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Learn/ 11th Class Mathematics/ Unit 5 /Numericals

Unit 5: Partial Fractions — Exercise 5 1

11th Class Mathematics · Unit 5: Partial Fractions

1.Resolve 2/(x²-1) into partial fractions
Given
Expression 2x2-1
Formula
Factor denominator 2(x-1)(x+1)
Working
Partial fraction form Ax-1 + Bx+1
Setting x=1 A = 1
Setting x=-1 B = -1
Result
Partial fractions 1x-1 - 1x+1
2.Resolve (a-b)/((x-a)(x-b)) into partial fractions
Given
Expression a-b(x-a)(x-b)
Working
Partial fraction form Ax-a + Bx-b
Setting x=a A = 1
Setting x=b B = -1
Result
Partial fractions 1x-a - 1x-b
3.Resolve (x²+1)/((x+1)(x-1)) into partial fractions
Given
Expression x2+1(x+1)(x-1)
Formula
Improper fraction, divide first x2+1x2-1 = 1 + 2x2-1
Working
Partial fraction form 1 + Ax-1 + Bx+1
Setting x=1 A = 1
Setting x=-1 B = -1
Result
Partial fractions 1 + 1x-1 - 1x+1
4.Resolve (2x+3)/((x+1)(x+2)(x+3)) into partial fractions
Given
Expression 2x+3(x+1)(x+2)(x+3)
Working
Partial fraction form Ax+1 + Bx+2 + Cx+3
Setting x=-1 A = 12
Setting x=-2 B = 1
Setting x=-3 C = -32
Result
Partial fractions 12(x+1) + 1x+2 - 32(x+3)
5.Resolve (x²+4x+5)/((x+1)(x+2)(x+3)) into partial fractions
Given
Expression x2+4x+5(x+1)(x+2)(x+3)
Working
Partial fraction form Ax+1 + Bx+2 + Cx+3
Setting x=-1 A = 1
Setting x=-2 B = -1
Setting x=-3 C = 1
Result
Partial fractions 1x+1 - 1x+2 + 1x+3
6.Resolve (4x³+5x²-3x-2)/(x²-1) into partial fractions
Given
Expression 4x3+5x2-3x-2x2-1
Formula
Improper fraction, divide first 4x + 5 + x+3x2-1
Working
Partial fraction form 4x + 5 + Ax-1 + Bx+1
Setting x=1 A = 2
Setting x=-1 B = -1
Result
Partial fractions 4x + 5 + 2x-1 - 1x+1
7.Resolve (3x²-12x+11)/((x-1)(x-2)(x-3)) into partial fractions
Given
Expression 3x2-12x+11(x-1)(x-2)(x-3)
Working
Partial fraction form Ax-1 + Bx-2 + Cx-3
Setting x=1 A = 1
Setting x=2 B = 1
Setting x=3 C = 1
Result
Partial fractions 1x-1 + 1x-2 + 1x-3
8.Resolve ((x-1)(x-2)(x-3))/((x-4)(x-5)(x-6)) into partial fractions
Given
Expression (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)
Formula
Improper fraction, divide first 1 + 9x2-63x+114(x-4)(x-5)(x-6)
Working
Partial fraction form 1 + Ax-4 + Bx-5 + Cx-6
Setting x=4 A = 3
Setting x=5 B = -24
Setting x=6 C = 30
Result
Partial fractions 1 + 3x-4 - 24x-5 + 30x-6
9.Resolve x²/((x²+a)(x²+b)(x²+c)) into partial fractions
Given
Expression x2(x2+a)(x2+b)(x2+c)
Working
Substitution x²=y y(y+a)(y+b)(y+c)
Partial fractions in y Ay+a + By+b + Cy+c
Coefficient A A = a(a-b)(a-c)
Coefficient B B = b(b-a)(b-c)
Coefficient C C = c(c-a)(c-b)
Result
Partial fractions a(a-b)(a-c)(x2+a) + b(b-a)(b-c)(x2+b) + c(c-a)(c-b)(x2+c)
10.Resolve (x+1)/(x-1)² into partial fractions
Given
Expression x+1(x-1)2
Working
Partial fraction form Ax-1 + B(x-1)2
Multiply by (x-1)² x+1 = A(x-1) + B
Comparing coefficients A = 1, B = 2
Result
Partial fractions 1x-1 + 2(x-1)2
11.Resolve (x²+x)/(x²-1)² into partial fractions
Given
Expression x2+x(x2-1)2
Formula
Factor denominator x(x+1)(x-1)2(x+1)2
Working
Partial fraction form Ax-1 + B(x-1)2 + Cx+1
Setting x=1 B = 12
Setting x=-1 C = -14
Coefficient A A = 14
Result
Partial fractions 14(x-1) + 12(x-1)2 - 14(x+1)
12.Resolve (3x²+4x-5)/(x-1)³ into partial fractions
Given
Expression 3x2+4x-5(x-1)3
Working
Partial fraction form Ax-1 + B(x-1)2 + C(x-1)3
Setting x=1 C = 2
Expanding and comparing A = 3, B = 10
Result
Partial fractions 3x-1 + 10(x-1)2 + 2(x-1)3
13.Resolve 1/(x(x+1)³) into partial fractions
Given
Expression 1x(x+1)3
Working
Partial fraction form Ax + Bx+1 + C(x+1)2 + D(x+1)3
Setting x=0 A = 1
Setting x=-1 D = -1
Expanding and comparing B = -1, C = -1
Result
Partial fractions 1x - 1x+1 - 1(x+1)2 - 1(x+1)3
14.Resolve (4x²-3x+1)/((x+1)(x-1)²) into partial fractions
Given
Expression 4x2-3x+1(x+1)(x-1)2
Working
Partial fraction form Ax+1 + Bx-1 + C(x-1)2
Setting x=-1 A = 2
Setting x=1 C = 1
Comparing coefficients B = 2
Result
Partial fractions 2x+1 + 2x-1 + 1(x-1)2
15.Resolve (12x²-48)/((x-2)²(x+2)²) into partial fractions
Given
Expression 12x2-48(x-2)2(x+2)2
Formula
Factor numerator 12(x2-4)(x-2)2(x+2)2 = 12(x-2)(x+2)(x-2)2(x+2)2
Simplify 12(x-2)(x+2)
Working
Partial fraction form Ax-2 + Bx+2
Setting x=2 A = 3
Setting x=-2 B = -3
Result
Partial fractions 3x-2 - 3x+2