Aussie Maths & Science Teachers: Save your time with SmarterEd
Use the data sheet provided and the information in the table to answer this question.
Consider the molecule shown.
For each of the following instrumental techniques, predict the expected features of the spectra produced.
Refer to the structural features of the molecule in your answer.
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A purple solution at 25°C contains a mixture of two different cobalt\(\text{(II)}\) complexes which are at equilibrium.
\(\underset{\text{(blue)}}{\ce{CoCl4^{2-}(aq)}} \ce{+ 6H2O(l)} \rightleftharpoons
\underset{\text{(pink)}}{\ce{Co(H2O)6^{2+}(aq)}} \ce{+ 4Cl^{-}(aq)}\)
The results of heating and cooling a sample of this solution are given in the table.
\begin{array}{|l|c|c|}
\hline\rule{0pt}{2.5ex} \textit{Temperature} \ \text{(°C)} \rule[-1ex]{0pt}{0pt}& 80 & 0 \\
\hline \rule{0pt}{2.5ex}\textit{Colour of solution} \rule[-1ex]{0pt}{0pt} & \quad \text{blue} \quad & \quad \text{pink} \quad \\
\hline
\end{array}
The energy profile diagram for this reaction is shown.
How do collision theory and Le Chatelier's principle account for the colour change to pink when the solution is cooled? Refer to the energy profile diagram in your answer. (5 marks)
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A 0.010 L aliquot of an acid was titrated with 0.10 mol L\(^{-1} \ \ce{NaOH}\), resulting in the following titration curve.
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a. \(\text{Strategy 1:}\)
\(\text{From the shape of the titration curve, the acid was weak.}\)
\(\text{Equivalence point}\ \ \Rightarrow \ \ \ce{NaOH}\ \text{added}\ = 0.24\ \text{L} \)
\(\text{Halfway to the equivalent point = 0.012 L}, \ce{[ HA ]=\left[ A^{-}\right]}\)
\(\text{Here the pH} \approx 4.4 , \text{or}\ \ce{\left[H^{+}\right] is 4.0 \times 10^{-5}}\).
\(K_a=10^{-\text{pH}}=\ce{\left[H+\right]} \times \dfrac{\ce{\left[A^{-}\right]}}{\ce{[HA]}}\)
\(\text{At this pH,} \ \ce{\left[A^{-}\right]=[HA] so} \ K_a=4.0 \times 10^{-5}\).
\(\text{Strategy 2:}\)
\(\text{Equivalence point is at} \ \ce{0.024 L NaOH added}\).
\(\text{Shape of curve shows acid is monoprotic.}\)
\(\ce{[HA] \times 0.010=0.1 \times 0.024}\)
\(\ce{[HA]=0.24 mol L^{-1}}\)
\(\text{pH at start is approximately 2.5}\)
\(\text{So,} \ \ce{\left[H+\right]=3.16 \times 10^{-3}}\)
\(K_a=\dfrac{\ce{\left[H^{+}\right]\left[A^{-}\right]}}{\ce{[HA]}} \quad \ce{\left[A^{-}\right]=\left[H^{+}\right]}\)
\(\ce{[HA]=0.24-3.16 \times 10^{-3}=0.237}\)
\(K_a=\dfrac{\left(3.16 \times 10^{-3}\right)^2}{0.237}=4.2 \times 10^{-5}\)
b. pH of the final solution < 13:
a. \(\text{Strategy 1:}\)
\(\text{From the shape of the titration curve, the acid was weak.}\)
\(\text{Equivalence point}\ \ \Rightarrow \ \ \ce{NaOH}\ \text{added}\ = 0.24\ \text{L} \)
\(\text{Halfway to the equivalent point = 0.012 L}, \ce{[ HA ]=\left[ A^{-}\right]}\)
\(\text{Here the pH} \approx 4.4 , \text{or}\ \ce{\left[H^{+}\right] is 4.0 \times 10^{-5}}\).
\(K_a=10^{-\text{pH}}=\ce{\left[H+\right]} \times \dfrac{\ce{\left[A^{-}\right]}}{\ce{[HA]}}\)
\(\text{At this pH,} \ \ce{\left[A^{-}\right]=[HA] so} \ K_a=4.0 \times 10^{-5}\).
\(\text{Strategy 2:}\)
\(\text{Equivalence point is at} \ \ce{0.024 L NaOH added}\).
\(\text{Shape of curve shows acid is monoprotic.}\)
\(\ce{[HA] \times 0.010=0.1 \times 0.024}\)
\(\ce{[HA]=0.24 mol L^{-1}}\)
\(\text{pH at start is approximately 2.5}\)
\(\text{So,} \ \ce{\left[H+\right]=3.16 \times 10^{-3}}\)
\(K_a=\dfrac{\ce{\left[H^{+}\right]\left[A^{-}\right]}}{\ce{[HA]}} \quad \ce{\left[A^{-}\right]=\left[H^{+}\right]}\)
\(\ce{[HA]=0.24-3.16 \times 10^{-3}=0.237}\)
\(K_a=\dfrac{\left(3.16 \times 10^{-3}\right)^2}{0.237}=4.2 \times 10^{-5}\)
b. pH of the final solution < 13:
Chalk is predominantly calcium carbonate. Different brands of chalk vary in their calcium carbonate composition.
The table shows the composition of three different brands of chalk.
\begin{array}{|l|c|c|c|}
\hline \rule{0pt}{2.5ex}\rule[-1ex]{0pt}{0pt}& \ \ \textit{Brand X} \ \ & \ \ \textit{Brand Y} \ \ & \ \ \textit{Brand Z} \ \ \\
\hline \rule{0pt}{2.5ex}\ce{CaCO3(\%)} \rule[-1ex]{0pt}{0pt}& 85.5 & 83.9 & 82.4 \\
\hline
\end{array}
The following procedure was used to determine the calcium carbonate composition of a chalk sample.
The results of the titrations are recorded.
\begin{array}{|l|c|c|c|c|}
\hline
\rule{0pt}{2.5ex}\textit{Burette volume}\text{(mL)} \rule[-1ex]{0pt}{0pt}& \textit{Trial 1} & \textit{Trial 2} & \textit{Trial 3} & \textit{Trial 4} \\
\hline
\rule{0pt}{2.5ex}\text{Final} \rule[-1ex]{0pt}{0pt}& 7.80 & 14.90 & 22.10 & 29.25 \\
\hline
\rule{0pt}{2.5ex}\text{Initial} \rule[-1ex]{0pt}{0pt}& 0.00 & 7.80 & 14.90 & 22.10 \\
\hline
\rule{0pt}{2.5ex}\text{Total used} \rule[-1ex]{0pt}{0pt}& 7.80 & 7.10 & 7.20 & 7.15 \\
\hline
\end{array}
Determine the brand of the chalk sample. Include a relevant chemical equation in your answer. (7 marks)
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The following three solids were added together to 1 litre of water:
Which precipitate(s), if any, will form? Justify your answer with appropriate calculations. (5 marks)
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A system consists of a sealed glass jar containing some oxygen and a small strip of magnesium.
The magnesium reacts with the oxygen to produce magnesium oxide as a product. Energy is released from the system in this reaction.
The mass of the system will
The escape velocity from the surface of a planet, which has no atmosphere, is \(v\). A mass is launched at 45° to the planet's surface at \(v\).
What will be the subsequent motion of the mass?
A satellite with velocity \(v\), is in a geostationary orbit as shown in Figure 1.
At point \(Y\), the satellite explodes and splits into two pieces \(m_{ a }\) and \(m_{ b }\), of identical mass. As a result of the explosion, the velocity of one piece, \(m_{ a }\), changes from \(v\) to \(2 v\) as shown in Figure 2.
Analyse the subsequent motion of BOTH \(m_{ a }\) and \(m_{ b }\) after the explosion. Include reference to relevant conservation laws and formulae in your answer. (8 marks)
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A hollow copper pipe is placed upright on an electronic balance, which shows a reading of 300 g. A 50 g magnet is suspended inside the pipe and subsequently released.
It was observed that the readings on the balance began to increase after the magnet began to fall, and that the reading reached a constant maximum of 350 g before the magnet reached the bottom of the tube.
Explain these observations. (4 marks)
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The diagram shows a model of the orbits of Earth, Jupiter and Io, including their orbital direction and periods of orbit. In this model, it is assumed that the orbits of Earth, Jupiter and Io are circular.
A method to determine the speed of light using this model is described below.
When Earth was at position \(P\), the orbital period of Io was measured, and the time that Io was at position \(R\) was recorded.
Six months later, Io had orbited Jupiter 103 times, and Earth had reached position \(Q\). The orbital period of Io was used to predict when it would be at position \(R\). Assume that Jupiter has not moved significantly in its orbit around the Sun.
The time for Io to reach position \(R\) was measured to be \(1.000 \times 10^3\) seconds later than predicted, due to the time it takes light to cross the diameter of Earth's orbit from \(P\) to \(Q\).
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The amount of water (\(W\)) in litres used by a garden irrigation system varies directly with the time (\(t\)) in minutes that the system operates.
This relationship is modelled by the formula \(W=kt\), where \(k\) is a constant.
The irrigation system uses 96 litres of water when it operates for 24 minutes.
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The cost (\(C\)) of copper wire varies directly with the length (\(L\)) in metres of the wire.
This relationship is modelled by the formula \(C = kL\), where \(k\) is a constant.
A 250 metre roll of copper wire costs $87.50.
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The amount of paint (\(P\)) needed to cover a wall varies directly with the area (\(A\)) of the wall.
A painter uses 3.5 litres of paint to cover a wall with an area of 28 square metres.
How much paint is needed to cover a wall with an area of 42 square metres?
A circular loop of wire is placed at position \(X\), next to a straight current-carrying wire with the current direction shown.
The loop is moved to position \(Y\) at a constant speed.
Which row in the table best describes the induced electromotive force (emf) in the loop as it moves from \(X\) to \(Y\) ?
A neutron is absorbed by a nucleus, \(X\).
The resulting nucleus undergoes alpha decay, producing lithium-7.
What is nucleus \(X\) ?
Which graph correctly represents Malus' Law?
A DC motor connected to a 12 V supply is maintaining a constant rotational speed. An ammeter in the circuit reads 0.5 A .
Some material falls into the running motor, causing it to slow down.
What is the subsequent ammeter reading likely to be?
The diagram shows four lines, \(W, X, Y\) and \(Z\), depicting radioactivity varying with time.
Which of the four lines is consistent with a decay graph with the smallest decay constant \((\lambda)\) ?
A satellite in a circular orbit around Earth at an altitude of 500 km is moved to a new circular orbit at a higher altitude of 35 800 km .
Which statement correctly compares properties of the satellite in the higher orbit with its properties in the lower orbit?
A beam of electrons travelling at \(4 \times 10^3 \ \text{m s}^{-1}\) exits an electron gun and is directed toward two narrow slits with a separation, \(d\), of 1 \(\mu\text{m}\). The resulting interference pattern is detected on a screen 50 cm from the slits.
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Analyse the role of experimental evidence and theoretical ideas in developing the Standard Model of matter. (6 marks)
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Analyse the consequences of the theory of special relativity in relation to length, time and motion. Support your answer with reference to experimental evidence. (8 marks)
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Experiments have been carried out by scientists to investigate cathode rays.
Assess the contribution of the results of these experiments in developing an understanding of the existence and properties of electrons. (5 marks)
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A mass moves around a vertical circular path of radius \(r\), in Earth's gravitational field, without loss of mechanical energy. A string of length \(r\) maintains the circular motion of the mass.
When the mass is at its highest point \(B\), the tension in the string is zero.
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The starting position of a simple AC generator is shown. It consists of a single rectangular loop of wire in a uniform magnetic field of 0.5 T. This loop is connected to two slip rings and the slip rings are connected via brushes to a voltmeter.
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a. \(\text{Using}\ \ \phi=BA \ \ \text{and} \ \ \varepsilon=\dfrac{\Delta \phi}{\Delta t}\)
\(\varepsilon=\dfrac{\Delta(B A)}{\Delta t}=\dfrac{B \Delta A}{\Delta t}=\dfrac{0.5 \times 0.4 \times 0.3}{0.1}=0.6\ \text{V}\)
b.
a. \(\text{Using}\ \ \phi=BA \ \ \text{and} \ \ \varepsilon=\dfrac{\Delta \phi}{\Delta t}\)
\(\varepsilon=\dfrac{\Delta(B A)}{\Delta t}=\dfrac{B \Delta A}{\Delta t}=\dfrac{0.5 \times 0.4 \times 0.3}{0.1}=0.6\ \text{V}\)
b.
The following flow chart represents the control of body temperature in humans. --- 0 WORK AREA LINES (style=lined) --- --- 4 WORK AREA LINES (style=lined) ---
Congenital amegakaryocytic thrombocytopenia (CAMT) is a rare, inherited disorder where bone marrow no longer makes platelets that are important for clotting and preventing bleeding. The pedigree below shows the inheritance of CAMT in a family.
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The following diagram shows the cell division processes occurring in two related individuals.
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The following graph shows the changes in allele frequencies in two separate populations of the same species. Each line represents an introduced allele.
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The diagram shows the steps in LASIK (Laser-Assisted In Situ Keratomileusis) surgery. Compare the LASIK technology shown with ONE other technology that can be used to treat a named visual disorder. (4 marks) --- 10 WORK AREA LINES (style=lined) ---
Visual disorder:
Alpha-gal syndrome (AGS) is a tick-borne allergy to red meat caused by tick bites. Alpha-gal is a sugar molecule found in most mammals but not humans, and can also be found in the saliva of ticks. The diagram shows how a tick bite might cause a person to develop an allergic reaction to red meat.
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Jacinta buys several items at the supermarket. The docket for her purchases is shown below.
What is the amount of GST included in the total?
The sheets of paper Jenny uses in her photocopier are 21 cm by 30 cm. The paper is 80 gsm, which means that one square metre of this paper has a mass of 80 grams. Jenny has a pile of this paper weighing 25.2 kg.
How many sheets of paper are in the pile?
Hydrazine is a compound of hydrogen and nitrogen. The complete combustion of 1.0 L of gaseous hydrazine requires 3.0 L of oxygen, producing 2.0 L of nitrogen dioxide gas and 2.0 L of water vapour. All volumes are measured at 400°C.
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Consider the following reaction.
\(\ce{2NO2(g) \rightleftharpoons N2O4(g) \quad \quad \Delta H= -57.2 \ \text{kJ mol}^{-1}}\)
A sealed reaction vessel of fixed volume contains a mixture of \(\ce{NO2}\) and \(\ce{N2O4}\) gases at equilibrium.
Explain the impact of the addition of argon, an inert gas, on the temperature of the system. (4 marks)
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Kevlar and polystyrene are two common polymers.
A section of their structures is shown.
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a.
b. The physical differences between the two polymers are:
a.
b. The physical differences between the two polymers are:
Mixtures of hydrocarbons can be obtained from crude oil by the process of fractional distillation. Examples include petrol, diesel and natural gas.
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a. Environmental implication:
b.
c.
a. Environmental implication:
b.
c.
The solubility constant for silver\(\text{(I)}\) oxalate \(\ce{(Ag2C₂O4)}\) was determined using the following method.
The calibration curve for the AAS is provided below.
The absorbance of the diluted sample was 0.055.
What is the \(K_{s p}\) for silver oxalate?
The concentration of silver ions in a solution is determined by titrating it with aqueous sodium chloride, using yellow potassium chromate as the indicator.
Which row of the table correctly identifies the colour change at the endpoint and the more soluble salt?
\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}\ \ \text{Colour change} \ \ & \text{More soluble salt} \\
\text{at endpoint}\rule[-1ex]{0pt}{0pt}& \text{} \\
\hline
\rule{0pt}{2.5ex}\text{Red to yellow}\rule[-1ex]{0pt}{0pt}& \ce{Ag2CrO4}\\
\hline
\rule{0pt}{2.5ex}\text{Yellow to red}\rule[-1ex]{0pt}{0pt}& \ce{Ag2CrO4}\\
\hline
\rule{0pt}{2.5ex}\text{Red to yellow}\rule[-1ex]{0pt}{0pt}& \ce{AgCl} \\
\hline
\rule{0pt}{2.5ex}\text{Yellow to red}\rule[-1ex]{0pt}{0pt}& \ce{AgCl} \\
\hline
\end{array}
\end{align*}
The chemical environment of an atom depends on the species surrounding that atom within a molecule.
In which of the following compounds does the number of carbon chemical environments equal the number of proton chemical environments?
\(\Rightarrow A\)
A single straight strand of polyester was produced through a condensation reaction of 1000 molecules of 3-hydroxypropanoic acid, \(\ce{HOCH2CH2COOH}\).
What is the approximate molar mass of the strand (in g mol\(^{-1}\) )?
Consider the following sequence of reactions.
This process has been presented in the flow chart below.
Which option correctly identifies the structures for \(X\), \(Y\) and \(Z\)?
The equation for the decomposition of hydrogen iodide is shown.
\(\ce{2HI(g)\rightleftharpoons I2(g) + H2(g)} \quad \quad \Delta H=+52 \ \text{kJ mol}^{-1}\)
The equilibrium formed during this reaction was investigated in two experiments carried out at different temperatures. The initial and equilibrium concentrations for both experiments are shown in the table, with only the \(K_{e q}\) for Experiment 1 shown.
Which row in the table correctly compares features of the two experiments?
\begin{align*}
\begin{array}{l}
\rule{0pt}{2.5ex} \ \rule[-1ex]{0pt}{0pt}& \\
\ \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 \quad K_{e q}\quad \quad\quad& \ \ \text{Temperature of}\ \ \\
\ \rule[-1ex]{0pt}{0pt}& \textit{experiment} \\
\hline
\rule{0pt}{2.5ex}\text{Lower in 1}\rule[-1ex]{0pt}{0pt}&\text{Lower in 2}\\
\hline
\rule{0pt}{2.5ex}\text{Lower in 1}\rule[-1ex]{0pt}{0pt}& \text{Higher in 2}\\
\hline
\rule{0pt}{2.5ex}\text{Higher in 1}\rule[-1ex]{0pt}{0pt}& \text{Lower in 2} \\
\hline
\rule{0pt}{2.5ex}\text{Higher in 1}\rule[-1ex]{0pt}{0pt}& \text{Higher in 2} \\
\hline
\end{array}
\end{align*}
Consider the point \(B\) with three-dimensional position vector \(\underset{\sim}{b}\) and the line \(\ell: \underset{\sim}{a}+\lambda \underset{\sim}{d}\), where \(\underset{\sim}{a}\) and \(\underset{\sim}{d}\) are three-dimensional vectors, \(\abs{\underset{\sim}{d}}=1\) and \(\lambda\) is a parameter.
Let \(f(\lambda)\) be the distance between a point on the line \(\ell\) and the point \(B\).
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The structures of two substances, \(\text{X}\) and \(\text{Y}\), are shown.
Which row of the table correctly classifies these substances as a Brønsted-Lowry acid or a Brønsted-Lowry base?
\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}\textit{Brønsted-Lowry} & \textit{Brønsted-Lowry} \\
\textit{acid}\rule[-1ex]{0pt}{0pt}& \textit{base} \\
\hline
\rule{0pt}{2.5ex}\text{-}\rule[-1ex]{0pt}{0pt}&\text{X and Y}\\
\hline
\rule{0pt}{2.5ex}\text{X and Y}\rule[-1ex]{0pt}{0pt}& \text{-}\\
\hline
\rule{0pt}{2.5ex}\text{Y}\rule[-1ex]{0pt}{0pt}& \text{X} \\
\hline
\rule{0pt}{2.5ex}\text{X}\rule[-1ex]{0pt}{0pt}& \text{Y} \\
\hline
\end{array}
\end{align*}
A particle of mass 1 kg is projected from the origin with a speed of 50 ms\(^{-1}\), at an angle of \(\theta\) below the horizontal into a resistive medium.
The position of the particle \(t\) seconds after projection is \((x, y)\), and the velocity of the particle at that time is \(\underset{\sim}{v}=\displaystyle \binom{\dot{x}}{\dot{y}}\).
The resistive force, \(\underset{\sim}{R}\), is proportional to the velocity of the particle, so that \(\underset{\sim}{R}=-k \underset{\sim}{v}\), where \(k\) is a positive constant.
Taking the acceleration due to gravity to be 10 ms\(^{-2}\), and the upwards vertical direction to be positive, the acceleration of the particle at time \(t\) is given by:
\(\underset{\sim}{a}=\displaystyle \binom{-k \dot{x}}{-k \dot{y}-10}\). (Do NOT prove this.)
Derive the Cartesian equation of the motion of the particle, given \(\sin \theta=\dfrac{3}{5}\). (5 marks)
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The complex number \(z\) lies on the unit circle.
What is the range of \(\operatorname{Arg}(z-2 i)\) ?
\(\text{The limits of Arg}(z-2i)\ \where it intersects the unit circle are:}\)
\(\sin \theta=\dfrac{1}{2} \ \Rightarrow \ \theta=\dfrac{\pi}{6}\)
\(\text {Range Arg}(z-2 i):\)
\(-\dfrac{\pi}{2}-\dfrac{\pi}{6} \leqslant \operatorname{Arg}(z-2 i) \leqslant-\dfrac{\pi}{2}+\dfrac{\pi}{6}\)
\(-\dfrac{2 \pi}{3} \leqslant \operatorname{Arg}(2-2 i) \leqslant-\dfrac{\pi}{3}\)
\(\Rightarrow D\)
The complex numbers \(z\) and \(w\) lie on the unit circle. The modulus of \(z+w\) is \(\dfrac{3}{2}\).
What is the modulus of \(z-w\) ?
Which of the following gives the same curve as \(\left(\begin{array}{c}\cos (t) \\ -t \\ \sin (t)\end{array}\right)\) for \(t \in \mathbb{R}\) ?
The points \(U, V, W\) and \(Z\) represent the complex numbers \(u, v, w\) and \(z\) respectively. It is given that \(v+z=u+w\) and \(u+k i z=w+k i v\) where \(k \in \mathbb{R} , k>1\).
Which quadrilateral best describes \(UVWZ\) ?
Consider the equation
\(z^n \cos\left[n \theta\right]+z^{n-1} \cos \left[(n-1) \theta\right]+z^{n-2} \cos \left[(n-2) \theta\right]+\cdots+z\, \cos\left[\theta\right]=1\)
where \(z \in \mathbb{C} , \theta \in \mathbb{R} \), and \(n\) is a positive integer.
Using a proof by contradiction and the triangle inequality, or otherwise, prove that all the solutions to the equation lie outside the circle \(\abs{z}=\dfrac{1}{2}\) on the complex plane. (4 marks)
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A particle moves in simple harmonic motion about the origin with amplitude \(A\), and it completes two cycles per second. When it is \(\dfrac{1}{4}\) metres from the origin, its speed is half its maximum speed.
Find the maximum positive acceleration of the particle during its motion. (4 marks)
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The adjacent sides of a parallelogram are represented by the vectors \(\underset{\sim}{a}=4 \underset{\sim}{i}+3 \underset{\sim}{j}-\underset{\sim}{k}\) and \(\underset{\sim}{b}=2 \underset{\sim}{i}-\underset{\sim}{j}+3 \underset{\sim}{k}\).
Show that the area of the parallelogram is \(6 \sqrt{10}\) square units. (4 marks)
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\(A=\dfrac{1}{2}\abs{a}\abs{b} \sin \theta\)
\(\underset{\sim}{a}=4\underset{\sim}{i}+3\underset{\sim}{j}-\underset{\sim}{k} \ \Rightarrow \ \abs{\underset{\sim}{a}}=\sqrt{16+9+1}=\sqrt{26}\)
\(\underset{\sim}{b}=2\underset{\sim}{i}-\underset{\sim}{j}+3 \underset{\sim}{k} \ \Rightarrow \ \abs{\underset{\sim}{b}}=\sqrt{4+1+9}=\sqrt{14}\)
\(\underset{\sim}{a} \cdot \underset{\sim}{b}=\left(\begin{array}{c}4 \\ 3 \\ -1\end{array}\right)\left(\begin{array}{c}2 \\ -1 \\ 3\end{array}\right)=8-3-3=2\)
\(\cos \theta=\dfrac{\underset{\sim}{a} \cdot \underset{\sim}{b}}{\abs{\underset{\sim}{a}}\abs{\underset{\sim}{b}}}=\dfrac{2}{\sqrt{26} \sqrt{14}}=\dfrac{1}{\sqrt{91}}\)
\(\sin \theta=\dfrac{\sqrt{90}}{\sqrt{91}}\)
\(\therefore A=\dfrac{1}{2} \times \sqrt{26} \times \sqrt{14} \times \dfrac{\sqrt{90}}{\sqrt{91}}=6 \sqrt{10} \ \text{units}^2\)
\(A=\dfrac{1}{2}\abs{a}\abs{b} \sin \theta\)
\(\underset{\sim}{a}=4\underset{\sim}{i}+3\underset{\sim}{j}-\underset{\sim}{k} \ \Rightarrow \ \abs{\underset{\sim}{a}}=\sqrt{16+9+1}=\sqrt{26}\)
\(\underset{\sim}{b}=2\underset{\sim}{i}-\underset{\sim}{j}+3 \underset{\sim}{k} \ \Rightarrow \ \abs{\underset{\sim}{b}}=\sqrt{4+1+9}=\sqrt{14}\)
\(\underset{\sim}{a} \cdot \underset{\sim}{b}=\left(\begin{array}{c}4 \\ 3 \\ -1\end{array}\right)\left(\begin{array}{c}2 \\ -1 \\ 3\end{array}\right)=8-3-3=2\)
\(\cos \theta=\dfrac{\underset{\sim}{a} \cdot \underset{\sim}{b}}{\abs{\underset{\sim}{a}}\abs{\underset{\sim}{b}}}=\dfrac{2}{\sqrt{26} \sqrt{14}}=\dfrac{1}{\sqrt{91}}\)
\(\sin \theta=\dfrac{\sqrt{90}}{\sqrt{91}}\)
\(\therefore A=\dfrac{1}{2} \times \sqrt{26} \times \sqrt{14} \times \dfrac{\sqrt{90}}{\sqrt{91}}=6 \sqrt{10} \ \text{units}^2\)
Let \(w\) be a complex number such that \(1+w+w^2+\cdots+w^6=0\).
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The complex number \(\alpha=w+w^2+w^4\) is a root of the equation \(x^2+b x+c=0\), where \(b\) and \(c\) are real and \(\alpha\) is not real.
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Let \(\displaystyle I_n=\large{\int}_{\small{\dfrac{\pi}{4}}}^{\small{\dfrac{\pi}{2}}}\)\(\cot ^{2 n} \theta \, d \theta\) for integers \(n \geq 0\).
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Evaluate \(\displaystyle \large{\int_0^{\small{\dfrac{\pi}{2}}}}\)\(\dfrac{u}{1+\sin u+\cos u} \, du\), by first using the substitution \(u=\dfrac{\pi}{2}-x\). (4 marks)
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A population lives across three regions, \(A,\ B\) and \(C\).
People in community \(B\) developed an environmental disease. An epidemiological study was carried out to determine the risk of developing the disease due to age at exposure. The results of this study are shown in the graph.
Design an epidemiological study that could be used to produce the results shown in the graph. Justify the features of your design. (7 marks)
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PAI-1 protein is encoded by the SERPINE 1 gene in humans. Anopheles mosquitoes have been genetically modified to express PAI-1, which blocks the entry of the malarial Plasmodium into the mosquito gut. This disrupts the Plasmodium life cycle, resulting in reduced transmission of malaria.
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The table shows the number of eggs produced by females in TWO different groups of animals. --- 18 WORK AREA LINES (style=lined) ---
\(\textit{Group A}\)
\(\textit{Group B}\)
\(\quad\textit{Animal}\quad\)
\(\quad\textit{Number of Eggs}\quad\)
\(\quad\textit{Animal}\quad\)
\(\quad\textit{Number of Eggs}\quad\)
\(\text{Monotremes}\quad\)
\(1-3\)
\(\text{Crabs}\)
\(1000-2000\)
\(\text{Snakes}\)
\(1-100\)
\(\text{Sea Urchins}\quad\)
\(100\, 000\ \text{to 2 million}\)
\(\text{Birds}\)
\(1-17\)
\(\text{Squid}\)
\(2000-3000\)
Compare the types of fertilisation that occur in group A and group B animals with reference to the data provided. (4 marks)
