{(x+6)(2 x+5)}\) = \(\frac{A}{x+6}\) + \(\frac{B}{2x+5}\)
∴ 13x + 43 = A(2x + 5) + B(x + 6)
putting x = -6
-78 + 43 = A(-12 + 5)
-35 = -7A
⇒ A = 5
Putting x = \(\frac{-5}{2}\)
13\(\left(\frac{-5}{2}\right)\) + 43 = \(B\left(\frac{-5}{2}+6\right)\)
⇒ -65 + 86 = B(—5 + 12)
⇒ 21 = 7B
⇒ B = 3
∴ \(\frac{13 x+43}{2 x^{2}+17 x+30}\) = \(\frac{5}{x+6}\) + \(\frac{3}{2 x+5}\)

Question 10.
Resolve \(\frac{x^{2}+5 x+7}{(x-3)^{3}}\) into partial fractions.
Solution:
Let x – 3 = y ⇒ x = y + 3
\(\frac{x^{2}+5 x+7}{(x-3)^{3}}\) = \(\frac{(y+3)^{2}+5(y+3)+7}{y^{3}}\)
Inter 2nd Year Maths 2A Partial Fractions Important Questions 4

Question 11.
Resolve \(\frac{x^{2}+13 x+15}{(2 x+3)(x+3)^{2}}\) into partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 5

Question 12.
Resolve \(\frac{1}{(x-1)^{2}(x-2)}\) into Partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 6
Put x = 1 in (1)
1 = A (0) + B(1 – 2) + C(0)
⇒ -B = 1 ⇒ B = -1
Put x = 2 in (1)
1 = A(0) + B(0) + C(2 – 1)2
⇒ C = 1
Equating the coefficients of x2 in (1)
0 = A + C ⇒ A = -C = -1
A = -1
∴ \(\frac{1}{(x-1)^{2}(x-2)}\) = \(\frac{-1}{x-1}\) – \(\frac{1}{(x-1)^{2}}\) + \(\frac{1}{x-2}\)

Inter 2nd Year Maths 2A Partial Fractions Important Questions

Question 13.
Resolve \(\frac{3 x-18}{x^{3}(x+3)}\) into Partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 7
Put x = -3 in(1)
3(-3) – 18 = A(0) + B(0) + C(O) + D (-3)3
⇒ -27D = -27 ⇒ D = 1
Put x = 0 in (1)
3(0) – 18 = A (0) + B(0) + C(0 + 3) + D(0)
⇒ 3C = -18 ⇒ C = -6
Equating the coefficients of x3 in (1)
0 = A + D
⇒ A = -D = -1 ⇒ A = -1
Equating the coefficients of x2 in (1)
0 = 3A + B
⇒ B = -3A = -3(-1) = 3 ⇒ B = 31
∴ \(\frac{3 x-18}{x^{3}(x+3)}\) = \(\frac{-1}{x}\) + \(\frac{3}{x^{2}}\) – \(\frac{6}{x^{3}}\) + \(\frac{1}{x+3}\)

Question 14.
Resolve \(\frac{x-1}{(x+1)(x-2)^{2}}\) into Partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 8
Inter 2nd Year Maths 2A Partial Fractions Important Questions 9

Question 15.
Resolve \(\frac{2 x^{2}+1}{x^{3}-1}\) into partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 10
Put x = 1 in (1)
2(1) + 1 = A(1 + 1 + 1) + (B + C)(0)
⇒ 3A = 3 ⇒ A = 1
Put x = 0 in (1)
0 + 1 = A(1) + (0 + C)(0 – 1)
⇒ 1 = A – C
⇒ 1 = 1 – C ⇒ C = 0
Equating the coefficients of x2 in (1)
2 = A + B
⇒ 2 = 1 + B ⇒ B = 1
Inter 2nd Year Maths 2A Partial Fractions Important Questions 11

Question 16.
Resolve \(\frac{x^{3}+x^{2}+1}{\left(x^{2}+2\right)\left(x^{2}+3\right)}\) into partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 12
Comparing the coefficients of x3, x2, x and constant terms
A + C = 1, B + D = 1, 3A + 2C = 0, 3B + 2D = 1
Solve
Inter 2nd Year Maths 2A Partial Fractions Important Questions 13

Question 17.
Resolve \(\frac{3 x^{3}-2 x^{2}-1}{x^{4}+x^{2}+1}\) into Partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 14
A – B + C + D = 0 —— (3)
B + D = -1 ——- (4) D = -1 – B
Substitute C, D in (2)
-A + B + 3 – A – 1 – B = -2
⇒ -2A = -4 ⇒ A = 2
Substitute C, D in (3)
A – B + 3 – A – 1 – B = 0 ⇒ 2 = 2B ⇒ B = 1
∴ C = 3 – 2 = 1, D = -1 – 1 = -2
Ax + B = 2x + 1, Cx + D = x – 2
Inter 2nd Year Maths 2A Partial Fractions Important Questions 15

Inter 2nd Year Maths 2A Partial Fractions Important Questions

Question 18.
Resolve \(\frac{x^{4}+24 x^{2}+28}{\left(x^{2}+1\right)^{3}}\) into partial fractions.
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 16

Question 19.
Resolve \(\frac{x+3}{(1-x)^{2}\left(1+x^{2}\right)}\) into partial fractions.
Solution:
Let \(\frac{x+3}{(1-x)^{2}\left(1+x^{2}\right)}\)
= \(\frac{\mathrm{A}}{1-x}\) + \(\frac{B}{(1-x)^{2}}\) + \(\frac{C x+D}{1+x^{2}}\)
= x + 3 = A(1 – x) (1 – x2) + B(1 + x2) + (Cx + D)(1 – x2)
Comparing the coefficients of like powers of x, we get
A + B + D = 3 → (1)
-A + C – 2D = 1 → (2)
A + B – 2C + D = 0 → (3)
-A + C = 0 → (4)
Solving these equations, we get
Inter 2nd Year Maths 2A Partial Fractions Important Questions 17

Question 20.
Resolve \(\frac{x^{3}}{(2 x-1)(x+2)(x-3)}\) into partial fractions.
Solution:
\(\frac{x^{3}}{(2 x-1)(x+2)(x-3)}\)
= \(\frac{1}{2}\) + \(\frac{A}{2 x-1}\) + \(\frac{B}{x+2}\) + \(\frac{c}{x-3}\)
Multiplying with 2(2x – 1) (x + 2) (x – 3)
2x3 = (2x – 1) (x + 2) (x – 3) + 2A(x + 2)
(x – 3) + 2B (2x – 1) (x – 3) + 2C (2x -1) (x + 2)
Inter 2nd Year Maths 2A Partial Fractions Important Questions 18

Question 21.
Resolve \(\frac{x^{4}}{(x-1)(x-2)}\) into partial fractions. (T.S. Mar. ’16, March – 2013)
Solution:
Inter 2nd Year Maths 2A Partial Fractions Important Questions 24
Equating the coefficients of (x – 1) (x – 2),
15x – 14 = A(x – 2) + B(x – 1)
Put x = 1, 15 – 14 = A(-1) ⇒ A = -1
Put x = 2, 30 – 14 = B(1) ⇒ B = 16
∴ \(\frac{x^{4}}{(x-1)(x-2)}\) = x2 + 3x + 7 – \(\frac{1}{x-1}\) + \(\frac{16}{x-2}\)

Question 22.
Find the coefficient of x4 in the expansion of \(\frac{3 x}{(x-2)(x+1)}\) in powers of x specifying the interval in which the expansion is valid.
Solution:
\(\frac{3 x}{(x-2)(x+1)}\) = \(\frac{A}{x-2}\) + \(\frac{B}{x+1}\)
Multiplying with (x – 2) (x + 1)
3x = A(x + 1) + B(x – 2)
Put x = -1, -3 = B(-3) ⇒ B = 1
Put x = 2, 6 = A(3) ⇒ A = 2
Inter 2nd Year Maths 2A Partial Fractions Important Questions 19
Inter 2nd Year Maths 2A Partial Fractions Important Questions 20

Inter 2nd Year Maths 2A Partial Fractions Important Questions

Question 23.
Find the coefficient of xn in the power series expansion of \(\frac{x}{(x-1)^{2}(x-2)}\) specifying the region in which the expansion is valid.
Solution:
\(\frac{x}{(x-1)^{2}(x-2)}\) = \(\frac{A}{x-1}\) + \(\frac{B}{(x-1)^{2}}\) + \(\frac{C}{x-2}\)
Multiplying with (x – 1)2 (x – 2)
x = A(x – 1) (x – 2) + B(x – 2) + C(x – 1)2
Put x = 1, 1 = B(-1) ⇒ B = -1
Put x = 2, 2 = C(1) ⇒ C = 2
Put x = 0, 0 = 2A – 2B + C ⇒ 2A = 2B – C
= -2 – 2 = -4 ⇒ A = -2
Inter 2nd Year Maths 2A Partial Fractions Important Questions 21