Mathematics Extension 2

Which of the following is not equivalent to $P \Rightarrow Q$?

(A) $P$ is sufficient for $Q$

(B) $Q$ is necessary for $P$

(C) If $Q$ is false, then $P$ is false

(D) None of the above

In which quadrant is the complex number $(-3+3i)^3$ located on the Argand plane?

(A) The first quadrant

(B) The second quadrant

(C) The third quadrant

(D) The fourth quadrant

Consider the following statement: “If an animal is a herbivore, then it does not eat meat.” Which of the following is the converse of this statement?

A. If an animal is a herbivore, then it eats meat.
B. If an animal is not a herbivore, then it eats meat.
C. If an animal eats meat, then it is not a herbivore.
D. If an animal does not eat meat, then it is a herbivore.

A complex number $z$ lies on the unit circle in the complex plane, as shown in the diagram. Which of the following complex numbers is equal to $\bar z$?

A. $-z$
B. $z^2$
C. $-z^3$
D. $z^4$

A line has equation

Which of the following is parallel to this line?

(A) $\displaystyle \vec r=\begin{pmatrix}2\\1\\0\end{pmatrix}+\lambda\begin{pmatrix}3\\2\\-15\end{pmatrix}$

(B) $\displaystyle \vec r=\begin{pmatrix}1\\0\\1\end{pmatrix}+\lambda\begin{pmatrix}3\\-6\\-15\end{pmatrix}$

(C) $\displaystyle \vec r=\begin{pmatrix}0\\-1\\2\end{pmatrix}+\lambda\begin{pmatrix}1\\-2\\1\end{pmatrix}$

(D) $\displaystyle \vec r=\begin{pmatrix}-1\\2\\5\end{pmatrix}+\lambda\begin{pmatrix}1\\2\\-5\end{pmatrix}$

Consider the following statement about real numbers: “Whichever positive number $r$ you pick, it is possible to find a number $x$ greater than $1$ such that $\frac{\ln x}{x^3}<r$.” When this statement is written in the formal language of proof, which of the following is obtained?

A. $\forall x>1\;\exists r>0\;\frac{\ln x}{x^3}<r$ B. C. D.

Which of the following is an expression for

(A) $\displaystyle \sin^{-1}\left(\frac{x-3}{2}\right)+C$

(B) $\displaystyle \sin^{-1}\left(\frac{x+3}{2}\right)+C$

(C) $\displaystyle \sin^{-1}\left(\frac{x-3}{4}\right)+C$

(D) $\displaystyle \sin^{-1}\left(\frac{x+3}{4}\right)+C$

Which of the following is a true statement about the lines
$\ell_1=\begin{pmatrix}-1\\2\\5\end{pmatrix}+\lambda\begin{pmatrix}-1\\3\\1\end{pmatrix},\qquad \ell_2=\begin{pmatrix}3\\-10\\1\end{pmatrix}+\mu\begin{pmatrix}1\\-3\\-1\end{pmatrix}?$

A. $\ell_1$ and $\ell_2$ are the same line.
B. $\ell_1$ and are not parallel and they intersect. C. and are parallel and they do not intersect. D. and are not parallel and they do not intersect.

In the diagram below, $z$ is any complex number which lies on the line $S$. Which equation best describes the locus of $z$?

[Diagram: Argand plane showing the line $S$ and its geometric construction.]

(A) $\displaystyle \arg z=\frac{\pi}{4}$

(B) $\displaystyle \arg z=\frac{3\pi}{4}$

(C) $\lvert z-2\rvert=\lvert z-2i\rvert$

(D) $\lvert z-2\rvert=\lvert z+2i\rvert$

Which of the following functions does not describe simple harmonic motion?

A. $x=\cos^2t-\sin 2t$
B. $x=\sin 4t+4\cos 2t$
C. $x=2\sin 3t-4\cos 3t+5$
D. $x=4\cos\left(2t+\frac\pi2\right)+5\sin\left(2t-\frac\pi4\right)$

The polynomial $P(z)$ has real coefficients and $P(0)=-1$. The imaginary number $\alpha$ and the real number $\beta$ satisfy $P(\alpha)=0$, $P(\beta)=0$ and $P'(\beta)=0$.

The degree of $P(z)$ is at least:

(A) $2$

(B) $3$

(C) $4$

(D) $5$

Which of the following statements about complex numbers is true?

A. For all real numbers $x,y,\theta$ with $x\ne0$, $\tan\theta=\frac yx\Rightarrow x+iy=re^{i\theta}$ for some real number $r$.
B. For all non-zero complex numbers $z_1,z_2$, if $\operatorname{Arg}(z_1)=\theta_1$ and $\operatorname{Arg}(z_2)=\theta_2$, then $\operatorname{Arg}(z_1z_2)=\theta_1+\theta_2$.
C. For all real $r_1,r_2,\theta_1,\theta_2$ with $r_1,r_2>0$, $r_1e^{i\theta_1}=r_2e^{i\theta_2}\Rightarrow r_1=r_2$ and $\theta_1=\theta_2$.
D. For all real $x,y,r,\theta$ with $r>0$ and $x\ne0$, $x+iy=re^{i\theta}\Rightarrow\theta=\arctan(y/x)$.

Which expression is equal to

(A) $\displaystyle 2x\sqrt{x}\left(\ln x-\frac{2}{3}\right)+c$

(B) $\displaystyle 2x\sqrt{x}\left(\ln x+\frac{2}{3}\right)+c$

(C) $\displaystyle \frac{1}{\sqrt{x}}\left(\frac{3}{2}\ln x-1\right)+c$

(D) $\displaystyle \frac{1}{\sqrt{x}}\left(\frac{3}{2}\ln x+1\right)+c$

A particle is travelling in simple harmonic motion such that its velocity, in metres per second, is given by the equation

where $a,b\ne0$. What is the period of motion?

(A) $\displaystyle \frac{2b\pi}{a}$

(B) $\displaystyle \frac{2a\pi}{b}$

(C) $\displaystyle \frac{2\pi}{a}$

(D) $\displaystyle \frac{2\pi}{b}$

A shaded region on a complex plane is shown. Which relation best describes the region shaded on the complex plane?

A. $|z-i|>2|z-1|$
B. $|z-i|<2|z-1|$
C. $|z-1|>2|z-i|$
D. $|z-1|<2|z-i|$

Which of the equations best represent the curve below?

[Diagram: A three-dimensional spiral around the $z$-axis, with radius increasing as $|t|$ increases.]

(A) $\displaystyle \vec r(t)=(\cos t)\vec i+(\sin t)\vec j+t\vec k$

(B) $\displaystyle \vec r(t)=(t\cos t)\vec i+(t\sin t)\vec j+t\vec k$

(C) $\displaystyle \vec r(t)=(t\cos t)\vec i+(t\sin t)\vec j+\frac{1}{t}\vec k$

(D) $\displaystyle \vec r(t)=(\cos t)\vec i+(\sin t)\vec j+\frac{1}{t}\vec k$

A particle travels along a curve from $O$ to $E$ in the $xy$-plane, as shown in the diagram. The position vector is $\mathbf r$, velocity is $\mathbf v$, and acceleration is $\mathbf a$. While travelling from $O$ to $E$, the particle is always slowing down. Which of the following is consistent with the motion?

A. $\mathbf r\cdot\mathbf v\le0$ and $\mathbf a\cdot\mathbf v\ge0$
B. $\mathbf r\cdot\mathbf v\le0$ and C. and D. and

A particle is moving along a straight line. At time $t$, its velocity is $v$ and its displacement from a fixed origin is $x$.

If

which of the following best describes the particle’s acceleration and velocity?

(A) Constant acceleration and constant velocity

(B) Constant acceleration and decreasing velocity

(C) Constant acceleration and increasing velocity

(D) Increasing acceleration and increasing velocity

Consider three-dimensional vectors $\mathbf a=\overrightarrow{OA}$, $\mathbf b=\overrightarrow{OB}$ and $\mathbf c=\overrightarrow{OC}$ satisfying $\mathbf a\cdot\mathbf b=1$, $\mathbf b\cdot\mathbf c=2$, and $\mathbf c\cdot\mathbf a=3$. Which statement is true?

A. Two of $\mathbf a,\mathbf b,\mathbf c$ could be unit vectors.
B. The points $A,B,C$ could lie on a sphere centred at $O$.
C. For any three-dimensional vector $\mathbf a$, vectors $$ can be found satisfying the conditions. D. For all such vectors, there exist positive reals such that .

Find a vector equation of the line through the points $A(-3,1,5)$ and $B(0,2,3)$.

Find the angle between the vectors $\mathbf a=\mathbf i+2\mathbf j-3\mathbf k$ and $\mathbf b=-\mathbf i+4\mathbf j+2\mathbf k$, giving your answer to the nearest degree.

Solve the quadratic equation $z^2-3z+4=0$, where $z$ is a complex number. Give your answers in Cartesian form.

A particle moves in simple harmonic motion described by $\ddot x=-9(x-4)$. Find the period and the central point of motion.

By first writing $\sqrt{3}+i$ and $\sqrt{3}-i$ in exponential form, express

in the form of $a^b$, where $a$ and $b$ are integers.