Mathematics Extension 2

A projectile is launched from $O$ at $V\,\mathrm{m/s}$ so that it will hit the point $A(a,b)$ on the slope. Air resistance is neglected and the acceleration due to gravity is $g\,\mathrm{m/s^2}$. Given that
$y=-\frac{gx^2}{2V^2}(1+\tan^2\alpha)+x\tan\alpha,$
and that $\alpha_1<\alpha_2$, show that
$\tan\alpha_1=\frac{V^2-\sqrt{V^4-g^2a^2-2V^2gb}}{ga}$
and
$\tan\alpha_2=\frac{V^2+\sqrt{V^4-g^2a^2-2V^2gb}}{ga}.$

By showing that
$\tan(\alpha_1+\alpha_2)=\tan(90^\circ+\theta),$
show that
$\angle COY=\angle BOA.$

Find all the complex numbers $z_1,z_2,z_3$ that satisfy the following three conditions simultaneously:
$|z_1|=|z_2|=|z_3|,$
$z_1+z_2+z_3=1,$
$z_1z_2z_3=1.$

Find
$\int\frac{\ln x-1}{(x+\ln x)^2}\,dx.$

Given
$S_n=\frac1{3^3}+\frac1{4^3}+\frac1{5^3}+\cdots+\frac1{n^3},\qquad n\ge3,$
use part (i), or otherwise, to prove that
$S_n<\frac1{12}.$

Four cubes are placed in a line as shown in the diagram. Which vector is equal to $\overrightarrow{AB}+\overrightarrow{CQ}$?

A. $\overrightarrow{AQ}$
B. $\overrightarrow{CP}$ C. D.

Which expression is equal to $\int x^5e^{7x}\,dx$?

A. $\frac17x^5e^{7x}-\frac57\int x^4e^{7x}\,dx$
B. $\frac17x^5e^{7x}-\frac57\int x^5e^{7x}\,dx$
C. $\frac57x^4e^{7x}-\frac57\int x^4e^{7x}\,dx$
D. $\frac57x^4e^{7x}-\frac57\int x^5e^{7x}\,dx$

Which of the following is a vector equation of the line joining $A(4,2,5)$ and $B(-2,2,1)$?

A. $\vec r=\begin{pmatrix}4\\2\\5\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\3\end{pmatrix}$
B. $\vec r=\begin{pmatrix}4\\2\\5\end{pmatrix}+\lambda\begin{pmatrix}3\\0\\2\end{pmatrix}$
C. $\vec r=\begin{pmatrix}1\\2\\3\end{pmatrix}+\lambda\begin{pmatrix}4\\2\\5\end{pmatrix}$
D. $\vec r=\begin{pmatrix}3\\0\\2\end{pmatrix}+\lambda\begin{pmatrix}4\\2\\5\end{pmatrix}$

Consider the statement: “For all integers $n$, if $n$ is a multiple of $6$, then $n$ is a multiple of $2$.” Which is its contrapositive?

A. There exists an integer $n$ such that $n$ is a multiple of $6$ and not a multiple of $2$.
B. There exists an integer $n$ such that $n$ is a multiple of $2$ and not a multiple of $6$.
C. For all integers $n$, if $n$ is not a multiple of $2$, then $n$ is not a multiple of $6$.
D. For all integers $n$, if $n$ is not a multiple of $6$, then $n$ is not a multiple of $2$.

Which of the following statements is FALSE?

A. $\forall a,b\in\mathbb R$, $a<b\Rightarrow a^3<b^3$.
B. $\forall a,b\in\mathbb R$, $a<b\Rightarrow e^{-a}>e^{-b}$.
C. $\forall a,b\in(0,+\infty)$, $a<b\Rightarrow\ln a<\ln b$.
D. $\forall a,b\in\mathbb R$, with $a,b\ne0$, $a<b\Rightarrow\frac1a>\frac1b$.

Which polynomial could have $2+i$ as a zero, given that $k$ is real?

A. $x^3-4x^2+kx$
B. $x^3-4x^2+kx+5$
C. $x^3-5x^2+kx$
D. $x^3-5x^2+kx+5$

Which diagram best shows the curve described by $\vec r(t)=-5\cos(t)\,\vec i+5\sin(t)\,\vec j+t\,\vec k$ for $0\le t\le4\pi$?

A particle is travelling from $A$ on the curve joining $A$ to $B$. At a particular time, the particle is at $P$ and has velocity $\vec v$, as shown. The speed is increasing. Which diagram shows an acceleration $\vec a$ that would allow the particle to follow the curve to $B$?

Four cards have either RED or BLACK on one side and either WIN or TRY AGAIN on the other side. The visible cards are: Card 1 WIN; Card 2 BLACK; Card 3 TRY AGAIN; Card 4 RED.
For the statement “If a card is RED, then it has WIN written on the other side.”

What is the minimum number of cards Sam should turn over to check if the statement is true and which cards should be turned over?

A. 1 and 4
B. 3 and 4
C. 1, 2 and 4
D. 1, 3 and 4

Consider the two non-zero complex numbers $z$ and $w$ as vectors. Which expression is the projection of $z$ onto $w$?

A. $\frac{\operatorname{Re}(zw)}{|w|}w$
B. $\left|\frac zw\right|w$
C. $\operatorname{Re}\left(\frac zw\right)w$
D. $\frac{\operatorname{Re}(z)}{|w|}w$

Express $\frac{3x^2-5}{(x-2)(x^2+x+1)}$ as a sum of partial fractions over $\mathbb R$.

The complex numbers $z=5+i$ and $w=2-4i$ are given. Find $\frac{\bar z}{w}$, giving your answer in Cartesian form.

Hence, or otherwise, solve $z^2+2z+1+i=0$, giving your solutions in the form $a+ib$ where $a$ and $b$ are real numbers.

Find the two square roots of $-i$, giving the answers in the form $x+iy$, where $x$ and $y$ are real numbers.

Find the angle between the vectors $\vec a=\begin{pmatrix}2\\0\\4\end{pmatrix}$ and $\vec b=\begin{pmatrix}-3\\1\\2\end{pmatrix}$, giving the angle in degrees correct to 1 decimal place.

The complex numbers $z=2e^{i\pi/2}$ and $w=6e^{i\pi/6}$ are given. Find $zw$, giving the answer in the form $re^{i\theta}$.

Find $\int\frac{2x+3}{x^2+2x+2}\,dx$.

Consider Statement A: “If $n^2$ is even, then $n$ is even.” What is the converse of Statement A?

The diagram shows pyramid $ABCDS$, where $ABCD$ is a square. The diagonals of the square bisect each other at $H$. Show that
$\overrightarrow{HA}+\overrightarrow{HB}+\overrightarrow{HC}+\overrightarrow{HD}=\vec0.$