(a) Expand the following in ascending powers of x up to and including the term in x².

(i) \((1 + 2x)^5\).

(ii) \((1 - ax)^6\), where a is a constant.

In the expansion of \((1 + 2x)^5(1 - ax)^6\), the coefficient of x² is -5.

(b) Find the possible values of a.

(a) Find the first three terms, in ascending powers of \(x\), in the expansion of \((1 + ax)^6\).

(b) Given that the coefficient of \(x^2\) in the expansion of \((1 - 3x)(1 + ax)^6\) is \(-3\), find the possible values of the constant \(a\).

(a) It is given that in the expansion of \((4 + 2x)(2 - ax)^5\), the coefficient of \(x^2\) is \(-15\). Find the possible values of \(a\).

(b) It is given instead that in the expansion of \((4 + 2x)(2 - ax)^5\), the coefficient of \(x^2\) is \(k\). It is also given that there is only one value of \(a\) which leads to this value of \(k\). Find the values of \(k\) and \(a\).

The coefficient of x in the expansion of \(\left(4x + \frac{10}{x}\right)^3\) is p. The coefficient of \(\frac{1}{x}\) in the expansion of \(\left(2x + \frac{k}{x^2}\right)^5\) is q.

\(Given that p = 6q, find the possible values of k.\)

The coefficient of \(x^3\) in the expansion of \((1 + kx)(1 - 2x)^5\) is 20.

Find the value of the constant \(k\).

In the expansion of \((2x^2 + \frac{a}{x})^6\), the coefficients of \(x^6\) and \(x^3\) are equal.

(a) Find the value of the non-zero constant \(a\).

(b) Find the coefficient of \(x^6\) in the expansion of \((1-x^3)(2x^2 + \frac{a}{x})^6\).

The coefficient of \(\frac{1}{x}\) in the expansion of \(\left( kx + \frac{1}{x} \right)^5 + \left( 1 - \frac{2}{x} \right)^8\) is 74.

Find the value of the positive constant \(k\).

The coefficient of \(\frac{1}{x}\) in the expansion of \(\left( 2x + \frac{a}{x^2} \right)^5\) is 720.

(a) Find the possible values of the constant \(a\).

(b) Hence find the coefficient of \(\frac{1}{x^7}\) in the expansion.

The coefficient of \(x^2\) in the expansion of \((4 + ax)\left(1 + \frac{x}{2}\right)^6\) is 3. Find the value of the constant \(a\).

The term independent of x in the expansion of \(\left( 2x + \frac{k}{x} \right)^6\), where k is a constant, is 540.

(i) Find the value of k.

(ii) For this value of k, find the coefficient of x² in the expansion.

The coefficient of \(x^3\) in the expansion of \((1 - px)^5\) is \(-2160\). Find the value of the constant \(p\).

The coefficient of \(x^3\) in the expansion of \((3 + 2ax)^5\) is six times the coefficient of \(x^2\) in the expansion of \((2 + ax)^6\).

Find the value of the constant \(a\).

The coefficient of \(x^2\) in the expansion of \(\left( 2 + \frac{x}{2} \right)^6 + (a + x)^5\) is 330. Find the value of the constant \(a\).

The coefficients of x and x² in the expansion of (2 + ax)⁷ are equal. Find the value of the non-zero constant a.

The coefficients of x² and x³ in the expansion of (3 - 2x)⁶ are a and b respectively. Find the value of \(\frac{a}{b}\).

In the expansion of \(\left( \frac{1}{ax} + 2ax^2 \right)^5\), the coefficient of \(x\) is 5. Find the value of the constant \(a\).

The coefficient of \(x^3\) in the expansion of \((1 - 3x)^6 + (1 + ax)^5\) is 100. Find the value of the constant \(a\).

In the expansion of \((3 - 2x)\left(1 + \frac{x}{2}\right)^n\), the coefficient of \(x\) is 7. Find the value of the constant \(n\) and hence find the coefficient of \(x^2\).

In the expansion of \((x + 2k)^7\), where \(k\) is a non-zero constant, the coefficients of \(x^4\) and \(x^5\) are equal. Find the value of \(k\).

In the expansion of \((2 + ax)^6\), the coefficient of \(x^2\) is equal to the coefficient of \(x^3\). Find the value of the non-zero constant \(a\).

In the expansion of \((2 + ax)^7\), the coefficient of \(x\) is equal to the coefficient of \(x^2\). Find the value of the non-zero constant \(a\).

In the expansion of \(\left(x^2 - \frac{a}{x}\right)^7\), the coefficient of \(x^5\) is \(-280\). Find the value of the constant \(a\).

(a) Give the complete expansion of \(\left( x + \frac{2}{x} \right)^5\).

(b) In the expansion of \((a + bx^2) \left( x + \frac{2}{x} \right)^5\), the coefficient of \(x\) is zero and the coefficient of \(\frac{1}{x}\) is 80. Find the values of the constants \(a\) and \(b\).

The coefficient of \(x^3\) in the expansion of \((a + x)^5 + (2 - x)^6\) is 90. Find the value of the positive constant \(a\).

The coefficient of \(x^2\) in the expansion of \(\left( k + \frac{1}{3}x \right)^5\) is 30. Find the value of the constant \(k\).

The coefficient of \(x^3\) in the expansion of \((a+x)^5 + (1-2x)^6\), where \(a\) is positive, is 90. Find the value of \(a\).

The coefficient of \(x^4\) in the expansion of \((x + a)^6\) is \(p\) and the coefficient of \(x^2\) in the expansion of \((ax + 3)^4\) is \(q\). It is given that \(p + q = 276\).

Find the possible values of the constant \(a\).

In the expansion of \(\left( \frac{x}{a} + \frac{a}{x^2} \right)^7\), it is given that

\(\frac{\text{the coefficient of } x^4}{\text{the coefficient of } x} = 3.\)

Find the possible values of the constant \(a\).

The coefficient of \(x^2\) in the expansion of \(\left( 1 + \frac{2}{p} x \right)^5 + (1 + px)^6\) is 70.

Find the possible values of the constant \(p\).

The coefficient of \(x^3\) in the expansion of \(\left(p + \frac{1}{p}x\right)^4\) is 144.

Find the possible values of the constant \(p\).

The coefficient of \(x^4\) in the expansion of \((3 + x)^5\) is equal to the coefficient of \(x^2\) in the expansion of \(\left(2x + \frac{a}{x}\right)^6\).

Find the value of the positive constant \(a\).

The coefficient of \(x^4\) in the expansion of \(\left( 2x^2 + \frac{k^2}{x} \right)^5\) is \(a\). The coefficient of \(x^2\) in the expansion of \((2kx - 1)^4\) is \(b\).

(a) Find \(a\) and \(b\) in terms of the constant \(k\).

(b) Given that \(a + b = 216\), find the possible values of \(k\).