In the expansion of \(\left( 1 - \frac{2x}{a} \right)(a + x)^5\), where \(a\) is a non-zero constant, show that the coefficient of \(x^2\) is zero.
(i) Find the first 3 terms in the expansion of \((2-y)^5\) in ascending powers of \(y\).
(ii) Use the result in part (i) to find the coefficient of \(x^2\) in the expansion of \((2-(2x-x^2))^5\).
(i) Find the terms in \(x^2\) and \(x^3\) in the expansion of \((1 - \frac{3}{2}x)^6\).
(ii) Given that there is no term in \(x^3\) in the expansion of \((k + 2x)(1 - \frac{3}{2}x)^6\), find the value of the constant \(k\).
In the expansion of \((1 + ax)^6\), where \(a\) is a constant, the coefficient of \(x\) is \(-30\). Find the coefficient of \(x^3\).
(i) Find, in terms of the non-zero constant \(k\), the first 4 terms in the expansion of \((k + x)^8\) in ascending powers of \(x\).
(ii) Given that the coefficients of \(x^2\) and \(x^3\) in this expansion are equal, find the value of \(k\).
Let \(f(x) = \frac{17x^2 - 7x + 16}{(2 + 3x^2)(2 - x)}\).
(a) Express \(f(x)\) in partial fractions.
(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^3\).
(c) State the set of values of \(x\) for which the expansion in (b) is valid. Give your answer in an exact form.
Let \(f(x) = \frac{16 - 17x}{(2 + x)(3 - x)^2}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{12 + 12x - 4x^2}{(2+x)(3-2x)}\).
(i) Express \(f(x)\) in partial fractions. [5]
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
Let \(f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{x - 4x^2}{(3-x)(2+x^2)}\).
(i) Express \(f(x)\) in the form \(\frac{A}{3-x} + \frac{Bx+C}{2+x^2}\).
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^3\).
Let \(f(x) = \frac{12x^2 + 4x - 1}{(x-1)(3x+2)}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{8x^2 + 9x + 8}{(1-x)(2x+3)^2}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{x(6-x)}{(2+x)(4+x^2)}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{3x^2 + x + 6}{(x+2)(x^2+4)}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{10x - 2x^2}{(x+3)(x-1)^2}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{24x + 13}{(1 - 2x)(2 + x)^2}\).
(a) Express \(f(x)\) in partial fractions.
(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
(c) State the set of values of \(x\) for which the expansion in (b) is valid.
Let \(f(x) = \frac{4x^2 + 12}{(x+1)(x-3)^2}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{5x^2 + x + 6}{(3 - 2x)(x^2 + 4)}\).
(i) Express \(f(x)\) in partial fractions. [5]
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]
Let \(f(x) = \frac{x^2 - 8x + 9}{(1-x)(2-x)^2}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).