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BIOLOGY, M5 EQ-Bank 27

Explain why internal fertilisation is a more robust process than external fertilisation in maintaining a species generation after generation.   (5 marks)

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→ In order for a species to successfully continue and avoid extinction, a sufficient number of offspring must be able to survive until gestational maturity and produce offspring themselves, hence continuing the cycle.

→ External fertilisation is used mainly in sea animals such as fish where the survival rate of offspring is low due to the plentiful numbers of predators in such a vast environment.

→ In response to this environment, external fertilisation methods must produce large numbers of sperm and eggs to increase the number of fertilised embryos. This process requires of a significant amount of energy.

→ Fertilised embryos typically experience minimal or no parental care, further lowering the survival rate of offspring. Therefore, external fertilisation has numerous disadvantages to overcome in achieving the continuity of the species.

→ Internal fertilisation is generally more common amongst varying species (such as humans). Despite often only producing one child at a time, internal fertilisation as a process protects the developing embryo inside the female uterus.

→ Although the number of fertilised embryos is lower, greater parental care throughout most of early life and a significant decrease in predation creates a higher survival rate.

→ In this way, internal fertilisation is more advantageous to ensure the continuity of a species.

Show Worked Solution

→ In order for a species to successfully continue and avoid extinction, a sufficient number of offspring must be able to survive until gestational maturity and produce offspring themselves, hence continuing the cycle.

→ External fertilisation is used mainly in sea animals such as fish where the survival rate of offspring is low due to the plentiful numbers of predators in such a vast environment.

→ In response to this environment, external fertilisation methods must produce large numbers of sperm and eggs to increase the number of fertilised embryos. This process requires of a significant amount of energy.

→ Fertilised embryos typically experience minimal or no parental care, further lowering the survival rate of offspring. Therefore, external fertilisation has numerous disadvantages to overcome in achieving the continuity of the species.

→ Internal fertilisation is generally more common amongst varying species (such as humans). Despite often only producing one child at a time, internal fertilisation as a process protects the developing embryo inside the female uterus.

→ Although the number of fertilised embryos is lower, greater parental care throughout most of early life and a significant decrease in predation creates a higher survival rate.

→ In this way, internal fertilisation is more advantageous to ensure the continuity of a species.

BIOLOGY, M5 EQ-Bank 22

  1. The following data shows the average amount of Human Chorionic Gonadotropin (hCG) produced by pregnant women.
      
    \begin{array}{|c|c|}
    \hline \rule{0pt}{2.5ex}\text {Weeks of pregnancy} \rule[-1ex]{0pt}{0pt} & h C G\,\text{(ng/mL)} \\
    \hline \rule{0pt}{2.5ex}0 \rule[-1ex]{0pt}{0pt}& 0 \\
    \hline \rule{0pt}{2.5ex}4 \rule[-1ex]{0pt}{0pt}& 85 \\
    \hline \rule{0pt}{2.5ex}8 \rule[-1ex]{0pt}{0pt}& 185 \\
    \hline \rule{0pt}{2.5ex}12 \rule[-1ex]{0pt}{0pt}& 185 \\
    \hline \rule{0pt}{2.5ex}16 \rule[-1ex]{0pt}{0pt}& 80 \\
    \hline \rule{0pt}{2.5ex}20 \rule[-1ex]{0pt}{0pt}& 65 \\
    \hline \rule{0pt}{2.5ex}24 \rule[-1ex]{0pt}{0pt}& 60 \\
    \hline \rule{0pt}{2.5ex}28 \rule[-1ex]{0pt}{0pt}& 65 \\
    \hline \rule{0pt}{2.5ex}32 \rule[-1ex]{0pt}{0pt}& 75 \\
    \hline \rule{0pt}{2.5ex}36 \rule[-1ex]{0pt}{0pt}& 65 \\
    \hline \rule{0pt}{2.5ex}40 \rule[-1ex]{0pt}{0pt}& 35 \\
    \hline
    \end{array}
      
    Use the information provided to graph the levels of hCG in a normal pregnancy.   (3 marks)
     
  2. Describe the role and changes in levels of a hormone in pregnancy.   (3 marks)

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a.   

b.   The role and changes in levels of hormones in pregnancy.

→ Progesterone is a sex hormone which is initially secreted by the corpus leuteum after ovulation to thicken the endometrium for implantation.

→ Once the placenta has developed, it then increases the progesterone levels which maintain the pregnancy.

→ Progesterone levels eventually drop in the last stage of pregnancy to facilitate delivery.
  

Answers could also include information about other hormones including

→ Oestrogen

→ hCG

Show Worked Solution

a.   

b.   The role and changes in levels of hormones in pregnancy.

→ Progesterone is a sex hormone which is initially secreted by the corpus leuteum after ovulation to thicken the endometrium for implantation.

→ Once the placenta has developed, it then increases the progesterone levels which maintain the pregnancy.

→ Progesterone levels eventually drop in the last stage of pregnancy to facilitate delivery.
  

Answers could also include information about other hormones including

→ Oestrogen

→ hCG

BIOLOGY, M5 EQ-Bank 9 MC

Muscular Distrophy is a sex-linked recessive trait.

Nicole does not have Muscular Distrophy but her father does. Nicole is married to Keith who does not have Muscular Distrophy. Nicole and Keith are expecting twins, a boy and a girl.

Which row in the table shows the probability of Muscular Distrophy in the boy and the girl?

\begin{align*}
\begin{array}{l}
\rule{0pt}{2.5ex} \ \rule[-1ex]{0pt}{0pt}& \\
\rule{0pt}{2.5ex} \textbf{A.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex} \textbf{B.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex} \textbf{C.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex} \textbf{D.}\rule[-1ex]{0pt}{0pt}\\
\end{array}
\begin{array}{|c|c|}
\hline \rule{0pt}{2.5ex} \quad \quad\  \textit{Boy}\  \quad\quad & \quad \quad\textit{Girl} \quad\quad  \\
\hline \rule{0pt}{2.5ex}  0 \% \rule[-1ex]{0pt}{0pt}& 0 \% \\
\hline \rule{0pt}{2.5ex} 50 \% \rule[-1ex]{0pt}{0pt}& 0 \% \\
\hline \rule{0pt}{2.5ex} 0 \% \rule[-1ex]{0pt}{0pt}& 50 \% \\
\hline \rule{0pt}{2.5ex} 50 \% \rule[-1ex]{0pt}{0pt}& 50 \% \\
\hline
\end{array}
\end{align*}

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\(B\)

Show Worked Solution

→ The probability can be predicted using the punnet square below.

→ Since Nicole does not have Muscular Dystrophy but her father does, she must be heterozygous.

→ Nicole : \(\text{X}^\text{B} \text{X}^\text{b}\)   Keith : \(\text{X}^\text{b} \text{Y}\)

\begin{array} {|l|c|c|}
\hline  & \text{X}^\text{b} & \text{Y} \\
\hline \text{X}^\text{B} & \text{X}^\text{B}\text{X}^\text{b} & \text{X}^\text{B}\text{Y}\\
\hline \text{X}^\text{b} & \text{X}^\text{b}\text{X}^\text{b} & \text{X}^\text{b}\text{Y} \\
\hline \end{array}

→ This shows that there is a 50% chance for boys to be \(\text{X}^{b} \text{Y}\) and therefore a 50% chance to have Muscular Dystrophy, while the girls will always have an \(\text{X}^\text{B}\) allele and therefore cannot have Muscular Dystrophy.

\(\Rightarrow B\)

BIOLOGY, M5 EQ-Bank 7 MC

In which of the following do both processes result in genetic variation of offspring?

  1. DNA mutation and polypeptide production
  2. Cell differentiation and gamete formation
  3. DNA mutation and gamete formation
  4. Cell differentiation and polypeptide production
Show Answers Only

`C`

Show Worked Solution

By Elimination

→ Cell differentiation is the specialisation of cells into a specific type, e.g. the differentiation of specific plasma B cells. This does not impact the individual’s genome or their offspring (Eliminate B and D).

→ Polypeptide production is a result of the individuals DNA and does not effect their genotype and thus has no effect on offspring (Eliminate A).

`=>C`

PHYSICS, M8 EQ-Bank 26

Observations and mathematical ideas are critical to the improvement of scientific models.

Discuss this statement with reference to scientific discoveries that have contributed to our understanding of the atom.   (8 marks)

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If scientific models are not consistent with observations and mathematical ideas, then they need to be changed. The historical progression of models of the atom is an example of this scientific method in action.

→ The Geiger-Marsden experiment involved shooting a beam of alpha particles at a thin sheet of gold foil. It was observed that most alpha particles passed through the gold however some bounced back.

→ Rutherford explained these observations in his model. He postulated that a small, dense positively charged nucleus caused the alpha particles to bounce back and that this nucleus was surrounded by orbiting electrons. The atom was mostly empty space so the majority of alpha particles passed through.

→ This model was limited however as an electron orbiting the nucleus would be accelerating and would radiate electromagnetic radiation, losing energy and spiralling into the nucleus.

→ Bohr overcame this limitation and improved the atomic model by analysing observations of the hydrogen spectrum. He postulated that electrons occupied specific energy orbits and could jump from one energy level to another and electrons travelling from a higher to lower energy orbit would emit a photon, corresponding to emission lines on hydrogen’s spectrum. 

→ Bohr developed a mathematical model explaining the specific wavelengths emitted which relies on Rydberg’s equation: `(1)/(lambda)=R((1)/(n_(f)^2)-(1)/(n_(i)^2))`.

→ This model was limited as it was only able to explain the hydrogen atom and wasn’t able to explain why electrons were quantised in such a way.

→ De Broglie built upon this model by postulating that electrons behaved as a wave. He stated that electrons behaved as standing waves with an integer number of wavelengths fitting the circumference of an orbit.

→ De Broglie’s contribution was critically supported by his mathematical model for calculating the associated wavelength `lambda=(h)/(mv)`

→ From these examples, it is clear that our understanding of the atom has been increased following observations and improved mathematical models.
 

Answers could also reference:

→ Thomson’s discovery of the electron.

→ Further contributions from Davisson-Germer as well as Schrodinger and Heisenberg.

Show Worked Solution

If scientific models are not consistent with observations and mathematical ideas, then they need to be changed. The historical progression of models of the atom is an example of this scientific method in action.

→ The Geiger-Marsden experiment involved shooting a beam of alpha particles at a thin sheet of gold foil. It was observed that most alpha particles passed through the gold however some bounced back.

→ Rutherford explained these observations in his model. He postulated that a small, dense positively charged nucleus caused the alpha particles to bounce back and that this nucleus was surrounded by orbiting electrons. The atom was mostly empty space so the majority of alpha particles passed through.

→ This model was limited however as an electron orbiting the nucleus would be accelerating and would radiate electromagnetic radiation, losing energy and spiralling into the nucleus.

→ Bohr overcame this limitation and improved the atomic model by analysing observations of the hydrogen spectrum. He postulated that electrons occupied specific energy orbits and could jump from one energy level to another and electrons travelling from a higher to lower energy orbit would emit a photon, corresponding to emission lines on hydrogen’s spectrum. 

→ Bohr developed a mathematical model explaining the specific wavelengths emitted which relies on Rydberg’s equation: `(1)/(lambda)=R((1)/(n_(f)^2)-(1)/(n_(i)^2))`.

→ This model was limited as it was only able to explain the hydrogen atom and wasn’t able to explain why electrons were quantised in such a way.

→ De Broglie built upon this model by postulating that electrons behaved as a wave. He stated that electrons behaved as standing waves with an integer number of wavelengths fitting the circumference of an orbit.

→ De Broglie’s contribution was critically supported by his mathematical model for calculating the associated wavelength `lambda=(h)/(mv)`

→ From these examples, it is clear that our understanding of the atom has been increased following observations and improved mathematical models.
 

Answers could also reference:

→ Thomson’s discovery of the electron.

→ Further contributions from Davisson-Germer as well as Schrodinger and Heisenberg.

PHYSICS, M8 EQ-Bank 27

Explain how the analysis of quantitative observations contributed to the development of the concept that certain matter and energy are quantised.   (9 marks)

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Experiments such as Millikan’s oil drop experiment and others testing the photoelectric effect have demonstrated that certain quantities of matter and energy are quantised which means they are multiples of some fundamental value.

Millikan’s Oil Drop Experiment

→ Millikan’s oil drop experiment was able to show that charge is quantised. 

→ Millikan levitated oil drops in an electric field by balancing the electric and gravitational forces on them. This allowed him to find the electric force acting on each oil drop, and using the mass of the oil drop he found its charge.

→ Analysing his results, he found that the charge on every oil drop was an integer multiple of `1.602 xx10^(-19) C`. This was determined to be the fundamental charge on an electron.

→ Further, with Thompson’s later discovery of the charge to mass ratio of an electron, its mass could be determined.
 

Photoelectric Effect

→ Photoelectric effect experiments showed the quantum properties of light which seemingly contradicted the view of light as a wave.

→ It was found that there was a minimum frequency (energy) of light that would cause photoemission when it was incident upon a metal plate, and no photoemission occurred with light lower than this frequency, regardless of intensity.

→ As one photon would strike one electron on the metal surface, the electron would receive a discrete amount of energy from that photon determined by its frequency `E=hf`. If a photon didn’t have enough energy, an electron couldn’t be removed.

→ This experimental evidence changed the conceptual understanding of energy within physics and provided a basis for the quantisation of the energy of light.
 

Other quantitative experiments that could be explored include:

→ Bohr’s analysis of emission spectra to demonstrate the existence of quantised energy levels in atoms.

→ Cathode ray experiments showing the particle nature of electrons.

→ Blackbody radiation experiments. 

Show Worked Solution

Experiments such as Millikan’s oil drop experiment and others testing the photoelectric effect have demonstrated that certain quantities of matter and energy are quantised which means they are multiples of some fundamental value.

Millikan’s Oil Drop Experiment

→ Millikan’s oil drop experiment was able to show that charge is quantised. 

→ Millikan levitated oil drops in an electric field by balancing the electric and gravitational forces on them. This allowed him to find the electric force acting on each oil drop, and using the mass of the oil drop he found its charge.

→ Analysing his results, he found that the charge on every oil drop was an integer multiple of `1.602 xx10^(-19) C`. This was determined to be the fundamental charge on an electron.

→ Further, with Thompson’s later discovery of the charge to mass ratio of an electron, its mass could be determined.
 

Photoelectric Effect

→ Photoelectric effect experiments showed the quantum properties of light which seemingly contradicted the view of light as a wave.

→ It was found that there was a minimum frequency (energy) of light that would cause photoemission when it was incident upon a metal plate, and no photoemission occurred with light lower than this frequency, regardless of intensity.

→ As one photon would strike one electron on the metal surface, the electron would receive a discrete amount of energy from that photon determined by its frequency `E=hf`. If a photon didn’t have enough energy, an electron couldn’t be removed.

→ This experimental evidence changed the conceptual understanding of energy within physics and provided a basis for the quantisation of the energy of light.
 

Other quantitative experiments that could be explored include:

→ Bohr’s analysis of emission spectra to demonstrate the existence of quantised energy levels in atoms.

→ Cathode ray experiments showing the particle nature of electrons.

→ Blackbody radiation experiments. 

PHYSICS, M8 EQ-Bank 25

Compare the features of emission and absorption spectra in terms of how they are produced.   (4 marks)

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→ Emission and absorption spectra are both produced by the transition of electrons between energy levels in an atom.

→ Emission spectra involve a series of coloured lines corresponding to “emitted” light of discrete wavelengths on a black background. Absorption spectra differ by presenting a continuous spectrum with some specific wavelengths appearing with lower intensities (having been “absorbed”). In this way, the emission and absorption spectra of a specific element or compound appear as negatives of each other.

→ Emission spectra are produced when electrons are in higher energy shells in which they are unstable and fall back to lower, more stable levels. They emit a photon of light with energy equal to the difference in energy between the two shells. These photons correspond to emission lines.

→ Absorption spectra are produced when a continuous spectrum is passed through a cool gas. Certain wavelengths of light with energies corresponding to a difference in energy levels are absorbed by the electron and it jumps to a higher energy level. This results in a dark line.

Show Worked Solution

→ Emission and absorption spectra are both produced by the transition of electrons between energy levels in an atom.

→ Emission spectra involve a series of coloured lines corresponding to “emitted” light of discrete wavelengths on a black background. Absorption spectra differ by presenting a continuous spectrum with some specific wavelengths appearing with lower intensities (having been “absorbed”). In this way, the emission and absorption spectra of a specific element or compound appear as negatives of each other.

→ Emission spectra are produced when electrons are in higher energy shells in which they are unstable and fall back to lower, more stable levels. They emit a photon of light with energy equal to the difference in energy between the two shells. These photons correspond to emission lines.

→ Absorption spectra are produced when a continuous spectrum is passed through a cool gas. Certain wavelengths of light with energies corresponding to a difference in energy levels are absorbed by the electron and it jumps to a higher energy level. This results in a dark line.

PHYSICS, M8 EQ-Bank 23

Describe TWO processes by which stars produce energy.   (4 marks)

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→ At the most basic level, stars produce energy through the process of nuclear fusion.

→ In less massive, lower temperature stars (like our Sun), the proton-proton chain reaction generates energy through nuclear fusion via the fusion of four hydrogen atoms to one helium atom. 

→ In more massive, hotter stars, the CNO cycle is the dominant means by which energy is produced. This process involves carbon, nitrogen and oxygen acting as catalysts for the similar nuclear fusion of four hydrogen atoms into one helium atom and high energy gamma ray photons.

Show Worked Solution

→ At the most basic level, stars produce energy through the process of nuclear fusion.

→ In less massive, lower temperature stars (like our Sun), the proton-proton chain reaction generates energy through nuclear fusion via the fusion of four hydrogen atoms to one helium atom. 

→ In more massive, hotter stars, the CNO cycle is the dominant means by which energy is produced. This process involves carbon, nitrogen and oxygen acting as catalysts for the similar nuclear fusion of four hydrogen atoms into one helium atom and high energy gamma ray photons.

PHYSICS, M5 EQ-Bank 23

A horizontal disc is rotating clockwise on a table when viewed from above. Two small blocks are attached to the disc at different radii from the centre.

Using the diagram below, draw vector arrows to show the relative linear velocities and centripetal forces for each block as the disc rotates.   (3 marks)
 

Show Answers Only

Show Worked Solution

Points to note on the image

→ Centripetal force and velocity vectors are perpendicular

→ Vector arrows should show relative magnitudes of velocities and centripetal forces.

PHYSICS, M8 EQ-Bank 22

Einstein's equation `E = mc^(2)`  is one of the most important equations in the history of physics.

Justify this statement.   (7 marks)

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`E=mc^(2)` at its most basic level, expresses the concept that mass and energy are interchangeable.

Its importance is due to its critical role in explaining and analysing a broad range of ideas, processes and phenomena. These include:

→  The energy sourced from the processes of nuclear fission and fusion, including the energy associated with radioactive decay.

→ The relationship between the binding energy of atoms and their mass defect.

→ Mass dilation of objects approaching the speed of light.

→ The fundamental principles and technology upon which nuclear bombs and nuclear reactors operate.

→ Processes which source energy in stars through the conversion of mass into energy.

→ Processes which allow for particle accelerators to operate, allowing us to investigate into the fundamental structure and properties of matter.

Other equations, such as Newton’s Universal Law of Gravity, do not have the same myriad of applications in fields that make up our current understanding of physics.

It is therefore justified in being called one of the most important equations in the history of physics

Show Worked Solution

`E=mc^(2)` at its most basic level, expresses the concept that mass and energy are interchangeable.

Its importance is due to its critical role in explaining and analysing a broad range of ideas, processes and phenomena. These include:

→  The energy sourced from the processes of nuclear fission and fusion, including the energy associated with radioactive decay.

→ The relationship between the binding energy of atoms and their mass defect.

→ Mass dilation of objects approaching the speed of light.

→ The fundamental principles and technology upon which nuclear bombs and nuclear reactors operate.

→ Processes which source energy in stars through the conversion of mass into energy.

→ Processes which allow for particle accelerators to operate, allowing us to investigate into the fundamental structure and properties of matter.

Other equations, such as Newton’s Universal Law of Gravity, do not have the same myriad of applications in fields that make up our current understanding of physics.

It is therefore justified in being called one of the most important equations in the history of physics

PHYSICS, M8 EQ-Bank 13 MC

The table lists the first generation of quarks and antiquarks.

   

The Standard Model of matter states that baryons, such as protons and neutrons, consist of three quarks.

Using the table, which of the following represents the quark composition for a neutron and an antineutron, respectively?

  1. `text{u}\text{u}\text{d} and bar text{u}\bar text{u}\bar text{d}`
  2. `bar text{u}\bar text{u}\bar text {d} and text{u}\text{u}\text{d}`
  3. `text{u}\text{d}\text{d} and bar text{u}\bar text{d}\bar text{d}`
  4. `bar text{u}\bar text{d}\bar text{d} and text{u}\text{d}\text{d}`
Show Answers Only

`C`

Show Worked Solution

→ A neutron has a charge of zero, so it is made up of one up quark and two down quarks.

→ An antineutron is composed of the corresponding antiquarks. So, it is made up of one up antiquark and two down antiquarks.

`=>C`

PHYSICS, M8 EQ-Bank 7 MC

The following equation describes the natural decay process of uranium-238.

\({ }_{92}^{238} \text{U} \rightarrow{ }_{90}^{234}\text{Th} +{ }_2^4 \text{He}\)

Which row of the table describes the changes in total mass and total binding energy in the decay of uranium-238?

\begin{align*}
\begin{array}{l}
\rule{0pt}{2.5ex} \ \rule[-1ex]{0pt}{0pt}& \\
\rule{0pt}{2.5ex}\textbf{A.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{B.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{C.}\rule[-1ex]{0pt}{0pt}\\
\rule{0pt}{2.5ex}\textbf{D.}\rule[-1ex]{0pt}{0pt}\\
\end{array}
\begin{array}{|c|c|}
\hline
\rule{0pt}{2.5ex}\quad \quad \textit{Total mass}\quad  \quad \rule[-1ex]{0pt}{0pt}& \textit{Total binding energy} \\
\hline
\rule{0pt}{2.5ex}\text{Decreases}\rule[-1ex]{0pt}{0pt}&\text{Increases}\\
\hline
\rule{0pt}{2.5ex}\text{Decreases}\rule[-1ex]{0pt}{0pt}& \text{Decreases}\\
\hline
\rule{0pt}{2.5ex}\text{Increases}\rule[-1ex]{0pt}{0pt}& \text{Increases} \\
\hline
\rule{0pt}{2.5ex}\text{Increases}\rule[-1ex]{0pt}{0pt}& \text{Decreases} \\
\hline
\end{array}
\end{align*}

Show Answers Only

\(A\)

Show Worked Solution

→ Mass is converted into energy in this nuclear reaction, so total mass decreases.

→ When elements heavier than iron undergo nuclear fission, the daughter nuclei produced are more stable with a greater binding energy per nucleon. As there are the same number of nucleons before and after the decay, total binding energy increases.

\(\Rightarrow A\)

PHYSICS, M8 EQ-Bank 5 MC

In an experiment, an electrically charged oil drop was suspended in air by an electric field. The electric field could be adjusted to balance the weight of the oil drop.

If more drops were suspended and measurements taken, which of the following properties would all of the oil drops be observed to have in common?

  1. The mass of each drop would be a multiple of a fundamental mass.
  2. The mass of each drop would be the same as each of the other drops.
  3. The charge of each drop would be a multiple of a fundamental charge.
  4. The charge of each drop would be the same as each of the other drops.
Show Answers Only

`C`

Show Worked Solution

→ Each oil drop would have an integer number of electrons.

→ Therefore, the charge on an oil drop would be a multiple of the charge on an electron which is a fundamental charge.

`=>C`

PHYSICS, M8 EQ-Bank 6 MC

The Hertzsprung-Russell diagram shown is used to classify stars.
 

Stars in region `S` of the diagram are much dimmer than other stars in the same spectral class.

What property of the stars in region `S` explains their relatively low luminosity?

  1. They are cooler than other stars.
  2. They have a smaller mass than other stars.
  3. They have a smaller surface area than other stars.
  4. They are further away from Earth than other stars.
Show Answers Only

`C`

Show Worked Solution

By elimination:

→ All stars in the same spectral class have similar temperatures (eliminate A).

→ Stars become white dwarves when they no longer are able to fuse elements to generate energy. The outer gaseous layers are expelled and only the hot, dense core is left. In this way, white dwarves do not have significantly lower masses compared to other stars (eliminate B).

→ Luminosity refers to the total energy radiated by a star per second and is independent on a star’s distance from Earth (eliminate D).

`=>C`

PHYSICS, M5 EQ-Bank 27

A toy car was placed facing outwards on a rotating turntable. The car was held in place by a force sensor connected to the centre of the turntable. The centre of mass of the car was 0.25 metres from the centre of the turntable. The reading from the force sensor was recorded at varying speeds of rotation. A stopwatch was used to time the rotation of the turntable. The linear velocity was calculated from the period of rotation. The graph shows the force on the car versus the square of the linear velocity of the car.
 


 

  1. Use the graph to determine the mass of the car.   (3 marks)
  1. Identify possible errors in the data and outline how to reduce their effects on the estimation of the mass of the car.   (4 marks)
Show Answers Only

a.   0.034 kg

b.   Errors and reduction strategies

→ If the sensor does not produce accurate readings then systemic errors will result. 

→ Inaccurate readings can be minimised by calibrating the sensor. This can be done against a known force such as the force of gravity on a 1 kg mass.

→ If the stopwatch used to time rotations is manually operated, then error due to human inaccuracy and varying reaction times is introduced. Inaccurate calculations of the motion’s period (i.e. timing of one rotation) makes linear velocity estimations (`v=romega`) less accurate.

→ A strategy for mitigating this inaccuracy is by using the stopwatch to measure several rotations at a time and then dividing the result by the total number of rotations.

Show Worked Solution

a.   Graph passes through (2,0) and (25, 3.1):

`text{Gradient}` `=(3.1-0)/(25-2)`  
  `=0.135`  

 
→ `text{Gradient}\ =(F)/(v^2)=0.135`

→ As the car is undergoing uniform circular motion:

`F_(c)` `=(mv^2)/(r)`  
`(F_(c))/(v^2)` `=(m)/(r)`  
`0.135` `=(m)/(r)`  
`:. m` `=0.135 xx 0.25`  
  `=0.03375`  
  `=0.034\ text{kg}`  

 

b.   Errors and reduction strategies

→ If the sensor does not produce accurate readings then systemic errors will result. 

→ Inaccurate readings can be minimised by calibrating the sensor. This can be done against a known force such as the force of gravity on a 1 kg mass.

→ If the stopwatch used to time rotations is manually operated, then error due to human inaccuracy and varying reaction times is introduced. Inaccurate calculations of the motion’s period (i.e. timing of one rotation) makes linear velocity estimations (`v=romega`) less accurate.

→ A strategy for mitigating this inaccuracy is by using the stopwatch to measure several rotations at a time and then dividing the result by the total number of rotations.

PHYSICS, M5 EQ-Bank 26

A baseball is hit with a velocity of 28 m s ¯1 at an angle of 30° to the horizontal at an initial height of 1.0 m above the plate. Ignore air resistance in your calculations.
 

  1. How long does it take the ball to return to the initial height above the ground?   (3 marks)
  1. The ball is hit directly towards a stationary outfielder who is 85 m from the plate. At the instant the ball is hit, the outfielder begins to run towards the plate with constant acceleration.
  2. What is the magnitude of her acceleration if she catches the ball when it is 0.50 m above the ground?  (4 marks)
Show Answers Only
  1. `t=2.9\ text{seconds}`
  2. `a=3.6  text{m s}^(-2)`
Show Worked Solution

a.   At initial height: `s_y=0`

Initial vertical velocity:  `u_y=28sin30°=14\ text{m s}^(-1)`

Vertical acceleration: `-9.8\ text{m s}^(-2)`

`s_(y)` `=u_(y)t+(1)/(2)a_(y)t^2`  
`0` `=14 t+(1)/(2)xx(-9.8)xxt^(2)`  
  `=14 t-4.9 xxt^(2)`  
  `=t(14-4.9 t)`  
`t` `=(14)/(4.9),\ \ (t>0)`  
  `=2.9\ text{seconds}`  

 

b.   At  `s_y=0.50\ text{m}`:

`v_(y)^2` `=u_(y)^2+2a_(y)s_(y)`  
  `=14^(2)+2(-9.8)(-0.50)`  
`v_(y)` `=sqrt(205.8)`  
  `~~14.3457\ text{m s}^(-1)`  

 
Using  `v_y=u_(y)+a_(y)t:`

`-14.3457` `=14-9.8t`  
`t` `=(14+14.3457)/(9.8)`  
  `=2.892\ text{s}`  

 
Horizontal range travelled (`s_x`):

`s_(x)` `=u_(x)t`  
  `=28 cos 30° xx 2.892`  
  `=70.127\ text{m}`  

COMMENT: Note `v_y` and `u_y` are in opposite directions and have opposite signs.

  Distance fielder travelled `=85-70.127 = 14.873\ text{m}`

`s` `=ut+(1)/(2)at^(2)`  
`14.873` `=0+(1)/(2)xx a xx2.892^(2)`  
`:.a` `=(2 xx 14.873)/2.892^(2)`  
  `=3.6\ text{m s}^(-2)\ \ text{(to 1 d.p.)}`  

PHYSICS, M5 EQ-Bank 21

Planet `X` has a mass eight times that of Earth. The acceleration due to gravity on the surface of this planet is half that on the surface of Earth.

If Earth has a radius of 1, what is the radius of Planet `X` ?   (3 marks)

Show Answers Only

`4`

Show Worked Solution

Radius of Earth:

`F_(g)` `=(GM_em)/(r_e)^2`  
`g` `=(F)/(m)`  
  `=(GM_e)/(r_e)^2`  
`:.r_(e)` `=sqrt((GM_e)/(g))`  

 
Radius of planet `X`:

`(g)/(2)` `=(G xxM_x)/(r_x)^2`  
  `=(G xx8M_e)/(r_x)^2`  
`:.r_(x)` `=sqrt((16GM_e)/(g))`  
  `=4r_e`  

 
`:.` Since Earth has a radius of 1 `=>\ r_(x)=4`

PHYSICS, M5 EQ-Bank 8 MC

A baseball training aid swings a 150 gram baseball around a pole in a horizontal circular path with radius 1.8 m.

The angular velocity of the motion is `5 pi\ text {rad s}^(-1)`.
  


 

What is the magnitude of the centripetal force required to maintain the motion of the ball?

  1. 23 N
  2. 37 N
  3. 67 N
  4. 74 N
Show Answers Only

`C`

Show Worked Solution
`v` `=omegar`  
  `=5pi xx1.8`  
  `=9pi\ text{ms}^(-1)`  

 

`F_(c)` `=(mv^2)/(r)`  
  `=(0.15 xx(9pi)^2)/(1.8)`  
  `~~67\ text{N}`  

 
`=>C`

PHYSICS, M5 EQ-Bank 6 MC

A researcher ties a small lead weight to a string and swings it in a horizontal circle above her head.

She uses the results to evaluate the relationship between centripetal force and speed. 

Which of the following needs to be kept constant to ensure a valid experiment?

  1. The mass of the lead weight and the length of the string
  2. The angular velocity of the lead weight
  3. The mass of the lead weight only
  4. The length of the string only
Show Answers Only

`A`

Show Worked Solution

→ `F_(c)=(mv^2)/(r)`

→ The factors that affect centripetal force are the mass of the lead weight, `m`, the speed,`v`, and the radius of rotation (length of string), `r`.

→ If the student aims to measure the relationship between centripetal force and speed, then to ensure a valid experiment the mass of the lead weight and length of the string must be kept constant to ensure these factors do not impact results.

`=>A`

PHYSICS, M5 EQ-Bank 2 MC

The horizontal and vertical components of the velocity of a projectile are respectively `v_x` and `v_y`.

Which pair of graphs best represents the velocity of the projectile?
 

Show Answers Only

`B`

Show Worked Solution

→ There are no horizontal forces on a projectile so its horizontal velocity will remain constant.

→ The projectile experiences a constant downwards vertical acceleration due to gravity. So, its vertical velocity will decrease at a constant rate.

`=>B`

PHYSICS, M5 EQ-Bank 3 MC

A ball is launched at speed `u` and angle `theta` from the horizontal as shown. `P` is the highest point reached by the ball.
 

Ignoring air resistance, what is the speed of the ball at point `P` ?

  1. Zero
  2. `u`
  3. `u\ cos theta`
  4. `u\ sin theta`
Show Answers Only

`C`

Show Worked Solution

At point `P`, the ball has no vertical velocity.

→ The ball’s speed is only due to its horizontal speed which is constant at  `v_(x)=u\ costheta`  throughout its entire flight.

`=>C`

Proof, EXT2 P2 EQ-Bank 9

Using mathematical induction, show

  `sum_(r=1)^n r^2=(n(n+1)(2n+1))/6`  for  `n>=1.`   (3 marks)

Show Answers Only

`text{Proof (See Worked Solution)}`

Show Worked Solution

`text{Prove true for}\ \ n=1:`

`text{LHS}\ =sum_(r=1)^1 n^2=1^2=1`

`text{RHS}\ =(1xx2xx3)/6=1=\ text{LHS}`

`:.\ text{True for}\ \ n=1`
 

`text{Assume true for}\ \ n=k:`

`sum_(r=1)^k r^2=(k(k+1)(2k+1))/6`
 

`text{Prove true for}\ \ n=k+1:`

`text{i.e.}\ sum_(r=1)^(k+1) r^2=((k+1)(k+2)(2k+3))/6`

`sum_(r=1)^(k+1) r^2` `=sum_(r=1)^k r^2+(k+1)^2`  
  `=(k(k+1)(2k+1))/6+(k+1)^2`  
  `=((k+1)[2k^2+k+6k+6])/6`  
  `=((k+1)(2k^2+7k+6))/6`  
  `=((k+1)(k+2)(2k+3))/6`  
  `=\ text{RHS}`  

COMMENT: Required proof – preserve the `(k+1)` factor in all algebra!

`:.\ text{True for}\ \ n=k+1`

`:.\ text{Since true for} \ n=1,\ text{by PMI, true for integers} \ n>=1.`

Proof, EXT2 P2 EQ-Bank 7

Using mathematical induction, show

  `1/4n^4<sum_(r=1)^n r^3<=n^4`  for  `n>=1.`   (4 marks)

Show Answers Only

`text{Proof (See Worked Solution)}`

Show Worked Solution

`text{Prove true for}\ \ n=1:`

`text{LHS}\ =1/4`

`text{Middle}\ =1^3=1`

`text{RHS}\ =1^4=1`

`1/4<1<=1`

`:.\ text{True for}\ \ n=1`
 

`text{Assume true for}\ \ n=k:`

`text{i.e.}\ 1/4k^4<sum_(r=1)^k r^3<=k^4`
 

`text{Prove true for}\ \ n=k+1:`

`text{i.e.}\ 1/4(k+1)^4<sum_(r=1)^(k+1) r^3<=(k+1)^4`

`text{LHS}\ =1/4(k+1)^4=1/4(k^4+4k^3+6k^2+4k+1)`

`text{RHS}\ =k^4+4k^3+6k^2+4k+1`

`text{Middle}\ =sum_(r=1)^(k+1) r^3=sum_(r=1)^k r^3+(k+1)^3`

`text{Consider LHS to show}\ \ 1/4(k+1)^4<sum_(r=1)^(k+1) r^3`

`text{Middle}` `=sum_(r=1)^k r^3+(k+1)^3`  
  `>1/4k^4+(k^3+3k^2+3k+1)`  
  `>1/4(k^4+4k^3+12k^2+12k+4)`  
  `>1/4(k^4+4k^3+6k^2+4k+1), \ \ (k>=1)`  
  `>\ text{LHS}`  

 

`text{Consider RHS to show}\ \ sum_(r=1)^(k+1) r^3<=k^4`

`text{Middle}` `=sum_(r=1)^k r^3+(k+1)^3`  
  `<=k^4+k^3+3k^2+3k+1`  
  `<=k^4+4k^3+6k^2+4k+4,\ \ (k>=1 => 4k^3>k^3, 6k^2>3k^2, 4k>3k)`  
  `<=(k+1)^4`  
  `<=\ text{RHS}`  

 

`:.\ text{True for}\ \ n=k+1`

`:.\ text{Since true for} \ n=1,\ text{by PMI, true for integers} \ n>=1.`

Calculus, EXT1 C3 EQ-Bank 15

Find the particular solution to the differential equation  `(dy)/(dx)=e^(2x+3y)`  that passes through the point `(0,0)`.  (3 marks)

Show Answers Only

`y=-1/3ln((5-3e^(2x))/2)`

Show Worked Solution
`(dy)/(dx)` `=e^(2x+3y)`  
`dy/dx` `=e^(2x)*e^(3y)`  
`e^(-3y)\ dy` `=e^(2x)\ dx`  
`int e^(-3y)\ dy` `=int e^(2x)\ dx`  
`-1/3 e^(-3y)` `=1/2 e^(2x)+c`  

 
`text{Passes through}\ (0,0):`

`-1/3e^0` `=1/2e^0+c`  
`c` `=-5/6`  

 

`-1/3 e^(-3y)` `=1/2 e^(2x)-5/6`  
`2e^(-3y)` `=5-3e^(2x)`  
`e^(-3y)` `=(5-3e^(2x))/2`  
`ln (e^(-3y))` `=ln((5-3e^(2x))/2)`  
`-3y` `=ln((5-3e^(2x))/2)`  
`y` `=-1/3ln((5-3e^(2x))/2)`  

Calculus, EXT1 C3 EQ-Bank 13

Find an expression for `y` in terms of `x` given

  `dy/dx=4y-3`  and  when  `x=-2, \ y=1`.   (3 marks)

Show Answers Only
  1. `y=(e^(4(x+2))+3)/4`
Show Worked Solution
`dy/dx` `=4y-3`  
`(dy)/(4y-3)` `=1\ dx`  
`int 1/(4y-3)\ dy` `=int 1\ dx`  
`1/4ln abs(4y-3)` `=x+c`  

 
`text{When}\ \ y=1, x=-2:`

`1/4ln(4-3)` `=-2+c`  
`c` `=2`  

 

`1/4ln abs(4y-3)` `=x+2`  
`ln abs(4y-3)` `=4(x+2)`  
`4y-3` `=+-e^(4(x+2))`  
`4y-3` `=e^(4(x+2)),\ \ (text{since}\ y(-2)=1)`  
`4y` `=e^(4(x+2))+3`  
`y` `=(e^(4(x+2))+3)/4`  

Calculus, EXT1 C3 EQ-Bank 10

A tank initially contains 300 grams of salt that is dissolved in 50 L of water. A solution containing 15 grams of salt per litre of water is poured into the tank at a rate of 2 litres per minute and the mixture in the tank is kept well stirred.

At the same time, 2 litres of the mixture flows out of the tank per minute.

  1. Find the differential equation, `(dm)/(dt)`, where `m` represents the mass, in grams, of salt in the tank at time `t` minutes.   (2 marks)

  2. Express `m` in terms of `t`.   (3 marks)

  3. Find the concentration of salt in the liquid in the longer term.   (1 mark)

Show Answers Only
  1. `(dm)/(dt) = 30-m/25`
  2. `m=750-450e^(-t/25)`
  3. `15\ text{g/L}`
Show Worked Solution

i.    `V = 50\ text{L}`

`(dm)/(dt) text{(in)} = 15 xx 2 = 30\ text(g/min)`

`(dm)/(dt) text{(out)} = 2 xx m/50 = m/25\ text(g/min)`

`:. (dm)/(dt) = 30-m/25`
 

ii.  `(dm)/(dt)=(750-m)/25`

`(dt)/(dm)` `=25/(750-m)`  
`t` `=int 25/(750-m)\ dm`  
  `=-25ln(750-m)+c`  

 
`text{When}\ \ t=0, m=300:`

`0` `=-25ln(750-300)+c`  
`c` `=25ln450`  

 

`t` `=25ln450-25ln(750-m)`  
  `=25ln(450/(750-m))`  
`t/25` `=ln(450/(750-m))`  
`e^(t/25)` `=450/(750-m)`  
`750-m` `=450e^(-t/25)`  
`m` `=750-450e^(-t/25)`  

 

iii.  `text{Method 1}`

`text{As}\ t->oo:`

`m->750-450e^(-oo)=750`

`text{Concentration} (m/V)->750/50=15\ text{g/L}`
 

`text{Method 2}`

`text{Concentration}\ -> (dm)/(dt) text{(in)} = 15\ text{g/L}`

Calculus, EXT1 C3 EQ-Bank 8

A researcher estimates the number of brumbies in a National Park after `t` years can be modelled by the equation

`B(t)=(18\ 000)/(1+4e^(-t))`

  1. Sketch the function `B(t)` over the first four years of the research.   (2 marks)

  2. Calculate when the brumby population should reach 13 000, giving your answer to 2 decimal places.   (1 mark)

  3. Show that  `B^(′)(t)=(72\ 000e^t)/(e^t+4)^2`   (2 marks)

  4. What is the maximum growth rate of the brumby population?   (3 marks)

Show Answers Only
  1.  
     
  2. `2.34\ text{years}`
  3. `text{Proof (See Worked Solutions}`
  4. `4500\ text{brumbies per year}`
Show Worked Solution

i.   

ii.   `text {Find}\ t\ text{when}\ \ B(t)=13\ 000`

`13\ 000` `=(18\ 000)/(1+4e^(-t))`  
`13\ 000(1+4e^(-t))` `=18\ 000`  
`1+4e^(-t)` `=18/13`  
`4e^(-t)` `=5/13`  
`e^(-t)` `=5/52`  
`-t` `=ln(5/52)`  
`t` `=2.34\ text{years (2 d.p.)}`  

 

 iii.   `text{Show}\ \ B^(′)(t)=(72\ 000e^t)/(e^t+4)^2`

`B(t)=18\ 000(1+4e^(-t))^(-1)`

`B^(′)(t)` `=-1*-1*4e^(-t)*18\ 000(1+4e^(-t))^-2`  
  `=(72\ 000)/(e^t(1+4e^(-t))^2)`  
  `=(72\ 000)/(e^t(1+4/e^t)^2)`  
  `=(72\ 000)/(e^t((e^t+4)/e^t)^2)`  
  `=(72\ 000)/(e^t/(e^t)^2*(e^t+4)^2)`  
  `=(72\ 000e^t)/(e^t+4)^2\ \ text{… as required}`  

 

iv.  `B^(′)(t)=72\ 000e^t(e^t+4)^(-2)`

`text{Using product rule:}`

`B^(′′)(t)` `=72\ 000e^t(e^t+4)^(-2)+(-2e^t)(e^t+4)^(-3)72\ 000e^t`  
  `=72\ 000e^t(1/(e^t+4)^2-(2e^t)/(e^t+4)^3)`  
  `=72\ 000e^t((e^t+4-2e^t)/(e^t+4)^3)`  
  `=72\ 000e^t((4-e^t)/(e^t+4)^3)`  

 
`text{Find}\ t\ text{when}\ \ B^(′′)(t)=0:`

`4-e^t` `=0`  
`e^t` `=4`  
`t` `=ln4`  
  `=1.386…\ text{years}`  

 
`text{Checking concavity changes:}`

`text{Since}\ e^t>0, (e^t+4)^3>0\ \ text{for all}\ t:`

`text{At}\ t=1, 4-e^1=1.28>0\ \ =>\ \ B^(′′)(1)>0`

`text{At}\ t=2, 4-e^2=-3.4<0\ \ =>\ \ B^(′′)(2)<0`

 
`B^(′)(ln4)=\ text{Max growth rate}`

`B^(′)(ln4)` `=(72\ 000e^(ln4))/(e^(ln4)+4)^2`  
  `=(72\ 000xx4)/((4+4)^2)`  
  `=4500\ text{brumbies per year}`  

Calculus, EXT1 C3 EQ-Bank 7

The rate of weekly sales of a product that has just been released can be modelled by the following differential equation

`S^(′)(t)=500/(t+1)^3-250/(t+1)^2`

where `S` is the number of sales in thousands and `t` is the number of weeks since launch.

Good quality products will result in increasing sales.

  1. Find the function that describes the weekly sales.   (2 marks)

  2. By calculating the number of sales for the first and fifth week, comment on the results of the launch with respect to the given information.  (2 marks)

Show Answers Only
  1. `S(t)=250/(t+1)-250/((t+1)^2)`
  2. `S(1) = 62\ 500, \ S(5) = 34\ 722`
  3. `text{Sales drop significantly between week 1 and week 5, suggesting}`
  4. `text{the product is of poor quality.}`
Show Worked Solution

i.   `S^(′)(t)=500/(t+1)^3-250/(t+1)^2`

`S(t)` `=int 500/(t+1)^3-250/(t+1)^2\ dt`  
  `=int 500(t+1)^(-3)-250(t+1)^(-2)\ dt`  
  `=(500(t+1)^(-2))/(-2)-(250(t+1)^(-1))/(-1)+c`  
  `=250/(t+1)-250/((t+1)^2)+c`  

 
`text{When}\ \ t=0, S(t)=0:`

`0=250/1-250/(1^2)+c\ \ =>\ \ c=0`

`:.S(t)=250/(t+1)-250/((t+1)^2)`
 

 ii.   `S(1)=250/(1+1)-250/((1+1)^2)=62.5`

`=>\ text{Sales in week 1 = 62 500}`
 

`S(5)=250/(1+5)-250/((1+5)^2)=34.7222`

`=>\ text{Sales in week 5 = 34 722}`
 

`text{Sales drop significantly between week 1 and week 5, suggesting}`

`text{the product is of poor quality.}`

Proof, EXT2 P2 EQ-Bank 4

Using mathematical induction, prove that the maximum number of diagonals of an `n`-sided plane convex polynomial is `(n(n-3))/2` for `n>=4`.  (3 marks)

Show Answers Only

`text{See Worked Solutions}`

Show Worked Solution

`text{Let}\ \ D_n=\ text{maximum number of diagonals of}\ n text{-sided polygon}`

`text{Prove true for}\ \ n=4:`

`D_4= 2\ \ text{(quadrilateral only has 2 possible diagonals)}`

`(4(4-3))/2 = 2`

`:.\ text{True for}\ \ n=4.`
 

`text{Assume true for}\ \ n=k:`

`text{i.e.}\ D_k=(k(k-3))/2`
 

`text{Prove true for}\ \ n=k+1:`

`text{i.e.}\ D_(k+1)=((k+1)(k+1-3))/2=((k+1)(k-2))/2`
 

`text{Let}\ ktext{-sided polygon be defined by points}\ P_1,P_2,…,P_k`

`text{Let}\ (k+1)text{-sided polygon be defined by points}\ P_1,P_2,…,P_k,P_(k+1)`

`=>\ text{The extra point adds}\ (k-1)\ text{diagonals}`
 

`D_(k+1)` `=D_k+(k-1)`  
  `=(k(k-3))/2+ (k-1)`  
  `=(k^2-3k+2k-2)/2`  
  `=(k^2-k-2)/2`  
  `=((k+1)(k-2))/2`  
  `=\ text{RHS}`  

 
`:.\ text{True for}\ \ n=k+1`

`:.\ text{Since true for}\ n=4,\ text{by PMI, true for integers}\ n>=4.`

Proof, EXT2 P2 EQ-Bank 3

Using mathematical induction, prove that the sum of the internal angles of an `n`-sided polynomial is  `180(n-2)°`  for  `n>=3`.  (3 marks)

Show Answers Only

`text{See Worked Solutions}`

Show Worked Solution

`text{Let}\ \ S_n=\ text{sum of interior angles of}\ n text{-sided polygon}`

`text{Prove true for}\ \ n=3:`

`S_3= 180°\ \ text{(sum of internal angles of a Δ)}`

`180(3-2) = 180°`

`:.\ text{True for}\ \ n=3.`
 

`text{Assume true for}\ \ n=k:`

`text{i.e.}\ S_k=(k-2)180°`
 

`text{Prove true for}\ \ n=k+1:`

`text{i.e.}\ S_(k+1)=(k+1-2)180°=(k-1)180°`
 

`text{Let}\ ktext{-sided polygon be defined by points}\ P_1,P_2,…,P_k`

`text{Let}\ (k+1)text{-sided polygon be defined by points}\ P_1,P_2,…,P_k,P_(k+1)`

`S_(k+1)` `=S_k+\ text{angle sum}\ ΔP_1P_kP_(k+1)`  
  `=(k-2)180° + 180°`  
  `=(k-1)180°`  
  `=\ text{RHS}`  

 
`:.\ text{True for}\ \ n=k+1`

`:.\ text{Since true for}\ n=3,\ text{by PMI, true for integers}\ n>=3.`

Calculus, EXT1 C3 EQ-Bank 6

The population of Myna birds in a national park is decreasing at a rate proportional to the population at that time.

  1. Write a differential equation that describes the situation.   (1 mark)

  2. If the population was originally 1300 and decreased to 1040 after 5 years, find the expected population after 10 years.  (3 marks)

Show Answers Only
  1. `(dN)/(dt) = -kN`
  2. `832`
Show Worked Solution

i.   `text{Let}\ (dN)/(dt)=\ text{rate of change of bird population at time}\ t\ text{years.}`

`(dN)/(dt) prop N`

`(dN)/(dt) = -kN\ \ (k>0,\ text{decreasing population)}`
 

ii.    `(dN)/(dt)` `=-kN`
  `1/N* (dN)/(dt)` `=-k`
  `int1/N\ dN` `=-intk\ dt`
  `ln absN` `=-kt+c`
  `N` `=e^(-kt+c)`
    `=e^(-kt)*e^c`
    `=Ae^(-kt)\ \ \ text{(where}\ A=e^c)`

 
`text{When}\ \ t=0, N=1300:`

`1300=Ae^0\ \ =>\ \ A=1300`

`N=1300e^(-kt)`
 

`text{When}\ \ t=5, N=1040:`

`1040` `=1300e^(-5k)`  
`e^(-5k)` `=1040/1300`  
`-5k` `=ln(0.8)`  
`k` `=-(ln(0.8))/(5)`  
  `=0.04462…`  

 
`text{Find}\ N\ text{when}\ \ t=10:`

`N` `=1300e^(-0.04462 xx 10)`  
  `=832\ \ text{myna birds}`  

Proof, EXT2 P2 EQ-Bank 2

`n` lines are drawn in a 2-dimensional plane such that no three lines are concurrent and no two lines are parallel.

`S_n` is the number of regions into which these lines divide the plane with the diagram illustrating that `S_3=7`
 

  1. In a similar way, draw diagrams that illustrate `S_1, S_2` and `S_4`.  (1 mark)

  2. Using the results in part (i), make a conjecture about the general formula for `S_n`.  (2 marks)

  3. Use mathematical induction to prove the formula from part (ii).  (3 marks)

Show Answers Only
  1. `text{See Worked Solutions}`
  2. `S_n=(n^2+n+2)/2`
  3. `text{See Worked Solutions}`
Show Worked Solution

i.       `S_1=2`                        `S_2=4`
 
             
 

   `S_4=11`


 

ii.   `text{Consider the pattern above:}`

`S_1=2,\ \ S_2=S_1+2`

`S_3=S_2+3,\ …\ , S_n=S_(n-1)+n`

`S_n` `=S_(n-1)+n`  
  `=S_(n-2)+(n-1)+n`  
  `=S_(n-3)+(n-2)+(n-1)+n`  
  `vdots`  
  `=S_1+(2+3+…+(n-1)+n)`  
  `=S_0+(1+2+…+(n-1)+n)`  
  `=1+(1+2+…+(n-1)+n)`  

 
`=>text{AP where}\ \ a=1, l=n, n=n`

`S_n` `=1+n/2(1+n)`  
  `=(2+n(n+1))/2`  
  `=(n^2+n+2)/2`  

 

iii.   `text{Prove}\ \ S_n=(n^2+n+2)/2\ \ text{for}\ \ n>=0`

`text{If}\ \ n=1`

`text{LHS}\ =S_1=2`

`text{RHS}\ = (1^2+1+2)/2=2=\ text{LHS}`

`:.\ text{True for}\ n=1.`
 

`text{Assume true for}\ \ n=k:`

`text{i.e.}\ S_k=(k^2+k+2)/2`

`text{Prove true for}\ \ n=k+1:`

`text{i.e.}\ S_(k+1)=((k+1)^2+(k+1)+2)/2=(k^2+3k+4)/2`
 

`text{Consider the line}\ d_4\ text{added below that crosses 3 existing lines}`

`text{and creates 4 new regions.}`

`text{Similarly, the line}\ d_(k+1)\ text{will cross}\ k\ text{existing lines and}`

`text{create}\ (k+1)\ text{new regions.}`

`S_(k+1)` `=S_k+k+1`  
  `=(k^2+k+2)/2+k+1`  
  `=(k^2+k+2+2k+2)/2`  
  `=(k^2+3k+4)/2`  
  `=\ text{RHS}`  

 
`:.\ text{True for}\ \ n=k+1`

`:.\ text{Since true for}\ n=1,\ text{by PMI, true for integers}\ n>=1`

Calculus, EXT1 C3 EQ-Bank 4

A varroa virus is infecting commercial beehives in a regional NSW town.

All infected hives detected so far lie within a circular region with radius 16 km and researchers believe that the increase of the radius `r` km can be modelled by a differential equation, where `(dr)/(dt)=2/5sqrtr` where `t` denotes the time in months.

What does this model predict for the radius of the region affected by the pest after `t` months?   (3 marks)

Show Answers Only

`r=((t+20)/5)^2`

Show Worked Solution
`(dr)/(dt)` `=2/5sqrtr`  
`5/(2sqrtr)\ dr` `=1\ dt`  
`5/2r^(-1/2)\ dr` `=1\ dt`  
`5/2intr^(-1/2)\ dr` `=int1\ dt`  
`5r^(1/2)` `=t+C`  

 

`text{When}\ \ t=0, r=16`

`5sqrt16=C\ \ =>\ \ C=20`

`5r^(1/2)` `=t+20`  
`r^(1/2)` `=(t+20)/5`  
`:.r` `=((t+20)/5)^2`  

Vectors, EXT2 V1 EQ-Bank 10

Given  `underset~a=(3,-2,1)`  and  `underset~b=(4,3,-4)`, verify numerically that 

`underset~a*underset~b=abs(underset~a)abs(underset~b)cos theta=x_1x_2+y_1y_2+z_1z_2`

where  `underset~a=(x_1,y_1,z_1)`  and  `underset~b=(x_2,y_2,z_2)`   (4 marks)

Show Answers Only

`text{See Worked Solution}`

Show Worked Solution

`text{Scalar Product result 1:}`

`underset~a*underset~b` `=x_1x_2+y_1y_2+z_1z_2`  
  `=3xx4+(-2)xx3+1xx(-4)`  
  `=2`  

 
`text{Scalar Product result 2:}`

`underset~a*underset~b=abs(underset~a)abs(underset~b)cos theta`

`text{Let}\ \ underset~c=underset~b-underset~a`

`underset~c=((4),(3),(-4))-((3),(-2),(1))=((1),(5),(-5))`

`abs(underset~c)=sqrt(1^2+5^2+(-5)^2)=sqrt(51)`

`abs(underset~a)=sqrt(3^2+(-2)^2+1^2)=sqrt(14)`

`abs(underset~b)=sqrt(4^2+3^2+(-4)^2)=sqrt(41)`
 

`text{Using cosine rule:}\ \ c^2=a^2+b^2-2ab\ cosC`

`=>\ \ ab\ cosC=(a^2+b^2-c^2)/2`

`underset~a*underset~b` `=abs(underset~a)abs(underset~b)cos theta`  
  `=(abs(underset~a)^2+abs(underset~b)^2-abs(underset~c)^2)/2`  
  `=(14+41-51)/2`  
  `=2`  

 
`:.underset~a*underset~b=abs(underset~a)abs(underset~b)cos theta=x_1x_2+y_1y_2+z_1z_2`

Vectors, EXT2 V1 EQ-Bank 13

In a rectangular prism, `M` is the midpoint of `AD`.

Use vector methods to find

  1. `angle HBD` to 1 decimal place  (2 marks)

  2. `angle HBM` to 1 decimal place  (2 marks)

Show Answers Only
  1. `40.5°`
  2. `51.5°`
Show Worked Solution

i.   `text{Consider the position vectors of points using}\ B\ text{as origin:}`

`vec(BH)=((0),(9),(6)), \ abs(vec(BH))=sqrt(9^2+6^2)=sqrt117`

`vec(BD)=((4),(9),(0)), \ abs(vec(BD))=sqrt(4^2+9^2)=sqrt97`

`cos angleHBD` `=(vec(BH)*vec(BD))/(abs(vec(BH))*abs(vec(BD)))`  
  `=(0xx4+9xx9+6xx0)/(sqrt117 xx sqrt97)`  
`angleHBD` `=cos^(-1)(81/(sqrt117 xx sqrt97))`  
  `=40.5°`  

 

ii.   `vec(BM)=((4),(9/2),(0)), \ abs(vec(BM))=sqrt(4^2+(9/2)^2)=sqrt(145/4)`

`cos angleHBM` `=(vec(BH) * vec(BM))/(abs(vec(BH))*abs(vec(BM)))`  
  `=(0xx4+9xx9/2+6xx0)/(sqrt117 xx sqrt(145/4))`  
`angleHBM` `=cos^(-1)(40.5/(sqrt117 xx sqrt(145/4)))`  
  `=51.5°\ \text{(1 d.p.)}`  

Vectors, EXT2 V1 EQ-Bank 8

Classify the triangle formed by joining the points  `A(3,1,0), B(-2,4,3)` and `C(3,3,-2)`.  (4 marks)

Show Answers Only

`text{ΔABC is a right-angled (scalene) triangle with a right-angle at B.}`

`text{(See Worked Solutions)}`

Show Worked Solution

`text{Calculating the side lengths:}`

`abs(AB)=sqrt((3+2)^2+(1-4)^2+(0-3)^2)=sqrt43`

`abs(BC)=sqrt((-2-3)^2+(4-3)^2+(3+2)^2)=sqrt51`

`abs(AC)=sqrt((3-3)^2+(1-3)^2+(0+2)^2)=sqrt8`

`=>\ text{Triangle is scalene.}`
 

`text{From above,}`

`abs(BC)^2=abs(AB)^2+abs(AC)^2\ \ (angleBAC=90°)`
 

`text{Consider scalar product of direction vectors:}`

`vec(AB)=((3),(1),(0))+lambda((-2-3),(4-1),(3-0))=((3),(1),(0))+lambda((-5),(3),(3))`
 

`vec(AC)=((3),(1),(0))+mu((3-3),(3-1),(-2-0))=((3),(1),(0))+mu((0),(2),(-2))`
 

`vec(AB)*vec(AC)=((-5),(3),(3))((0),(2),(-2))=0+6-6=0`

`:.vec(AB) ⊥ vec(AC)`

`:.\ text{ΔABC is a right-angled (scalene) triangle with a right-angle at A.}`

Vectors, EXT2 V1 EQ-Bank 4

Find all possible values of  `b`  given  `absvec(PQ)=sqrt76`  where  `P(-5,b,-2)`  and  `Q(-3,-2,4)`.   (3 marks)

Show Answers Only

`b=4\ \ text(or)\ \ -8`

Show Worked Solution

`vec(PQ)=((-3),(-2),(4))-((-5),(b),(-2))=((2),(-2-b),(6))`

`absvec(PQ)` `=sqrt(2^2+(-2-b)^2+6^2)`  
`76` `=(-2-b)^2+40`  
`36` `=(-2-b)^2`  
`6` `=+-(-2-b)`  

 
`-2-b=6\ \ =>\ \ b=-8`

`2+b=6\ \ =>\ \ b=4`

`:.b=4\ \ text(or)\ \ -8`

BIOLOGY, M7 EQ-Bank 24

Scientific advances have resulted in new methods of managing plant diseases and insect pests.

Describe how TWO of these methods have changed the management of plant diseases and/or insect pests.   (5 marks)

Show Answers Only

Transgenic Organisms

→ The development transgenic organisms has lead to the creation of GMO’s with ‘built-in’ pesticides. 

→ The modified plant contains a gene from another genus which allows it to be partially or fully resistant to a pest or disease.

→ For example, Bt Cotton is a type of transgenic cotton which has been modified with a gene which makes it almost immune to Bollworm, the number one burden of cotton plant death globally. This means that pesticides for Bollworm do not have to be used, saving time and expense.

Quarantine

→ The knowledge behind the transmission of specific disease has allowed societies to take adequate measures which will have the best chance for reducing the spread of said disease.

→ Quarantine is the process of isolating any imported crops for a certain amount of time dependent on the infection period of a given disease. This significantly reduces the chances of international diseases spreading into Australia.

Show Worked Solution

Transgenic Organisms

→ The development transgenic organisms has lead to the creation of GMO’s with ‘built-in’ pesticides. 

→ The modified plant contains a gene from another genus which allows it to be partially or fully resistant to a pest or disease.

→ For example, Bt Cotton is a type of transgenic cotton which has been modified with a gene which makes it almost immune to Bollworm, the number one burden of cotton plant death globally. This means that pesticides for Bollworm do not have to be used, saving time and expense.

Quarantine

→ The knowledge behind the transmission of specific disease has allowed societies to take adequate measures which will have the best chance for reducing the spread of said disease.

→ Quarantine is the process of isolating any imported crops for a certain amount of time dependent on the infection period of a given disease. This significantly reduces the chances of international diseases spreading into Australia.

ENGINEERING, CS 2017 HSC 13 MC

An image of an Allen truss bridge is shown.

Which of the following correctly identifies the loading conditions of an Allen truss bridge?

  1. Timber members in shear and steel members in torsion
  2. Timber members in torsion and steel members in shear
  3. Timber members in tension and steel members in compression
  4. Timber members in compression and steel members in tension
Show Answers Only

`D`

Show Worked Solution

→ Members of a truss can only be in either tension or compression.

→ Steel has very high tensile strength, and although timber has lower tensile strength it has relatively high compressive strength.

→ Therefore D is the only answer that would produce an effective truss.

`=>D`

BIOLOGY, M8 2014 HSC 31a

The biological structure shown is part of one of the systems in the body.
 

  1. Name the biological structure.   (1 mark)

  2. Outline the steps involved in this system's response to a stimulus.   (3 marks)

Show Answers Only

“

Show Worked Solution

i.   Nerve cell/neuron
  

ii.   Steps involved in system’s response to a stimulus

→ A stimulus is detected by a receptor nerve cell which convert the stimulus to an electrochemical impulse, which is then carried by sensory cells to the brain.

→ The signal reaches a certain segment on the brain depending on what signal was received, where an action is determined to be carried out.

→ Motor neurons carry the signal to an effector, which carries out the response.


♦ Mean mark (ii) 48%.

 

BIOLOGY, M7 2014 HSC 21

  1. The diagram shows a process that is a part of the immune response.
     

  1. What is the name of the process?   (1 mark)

  2. Outline how inflammation contributes to the immune response.   (2 marks)

Show Answers Only

“

Show Worked Solution

a.   Phagocytosis

→ The black circle represents foreign material such as microbes or toxins, while the outlined circular shape represents a phagocyte engulfing it.
  

b.   Answers can include

→ Inflammation increases blood flow, meaning more antibodies and white blood cells can reach the site faster.

→ The swelling confines the infection to a smaller area, allowing white blood cells and antibodies to eliminate it faster, using less resources.

→ The release of histamines and prostaglandins increases permeability of blood vessels, enabling more phagocytes to pass through to the infection site.

BIOLOGY, M7 2014 HSC 12-13 MC

Refer to the following information to answer Questions 12 and 13.

A student conducted a first-hand investigation using nutrient broth, beakers and an S-shaped delivery tube in an attempt to model Pasteur's experiment.

The equipment and data collected are shown.
 

Question 12

The data collected by the student are

  1. quantitative because data were collected for fourteen days.
  2. qualitative because the appearance of the broth is described.
  3. qualitative because this is a model of a past scientific experiment.
  4. quantitative because the results were recorded for two different beakers.

Question 13

The student's results were different from Pasteur's results.

Which of the following provides the best explanation for the difference?

  1. The nutrient broth was different from Pasteur's.
  2. The nutrient broth always goes cloudy as it ages.
  3. The nutrient broth was not boiled thoroughly on Day 1 .
  4. The nutrient broths were both exposed to oxygen from the outside air.
Show Answers Only

“

Show Worked Solution

Question 12

By Elimination

→ Quantitative data is one in which numerical values are obtained for comparison. (Eliminate A and D).

→ The data then must be qualitative as it includes a description of the broth’s appearance.

`=>B`

Question 13

→ The apparatus matches that of Pasteur’s, so the best explanation for flask 2 also going cloudy would be due to not thoroughly boiling the broth before the experiment, and therefore not eliminating any bacteria already present.

`=>C`


♦ Mean mark 53%.

BIOLOGY, M7 2014 HSC 11 MC

What is a likely adverse impact of the evolution of bacteria on the management of infectious disease?

  1. Viruses become resistant to antibiotics.
  2. Some treatments lose their effectiveness.
  3. Inherited diseases become more common.
  4. Quarantine of infected individuals no longer works.
Show Answers Only

`B`

Show Worked Solution

→ The evolution of bacteria coupled with their ability to allow horizontal gene transfer between themselves means that many treatments, particularly pharmaceuticals such as antibiotics, become ineffective.

`=>B`

BIOLOGY, M5 2014 HSC 10 MC

Why do co-dominant alleles NOT produce simple Mendelian ratios?

  1. Both alleles are expressed in the phenotype.
  2. Neither allele is expressed in the phenotype.
  3. The recessive allele is only expressed in the homozygous genotype.
  4. The expression of the dominant allele is affected by the recessive allele.
Show Answers Only

`A`

Show Worked Solution

→ In regular Mendelian allele ratios, the heterozygous genotype means the dominant allele is expressed.

→ In co-dominance, the heterozygous individuals have both alleles expressed, with single alleles only expressed in homozygous genotypes.

`=>A`

BIOLOGY, M6 2014 HSC 7 MC

Which of the following is essential in any model of natural selection?

  1. High reproductive rates
  2. Random selection of prey
  3. A population of predators
  4. Differences in the population
Show Answers Only

`D`

Show Worked Solution

→ For natural selection to be selective, there must be distinctive characteristic differences within the population which allows certain individuals to be favoured.

`=>D`

BIOLOGY, M8 2014 HSC 3 MC

Which of the following is a plant adaptation that minimises water loss?

  1. Small leaves
  2. Large flowers
  3. Large fleshy fruit
  4. Deep root systems
Show Answers Only

`A`

Show Worked Solution

→ Smaller leaves results in less stomata, which therefore leads to an overall reduction in water loss via transpiration.

`=>A`

ENGINEERING, PPT 2016 HSC 16 MC

Which row of the table correctly identifies the project responsibilities of a civil engineer when planning a new highway?
 

Show Answers Only

`D`

Show Worked Solution

By Elimination:

→ Engineer planning responsibilities do not extend to traffic management (eliminate `A` and `B`).

→ Engineers will be responsible for pavement alignment in their planning (eliminate `C`).

`=>D`

ENGINEERING, PPT 2017 HSC 19 MC

An engineering component is required to have the following features:

  • 30 mm spotface
  • 10 mm counterbore, 8 deep
  • 6 mm through.
     

Which of the following shows the correct dimensioning?
 

Show Answers Only

`A`

Show Worked Solution

→ The vertical arrow represents how deep the counterbore should be, which is 8mm (given in question), therefore the answer is A or C.

→ However C does not have the correct symbol for counterbore, instead just stating 2 diameters next to each other.

`=>A`

ENGINEERING, PPT 2017 HSC 15 MC

What can be used to minimise significant losses in electrical power transmission cables?

  1. Very high current
  2. Very high voltage
  3. Very high resistance
  4. Very high capacitance
Show Answers Only

`B`

Show Worked Solution

→ Power losses are proportional to square of the current in an electrical transmission.

→ Hence if the current is kept to a minimum then losses are minimised.

→ To keep the current low the transmission has to be at high voltage as per Ohms Law.

`=>B`

ENGINEERING, CS 2017 HSC 14 MC

A highly stressed steel plate develops a fatigue crack while in service.
 

Which of the following is the most effective temporary method to slow or stop the crack's growth?

  1. Weld the crack as it is
  2. Drill a hole ahead of the tip of the crack
  3. Drill a hole through the start of the crack
  4. Weld a small steel plate over the tip of the crack
Show Answers Only

`B`

Show Worked Solution

→ A and D are possible permanent solutions, but as the question requests the most effective temporary method it must be B or C.

→ Drilling a hole at the tip of the crack is a more effective solution as it prevents the crack from penetrating further, while drilling a hole at the start would not prevent the cracks growth.

`=>B`

ENGINEERING, CS 2017 HSC 9 MC

Which free-body diagram shows a beam subjected to a uniformly distributed load?
 

Show Answers Only

`A`

Show Worked Solution

→ As the name suggests, a uniformly distributed load has an equal force acting on the entire beam (as shown in A).

→ B and D show forces on specific points, and although C shows a load distributed along the bar, it is clearly not a uniform force across the whole beam.

`=>A`

ENGINEERING, AE 2016 HSC 25c

A taut cable and aerofoil of similar thickness are tested in a wind tunnel. The results show that the aerofoil produces significantly less drag than the cable.

Explain how the aerofoil produces less drag than the taut cable. Use the diagram to assist your response.   (3 marks)


 

 

Show Answers Only

→ Significant surface drag is created by the cable due to its rough surface.

→ Low pressure is created behind the cable due to its non aerodynamic shape, thus pulling the cable backwards and creating sizeable drag.

→ Vortices (turbulence) created behind the cable also produce drag.

→ The smooth surface, and aerodynamic shape of the symmetrical aerofoil, help minimise turbulence at the rear of the aerofoil, resulting in substantially less drag.

Show Worked Solution

→ Significant surface drag is created by the cable due to its rough surface.

→ Low pressure is created behind the cable due to its non aerodynamic shape, thus pulling the cable backwards and creating sizeable drag.

→ Vortices (turbulence) created behind the cable also produce drag.

→ The smooth surface, and aerodynamic shape of the symmetrical aerofoil, help minimise turbulence at the rear of the aerofoil, resulting in substantially less drag.

ENGINEERING, PPT 2016 HSC 27a

The table compares the `\text{CO}_2` emissions of three transport systems - car, train and aircraft.
 

The table shows that these transport systems produce different amounts of `\text{CO}_2` emissions per passenger.

Why are these different transport systems still required?   (2 marks)

Show Answers Only

Reasons for different transport systems

→ Long distance travel

→ Safety

→ Convenience

→ Access to diverse range of passengers

→ Trip times

Show Worked Solution

Reasons for different transport systems

→ Long distance travel

→ Safety

→ Convenience

→ Access to diverse range of passengers

→ Trip times

ENGINEERING, AE 2017 HSC 1 MC

Which of the following describes the primary responsibility of an engineer?

  1. To read the latest engineering research
  2. To explain engineering decisions to local media
  3. To manage successful delivery of engineering projects
  4. To ensure workplace compliance with work health and safety (WHS) requirements
Show Answers Only

`C`

Show Worked Solution

→ While workplace compliance and WHS are extremely important, they are not the primary responsibility of engineers.

`=>C`