Find $G^{\text{'}\text{'}}\left(x\right)$

Hence determine where y=G(x) is concave downwards.

Copy the graph of y = F'(x) shown below and sketch a possible graph of y=F(x) on the same number plane.

$$\begin{gathered} The numbers 1 to 20 are written on cards and placed in a bag. One card is drawn \\ at random. What is the probability that the number on the card is even or a multiple of 3? \\ \frac{1}{2} \\ \frac{4}{5} \\ \frac{13}{20} \\ \frac{7}{20} \end{gathered}$$

$$\begin{gathered} Geoff has found an idyllic beach house and needs to borrow \$800 000 from the bank to finance his purchase. The loan and interest is to be repaid in equal monthly instalments of \$M, at the end of each month, over a 25 year period. The reducible interest will be charged at 4·8\% p.a. and compounded monthly. As an extra inducement the bank agrees that the first six months of the loan can be interest free, although Geoff will begin making repayments from the end of the first month. \\ Let \$An be the amount owing to the bank at the end of n months. \\ Given A_6 = 800 000 - 6M, write down an expression for A_7 and show that the amount owing after eight months is given by \\ A_8 = (800 000 - 6M)1·{004}^2 -M(1·004 + 1). \\ Hence show \\ A_n = (800 000 - 6M)1·{004}^{n-6} - 250M(1·{004}^{n-6} - 1). \\ Calculate the monthly instalment, \$M, Geoff will need to pay in order to repay the loan on time. Give your answer correct to the nearest cent. \end{gathered}$$

Show that $3t^2-2t+3$is a positive definite quadratic by examining the discriminant.

Find F(t) given that F'(t) = $3t^2-2t+3 and F\left(1\right) =3.$

$$\begin{gathered} \mathrm{Atticus} \mathrm{makes} \mathrm{a} \mathrm{deposit} \mathrm{of} \$5000 \mathrm{at} \mathrm{the} \mathrm{start} \mathrm{of} \mathrm{each} \mathrm{year} \mathrm{into} \mathrm{a} \mathrm{savings} \mathrm{account}. \mathrm{He} \mathrm{earns} \mathrm{monthly} \mathrm{compound} \mathrm{interest} \mathrm{on} \mathrm{his} \mathrm{savings} \mathrm{account} \mathrm{at} 4.8\% \mathrm{per} \mathrm{annum}. \\ \mathrm{Let} {\mathrm{A}}_{\mathrm{n} }\mathrm{be} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{the} \mathrm{account} \mathrm{at} \mathrm{the} \mathrm{end} \mathrm{of} \mathrm{n} \mathrm{years}. \\ \left(\mathrm{i}\right) \mathrm{Show} \mathrm{that} \mathrm{\Lambda \_}1=\$5245.35. \\ \left(\mathrm{ii}\right) \mathrm{Show} \mathrm{that} \mathrm{A\_}2=\$5000(〖1.004〗^12+〖1.004〗^24 ). \\ \left(\mathrm{iii}\right) \mathrm{Show} \mathrm{that} \mathrm{A\_n}=(\$5000\times 1\cdot 004^12\times (1\cdot 004^12\mathrm{n}-1\left)\right)/(1\cdot 004^12-1) \\ \left(\mathrm{iv}\right) \mathrm{Find} \mathrm{the} \mathrm{amount} \mathrm{of} \mathrm{interest} \mathrm{Atticus} \mathrm{earns} \mathrm{on} \mathrm{his} \mathrm{savings} \mathrm{account} \mathrm{over} 10 \mathrm{years}. \end{gathered}$$

A particle is moving on the x-axis with displacement x metres after t seconds given by the function

 x = 2$t^2$  − 25t + 50.

 (i) What was the initial position of the particle? 

 (ii) What was the initial velocity of the particle? 
 (iii) At what times was the particle at the origin? 

 (iv) At what time was the particle instantaneously at rest? 

 (v) How far did the particle travel in between its visits to the origin?

$$\begin{gathered} A particle is moving on the x-axis and is initially at the origin. Its velocity, v metres per second, \\ at time t seconds is given by \\ v=\frac{4}{t+1}-2t \\ \left(i\right) What is the initial velocity of the particle? \\ \left(ii\right) Find the time when the particle changes direction. \\ \left(iii\right) Find the acceleration of the particle when t=3 \\ \left(iv\right) Find the dis\tan ce travelled by the particle in the third second. \\ Write your answer correct to 2 decimal places. \end{gathered}$$