1.04a Binomial expansion: (a+b)^n for positive integer n

The coefficient of \(x^2\) in the expansion of \((1-4x)^6\) is 12 times the coefficient of \(x^2\) in the expansion of \((2+ax)^5\). Find the value of the positive constant \(a\). [3]

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 \(a\). [4]

In the expansion of \(\left(kx+\frac{2}{x}\right)^4\), where \(k\) is a positive constant, the term independent of \(x\) is equal to 150. Find the value of \(k\) and hence determine the coefficient of \(x^5\) in the expansion. [4]

Find the term independent of \(x\) in the expansion of each of the following:

1. \(\left(x + \frac{3}{x^2}\right)^6\) [2]
2. \((4x^3 - 5)\left(x + \frac{3}{x^2}\right)^6\) [4]

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

The first three terms in the expansion of \((1 - 2x)^2(1 + ax)^6\), in ascending powers of \(x\), are \(1 - x + bx^2\). Find the values of the constants \(a\) and \(b\). [6]

The coefficients of \(x\) and \(x^2\) in the expansion of \((2 + ax)^7\) are equal. Find the value of the non-zero constant \(a\). [3]

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

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

1. Find the first 3 terms, in ascending powers of \(x\), in the expansion of \((1 + x)^5\). [2]

The coefficient of \(x^2\) in the expansion of \((1 + (px + x^2))^5\) is 95.

1. Use the answer to part (i) to find the value of the positive constant \(p\). [3]

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 \(a\). [4]

The coefficient of \(x^3\) in the expansion of \((1 - 3x)^6 + (1 + ax)^5\) is 100. Find the value of the constant \(a\). [4]

Find the coefficient of \(\frac{1}{x^2}\) in the expansion of \(\left(3x + \frac{2}{3x^2}\right)^7\). [4]

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 \(a\). [5]

Find the coefficient of \(\frac{1}{x}\) in the expansion of \(\left(x - \frac{2}{x}\right)^5\). [3]

It is given that \(y = e^x u\), where \(u\) is a function of \(x\). The \(r\)th derivatives \(\frac{\mathrm{d}^r y}{\mathrm{d}x^r}\) and \(\frac{\mathrm{d}^r u}{\mathrm{d}x^r}\) are denoted by \(y^{(r)}\) and \(u^{(r)}\) respectively. Prove by mathematical induction that, for all positive integers \(n\), $$y^{(n)} = e^x\left[\binom{n}{0}u + \binom{n}{1}u^{(1)} + \binom{n}{2}u^{(2)} + \ldots + \binom{n}{r}u^{(r)} + \ldots + \binom{n}{n}u^{(n)}\right].$$ [8] [You may use without proof the result \(\binom{k}{r} + \binom{k}{r-1} = \binom{k+1}{r}\).]

Find the first four terms, in ascending powers of \(x\), of the binomial expansion of $$\left(2 + \frac{3}{8}x\right)^{10}$$ Give each coefficient as an integer. [4]

Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \((3 + 2x)^5\), giving each term in its simplest form. [4]

1. Write down the first three terms, in ascending powers of \(x\), of the binomial expansion of \((1 + px)^{12}\), where \(p\) is a non-zero constant. [2]

Given that, in the expansion of \((1 + px)^{12}\), the coefficient of \(x\) is \((-q)\) and the coefficient of \(x^2\) is \(11q\),

1. find the value of \(p\) and the value of \(q\). [4]