GCSE / IGCSE Maths – Vectors and Proof

$\mathbf{a} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}$ and $\mathbf{b} = \begin{pmatrix} -2 \\ 4 \end{pmatrix}$.(a) Write $2\mathbf{a} + \mathbf{b}$ as a column vector.(b) Write $\mathbf{a} – 3\mathbf{b}$ as a column vector.(c) Find the magnitude of $\mathbf{a} – 3\mathbf{b}$, giving your answer in the form $k\sqrt{10}$.

In triangle $OAB$, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $M$ is the midpoint of $AB$.(a) Find $\overrightarrow{AB}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.(b) Show that $\overrightarrow{OM} = \frac{1}{2}(\mathbf{a} + \mathbf{b})$.

Prove that the sum of any three consecutive integers is a multiple of $3$.

Prove that $(n+3)^2 – (n-3)^2$ is a multiple of $12$ for all integers $n$.

Prove that the product of two odd numbers is always odd.

In triangle $OAB$, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. Point $C$ lies on $OA$ with $OC : CA = 2 : 1$ and point $D$ lies on $AB$ with $AD : DB = 1 : 2$.(a) Find $\overrightarrow{CD}$ in terms of $\mathbf{a}$ and $\mathbf{b}$, simplifying your answer.(b) What does your answer tell you about the lines $CD$ and $OB$?

$AB$ is a diameter of a circle with centre $O$, and $C$ is a point on the circumference. Prove that angle $ACB = 90^\circ$. You may use the fact that $OA = OB = OC$.

Prove that $n^2 – n$ is even for every integer $n$.

Prove that the difference between the squares of any two consecutive odd numbers is a multiple of $8$.

$\overrightarrow{OA} = 6\mathbf{a}$ and $\overrightarrow{OB} = 6\mathbf{b}$. $M$ is the point on $OA$ with $OM : MA = 1 : 2$ and $N$ is the point on $AB$ with $AN : NB = 2 : 1$.(a) Find $\overrightarrow{ON}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.(b) Show that $MN$ is parallel to $OB$ and find the ratio $MN : OB$.

$P$, $Q$ and $R$ have position vectors $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$, $\begin{pmatrix} 5 \\ 7 \end{pmatrix}$ and $\begin{pmatrix} k \\ 11 \end{pmatrix}$. Find the value of $k$ for which $P$, $Q$ and $R$ lie on a straight line.

(a) A two-digit number has tens digit $a$ and units digit $b$. Prove that the difference between the number and the number formed by reversing its digits is always a multiple of $9$.(b) Prove that $x^2 – 6x + 10$ is positive for all real values of $x$.