$A particle is moving in a straight line .The acceleration /time graph is shown below.$

$$\begin{gathered} If the particle was originally stationaryat the origin, which of the following statements \\ best describes the particle at point P? \\ \left(A\right) P is stationary at a point on the right of the origin. \\ \left(B\right) P has positive velocity and negative acceleration \\ \left(C\right) P has negative velocity and negative acceleration \\ \left(D\right) P has negative velocity and zero acceleration \end{gathered}$$

$$\begin{gathered} A Farmer is building a wheat silo in the shape of a closed cylinder \\ radius r metres and height h metres. The silo is to be made from \\ galvanised iron sheeting and is to have a capacity of 300m^3. \\ U\sin g the formula V=\pi r^{2}h and \\ S=2\pi r^2+2\pi rh \\ i. Find an expression for h in terms of r. \\ ii. Show that the surface area S, is given by S =\frac{2{\mathrm{\pi r}}^3+600}{r}. \\ iii. Hence, find the value of r, in exact form, that gives a \\ minimum area of galvanised iron sheeting to be used. \end{gathered}$$

$$\begin{gathered} A particle is moving in a straight line. Its displacement, x metres, from the origin, \\ 0, at time t seconds, where t > 0, is given by \mathcal{x}=1-\frac{7}{t+4} \\ i.Find the initial displacement of the particle. \\ ii.Find the velocity of the particle as it passes through the origin, \\ iii.Show that the acceleration of the particle is always negative. \\ iv.Sketch the graph of the displacement of the particle as a function of time. \end{gathered}$$

$$\begin{gathered} Consider a parabola \mathcal{y}={\mathcal{x}}^2-8\mathcal{x}+4. \\ Find the coordinates of the vertex. \\ Find the coordinates of the focus. \\ In march 1958 , 24 koalas were introduced on kangaroo island .By march 2018, \\ the no.of koalas had risen to 5000. \\ Assume that the number N of koalas is increa\sin g exponentially and satisfies the \\ equation N=24e^{kt} , where k is cons\tan t and t is meaured in years from march 1958. \\ Show that k=0.0890 , correct to 4 significant figures \\ Predict the no.of koalas that will be present on kangaroo island in march 2024. \end{gathered}$$

$$\begin{gathered} The area of a sector AOB of a circle centre 0 and radius length 6cm \\ is 27 c{m}^2 • \\ Find \\ \left(i\right) the angle AOB in radians \\ \left(ii\right) the length ofthe minor arc AB \\ The fifth term of an Arithmetic sequence is 14 and the sum ofthe first ten tenns is 165. \\ Find the first term and the common difference ofthe sequence. \\ In 1810 Lily, an early settler of Gosford, left a will in which she established a fund \\ of\$500 for its future citizens to spend on such things as schools, hospitals etc. \\ Her instructions were that this money was to be invested at 6\% p.a. compounded yearly. \\ \left(i\right)IfLily\text{'}s instructions were followed, how much would have been in the fund 100 years \\ after it was established? \\ \left(ii\right) Suppose that at the beginning of each subsequent year after the establishment, \\ a further \$500 had been added to the fund and had also earned 6\% interest, \\ compounded annually. Express the amount ofmoney (\$.M) in the fund after 100 years \\ as a Geometric series and hence derive the value ofM correct to the nearest dollar. \end{gathered}$$

$$\begin{gathered} the population of New south wale in 2009 was 7.13 million in 2021 the population had \\ grown to approximately 7.3 million people \\ Assuming that the growth rate is proportional to the population show that the annual growth \\ rate is approximatly 0.79\% \\ Calculate the expected population of New south wales in 20p19 u\sin g \\ this model give your answer near rounded to the hunred thposand \\ in what year will the population exceed 1p0 millions ? \\ Find the solutions to the equations \\ 4{\cos }^2\theta =6\sin \theta +6 in the domain 0\le \theta \le 2\mathrm{\pi } \\ \mathrm{show} \mathrm{that} \frac{1}{\sqrt{\mathrm{n}}+\sqrt{\mathrm{n}+1}}=\sqrt{\mathrm{n}+1}-\sqrt{\mathrm{n}} \mathrm{for} \mathrm{all} \mathrm{integers} \mathrm{n}\ge 1 \end{gathered}$$

$$\begin{gathered} At the start of the month , katrina opens a bank account and deposit \$300 into the \\ acount. \\ At the start of the subsequent month , katrina makes a deposit hich is 1.5\% more \\ than the previous deposit. \\ At the end of every month the bank pays katrina interets at a rate of 3\% per annum \\ on the balance of the account. \\ Show that the balance of the account at the end of the second month is \\ \$300{(1.0025)}^2+\$300(1.015)(1.0025) \\ Show that the balance of the account at the end of the nth month is given by \\ \$300{(1.0025)}^n\left(\frac{{\left({\displaystyle \frac{1.015}{1.0025}}\right)}^n-1}{\left(\frac{1.015}{1.0025}\right)-1}\right) \\ Calculate the balance of the amount at the end of the 60th month , correct \\ to nearest dollar. \end{gathered}$$

$$\begin{gathered} Which statement is true for an ungrouped data set with no outliers? \\ \left(A\right) The largest possible range is 2 times the interquartile range. \\ \left(8\right) The largest possible range is 3 times the interquartile range. \\ \left(C\right)The largest possible range is 4 times the interquartile range. \\ \left(D\right) The largest possible range is 5 times the interquartile range. \end{gathered}$$

$$\begin{gathered} Ten ki\log rams of chlorine are placed in water and begin \\ to dissolve. After t hours the amount A kg of undissolved \\ chlorine is given byA= 10e^{-k}. \\ What is the value of k given that A= 3.6 and t = 5? \\ \left(A\right) -0.717 \\ \left(B\right) -0.204 \\ \left(C\right) 0.204 \\ \left(D\right) 0.717 \end{gathered}$$

$$\begin{gathered} The length of steel rods produced by a \\ machine is normally distributed with a s\tan dard \\ deviation of 3 mm. It is found that 2.5\% of all rods are less than 25 mm long. \\ Find the mean length of rods produced by the machine. \end{gathered}$$