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5. (a) Find the values of the constants A, B and C - Edexcel - A-Level Maths Pure - Question 6 - 2010 - Paper 6
Question 6
5. (a) Find the values of the constants A, B and C. (b) Hence, or otherwise, expand \( \frac{2x^2+5x-10}{(x-1)(x+2)} \) in ascending powers of x, as far as the ter... show full transcript
Worked Solution & Example Answer:5. (a) Find the values of the constants A, B and C - Edexcel - A-Level Maths Pure - Question 6 - 2010 - Paper 6
Step 1
Find the values of the constants A, B and C.
Answer
To find the constants A, B, and C, we will substitute specific values of x into the equation:
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Let ( x = 1 ): [ 2(1)^2 + 5(1) - 10 = A(1-1)(1+2) + B(1-1) + C(1+2) ] This simplifies to ( 2 + 5 - 10 = 0 + 0 + 3C ) which gives us: [ C = -1. ]
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Let ( x = -2 ): [ 2(-2)^2 + 5(-2) - 10 = A(-3)(0) + B(-3) + C(0) ] This simplifies to ( 8 - 10 = -3B ) resulting in: [ B = \frac{2}{3}. ]
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Lastly, let ( x = 2 ): [ 2(2)^2 + 5(2) - 10 = A(1)(4) + B(1) + C(4) ] Thus, we find: [ 8 + 10 - 10 = 4A + B + 4C. ] Substituting C and B into this equation: [ 8 = 4A + \frac{2}{3} - 4 ] Solving gives us: [ A = 2. ]
Thus, the values are A = 2, B = -1, C = -1.
Step 2
Hence, or otherwise, expand \( \frac{2x^2+5x-10}{(x-1)(x+2)} \) in ascending powers of x, as far as the term in \( x^2 \). Give each coefficient as a simplified fraction.
Answer
To expand the expression ( \frac{2x^2+5x-10}{(x-1)(x+2)} ) in ascending powers of x:
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Rewrite the expression using partial fractions: [ \frac{2x^2+5x-10}{(x-1)(x+2)} = \frac{A}{x-1} + \frac{B}{x+2} ] where A = 2, B = -1.
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Therefore, we write: [ \frac{2}{x-1} - \frac{1}{x+2}. ]
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Expanding each fraction: [ \frac{2}{x-1} = 2(1 + x + x^2 + \ldots), ] [ \frac{1}{x+2} = \frac{1}{2}(1 - \frac{x}{2} + \frac{x^2}{4} - \ldots). ]
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Adding these together gives us the expansion as: [ 5 + 1 + \frac{3}{2} x^2 + \ldots = 2x^2 + 5x - 10. ]
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The coefficients for terms in x^2 become: [ \frac{3}{2} ] as the coefficient of ( x^2 ).