Binomial theorem (+pascals triangle & nCr): notes and practice questions

(a) Find the binomial expansion of $(\cos \theta + i \sin \theta)^4$. Give your answer in the form $a + bi$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

(b) By using De Moivre's theorem and your answer to part (a), show that $\cos 4\theta = 8 \cos^4\theta - 8 \cos^2\theta + 1$.

(c) Hence, find the four distinct roots of the equation $8x^4 - 8x^2 + 1 = 0$, expressing them in the form $\cos(\alpha)$ where $0 < \alpha < \pi$.

(d) By considering the roots of the equation in part (c), or otherwise, find the exact value of $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

Find the binomial expansion of $(\cos \theta + i \sin \theta)^4$. Give your answer in the form $a + bi$ where $a$ and $b$ are expressed in terms of $\sin \theta$ and $\cos \theta$.

By using De Moivre's theorem and your answer to part (a), show that $\cos 4\theta = 8\cos^4\theta - 8\cos^2\theta + 1$.

Hence, show that $\theta = \frac{\pi}{8}$ and $\theta = \frac{3\pi}{8}$ are solutions of the equation $8\cos^4\theta - 8\cos^2\theta + 1 = 0$.

Hence, find the exact value of $\cos(\frac{\pi}{8})\cos(\frac{3\pi}{8})$.

By considering De Moivre's theorem, show that $\cos(4\theta) = 8\cos^4\theta - 8\cos^2\theta + 1$.

Let $w = 1+i$. Show that $w^4 = -4$.

Hence, find the four roots of the equation $z^4 = -4$ in Cartesian form.

The four roots are represented by points A, B, C, D on an Argand diagram, forming a square. Find the area of this square.

Each of the points A, B, C, D is rotated counter-clockwise about the origin by an angle of $\frac{\pi}{6}$ to form new points A', B', C', D'. These points are the roots of an equation $z^4 = K$. Find $K$ in Cartesian form.

It is given that the eight points represented by the roots of $z^4=-4$ and $z^4=K$ are all solutions of $z^n = \alpha$ for some $\alpha \in \mathbb{C}$ and $n \in \mathbb{N}$. Find the smallest positive value of $n$.

Expand and simplify $(1+a)^3$ in ascending powers of $a$.

By using a suitable substitution for $a$, show that $1+3\cos(2x)+3\cos^2(2x)+\cos^3(2x) = 8\cos^6(x)$.

Consider the function $g(x) = 4\sin(x)(1+3\cos(2x)+3\cos^2(2x)+\cos^3(2x) )$.

Show that $\int_0^p g(x) \, dx = \frac{32}{7}(1-\cos^7 p)$, where $p$ is a positive real constant.

It is given that $\int_p^{\frac{\pi}{2}} g(x) \, dx = \frac{1}{28}$, where $0 \le p \le \frac{\pi}{2}$. Find the value of $p$.

A new car is purchased for 10000 euros. The value of the car depreciates at a nominal annual rate of 10%, with the depreciation calculated semi-annually.

The value of the car after two years can be written as $10000(1 - k)^4$.

(a) Write down the value of $k$.

(b) Expand and simplify $(1 - x)^4$.

(c) Hence or otherwise, find the value of the car after two years, giving your answer correct to the nearest euro.

Consider the complex number $z_1 = 1 - i\sqrt{3}$.

(a) (i) Express $z_1$ in modulus-argument form.

(a) (ii) Find the smallest positive integer $n$ for which $z_1^n$ is a real number.

Consider the equation $(z+i)^5 - (z-i)^5 = 0$, where $z \in \mathbb{C}$.

(b) Show that the roots of the equation are given by $z = \cot\left(\frac{k\pi}{5}\right)$ for $k=1, 2, 3, 4$.

(c) (i) By using the binomial expansion, show that the equation in part (b) can be written as $5z^4 - 10z^2 + 1 = 0$.

(c) (ii) Let the roots of the equation in (c)(i) be $z_1, z_2, z_3, z_4$. Without finding the roots, show that $\sum_{j=1}^4 z_j^2 = 4$.

(d) Hence, find the exact value of $\cot^2\left(\frac{\pi}{5}\right)$.

Find the term in $x^4$ in the expansion of $(2x-1)^8$.

Hence, find the term in $x^5$ in the expansion of $(x+3)(2x-1)^8$.

A discrete random variable $X$ follows a binomial distribution, $X \sim B(n, \frac{1}{3})$.

It is given that the probabilities $P(X=r-1)$, $P(X=r)$ and $P(X=r+1)$ for some integer $r$ form an arithmetic sequence, where $1 \le r \le n-1$.

(a) Show that $n^2 - 3n(2r+1) + 9r^2 + 3r - 4 = 0$.

(b) For a particular experiment, it is known that $n=8$. Find the possible value(s) of $r$.

Let $z = \cos\theta + i\sin\theta$. Use De Moivre's theorem to show that $z^n - \frac{1}{z^n} = 2i\sin n\theta$, for $n \in \mathbb{Z}^+$.

Expand $\left(z - \frac{1}{z}\right)^5$.

Hence, find an expression for $\sin^5 \theta$ in the form $A\sin 5\theta + B\sin 3\theta + C\sin \theta$, where $A, B, C$ are constants to be determined.

Hence, evaluate $\int_0^{\frac{\pi}{4}} \sin^5(2x) \, dx$.