IB Mathematics Analysis and Approaches Topic SL…

Find the coefficient of $x^2$ in the expansion of $(1 - 3x^2)(1 + 4x)^5$ .

Consider the expansion of $(1+ax)^{n}$ , where $a > 0$ and $n$ is an integer with $n \ge 4$ .

The coefficient of $x$ is $24$ and the coefficient of $x^{4}$ is $7920$ .

(a) Show that $\frac{(n-1)(n-2)(n-3)}{n^{3}} = \frac{55}{96}$ .

(b) Hence find the value of $n$ and the value of $a$ .

Find the coefficient of $x^8$ in the expansion of $\left(2x^3-\frac{1}{x^2}\right)^6$ .

(a) Obtain the first 3 terms, in ascending powers of x, in the expansion of $(2-3x)^5$ .

(b) Use this expansion to find an approximate value of $(1.94)^5$ .

Let $n$ be a positive integer and let

$S_{n} = \left(2+\sqrt{3}\right)^{n} + \left(2-\sqrt{3}\right)^{n}.$

(a) Use the binomial theorem to show that $S_{n}$ is an integer.

(b) Hence show that the integer part of $\left(2+\sqrt{3}\right)^{n}$ is $S_{n}-1$ .

(c) Find the exact value of $S_{4}$ , hence write down the integer part of $\left(2+\sqrt{3}\right)^{4}$ , and find $\left(2+\sqrt{3}\right)^{4} - 193$ in the form $a\sqrt{3}-b$ , where $a$ and $b$ are integers.

IB Mathematics Analysis and Approaches SL 1.9—binomial Theorem Where n Is an Integer Questionbank

IB Mathematics Analysis and Approaches SL 1.9—binomial Theorem Where n Is an Integer Questionbank

IB Mathematics Analysis and Approaches SL 1.9—binomial Theorem Where n Is an Integer Questionbank

IB Mathematics Analysis and Approaches SL 1.9—binomial Theorem Where n Is an Integer Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 1.9—binomial Theorem Where n Is an Integer Questionbank