1.6 Simple deductive proof questions for IB Maths AA

Show that $f(x)=1-\sin 2x$.

Hence, solve $f(x)={\displaystyle \frac{3}{2}}$, for $0\le x\le \pi$.

Use your GDC to find the values of $n$, where $1\le n\le 12$, for which the inequality is true.

Prove by mathematical induction that $2^n>n^3$ for all integers $n\ge 10$.

Consider a triangle ($n=3$). Show that ${\mathbf{p}}_1={\mathbf{m}}_1-{\mathbf{m}}_2+{\mathbf{m}}_3$.

The midpoints of the sides of a triangle are ${\mathrm{M}}_1(2,1,3)$, ${\mathrm{M}}_2(4,-1,5)$ and ${\mathrm{M}}_3(1,0,2)$. Find the coordinates of ${\mathrm{P}}_1$, ${\mathrm{P}}_2$ and ${\mathrm{P}}_3$.

For any triangle, show that $\overrightarrow{{\mathrm{P}}_1{\mathrm{P}}_2}=2\overrightarrow{{\mathrm{M}}_3{\mathrm{M}}_2}$.

State what this result tells you about the side $[{\mathrm{P}}_1{\mathrm{P}}_2]$ and the line segment $[{\mathrm{M}}_3{\mathrm{M}}_2]$.

Now consider a quadrilateral ($n=4$). Show that ${\mathbf{m}}_1-{\mathbf{m}}_2+{\mathbf{m}}_3-{\mathbf{m}}_4=\mathbf{0}$.

Hence show that ${\mathrm{M}}_1{\mathrm{M}}_2{\mathrm{M}}_3{\mathrm{M}}_4$ is a parallelogram.

The points ${\mathrm{M}}_1(1,2)$, ${\mathrm{M}}_2(4,3)$, ${\mathrm{M}}_3(5,1)$ and ${\mathrm{M}}_4(2,0)$ lie in a plane, with coordinates given in two dimensions.

Verify that these points satisfy the result in part (e).

These points are the midpoints of the sides of a quadrilateral ${\mathrm{P}}_1{\mathrm{P}}_2{\mathrm{P}}_3{\mathrm{P}}_4$, where ${\mathrm{P}}_1$ is the origin. Find the coordinates of ${\mathrm{P}}_2$, ${\mathrm{P}}_3$ and ${\mathrm{P}}_4$.

Explain why the vertices of a quadrilateral are not uniquely determined by the midpoints of its sides.

Now consider a pentagon ($n=5$). Find an expression for ${\mathbf{p}}_1$ in terms of ${\mathbf{m}}_1,{\mathbf{m}}_2,{\mathbf{m}}_3,{\mathbf{m}}_4$ and ${\mathbf{m}}_5$, justifying your answer.

Suggest, with a reason, the values of $n$ for which a polygon with $n$ sides can be reconstructed uniquely from the midpoints of its sides.

Show that ${\displaystyle \frac{1-\cos 2x}{\sin 2x}}\equiv \tan x$.

Hence, find the exact value of $\tan \left({\displaystyle \frac{3\pi }{8}}\right)$, giving your answer in the form $a+\sqrt{b}$, where $a,b\in {\mathbb{Z}}^{+}$.

Find the number of oranges in the $5$th layer from the top.

Prove by mathematical induction that the total number of oranges in a pyramid with $n$ complete layers is ${\displaystyle \frac{n(n+1)(n+2)}{6}}$, for $n\in {\mathbb{Z}}^{+}$.

The shopkeeper has $2000$ oranges. Find the greatest number of complete layers that she can build.

Find the number of oranges that are left over.

A second shopkeeper also has $2000$ oranges and builds a pyramid with a square base, in which the $r$th layer from the top has $r^2$ oranges. It is given that ${\displaystyle \sum _{r=1}^n{r}^2=\frac{n(n+1)(2n+1)}{6}}$.

Find how many fewer complete layers the second shopkeeper can build than the first.

Consider the case $p={\displaystyle \frac{1}{2}}$ and $q={\displaystyle \frac{1}{3}}$. Find $\overrightarrow{\mathrm{A}\mathrm{Q}}$ and $\overrightarrow{\mathrm{B}\mathrm{P}}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

By writing $\overrightarrow{\mathrm{O}\mathrm{X}}$ in two different ways, find $\overrightarrow{\mathrm{O}\mathrm{X}}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.

The line $(\mathrm{O}\mathrm{X})$ meets $[\mathrm{A}\mathrm{B}]$ at the point $\mathrm{R}$. For this case, find $\overrightarrow{\mathrm{O}\mathrm{R}}$ and hence find the ratio $\mathrm{A}\mathrm{R}:\mathrm{R}\mathrm{B}$.

For this case, show that ${\displaystyle \frac{\mathrm{O}\mathrm{P}}{\mathrm{P}\mathrm{A}}}\times {\displaystyle \frac{\mathrm{A}\mathrm{R}}{\mathrm{R}\mathrm{B}}}\times {\displaystyle \frac{\mathrm{B}\mathrm{Q}}{\mathrm{Q}\mathrm{O}}}=1$.

Now consider the general case. Show that

Hence show that ${\displaystyle \frac{\mathrm{A}\mathrm{R}}{\mathrm{R}\mathrm{B}}}={\displaystyle \frac{q(1-p)}{p(1-q)}}$.

Hence prove that ${\displaystyle \frac{\mathrm{O}\mathrm{P}}{\mathrm{P}\mathrm{A}}}\times {\displaystyle \frac{\mathrm{A}\mathrm{R}}{\mathrm{R}\mathrm{B}}}\times {\displaystyle \frac{\mathrm{B}\mathrm{Q}}{\mathrm{Q}\mathrm{O}}}=1$ for all $p$ and $q$.

By considering the case $p=q={\displaystyle \frac{1}{2}}$, deduce that the three medians of a triangle meet at a single point, and find the position vector of this point.

Show that $\mathrm{R}$ is the midpoint of $[\mathrm{A}\mathrm{B}]$ if and only if $(\mathrm{P}\mathrm{Q})$ is parallel to $(\mathrm{A}\mathrm{B})$.