Mathematics Extension 2

Find the value of $\lambda$ such that $\overrightarrow{HG}=\lambda\overrightarrow{HS}$.

Show that the converse of Statement A is true.

Region $A$ is bounded by $y=1$ and $x^2+y^2=1$ between $x=0$ and $x=1$. Region $B$ is bounded by $y=1$ and $y=\ln x$ between $x=1$ and $x=e$. Using part (i), or otherwise, show that the ratio of the volumes formed by rotating regions $A$ and $B$ about the $x$-axis is $V_A:V_B=1:3$.

An object is moving in simple harmonic motion along the $x$-axis. Its acceleration is $\ddot x=-4(x-3)$, where $x$ is its displacement from the origin in metres. Initially, it is $5.5$ metres to the right of the origin and moving towards the origin. Its speed is $8\,\mathrm{m\,s^{-1}}$ as it passes through the origin. Between which two values of $x$ is the particle oscillating?

For the simple harmonic motion described in part (i), find the first value of $t$ for which $x=0$, giving the answer correct to 2 decimal places.

There is one value of $v$ such that the object slides down the slope at constant speed. Find this value of $v$ in $\mathrm{m\,s^{-1}}$, correct to 1 decimal place, given that $g=10$.

By using part (i), or otherwise, show that
$\operatorname{Re}\left(e^{i\pi/10}\right)=\sqrt{\frac{5+\sqrt5}{8}}.$

Using part (i), or otherwise, show that for all non-negative real numbers $a,b,c$,
$\sqrt{abc}\le\frac{a^2+b^2+c^2+a+b+c}{6}.$

Prove that $2^n+3^n\ne5^n$ for all integers $n\ge2$.

For the resisted vertical motion described in part (i), find an expression for the maximum height reached by the object, in terms of $k$, $g$ and $u$.

An object of mass $1\,\mathrm{kg}$ is projected vertically upwards with initial velocity $u\,\mathrm{m\,s^{-1}}$. It experiences air resistance of magnitude $kv^2$ newtons, where $v$ is velocity in $\mathrm{m\,s^{-1}}$ and $k>0$. The height above its starting point is $x$ metres, time is $t$ seconds, and acceleration due to gravity is $g\,\mathrm{m\,s^{-2}}$. Show that the time to reach maximum height is
$\frac1{\sqrt{gk}}\arctan\left(u\sqrt{\frac{k}{g}}\right)\text{ seconds}.$

Show that the triangular numbers $t_2,t_4,t_6$, and so on, are not hexagonal numbers.

Show that the triangular numbers $t_1,t_3,t_5$, and so on, are also hexagonal numbers.

For all non-negative real numbers $x$ and $y$, $\sqrt{xy}\le\frac{x+y}{2}$. (Do NOT prove this.) Using this fact, show that for all non-negative real numbers $a,b,c$,
$\sqrt{abc}\le\frac{a^2+b^2+2c}{4}.$

As in part (i), $P(x,y,z)$ lies on the unit sphere centred at the origin. Using part (ii), or otherwise, show that
$|x|+|y|+|z|\le\sqrt3.$

The point $P(x,y,z)$ lies on the sphere of radius $1$ centred at the origin $O$. Using $\overrightarrow{OP}=x\vec i+y\vec j+z\vec k$ and the triangle inequality, or otherwise, show that
$|x|+|y|+|z|\ge1.$

Given vectors $\vec a=\begin{pmatrix}a_1\\a_2\\a_3\end{pmatrix}$ and $\vec b=\begin{pmatrix}b_1\\b_2\\b_3\end{pmatrix}$, show that
$|a_1b_1+a_2b_2+a_3b_3|\le\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}.$

What is the length of the vector $-\mathbf{i}+18\mathbf{j}-6\mathbf{k}$?

A. $5$
B. $19$
C. $25$
D. $361$

Given that $z=3+i$ is a root of $z^2+pz+q=0$, where $p$ and $q$ are real, what are the values of $p$ and $q$?

A. $p=-6$, $q=\sqrt{10}$
B. $$, C. , D. ,

What is the Cartesian equation of the line $\mathbf r=\begin{pmatrix}1\\3\end{pmatrix}+\lambda\begin{pmatrix}-2\\4\end{pmatrix}$?

A. $2y+x=7$
B. $y-2x=-5$
C. $y+2x=5$ D.

The diagram shows the complex number $z$ on the Argand diagram.

Which of the following diagrams best shows the position of $\dfrac{z^2}{|z|}$?

A particle undergoing simple harmonic motion has a maximum acceleration of $6\ \mathrm{m/s^2}$ and a maximum velocity of $4\ \mathrm{m/s}$. What is the period of the motion?

A. $\pi$
B. $\dfrac{2\pi}{3}$
C. $3\pi$
D. $\dfrac{4\pi}{3}$

Which expression is equal to $\displaystyle\int\frac{1}{x^2+4x+10}\,dx$?

A. $\dfrac1{\sqrt6}\tan^{-1}\left(\dfrac{x+2}{\sqrt6}\right)+c$ B. C. D.

Consider the proposition: “If $2^n-1$ is not prime, then $n$ is not prime”. Given that each of the following statements is true, which statement disproves the proposition?

A. $2^5-1$ is prime
B. $2^6-1$ is divisible by $9$
C. $2^7-1$ is prime
D. $2^{11}-1$ is divisible by $23$