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Financial Maths, 2ADV M1 2024 HSC 30

Suppose the geometric series \(x+x^2+x^3+\ \cdots\) has a limiting sum.

By considering the graph  \(y=-1-\dfrac{1}{x-1}\), or otherwise, find the range of possible values of \(S\).   (3 marks)

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\(S \in \Big( -\dfrac{1}{2}, \infty\Big ) \)

Show Worked Solution

\(x + x^2+x^3\ \cdots \Rightarrow a=x, r=x\)

\(S=\dfrac{a}{1-r} = \dfrac{x}{1-x}=\dfrac{-(1-x)+1}{1-x} = -1+\dfrac{1}{1-x}=-1-\dfrac{1}{x-1}\)
 

\(\text{If limiting sum}\ \Rightarrow \abs{r} \lt 1\ \Rightarrow \ \abs{x} \lt 1\)

\(\Rightarrow \text{Possible values of}\ S =\ \text{range of}\ \ y=-1-\dfrac{1}{x-1}\ \text{in domain}\ \abs{x} \lt 1\)

\(\text{Consider}\ \ y=-1-\dfrac{1}{x-1}:\)

\(\text{Asymptote at}\ \ x=1.\)

\begin{array} {|c|c|c|c|c|c|}
\hline
\rule{0pt}{2.5ex} x \rule[-1ex]{0pt}{0pt} & -2 & -1 & 0 & 1 & 2 \\
\hline
\rule{0pt}{2.5ex} y \rule[-1ex]{0pt}{0pt} & -\dfrac{2}{3} &  -\dfrac{1}{2} & \ \ 0\ \  & \infty & -2 \\
\hline
\end{array}

\(\text{As}\ x \rightarrow \infty, \ y \rightarrow -1\)

\(\text{As}\ x \rightarrow -\infty, \ y \rightarrow -1\)

\(\abs{x} \lt 1\ \Rightarrow \  y \in \Big( -\dfrac{1}{2}, \infty\Big ) \)

\(\therefore\ \text{Possible values of}\ S \in \Big( -\dfrac{1}{2}, \infty\Big ) \)

Trigonometry, 2ADV T2 2004 HSC 9a

Consider the geometric series  `1 − tan^2 theta + tan^4 theta − …`

  1. When the limiting sum exists, find its value in simplest form.  (2 marks)

  2. For what values of  `theta`  in the interval
     
        `−pi/2 < theta < pi/2`  does the limiting sum of the series exist?  (2 marks)

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  1. `cos^2 theta`
  2. `− pi/4 < theta < pi/4`
Show Worked Solution

i.   `1 − tan^2 theta + tan^4 theta − …`

`=>\ text(GP where)\ \ a=1,\ \ r=T_2/T_1= − tan^2 theta`

`:. S_∞` `= 1/(1 − (−tan^2 theta))`
  `= 1/(1 + tan^2 theta)`
  `= 1/(sec^2 theta)`
  `= cos^2 theta`

 

ii.   `text(Find)\ \ theta\ \ text(such that)\ \ \ |\ r\ |` `< 1`
  `|−tan^2 theta\ |` `< 1`
  ` tan^2 theta` `< 1`
  `−1 < tan theta` `< 1`
  `:. − pi/4 < theta` `< pi/4`

Trigonometry, 2ADV T1 2005 HSC 9b

Trig Ratios, 2UA 2005 HSC 9b
 

The triangle  `ABC`  has a right angle at `B, \ ∠BAC = theta` and  `AB = 6`. The line `BD` is drawn perpendicular to `AC`. The line `DE` is then drawn perpendicular to `BC`. This process continues indefinitely as shown in the diagram.

  1. Find the length of the interval `BD`, and hence show that the length of the interval  `EF`  is  `6 sin^3\ theta`.  (2 marks)

  2. Show that the limiting sum 
     
    `qquad BD + EF + GH + ···`
     
    is given by  `6 sec\ theta tan\ theta`.  (3 marks)

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  1. `text(See Worked Solutions)`
  2. `text(See Worked Solutions)`
Show Worked Solution
i.   Trig Ratios, 2UA 2005 HSC 9b Answer

`text(Show)\ EF = 6\ sin^3\ theta`

`text(In)\ ΔADB`

`sin\ theta` `= (DB)/6`
`DB` `= 6\ sin\ theta`
`∠ABD` `= 90 − theta\ \ \ text{(angle sum of}\ ΔADB)`
`:.∠DBE` `= theta\ \ \ (∠ABE\ text{is a right angle)}`

 

`text(In)\ ΔBDE:`

`sin\ theta` `= (DE)/(DB)`
  `= (DE)/(6\ sin\ theta)`
`DE` `= 6\ sin^2\ theta`
`∠BDE` `= 90 − theta\ \ \ text{(angle sum of}\ ΔDBE)`
`∠EDF` `= theta\ \ \ (∠FDB\ text{is a right angle)}`

 

`text(In)\ ΔDEF:`

`sin\ theta` `= (EF)/(DE)`
  `= (EF)/(6\ sin^2\ theta)`
`:.EF` `= 6\ sin^3\ theta\ \ …text(as required)`

 

ii. `text(Show)\ \ BD + EF + GH\ …`

`text(has limiting sum)\ =6 sec theta tan theta`

`underbrace{6\ sin\ theta + 6\ sin^3\ theta +\ …}_{text(GP where)\ \ a = 6 sin theta, \ \ r = sin^2 theta}`
 

`text(S)text(ince)\ \ 0 < theta < 90^@`

`−1` `< sin\ theta` `< 1`
`0` `< sin^2\ theta` `< 1`

 
`:. |\ r\ | < 1`
 

`:.S_∞` `= a/(1 − r)`
  `= (6\ sin\ theta)/(1 − sin^2\ theta)`
  `= (6\ sin\ theta)/(cos^2\ theta)`
  `= 6 xx 1/(cos\ theta) xx (sin\ theta)/(cos\ theta)`
  `= 6 sec\ theta\ tan\ theta\ \ …text(as required.)`

Probability, 2ADV S1 2013 HSC 15d

Pat and Chandra are playing a game. They take turns throwing two dice. The game is won by the first player to throw a double six. Pat starts the game.

  1. Find the probability that Pat wins the game on the first throw.     (1 mark)

  2. What is the probability that Pat wins the game on the first or on the second throw?     (2 marks)

  3. Find the probability that Pat eventually wins the game.     (2 marks)

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  1. `1/36`
  2. `(2521)/(46\ 656)\ \ text(or)\ \ 0.054`
  3. `36/71`
Show Worked Solutions

i.   `P\ text{(Pat wins on 1st throw)}=P(W)`

`P(W)` `=P\ text{(Pat throws 2 sixes)}`
  `=1/6 xx 1/6`
  `=1/36`

 

ii.  `text(Let)\ P(L)=P text{(loss for either player on a throw)}=35/36`

`P text{(Pat wins on 1st or 2nd throw)}` 

♦♦ Mean mark 33%
MARKER’S COMMENT: Many students did not account for Chandra having to lose when Pat wins on the 2nd attempt.

`=P(W) + P(LL W)`

`=1/36\ + \ (35/36)xx(35/36)xx(1/36)`

`=(2521)/(46\ 656)`

`=0.054\ \ \ text{(to 3 d.p.)}`

 

iii.  `P\ text{(Pat wins eventually)}`

`=P(W) + P(LL\ W)+P(LL\ LL\ W)+ … `

`=1/36\ +\ (35/36)^2 (1/36)\ +\ (35/36)^2 (35/36)^2 (1/36)\ +…`

 
`=>\ text(GP where)\ \ a=1/36,\ \ r=(35/36)^2=(1225)/(1296)`

♦♦♦ Mean mark 8%!
 COMMENT: Be aware that diminishing probabilities and `S_oo` within the Series and Applications are a natural cross-topic combination.

 
`text(S)text(ince)\ |\ r\ |<\ 1:`

`S_oo` `=a/(1-r)`
  `=(1/36)/(1-(1225/1296))`
  `=1/36 xx 1296/71`
  `=36/71`

 

`:.\ text(Pat’s chances to win eventually are)\  36/71`.