Mathematics Extension 1

A standard six-sided die is rolled 12 times.

Let $\hat p$ be the proportion of the rolls with an outcome of 2.

Which of the following expressions is the probability that at least 9 of the rolls have an outcome of 2?
A. $P\left(\hat p\geq\frac{3}{4}\right)$
B. $P\left(\hat p\geq\frac{1}{6}\right)$
C. $P\left(\hat p\leq\frac{3}{4}\right)$
D. $P\left(\hat p\leq\frac{1}{6}\right)$

The diagram shows the direction field of a differential equation. A particular solution to the differential equation passes through $(-2,1)$.

Where does the solution that passes through $(-2,1)$ cross the $y$-axis?
A. $y=1.12$
B. $y=1.34$
C. $y=1.56$
D. $y=1.78$

The diagram shows the graphs of the functions $f(x)$ and $g(x)$.

It is known that

What is the area between the curves $y=f(x)$ and $y=g(x)$ between $x=a$ and $$? A. B. C. D.

Which of the following is the value of $\sin^{-1}(\sin a)$ given that

A. $a-\pi$
B. $\pi-a$
C. $a$
D. $-a$

Given the two non-zero vectors $\vec a$ and $\vec b$, let $\vec c$ be the projection of $\vec a$ onto $\vec b$.

What is the projection of $10\vec a$ onto $2\vec b$? A. B. C. D.

Which statement is always true for real numbers $a$ and $b$ where $-1\leq a<b\leq1$?
A. $\sec a<\sec b$
B. $\sin^{-1}a<\sin^{-1}b$
C. $\arccos a<\arccos b$
D. $\cos^{-1}a+\sin^{-1}a<\cos^{-1}b+\sin^{-1}b$

The diagram shows the graph of a function.

[Graph of the function]

Which of the following is the equation of the function?
A. $y=\left|1-\left||x|-2\right|\right|$
B. $y=\left|2-\left||x|-1\right|\right|$
C. $y=\left|1-\left|x-2\right|\right|$
D. $y=\left|2-\left|x-1\right|\right|$

The graph of a cubic function, $y=f(x)$, is given below.

[Graph of the cubic function $y=f(x)$]

Which of the following functions has an inverse relation whose graph has more than 3 points with an $x$-coordinate of $1$?
A. $y=\sqrt{f(x)}$
B. $y=\dfrac{1}{f(x)}$
C. $y=f(|x|)$
D. $y=|f(x)|$

A group with 5 students and 3 teachers is to be arranged in a circle.

In how many ways can this be done if no more than 2 students can sit together?
A. $4!\times3!$
B. $5!\times3!$
C. $2!\times5!\times3!$
D. $2!\times2!\times2!\times3!$

Solve $\cos\theta+\sin\theta=1$ for $0\leq\theta\leq2\pi$.

A recent census found that 30% of Australians were born overseas.

A sample of 900 randomly selected Australians was surveyed.

Let $\hat p$ be the sample proportion of surveyed people who were born overseas.

A normal distribution is to be used to approximate $P(\hat p\leq0.31)$.

(i) Show that the variance of the random variable $\hat p$ is

The parametric equations of a line are given below.

Find the Cartesian equation of this line in the form $y=mx+c$.

In how many different ways can all the letters of the word CONDOBOLIN be arranged in a line?

Consider the polynomial

where $a$ and $b$ are real numbers.

It is given that $x+1$ is a factor of $P(x)$ and that, when $P(x)$ is divided by $x-2$, the remainder is .

Find $a$ and $b$.

Find

A recent census found that 30% of Australians were born overseas.

A sample of 900 randomly selected Australians was surveyed.

Let $\hat p$ be the sample proportion of surveyed people who were born overseas.

A normal distribution is to be used to approximate $P(\hat p\leq0.31)$.

(ii) Use the standard normal distribution and the information on page 16 to approximate $P(\hat p\leq0.31)$, giving your answer correct to two decimal places.

A gym has 9 pieces of equipment: 5 treadmills and 4 rowing machines.

On average, each treadmill is used 65% of the time and each rowing machine is used 40% of the time.

(i) Find an expression for the probability that, at a particular time, exactly 3 of the 5 treadmills are in use.

Evaluate

using the substitution $u=x-3$.

Use mathematical induction to prove that

for all integers $n\geq1$.

A gym has 9 pieces of equipment: 5 treadmills and 4 rowing machines.

On average, each treadmill is used 65% of the time and each rowing machine is used 40% of the time.

(ii) Find an expression for the probability that, at a particular time, exactly 3 of the 5 treadmills are in use and no rowing machines are in use.

It is known that

for all integers such that $1\le r\le n-1$. (Do NOT prove this.)

Find ONE possible set of values for $p$ and $q$ such that

The region, $R$, bounded by the hyperbola

the line $x=10$ and the coordinate axes is shown. [Diagram of the shaded region and rotation about the $y$-axis.]

Find the volume of the solid of revolution formed when the region $R$ is rotated about the $y$-axis. Leave your answer in exact form.

A hemispherical water tank has radius $R$ cm. The tank has a hole at the bottom which allows water to drain out.

Initially the tank is empty. Water is poured into the tank at a constant rate of $2kR\ \mathrm{cm^3\,s^{-1}}$, where $k$ is a positive constant.

After $t$ seconds, the height of the water in the tank is $h$ cm, as shown in the diagram, and the volume of water in the tank is $V\ \mathrm{cm^3}$.

It is known that

(Do NOT prove this.)

While water flows into the tank and also drains out of the bottom, the rate of change of the volume of water in the tank is given by

(i) Show that

A hemispherical water tank has radius $R$ cm. The tank has a hole at the bottom which allows water to drain out.

Initially the tank is empty. Water is poured into the tank at a constant rate of $2kR\ \mathrm{cm^3\,s^{-1}}$, where $k$ is a positive constant.

After $t$ seconds, the height of the water in the tank is $h$ cm, as shown in the diagram, and the volume of water in the tank is $V\ \mathrm{cm^3}$.

It is known that

(Do NOT prove this.)

While water flows into the tank and also drains out of the bottom, the rate of change of the volume of water in the tank is given by

(ii) Show that the tank is full of water after

seconds.