2.7 Solving quadratic equations and inequalities questions for IB Maths AA

Find the possible values of $k$.

For the integer value of $k$, find the coordinates of the midpoint of $[\mathrm{P}\mathrm{Q}]$.

Show that the equation can be written as $3{\sin }^{2}x+2\sin x-1=0$.

Hence, solve the equation $3\cos 2x=4\sin x+1$, for $0\le x\le 2\pi$.

Using the result $\mathbf{a}\cdot \mathbf{b}=\mathrm{\mid }\mathbf{a}\mathrm{\mid }\mathrm{\mid }\mathbf{b}\mathrm{\mid }\cos \theta$, where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$, show that $\mathrm{\mid }\mathbf{a}\cdot \mathbf{b}\mathrm{\mid }\le \mathrm{\mid }\mathbf{a}\mathrm{\mid }\mathrm{\mid }\mathbf{b}\mathrm{\mid }$, and state the condition on $\mathbf{a}$ and $\mathbf{b}$ for which equality holds.

Let $\mathbf{a}=\left(\begin{array}{c}2\\ 3\\ 6\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}x\\ y\\ z\end{array}\right)$, where $x^2+y^2+z^2=1$.

Show that $2x+3y+6z\le 7$.

Find the values of $x$, $y$ and $z$ for which $2x+3y+6z=7$.

The proof in part (a) assumes that the angle $\theta$ exists, which itself relies on ${\displaystyle \frac{\mathbf{a}\cdot \mathbf{b}}{\mathrm{\mid }\mathbf{a}\mathrm{\mid }\mathrm{\mid }\mathbf{b}\mathrm{\mid }}}$ lying between $-1$ and $1$. In the rest of the question, you will prove the inequality without using angles.

Let $f(t)=\mathrm{\mid }\mathbf{a}+t\mathbf{b}{\mathrm{\mid }}^2$, where $t\in \mathbb{R}$.

Show that $f(t)=\mathrm{\mid }\mathbf{b}{\mathrm{\mid }}^2{t}^2+2(\mathbf{a}\cdot \mathbf{b})t+\mathrm{\mid }\mathbf{a}{\mathrm{\mid }}^2$.

Explain why $f(t)\ge 0$ for all $t\in \mathbb{R}$, and hence show that $(\mathbf{a}\cdot \mathbf{b}{)}^2\le \mathrm{\mid }\mathbf{a}{\mathrm{\mid }}^2\mathrm{\mid }\mathbf{b}{\mathrm{\mid }}^2$.

Show that if $(\mathbf{a}\cdot \mathbf{b}{)}^2=\mathrm{\mid }\mathbf{a}{\mathrm{\mid }}^2\mathrm{\mid }\mathbf{b}{\mathrm{\mid }}^2$, then $\mathbf{a}$ and $\mathbf{b}$ are parallel.

Use the result of part (d) to prove that $\mathrm{\mid }\mathbf{a}+\mathbf{b}\mathrm{\mid }\le \mathrm{\mid }\mathbf{a}\mathrm{\mid }+\mathrm{\mid }\mathbf{b}\mathrm{\mid }$.

The real numbers $x$, $y$ and $z$ satisfy $x+2y-2z=12$.

By choosing suitable vectors $\mathbf{a}$ and $\mathbf{b}$, find the minimum value of $x^2+y^2+z^2$ and the values of $x$, $y$ and $z$ for which it occurs.

Show that $(p+q+1{)}^2\le 3(p^2+q^2+1)$ for all $p,q\in \mathbb{R}$.

Find the value of $x$.

Find $\mathrm{B}\mathrm{C}$, giving your answer in the form $\sqrt{p+q\sqrt{3}}$, where $p,q\in {\mathbb{Z}}^{+}$.

Find the possible values of $p$.

For the positive value of $p$, find the vector $\mathbf{c}$ such that $\mathrm{\mid }\mathbf{c}\mathrm{\mid }=15$ and $\mathbf{c}$ is in the same direction as $\mathbf{a}$.

In the first case, the ship's position vector is $\mathbf{r}=\left(\begin{array}{c}8\\ 0\end{array}\right)+t\left(\begin{array}{c}0\\ 6\end{array}\right)$, and the patrol boat travels at $10{\text{ km h}}^{-1}$.

Write down the speed of the ship.

Show that the patrol boat meets the ship when $t=1$.

Find the velocity vector of the patrol boat and the bearing on which it travels.

In general, let the ship's position vector be $\mathbf{r}=\mathbf{a}+t\mathbf{u}$, where $\mathrm{\mid }\mathbf{a}\mathrm{\mid }=d>0$ and $\mathrm{\mid }\mathbf{u}\mathrm{\mid }=w>0$. Let $\alpha$ be the angle between $\mathbf{a}$ and $\mathbf{u}$, where $0\le \alpha \le \pi$.

Show that the patrol boat can meet the ship at time $t>0$ if and only if

Suppose that $v>w$. Show that the patrol boat can always intercept the ship, and that there is exactly one possible time of interception.

Suppose now that $v<w$. Show that interception is only possible if $\cos \alpha <0$.

Show that interception is also only possible if $v\ge w\sin \alpha$.

Consider the case $v=w$. Find an expression for the time of interception in terms of $d$, $w$ and $\alpha$, and state the condition on $\alpha$ for interception to be possible.

In the second case, the ship's position vector is $\mathbf{r}=\left(\begin{array}{c}20\\ 15\end{array}\right)+t\left(\begin{array}{c}-10\\ 0\end{array}\right)$.

Find the minimum speed at which the patrol boat can intercept the ship.

The patrol boat travels at $8{\text{ km h}}^{-1}$. Find the earliest time at which it can intercept the ship.

The patrol boat travels at the minimum speed found in part (f)(i).

Find the time and the point of interception, and show that the velocity of the patrol boat is perpendicular to the initial position vector of the ship.