Mathematics Extension 2

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. $\frac{1}{7}{x}^5{e}^{7x}-\frac{5}{7}\int$ B. C. D.

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

A. $\vec{r}=$ B. C. D.

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

Which of the following statements is FALSE?

A. $\forall a,b\in\mathbb R$, $a<b\Rightarrow a^3<b^3$. B. , . C. , . D. , with , .

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

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

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 ?

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 that would allow the particle to follow the curve to ?

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. $\frac{\mathrm{Re}(zw)}{\mathrm{\mid }w\mathrm{\mid }}$ B. C. D.

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

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

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

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

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

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

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

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

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

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

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

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. C. D.

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

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. is divisible by C. is prime D. is divisible by

Consider the statement: “If $n$ is even, then if $n$ is a multiple of $3$, then $n$ is a multiple of $6$”. Which of the following is the negation of this statement?

A. $$ is odd and is not a multiple of or . B. is even and is a multiple of but not a multiple of . C. If is even, then is not a multiple of and is not a multiple of . D. If is odd, then if is not a multiple of then is not a multiple of .