Mathematics Extension 2

Let $A$, $B$, $P$ be three points in three-dimensional space with $A\ne B$.

If $P$ is on the line $AB$, then there exists a real number $\lambda$ such that $\overrightarrow{AP}=\lambda\overrightarrow{AB}$.

Which of the following is the contrapositive of this statement?

A. If for all real numbers $\lambda$, $\overrightarrow{AP}=\lambda\overrightarrow{AB}$, then is on the line . B. If for all real numbers , , then is not on the line . C. If there exists a real number such that , then is on the line . D. If there exists a real number such that , then is not on the line .

Given
$\overrightarrow{OA}=\begin{bmatrix}1\\1\\0\end{bmatrix},\quad \overrightarrow{OB}=\begin{bmatrix}2\\-1\\1\end{bmatrix},\quad \overrightarrow{OC}=\begin{bmatrix}3\\\mu\\\lambda\end{bmatrix},$
where $O$ is the origin, what are the values of $\mu$ and $\lambda$ if $A$, $B$ and $C$ are collinear?

A. $\mu=3,\ \lambda=2$

B. $\mu=-3,\ \lambda=2$

C. $\mu=3,\ \lambda=-2$

D. $\mu=-3,\ \lambda=-2$

Of the following expressions, which one need NOT contain a term involving a logarithm in its anti-derivative?

A. $\dfrac{x+2}{x^2+4x+5}$

B. $\dfrac{x+2}{x^2-4x-5}$

C. $\dfrac{x-1}{x^3-x^2+x-1}$

D. $\dfrac{x+1}{x^3-x^2+x-1}$

The equation of the curve in 3-dimensional space below is

A. $x=t^2$, $y=t$, $z=0$

B. $x=0$, $y=t^2$, $z=t$

C. $x=0$, $y=t$, $z=t^2$

D. $x=t^2$, $y=t^2$, $z=0$

One of the roots of $iz^2+3z+3-11i=0$ is $3+2i$. What is the other root?

A. $-2i$
B. $-3+i$
C. $3-2i$
D. $-5-3i$

If $\displaystyle\int_a^x f(t)\,dt=g(x)$, which of the following is a primitive of $f(x)g(x)$?

A. $\frac12[f(x)]^2$

B. $\frac12[f'(x)]^2$

C. $\frac12[g(x)]^2$

D. $\frac12[g'(x)]^2$

$\int\frac{1}{\sqrt{6x-x^2}}\,dx=$

A. $\sin^{-1}\left(\frac{x}{\sqrt6}\right)+C$

B. $\sin^{-1}\left(\frac{x}{6}\right)+C$

C. $\sin^{-1}\left(\frac{x-3}{\sqrt6}\right)+C$

D. $\sin^{-1}\left(\frac{x-3}{3}\right)+C$

It is known that a particular complex number $z$ is NOT a real number. Which of the following could be true for this number $z$?

A. $\bar z=iz$

B. $\bar z=|z^2|$

C. $\operatorname{Re}(iz)=\operatorname{Im}(z)$

D. $\operatorname{Arg}(z^3)=\operatorname{Arg}(z)$

Which of the following is the locus of $z$ such that
$\arg(z+3i)-\arg(z-i)=\frac\pi2?$

If $f(x)$ is even then
$\int_{-a}^{a}f(x)\,dx=$

A. $2\int_0^a f(a-x)\,dx$

B. $2\int_0^{-a} f(a-x)\,dx$

C. $2\int_a^0 f(x)\,dx$

D. All of the above.

$z^2+pz+q=0$, where $p,q$ are real, has one solution $z=3-2i$. The values of $p$ and $q$ are:

A. $p=6$, $q=13$

B. $p=-6$, $q=13$

C. $p=-6$, $q=-13$

D. $p=-6$, $q=-13$

Consider the statement $P$.

$P$: For all integers $n\ge1$, if $n$ is a prime number then $\dfrac{n(n+1)}2$ is a prime number.

Which of the following is true about this statement and its converse?

A. The statement $P$ and its converse are both true.
B. The statement $P$ and its converse are both false.
C. The statement $P$ is true and its converse is false.
D. The statement $P$ is false and its converse is true.

Which of the following definite integrals has a positive value?

A. $\displaystyle\int_{-1/2}^{1/2}x^3(1+\cos x)\,dx$

B. $\displaystyle\int_{-1/2}^{1/2}\frac{\sin^{-1}x}{1+x^4}\,dx$

C. $\displaystyle\int_{-1/2}^{1/2}\frac{\cos^{-1}x}{e^{-x^2}}\,dx$

D. $\displaystyle\int_{-1/2}^{1/2}\frac{\tan^{-1}x}{e^x+e^{-x}}\,dx$

As a projectile of mass $m$ kilograms travels through air, it experiences a frictional force. The magnitude of this force is proportional to the square of the speed $v$ of the projectile. The constant of proportionality is the positive number $k$. The position of the particle at time $t$ is denoted by $\begin{pmatrix}x\\y\end{pmatrix}$. The acceleration due to gravity is $g\,\mathrm{m\,s^{-2}}$.

Based on Newton’s laws of motion, which equation models the motion of this projectile?

A. $\begin{pmatrix}0\\-mg\end{pmatrix}+kv\begin{pmatrix}\dot x\\\dot y\end{pmatrix}=m\begin{pmatrix}\ddot x\\\ddot y\end{pmatrix}$

B. $\begin{pmatrix}0\\-mg\end{pmatrix}-kv\begin{pmatrix}\dot x\\\dot y\end{pmatrix}=m\begin{pmatrix}\ddot x\\\ddot y\end{pmatrix}$

C. $\begin{pmatrix}0\\-mg\end{pmatrix}+kv^2\begin{pmatrix}\dot x\\\dot y\end{pmatrix}=m\begin{pmatrix}\ddot x\\\ddot y\end{pmatrix}$

D. $\begin{pmatrix}0\\-mg\end{pmatrix}-kv^2\begin{pmatrix}\dot x\\\dot y\end{pmatrix}=m\begin{pmatrix}\ddot x\\\ddot y\end{pmatrix}$

If the complex number $z$ has $|z|>1$ and $\frac\pi2<\operatorname{Arg}(z)<\pi$, and $\overrightarrow{OA}=z$, then if the two square roots of $z$ are $\overrightarrow{OP}$ and $\overrightarrow{OQ}$, $P$ and $Q$ could be at:

Given that $z$ is a complex number such that $\operatorname{Re}(z)\ne0$, which of the following is equivalent to
$\frac{4z\bar z}{(z+\bar z)^2}?$

A. $1+\left(\frac{\operatorname{Im}(z)}{\operatorname{Re}(z)}\right)^2$ B. C. D.

Let $A$ and $B$ be two distinct points in three-dimensional space. Let $M$ be the midpoint of $AB$.

Let $S_1$ be the set of all points $P$ such that $\overrightarrow{AP}\cdot\overrightarrow{BP}=0$.

Let $S_2$ be the set of all points $N$ such that $|\overrightarrow{AN}|=|\overrightarrow{MN}|$.

The intersection of $S_1$ and $S_2$ is the circle $S$.

What is the radius of the circle $S$?

A. $\dfrac{|\overrightarrow{AB}|}{2}$
B. ${\displaystyle }$ C. D.

Which of the following is the negation of the statement:

“$\forall p\in P$, $p$ is of the form $4m+1\Rightarrow p$ can be written as a sum of two squares”?

A. $\forall p\in P$, $p$ is of the form $4m+1$ and $p$ cannot be written as a sum of two squares.
B. $\exists p\in P$, is not of the form and can be written as a sum of two squares. C. , is not of the form or cannot be written as a sum of two squares. D. , is of the form and cannot be written as a sum of two squares.

The acceleration of a particle is given by $\ddot{x}=\sec^2x$, where $x$ is the displacement in metres. The rule for the velocity could be:

A. $v=\sqrt{\tan x}$

B. $v=\sqrt{2\tan x}$

C. $v=\tan x$

D. $v=2\tan x$

A particle is moving vertically in a resistive medium under the influence of gravity. The resistive force is proportional to the velocity of the particle.

The initial speed of the particle is NOT zero.

Which of the following statements about the motion of the particle is always true?

A. If the particle is initially moving downwards, then its speed will increase.
B. If the particle is initially moving downwards, then its speed will decrease.
C. If the particle is initially moving upwards, then its speed will eventually approach a terminal speed.
D. If the particle is initially moving upwards, then its speed will not eventually approach a terminal speed.

In the diagram, the vectors $\overrightarrow{OP}$ and $\overrightarrow{OQ}$ represent the complex numbers $z_1$ and $z_2$ respectively. If
$\frac{z_2}{z_1}=\sqrt3\,i,$
what is the value of
$\left|\frac{z_1+z_2}{z_2-z_1}\right|?$

A. $1$
B. $\sqrt3$
C. $\frac1{\sqrt3}$ D.

A particle moves with simple harmonic motion with equation $x=a\sin nt$. It first returns to its starting point after $\frac\pi3$ seconds and is travelling at $15$ m/s forwards at this time. The displacement of the particle is given by

A. $x=\frac52\sin6t$

B. $x=-\frac52\sin6t$

C. $x=5\sin3t$

D. $x=-5\sin3t$

Using the points defined in part (d), what is the size of the acute angle between $\overrightarrow{AB}$ and $\overrightarrow{CB}$? Answer correct to the nearest degree.

Find
$\int\frac{1}{\sqrt{5+4x-x^2}}\,dx.$