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Probability, MET2 2012 VCAA 20 MC

A discrete random variable `X` has the probability function `text(Pr)(X = k) = (1 -p)^k p`, where `k` is a non-negative integer.

`text(Pr)(X > 1)` is equal to

`1 - p + p^2`
`1 - p^2`
`p - p^2`
`2p - p^2`
`(1 - p)^2`

Show Answers Only

`E`

Show Worked Solution

♦♦♦ Mean mark 19%.

`text(Pr)(X > 1)`	`= 1 – text(Pr)(X = 0) – text(Pr)(X = 1)`
	`= 1 – [(1 – p)^0p] – [(1 – p)p]`
	`= 1 – p – (p – p^2)`
	`= 1 – 2p + p^2`
	`= (1 – p)^2`

`=> E`

Polynomials, EXT2 2015 HSC 16b

Let `n` be a positive integer.

By considering `(cos alpha + i sin alpha)^(2n)`, show that
`cos(2n alpha) = cos^(2n) alpha - ((2n), (2)) cos^(2n - 2) alpha sin^2 alpha + ((2n), (4)) cos^(2n - 4) alpha sin^4 alpha - …`
1. `+ … + (-1)^(n - 1) ((2n), (2n - 2)) cos^2 alpha sin^(2n - 2) alpha + (-1)^n sin^(2n) alpha.`
Let `T_(2n) (x) = cos(2n cos^-1 x)`, for `-1 <= x <= 1.` (2 marks)
Show that
`T_(2n)(x) = x^(2n) - ((2n), (2)) x^(2n - 2)(1 - x^2) + ((2n), (4)) x^(2n - 4) (1 - x^2)^2 +`
1. `… + (-1)^n (1 - x^2)^n.` (2 marks)
By considering the roots of `T_(2n) (x)`, find the value of
1. `cos(pi/(4n)) cos((3 pi)/(4n)) …\ cos (((4n - 1) pi)/(4n)).` (3 marks)
Prove that
1. `1 - ((2n), (2)) + ((2n), (4)) - ((2n), (6)) + … + (-1)^n ((2n), (2n)) = 2^n cos ((n pi)/2).` (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`(-1)^n/2^(2n – 1)`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i) `text(By De Moivre)`

`(cos alpha + i sin alpha)^(2n)=cos (2n alpha) + i sin (2n alpha)`

`text(By the Binomial Theorem)`

`(cos alpha + i sin alpha)^(2n)`

`=cos^(2n) alpha + ((2n), (1)) cos^(2n – 1) alpha (i sin alpha) + ((2n), (2)) cos^(2n – 2) alpha (isin alpha)^2 +`

`… ((2n), (2n – 2)) cos^2 alpha (isin alpha)^(2n-2) + ((2n), (2n – 1)) cos alpha (i sin alpha)^(2n-1) + (i sin alpha)^(2n)`

`text(Equating real parts,)`

`cos(2n alpha) = cos^(2n) alpha – ((2n), (2)) cos^(2n – 2) alpha sin^2 alpha + … `

`+ (-1)^(n – 1) ((2n), (2n – 2)) cos^2 alpha sin^(2n – 2) alpha + (-1)^n sin^(2n) alpha`

(ii) `T_(2n) (x) = cos (2n cos^-1 x)\ \ text(for)\ \ -1 <= x <= 1`

`text(Given)\ \ alpha = cos^-1 x,\ \ =>x = cos alpha`

`text(Substituting)\ \ x = cos alpha\ \ text{into part (i),}`

`cos (2n cos^-1 x)`

`= x^(2n) – ((2n), (2)) x^(2n – 2) (1 – cos^2 alpha) + ((2n), (4)) x^(2n – 4) (1 – cos^2 alpha)^2 +`

`… + (-1)^n (1 – cos^2 alpha)^n`

`:. T_(2n) (x) = x^(2n) – ((2n), (2)) x^(2n – 2) (1 – x^2) + ((2n), (4)) x^(2n – 4)(1 – x^2)^2 +`

`… + (-1)^n (1 – x^2)^n`

♦♦♦ Mean mark 8%!
STRATEGY: Note that finding the correct roots gained 2 full marks in this part.

(iii) `T_(2n) (x) = 0\ \ text(when)`

`cos (2n cos^-1 x)`	`=0`
`2n cos^-1 x`	`=pi/2, (3 pi)/2, (5 pi)/2, … , ((2k + 1) pi)/2,\ \ k = 0, 1, 2 … , (2n – 1)`
`cos^-1 x`	`= ((2k + 1) pi)/(4n),\ \ k = 0, 1, 2 …, (2n – 1)`
`:. x`	`= cos(pi/(4n)), cos((3 pi)/(4n)), … , cos((4n – 1) pi)/(4n)`

`=>T_(2n) (x) = 0\ \ text(has degree)\ \ 2n`

`=>T_(2n) (x) = 0\ \ text(has)\ \ 2n\ \ text(distinct roots)`

`:.\ text(Product of roots of)\ \ T_(2n) (x) = 0\ \ text(is the constant term)`

`text(divided by the coefficient of)\ \ x^(2n).`

`text(Constant term is)\ \ T_(2n) (0)=(-1)^n`

`text(Coefficient of)\ \ x^(2n)\ \ text(is)\ \ (1 + ((2n), (2)) + ((2n), (4)) + … + 1)`

`:.cos(pi/(4n)) cos((3 pi)/(4n)) cos ((5 pi)/(4n)) …\ cos(((4n – 1) pi)/(4n))`

`=((-1)^n)/((1 + ((2n), (2)) + ((2n), (4)) + … + 1))`

`text(*The denominator can be further simplified to)\ \ 2^(2n-1)\ \ text(by)`

`text(using the binomial expansion of)\ \ (1+x)^(2n)`

♦♦ Mean mark 14%.

(iv) `cos(2n cos^-1 x)= cos ((n pi)/2)\ \ text(when)`

`cos^-1 x=pi/4\ \ \ =>x=1/sqrt2`

`text{Using part (ii)}`

`cos((n pi)/2)`	`=(1/sqrt 2)^(2n) – ((2n), (2)) (1/sqrt 2)^(2n – 2) (1/2) +`
	`((2n), (4)) (1/sqrt 2)^(2n – 4) (1/2)^2 – … + (-1)^n (1/2)^n`
	` = 1/2^n – ((2n), (2)) 1/2^(n-2) 1/2^2 + ((2n), (4)) 1/2^(n-4) 1/2^4 – … + (-1)^n 1/2^n`
	`=1/2^n – ((2n), (2)) 1/2^n + ((2n), (4)) 1/2^n – … + (-1)^n 1/2^n`
`:.2^n cos ((n pi)/2)`	`=1 – ((2n), (2)) + ((2n), (4)) – ((2n), (6)) + … + (-1)^n`

Harder Ext 1 Topics, EXT2 2015 HSC 16a

A table has `3` rows and `5` columns, creating `15` cells as shown.
Counters are to be placed randomly on the table so that there is one counter in each cell. There are `5` identical black counters and `10` identical white counters.
Show that the probability that there is exactly one black counter in each column is `81/1001.` (2 marks)
The table is extended to have `n` rows and `q` columns. There are `nq` counters, where `q` are identical black counters and the remainder are identical white counters. The counters are placed randomly on the table with one counter in each cell.
Let `P_n` be the probability that each column contains exactly one black counter.
Show that `P_n = n^q/(((nq), (q))).` (2 marks)
Find `lim_(n -> oo) P_n.` (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`(q!)/q^q`

Show Worked Solution

(i) `text(Arrangements of 5 black counters in 15 cells)`

`=\ ^15C_5`

`text(In each column, there are 3 cells to place a black counter.)`

`:.\ text(Ways to place a black counter in each column)`

`\ =3^5=243`

`:.P text{(1 black counter in each column)`

`=243/(\ ^15C_5`

`=81/1001`

(ii) `text{Similarly to part (i),}`

♦♦ Mean mark 16%.

`text(Arrangements of)\ \ q\ \ text(black counters in)\ \ nq\ \ text(cells)`

`=\ ^(nq)C_q`

`text(In each column, there are)\ \ n\ \ text(cells to place a black counter.)`

`:.\ text(Ways to place a black counter in each of)\ \ q\ \ text(columns)`

`\ =n^q`

`:.P text{(1 black counter in each column)`

`=n^q/(\ ^(nq)C_q)`

♦♦♦ Mean mark 5%.
COMMENT: Lowest State mean mark on the 2015 exam.

(iii)	`lim_(n -> oo) P_n`	`= lim_(n -> oo) n^q/(\ ^(nq)C_q)`
		`= lim_(n -> oo)[n^q xx (q!)/((nq)(nq – 1)…(nq – q + 1))]`
		`= lim_(n -> oo)[(n^q q!)/(n^q q(q – 1/n)(q – 2/n) … (q – (q – 1)/n))]`
		`= lim_(n -> oo)[(q!)/(q(q – 1/n)(q – 2/n) … (q – (q – 1)/n))]`
		`= (q!)/q^q`

Proof, EXT2 P1 2015 HSC 15b

Suppose that `x >= 0` and `n` is a positive integer.

Show that `1 - x <= 1/(1 + x) <= 1.` (2 marks)

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Hence, or otherwise, show that

`1 - 1/(2n) <= n ln (1 + 1/n) <= 1.` (2 marks)

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Hence, explain why

`lim_(n -> oo) (1 + 1/n)^n = e.` (1 mark)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i.	`1-x^2`	`<=1\ \ \ text(for)\ x>=0`
	`(1-x)(1+x)`	`<=1`
	`(1-x)`	`<=1/(1+x)`

`text(S)text(ince)\ \ 1 + x >= 1\ \ text(when)\ \ x >= 0`

`=>1/(1 + x) <= 1`

`:. 1 – x <= 1/(1 + x) <= 1.`

♦♦ Mean mark 21%.
STRATEGY: The conversion of the middle term from a fraction into a logarithm should flag the need for integration of each term.

ii.	`int_0^(1/n) (1 – x)\ dx`	`<= int_0^(1/n) (dx)/(1 + x) <= int_0^(1/n) 1\ dx`
	`[x – x^2/2]_0^(1/n)`	`<= [ln (1 + x)]_0^(1/n) <= [x]_0^(1/n)`
	`1/n-1/(2n^2)`	`<= ln (1 + 1/n) <= 1/n`
	`1 – 1/(2n) `	`<= n ln (1 + 1/n) <= 1,\ \ \ \ \ (n>=1)`

iii.	`lim_(n -> oo) (1 – 1/(2n))`	`<= lim_(n -> oo){n ln (1 + 1/n)} <= lim_(n -> oo) (1)`
	`1`	`<= lim_(n -> oo) {l ln (1 + 1/n)} <= 1`

♦♦ Mean mark 29%.

`:. lim_(n -> oo) (n ln (1 + 1/n))`	`=1`
`lim_(n -> oo) ln (1 + 1/n)^n`	`=1`
`:.lim_(n -> oo) (1 + 1/n)^n`	`=e`

Harder Ext1 Topics, EXT2 2006 HSC 8b

For `x > 0`, let `f(x) = x^n e^-x`, where `n` is an integer and `n >= 2.`

The two points of inflection of `f(x)` occur at `x = a` and `x = b`, where `0 < a < b`. Find `a` and `b` in terms of `n.` (4 marks)
Show that
1. `(f(b))/(f(a)) = ((1 + 1/sqrt n)/(1 - 1/sqrt n))^n e^(-2 sqrt n).` (2 marks)
Using the following
If `0<=x<=1/sqrt2` then `1 <= ((1 + x)/(1 - x)) e^(-2x) <= e^((4x^3)/3),` (DO NOT prove this)
2. show that `1 <= (f(b))/(f(a)) <= e^(4/(3 sqrt n)).` (2 marks)
What can be said about the ratio `(f(b))/(f(a))` as `n -> oo?` (1 mark)

Show Answers Only

`a = n – sqrt n,\ \ \ \ \ b = n + sqrt n`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`lim_(n -> oo) (f(b))/(f(a)) = 1`

Show Worked Solution

(i)	`f(x)`	`= x^n e^-x,\ \ \ \ n >= 2,\ \ \ x > 0`
	`f′(x)`	`= nx^(n – 1) e^-x – x^n e^-x`
	`f″(x)`	`= n[(n – 1) x^(n -2) e^-x – x^(n – 1) e^-x] – (nx^(n – 1) e^-x – x^n e^-x)`
		`= x^(n – 2) e^-x [n(n – 1) – nx – nx + x^2]`
		`= x^(n – 2) e^-x (x^2-2nx +n^2 – n)`

`text(P.I.’s occur when) \ \ f″(x)=0`

`text(i.e. when)\ \ \ x^2-2nx +n^2 – n = 0,\ \ \ \ (e^-x>0 and x>0)`

`:.x=`	`(2n+-sqrt(4n^2- 41(n^2-n)))/2`
`=`	`\ \ n+- sqrtn`

`text(S)text(ince)\ \ a<b`

`:. a = n – sqrt n\ and\ b = n + sqrt n.`

(ii) `(f(b))/(f(a)) =`	`((n + sqrt n)^n e^-(n + sqrt n))/((n – sqrt n)^n e^-(n – sqrt n))`
`=`	`(n^n(1 + 1/sqrt n)^n)/(n^n(1 – 1/sqrt n)^n) xx e^(-n – sqrt n + n – sqrt n)`
`=`	`((1 + 1/sqrt n)/(1 – 1/sqrt n))^n e^(-2 sqrt n)`

(iii) `text(S)text(ince)\ \ n>=2\ \ \ => 1/sqrt n <=1/sqrt2`

`text(Substituting)\ \ x=1/sqrt n\ \ text(into the inequality)`

`1`	`<= ((1 + 1/sqrt n)/(1 – 1/sqrt n)) e^((-2)/sqrt n) <= e^(4/(3(sqrt n)^3))`
`1^n`	`<= ((1 + 1/sqrt n)/(1 – 1/sqrt n))^n e^((-2)/sqrt n xx n) <= e^(4/(3(sqrt n)^3) xx n)`
`:.1`	`<= (f(b))/(f(a)) <= e^(4/(3 sqrt n))`

(iv)	`lim_(n -> oo) e^(4/(3 sqrt n))`	`=1`
	`:.lim_(n -> oo) (f(b))/(f(a))`	`=1`

Calculus, EXT2 C1 2006 HSC 7b

Let `I_n = int_0^x sec^n t\ dt`, where `0 <= x <= pi/2`.

Show that `I_n = (sec^(n - 2) x tan x)/(n - 1) + (n - 2)/(n - 1) I_(n - 2).` (3 marks)

--- 8 WORK AREA LINES (style=lined) ---
Hence find the exact value of

`int_0^(pi/3) sec^4 t\ dt.` (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`2 sqrt 3`

Show Worked Solution

STRATEGY: The choice of `u=sec^(n-2)t` and `v=sec^2 t` was extremely important in being able to answer this question. Take note!

i. `I_n`	`=int_0^x sec^n t\ dt,\ \ \ 0 <= x < pi/2`
	`=int_0^x sec^(n – 2)t sec^2 t\ dt`

`text(Integrating by parts)`

`u`	`=sec^(n-2)t,`	`u′`	`=(n-2) sec^(n-2)t tan\ t`
`v`	`=tan\ t,`	`v′`	`=sec^2 t`

`I_n`	`=[tan t sec^(n – 2)t]_0^x – int_0^x tan t (n – 2) sec^(n – 3) t sec t tan t\ dt`
	`=tan x sec^(n – 2) x – (n – 2) int_0^x sec^(n – 2) t tan^2 t\ dt`
	`=tan x sec^(n – 2) x – (n – 2) int_0^x sec^(n – 2) t (1+sec^2 t)\ dt`
	`=tan x sec^(n – 2) x – (n – 2) int_0^x sec^n t\ dt + (n – 2) int_0^x sec^(n – 2)t\ dt`
	`=tan x sec^(n – 2) x-(n-2)I_n+(n-2)I_(n-2)`

`I_n + (n – 2) I_n`	`= tan x sec^(n – 2) x + (n – 2) I_(n – 2)`
`(n – 1)I_n`	`= tan x sec^(n – 2) x + (n – 2) I_(n – 2)`
`:.I_n `	`= (tan x sec^(n – 2) x)/(n – 1) + ((n – 2)/(n – 1)) I_(n – 2)`

ii. `int_0^(pi/3) sec^4 t\ dt =`	`(tan\ pi/3 sec^2\ pi/3)/3 + 2/3 int_0^(pi/3) sec^2 t\ dt`
`=`	`(sqrt 3 xx 4)/3 + 2/3 [tan t]_0^(pi/3)`
`=`	`(4 sqrt 3)/3 + 2/3 (sqrt 3 – 0)`
`=`	`2 sqrt 3`

Harder Ext1 Topics, EXT2 2006 HSC 4d

In the acute-angled triangle `ABC, \ K` is the midpoint of `AB, \ L` is the midpoint of `BC` and `M` is the midpoint of `CA`. The circle through `K, L` and `M` also cuts `BC` at `P` as shown in the diagram.

Copy or trace the diagram into your writing booklet.

Prove that `KMLB` is a parallelogram. (1 mark)
Prove that `/_ KPB = /_ KML.` (1 mark)
Prove that `AP _|_ BC.` (2 marks)

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`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i)

`(AK)/(KB) = (AM)/(MC) = 1/1`

`:. KM\ text(\|\|)\ BC`	`\ \ \ text{(parallel lines cut in the}` `\ \ \ \ text{same proportion)`

`text(Similarly,)\ \ (CL)/(LB) = (CM)/(MA) = 1/1`

`:. ML\ text(||)\ AB`

`:. KMLB\ \ text(is a parallelogram)`

(ii)

`/_ BPK`

`= /_ KML`

`text{(exterior angle of a cyclic}`
`text{quadrilateral}\ \ KMLP text{)}`

(iii)	`/_ KBP`	`= /_ KML\ \ \ text{(opposite angles of a parallelogram)}`
	`:. /_ KBP`	`= /_ KPB\ \ \ text{(both equal}\ \ /_ KML text{)}`

`:. Delta BKP\ \ text(is isosceles)`

`text(S)text(ince)\ \ BK = KP=KA\ \ \ text{(given}\ K\ \ text(is the midpoint of)\ \ ABtext{)}`

`=>K\ \ text(is the centre of a circle, diameter)\ \ AB,`

`text(that passes through)\ \ A, B and P`

`/_ APB = 90^@\ \ \ \ text{(angle in semi-circle)}`

`:. AP _|_ BC.`

Harder Ext1 Topics, EXT2 2007 HSC 8c

The diagram shows a regular `n`-sided polygon with vertices `X_1, X_2, …, X_n`. Each side has unit length. The length `d_k` of the ‘diagonal’ `X_n X_k` where `k = 1, 2, …, n - 1` is given by

`d_k = (sin\ (k pi)/n)/(sin\ pi/n).` (Do NOT prove this.)

Show, using the identity,
`sin\ pi/n + sin\ (2 pi)/n + … + sin\ ((n - 1) pi)/n = cot\ pi/(2n),` (Do NOT prove this.)
1. that `d_1 + … + d_(n - 1) = 1/(2 sin^2\ pi/(2n)).` (2 marks)
Let `p` be the perimeter of the polygon and
`q = 1/n (d_1 + … + d_(n - 1))`.
Show that `p/q = 2 (n sin\ pi/(2n))^2.` (2 marks)
Hence calculate the limiting value of `p/q` as `n -> oo.` (1 mark)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`pi^2/2`

Show Worked Solution

(i) `d_k = (sin\ (k pi)/n)/(sin\ pi/n)`

`d_1 + d_2 + … + d_(n – 1)`

`=(sin\ pi/n)/(sin\ pi/n) + (sin\ (2 pi)/n)/(sin\ pi/n) + … + (sin\ ((n – 1) pi)/n)/(sin\ pi/n)`

`=1/(sin\ pi/n) xx cot\ pi/(2n)`

`=(cos\ pi/(2n))/(2 sin\ pi/(2n) cos\ pi/(2n) xx sin\ pi/(2n))`

`=1/(2 sin^2\ pi/(2n))`

(ii)	`p/q`	`=n/(1/n (d_1 + d_2 + … + d_(n – 1))`
		`=n^2 xx (2 sin^2\ pi/(2n))/1`
		`=2 (n sin\ pi/(2n))^2`

MARKER’S COMMENT: Applying the result `lim_(theta->0)\ sin theta/theta = 1` was poorly done.

(iii)	`lim_(n -> oo)\ p/q`	`= lim_(n -> oo)\ 2(n sin\ pi/(2n))^2`
		`=2n^2 xx pi^2/(4n^2) xx lim_(n -> oo)\ ((sin\ pi/(2n))^2)/(pi^2/(4n^2))`
		`=pi^2/2 xx lim_(n -> oo)\ (sin\ pi/(2n))/(pi/(2n)) xx lim_(n -> oo)\ (sin\ pi/(2n))/(pi/(2n))`
		`=pi^2/2 xx 1 xx 1\ \ \ \ \ \ text{(noting that as}\ \ n->oo, pi/(2n)->0text{)}`
		`=pi^2/2`

Complex Numbers, EXT2 N2 2007 HSC 8b

Let `n` be a positive integer. Show that if `z^2 != 1` then

`1 + z^2 + z^4 + … + z^(2n - 2) = ((z^n - z^-n)/(z - z^-1)) z^(n - 1)`. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---
By substituting `z = cos theta + i sin theta` where `sin theta != 0`, into part (i), show that

`1 + cos 2 theta + … + cos (2n - 2) theta + i[sin 2 theta + … + sin (2n - 2) theta]`

`= (sin n theta)/(sin theta) [cos (n - 1) theta + i sin (n - 1) theta].` (3 marks)

--- 8 WORK AREA LINES (style=lined) ---
Suppose `theta = pi/(2n)`. Using part (ii), show that

`sin\ pi/n + sin\ (2 pi)/n + … + sin\ ((n - 1) pi)/n = cot\ pi/(2n).` (2 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i. `1 + z^2 + z^4 + … + z^(2n – 2),\ z^2 != 1`

`text(GP where)\ a = 1,\ \ r = z^2,\ \ n\ text(terms):`

`S_n`	`=(1((z^2)^n – 1))/(z^2 – 1)`
	`=(z^(2n) – 1)/(z^2 – 1)`
	`=((z^n – z^-n))/(z – z^-1) xx z^n/z`
	`=((z^n – z^-n)/(z – z^-1))z^(n – 1)`

ii.	`z`	`= cos theta + i sin theta`
	`z^n`	`= cos n theta + i sin n theta\ \ …\ text(etc)\ \ \ \ text{(De Moivre)}`
	`z^-n`	`= cos( -n theta) + i sin (-n theta)`
		`= cos n theta – i sin n theta`

`text(LHS)`	`= 1 + (cos 2 theta + i sin 2 theta) + (cos 4 theta + i sin 4 theta) + `
	`… + (cos(2n – 2) theta + i sin (2n – 2) theta)`
	`= 1 + cos 2 theta + cos 4 theta + … + cos (2n – 2) theta + `
	`i (sin 2 theta + sin 4 theta + … + sin (2n – 2) theta)`

`text{Using part (i):}`

`text(LHS)`	`=((cos n theta + i sin n theta – cos n theta + i sin n theta))/(cos theta + i sin theta – cos theta + i sin theta) xx`
	`[cos (n – 1) theta + i sin (n – 1) theta]`
	`=(2 i sin n theta)/(2 i sin theta) [cos (n – 1) theta + i sin (n – 1) theta]`
	`=(sin n theta)/(sin theta) [cos (n – 1) theta + i sin (n – 1) theta]\ \ text(… as required.)`

iii. `text{Equating the imaginary parts in part (ii):}`

`sin 2 theta + sin 4 theta + … + sin 2 (n – 1) theta = (sin n theta sin (n – 1) theta)/(sin theta)`

`text(When)\ \ theta = pi/(2n),`

`sin\ (2 pi)/(2n) + sin\ (4 pi)/(2n) + … + sin\ (2(n – 1) pi)/(2 n) = (sin\ (n pi)/(2n) sin\ ((n – 1) pi)/(2n))/(sin\ pi/(2n))`

`:. sin\ pi/n + sin\ (2 pi)/n + … + sin\ ((n – 1) pi)/n`

`=(sin\ pi/2)/(sin\ pi/(2n)) xx sin\ ((n – 1) pi)/(2n)`

`=1/(sin\ pi/(2n)) sin (pi/2 – pi/(2n))`

`=(cos\ pi/(2n))/(sin\ pi/(2n))`

`=cot\ pi/(2n)`

Calculus, EXT2 C1 2007 HSC 8a

Using a suitable substitution, show that

`int_0^a f(x)\ dx = int_0^a f(a - x)\ dx.` (1 mark)

--- 6 WORK AREA LINES (style=lined) ---
A function `f(x)` has the property that `f(x) + f(a - x) = f(a).`

Using part (i), or otherwise, show that

`int_0^a f(x)\ dx = a/2\ f(a).` (2 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i. `text(Show)\ \ int_0^a f(x)\ dx = int_0^a f(a – x)\ dx.`

`text(Let)\ \ x = a – u,\ \ dx = -du`

`text(When)\ \ x = 0,\ \ u = a`

`text(When)\ \ x = a,\ \ u = 0`

`:. int_0^a f(x)\ dx`	`=int_a^0 f(a – u) (-du)`
	`=int_0^a f(a – u)\ du`
	`=int_0^a f(a – x)\ dx\ \ text{.. as required}`

MARKER’S COMMENT: Integrating `f(a)` was poorly done in part (ii). Pay careful attention to this in the Worked Solution.

ii. `f(x) = f(a) – f(a – x)`

`int_0^a f(x)\ dx`	`=int_0^a [f(a) – f(a – x)]\ dx`
	`=int_0^a f(a)\ dx – int_0^a f(a – x)\ dx`
	`=int_0^a f(a)\ dx – int_0^a f(x)\ dx`
`2 int_0^a f(x)\ dx`	`=int_0^a f(a)\ dx`
	`=[f(a) xx x]_0^a`
`int_0^a f(a)\ dx`	`=(f(a))/2 (a – 0)`
	`=a/2\ f(a)`

Proof, EXT2 P1 2007 HSC 7a

Show that `sin x < x` for `x > 0.` (2 marks)

--- 5 WORK AREA LINES (style=lined) ---
Let `f(x) = sin x - x + x^3/6`.

Show that the graph of `y = f(x)` is concave up for `x > 0.` (2 marks)

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By considering the first two derivatives of `f(x)`,show that `sin x > x - x^3/6` for `x > 0.` (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i.	`text(Let)\ \ g(x)`	`=sin x-x`
	`g′(x)`	`=cosx-1<=1\ \ \ text(for all)\ x>0`

MARKER’S COMMENT: A geometric proof using arc length and a right angled triangle caused problems as few students could deal with the case `x>pi`.

`=>g(x)\ \ text(is a decreasing function)`

`text(When)\ \ x=0,\ \ g(0)=0`

`text(Considering)\ \ g(x)\ \ text(when)\ \ x>0,`

`g(x)`	`<0`
`sinx -x`	`<0`
`sin x`	`<x\ \ \ text(for all)\ x>0`

ii.	`f(x)`	`=sin x – x + x^3/6`
	`f prime (x)`	`=cos x – 1 + x^2/2`
	`f ″ (x)`	`=x – sin x`
	`:.\ f″ (x)`	`> 0\ \ \ \ text{(using part (i))}`

`:. f(x)\ \ text(is concave up for)\ \ x > 0.`

iii. `f″(x)>0\ \ \ \ text{(part (ii))}`

`=>f′(x)\ \ text(is an increasing function)`

`text(When)\ \ x=0,\ \ f′(0)=0`

`=>f′(x)>0\ \ \ text(for)\ \ x>0`

`:. f(x)\ \ text(has a positive gradient that steepens)`

`text(for)\ \ x>0, and f(0)=0`

`f(x)`	`>0`
`sin x – x + x^3/6`	`>0`

`:.sin x > x – x^3/6\ \ \ text(for)\ \ x>0`

Calculus, EXT2 C1 2012 HSC 10 MC

Without evaluating the integrals, which one of the following integrals is greater than zero?

`int_(−pi/2)^(pi/2) x/(2 + cos x)\ dx`
`int_(−pi)^pi x^3 sin x\ dx`
`int_(−1)^1 (e^(−x^2) − 1)\ dx`
`int_(−2)^2 tan^(−1)(x^3)\ dx`

Show Answers Only

`B`

Show Worked Solution

`text{Consider (A) and (D)}`

`f(x)=-f(-x)\ \ =>\ text(ODD functions where)`

`int_(−a)^a f(x)\ dx = 0`

`text{Consider (C)}`

`e^(−x^2)<1\ \ text(for all)\ x\ \ => e^(−x^2) − 1<0`

`:. text(Its graph is below the)\ x text(-axis and any integral)`

`text(will be negative)`

`text{Consider (B)}`

`text{(B)}\ text(is an even function where,)`

`x^3 sinx>=0\ \ text(for)\ \ \ -pi<=x<=pi`

`=>B`

Calculus, EXT2 C1 2014 HSC 10 MC

Which integral is necessarily equal to `int_(−a)^a f(x)\ dx`?

`int_0^a f(x) − f(−x)\ dx`
`int_0^a f(x) − f(a − x)\ dx`
`int_0^a f(x − a) + f(−x)\ dx`
`int_0^a f(x − a) + f(a − x)\ dx`

Show Answers Only

`D`

Show Worked Solution

`int_(−a)^a f(x)\ dx= int_0^a f(x)\ dx + int_(−a)^0 f(x)\ dx`

`text(Consider)\ \ int_0^a f(x)\ dx`

♦♦ Mean mark 29%.

`text(Let)\ u = a − x\ \ => x = a − u\ \ text(and)\ \ dx = −du`

`int_0^a f(x)\ dx`	`= int_0^a f(a − u) − du`
	`=int_a^0 f(a-u)\ du`
	`= int_0^a f(a − x)\ dx`

`text(Consider)\ \ int_(−a)^0 f(x)\ dx`

`text(Let)\ u = x + a\ \ => x = u − a\ \ text(and)\ \ dx = du`

`int_(−a)^0 f(x)\ dx`	`= int_0^a f(u − a)\ du`
	`= int_0^a f(x − a)\ dx`
`:.int_(−a)^a f(x)\ dx`	`= int_0^a f(x − a)+ f(a − x)\ dx`

`=> D`

Harder Ext1 Topics, EXT2 2013 HSC 10 MC

A hostel has four vacant rooms. Each room can accommodate a maximum of four people.

In how many different ways can six people be accommodated in the four rooms?

4020
4068
4080
4096

Show Answers Only

`A`

Show Worked Solution

`text(If NO maximum)`

♦♦♦ Mean mark 15%.
STRATEGY: The group of 5 can be in any of the four rooms, and note that the remaining person can then be in any of the 3 remaining rooms.

`text(# Ways) = 4^6 = 4096`

`text(However, we can’t have 5 or 6 people in a room)`

`text(# Ways to have 6 people in 1 room) = 4`

`text(# Ways to have 5 people in 1 room)`

`=\ ^6C_5 ^4C_1\ ^3C_1`

`=72`

`:.\ text(Total ways with a maximum of 4)`

`=4096-4-72`

`=4020`

`=> A`

Harder Ext1 Topics, EXT2 2009 HSC 8c

A game is being played by `n` people, `A_1, A_2, ..., A_n`, sitting around a table. Each person has a card with their own name on it and all the cards are placed in a box in the middle of the table. Each person in turn, starting with `A_1`, draws a card at random from the box. If the person draws their own card, they win the game and the game ends. Otherwise, the card is returned to the box and the next person draws a card at random. The game continues until someone wins.

Let `W` be the probability that `A_1` wins the game.

Let `p = 1/n and q = 1 - 1/n`.

Show that `W = p + q^n W.` (1 mark)
Let `m` be a fixed positive integer and let `W_m` be the probability that `A_1` wins in no more than `m` attempts.
Use `e^(-n/(n - 1)) < (1 - 1/n)^n < e^-1,`
to show that, if `n` is large, `W_m/W` is approximately equal to `1 - e^-m.` (3 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i) `text(Chance of drawing own card is)\ \ p = 1/n`

`text(Chance of not drawing own card is)\ \ q = 1 – p = 1 – 1/n`

`P(A_1\ text{wins in round 1)}`	`= p`
`P(A_1\ text{wins in round 2)}`	`= q^n*p`
`P(A_1\ text{wins in round 3)}`	`= q^nq^np`
`:.P(A_1\ \ text{wins)}`	`= p + pq^n + pq^(2n) + …`

`=> text(G.P. where)\ \ a=p,\ \ r=q^n,\ \ and\ \ 0<q^n<1`

`W`	`=p/(1-q^n)`
`W-W q^n`	`=p`
`:.W`	`=p+W q^n`

(ii) `W_m`	`=p + q^n p + q^(2n) p + … + q^((m – 1)n) p`
	`=p ((1 – q^(nm)))/(1 – q^n)\ \ \ \ \ \ text{(G.P. with}\ m\ text{terms)}`
`W_m/W`	`=p ((1 – q^(nm)))/(1 – q^n) xx (1 – q^n)/p`
	`=1 – q^(nm)`
	`= 1 – (1 – 1/n)^(nm)`

`text(Considering)\ \ \ e^(-n/(n – 1)) < (1 – 1/n)^n < e^-1`

`text{As}\ \ n -> oo, \ \ n/(n-1)->1, \ \ e^(-n/(n – 1))->e^-1, and`

`:.(1-1/n)^n`	`~~e^-1`
`(1-1/n)^(mn)`	`~~e^-m`
`:.\ W_m/W`	`~~ 1 – e^-m`

Proof, EXT2 P2 2009 HSC 8a

Using the substitution `t = tan\ theta/2`, or otherwise, show that

`qquad cot theta + 1/2 tan\ theta/2 = 1/2 cot\ theta/2.` (2 marks)

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Use mathematical induction to prove that, for integers `n >= 1`,

`qquad sum_(r = 1)^n 1/2^(r - 1) tan x/2^r = 1/2^(n - 1) cot\ x/2^n - 2 cot x.` (3 marks)

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Show that `lim_(n -> oo) sum_(r = 1)^n 1/2^(r - 1) tan\ x/2^r = 2/x - 2 cot x.` (2 marks)

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Hence find the exact value of
`qquad tan\ pi/4 + 1/2 tan\ pi/8 + 1/4 tan\ pi/16 + ….` (2 marks)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`4/pi`

Show Worked Solution

i. `t=tan\ theta/2, \ \ \ sin theta=(2t)/(1+t^2),\ \ \ cos theta=(1-t^2)/(1+t^2)`

`text(Prove)\ \ cot theta + 1/2 tan\ theta/2 = 1/2 cot\ theta/2`

`text(LHS)`	`=cot theta + 1/2 tan\ theta/2`
	`=cos theta/sin theta+ 1/2 tan\ theta/2`
	`=(1-t^2)/(2t)+t/2`
	`=(1-t^2+t^2)/(2t)`
	`=1/(2t)`
	`=1/2 cot\ theta/2`
	`=\ text(RHS)`

ii. `text(If)\ \ n = 1`

`text(LHS)`	`=1/2^0 tan\ x/2^1=tan\ x/2`
`text(RHS)`	`=1/2^0 cot\ x/2 – 2 cot x`
	`=cot\ x/2 – 2 cot x`

`text{Using part (i)},\ \ 1/2 tan\ theta/2`	`= 1/2 cot\ theta/2 – cot theta,`
`:.tan\ theta/2`	`= cot\ theta/2 – 2 cot theta`

`text(RHS)`	`=tan\ x/2`
	`=\ text(LHS)`
`:.\ text(True for)\ \ n=1`

`text(Assume true for)\ \ n = k`

`text(i.e.)\ \ sum_(r = 1)^k 1/2^(r – 1) tan\ x/2^r = 1/2^(k – 1) cot\ x/2^k – 2 cot x.`

`text(Prove the result true for)\ \ n = k+1`

`text(i.e.)\ \sum_(r = 1)^(k + 1) 1/2^(r – 1) tan x/2^r = 1/2^k cot x/2^(k + 1) – 2 cot x`

`text(LHS)`	`=sum_(r = 1)^(k) 1/2^(r – 1) tan x/2^r + 1/2^k tan x/2^(k + 1)`
	`=1/2^(k – 1) cot x/2^k – 2 cot x + 1/2^k tan x/2^(k + 1)`
	`=1/2^(k – 1) (cot x/2^k + 1/2 tan x/2^(k +1)) – 2 cot x,\ \ \ \ text{(Let}\ \ theta=x/2^ktext{)}`
	`=1/2^(k – 1)(cot\ theta + 1/2 tan\ theta/2) – 2 cot x`
	`=1/2^(k – 1)(1/2 cot\ theta/2) – 2 cot x,\ \ \ \ text{(from part (i))}`
	`=1/2^k cot\ theta/2 – 2 cot x`
	`=1/2^k cot x/2^(k + 1) – 2 cot x`
	`=\ text(RHS)`

`=>text(True for)\ \ n = k + 1\ \ text(if it is true for)\ \ n = k.`

`:.text(S)text(ince true for)\ \ n = 1,\ text(by PMI, true for integral)\ \ n >= 1.`

iii. `lim_(n -> oo) sum_(r = 1)^n 1/2^(r – 1) tan x/2^r `

`=lim_(n -> oo) (1/2^(n-1) cot x/2^n – 2 cot x)`

`=lim_(n -> oo) (2/x * x/2^n * 1/(tan x/2^n) – 2 cot x)`

`=2/x xx lim_(n -> oo) ((x/2^n)/(tan x/2^n)) – 2 cot x`

`=>text(As)\ \ n -> oo,\ \ x/2^n=theta->0, and`

`=>lim_(theta-> 0) (theta)/(tan theta) =1`

`:.lim_(n -> oo) sum_(r = 1)^n 1/2^(r – 1) tan x/2^r =2/x – 2 cot x`

iv. `tan\ pi/4 + 1/2 tan\ pi/8 + 1/4 tan\ pi/16 + …`

`=lim_(n -> oo) sum_(r = 1)^n 1/2^(r – 1) tan (pi/2)/2^r`

`=(2/(pi/2)) – 2 cot pi/2`

`=4/pi`

Complex Numbers, EXT2 N2 2009 HSC 7b

Let `z = cos theta + i sin theta.`

Show that `z^n + z^-n = 2 cos n theta`, where `n` is a positive integer. (2 marks)

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Let `m` be a positive integer. Show that

`(2 cos theta)^(2m) = 2 [cos 2 m theta + ((2m), (1)) cos (2m - 2) theta + ((2m), (2)) cos (2m - 4) theta`

`+ … + ((2m), (m - 1)) cos 2 theta] + ((2m), (m)).` (3 marks)

--- 8 WORK AREA LINES (style=lined) ---
Hence, or otherwise, prove that

`int_0^(pi/2) cos^(2m) theta\ d theta = pi/(2^(2m + 1)) ((2m), (m))`

where `m` is a positive integer. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i.	`z`	`= cos theta + i sin theta`
	`z^n`	`= cos n theta + i sin n theta\ \ \ \ text{(De Moivre)}`
	`z^-n`	`= cos (-n theta) + i sin (-n theta)\ \ \ \ text{(De Moivre)}`
		`= cos n theta – i sin n theta`
	`z^n + z^-n`	`= cos n theta + i sin n theta + cos n theta – i sin n theta`
		`= 2 cos n theta,\ \ \ \ n > 0`

ii. `z + z^-1 = 2 cos theta`

`:.(2 cos theta)^(2m)`

`=(z + z^-1)^(2m)`

`=z^(2m) + ((2m), (1)) z^(2m – 1) z^-1 + ((2m), (2)) z^(2m – 2) z^-2+`

` … + ((2m), (2m – 1)) z^1 z^-(2m – 1) + z^-(2m)`

`=z^(2m) + ((2m), (1)) z^(2m – 2) + ((2m), (2)) z^(2m – 4)+`

` … + ((2m), (2m – 1)) z^-(2m – 2) + z^(-2m)`

`=z^(2m) + ((2m), (1)) z^(2m – 2) + ((2m), (2)) z^(2m – 4) + … + ((2m), (m)) z^(2m-2m) …`

`+ ((2m), (2)) z^-(2m – 4) + ((2m), (1)) z^-(2m – 2) + z^(-2m)`

`=(z^(2m) + z^(-2m)) + ((2m), (1)) (z^(2m – 2) + z^-(2m – 2)) + ((2m), (2))`

`(z^(2m – 4) + z^-(2m – 4)) + … + ((2m), (m – 1)) (z + z^-1) + ((2m), (m))`

`=2 [cos 2 m theta + ((2m), (1)) cos (2m – 2) theta + ((2m), (2)) cos (2m – 4) theta`

`+ … + ((2m), (m – 1)) cos 2 theta] + ((2m), (m))`

iii. `int_0^(pi/2) cos^(2m) d theta`

`=1/(2^(2m)) int_0^(pi/2) (2 cos theta)^(2m)`

`=1/(2^(2m)) int_0^(pi/2)[2(cos 2 m theta + ((2m), (1)) cos (2m – 2) theta + ((2m), (2))`

`cos (2m – 4) theta + … + ((2m), (m – 1)) cos 2 theta) + ((2m), (m))] d theta`

`=1/(2^(2m)) [2((sin 2 m theta)/(2m) + ((2m), (1)) (sin (2m – 2) theta)/(2m – 2)`

`+ … + ((2m), (m – 1)) (sin 2 theta)/2) + ((2m), (m)) theta]_0^(pi/2)`

`=1/(2^(2m)) [2(0 + 0 + … + 0) + ((2m), (m)) pi/2 – (0)]`

`=pi/(2^(2m + 1)) ((2m), (m))`

Harder Ext1 Topics, EXT2 2009 HSC 6c

The diagram shows a circle of radius `r`, centred at the origin, `O`. The line `PQ` is tangent to the circle at `Q`, the line `PR` is horizontal, and `R` lies on the line `x = c`.

Find the length of `PQ` in terms of `x, y and r.` (1 mark)
The point `P` moves such that `PQ = PR`.
Show that the equation of the locus of `P` is
1. `y^2 = r^2 + c^2 - 2cx.` (2 marks)
Find the focus, `S`, of the parabola in part (ii). (2 marks)
Show that the difference between the length `PS` and the length `PQ` is independent of `x.` (2 marks)

Show Answers Only

`PQ = sqrt (x^2 + y^2 – r^2)`
`text(Proof)\ \text{(See Worked Solutions)}`
`(r^2/(2c), 0)`
`text(Proof)\ \text{(See Worked Solutions)}`

Show Worked Solution

(i)	`OP`	`= sqrt (x^2 + y^2)`
	`PQ^2`	`= OP^2 – OQ^2`
	`PQ^2`	`= x^2 + y^2 – r^2`
	`:.\ PQ`	`= sqrt (x^2 + y^2 – r^2)`

(ii) `text(When)\ \ PQ = PR`

`sqrt (x^2 + y^2 – r^2)`	`=c-x`
`x^2 + y^2 – r^2`	`= (c – x)^2`
`x^2 + y^2 – r^2`	`= c^2 – 2cx + x^2`
`y^2 – r^2`	`= c^2 – 2cx`

`:.\ y^2 = r^2 + c^2 – 2cx\ \ text(is the locus of)\ \ P`

(iii) `text(Rearranging the locus of)\ \ P\ \ text(in the form)`

♦♦♦ “Few” students answered part (iii) correctly (exact data unavailable).

`(y-y_0)^2`	`=4a(x-x_0)`
`y^2`	`= r^2 + c^2 – 2cx`
	`= -2c(x – (r^2 + c^2)/(2c))`

`=>\ text(The parabola is lying on its side and opening to the left.)`

`text(Vertex) = ((r^2 + c^2)/(2c),0)`

`text(Focal length) \ \ \ \ 4a`	`=-2c`
`a`	`=- c/2`

`:.S\ text(has coordinates)\ \ ((r^2 + c^2)/(2c) – c/2, 0) -=(r^2/(2c), 0)`

(iv) `text(The directrix of the parabola has the equation)`

♦♦♦ “Few” students answered part (iv) correctly (exact data unavailable).
MARKER’S COMMENT: Very few used the efficient focus-directrix definition here.

`x`	`=(r^2+c^2)/(2c) + c/2`
	`=(r^2+2c^2)/(2c)`

`text(The definition of a parabola requires that)`

`PS`	`=PM`
	`=(r^2+2c^2)/(2c)-x`

`text(S)text(ince)\ \ \ PQ`	`=PR\ \ \ \ \ \ text{(from part (ii))}`
	`=c-x`

`=> PS-PQ`	`=(r^2+2c^2)/(2c)-x-(c-x)`
	`=r^2/(2c)`

`:. PS-PQ\ \ text(is independent of)\ x.`

Integration, EXT2 2010 HSC 8

Let

`A_n = int_0^(pi/2) cos^(2n) x\ dx` and `B_n = int_0^(pi/2) x^2cos^(2n)x\ dx`,

where `n` is an integer, `n ≥ 0`. (Note that `A_n > 0`, `B_n > 0`.)

Show that
`nA_n = (2n − 1)/2 A_(n − 1)` for `n ≥ 1`. (2 marks)
Using integration by parts on `A_n`, or otherwise, show that
1. `A_n = 2n int_0^(pi/2) x sin x cos^(2n − 1) x\ dx` for `n ≥ 1`. (1 mark)
Use integration by parts on the integral in part (ii) to show that
1. `(A_n)/(n^2) = ((2n − 1))/n B_(n − 1) − 2B_n` for `n ≥ 1`. (3 marks)
Use parts (i) and (iii) to show that
1. `1/(n^2) = 2((B_(n − 1))/(A_(n − 1)) − (B_n)/(A_n))` for `n ≥ 1`. (1 mark)
Show that
1. `sum_(k = 1)^n 1/(k^2) = (pi^2)/6 − 2 (B_n)/(A_n)`. (2 marks)
Use the fact that
`sin x ≥ 2/pi x` for `0 ≤ x ≤ pi/2` to show that
1. `B_n ≤ int_0^(pi/2) x^2(1 − (4x^2)/(pi^2))^n dx`. (1 mark)
Show that
`int_0^(pi/2) x^2(1 − (4x^2)/(pi^2))^n dx = (pi^2)/(8(n + 1)) int_0^(pi/2) (1 − (4x^2)/(pi^2))^(n + 1) dx`. (1 mark)
From parts (vi) and (vii) it follows that
1. `B_n ≤ (pi^2)/(8(n + 1)) int_0^(pi/2)(1 − (4x^2)/(pi^2))^(n + 1) dx`.
Use the substitution `x = pi/2 sin t` in this inequality to show that
1. `B_n ≤ (pi^3)/(16(n + 1)) int_0^(pi/2) cos^(2n + 3)t\ dt ≤ (pi^3)/(16(n + 1)) A_n`. (2 marks)
Use part (v) to deduce that
1. `(pi^2)/6 − (pi^3)/(8(n + 1)) ≤ sum_(k = 1)^n 1/(k^2) < (pi^2)/6`. (1 mark)
What is
`lim_(n → ∞) sum_(k = 1)^n 1/(k^2)`? (1 mark)

Show Answers Only

`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`(pi^2)/6`

Show Worked Solution

♦♦ Mean mark part (i) 21%.

(i)	`u`	`=cos^(2n − 1)x`
	`du`	`=-(2n-1) sinx cos^(2n − 2) x\ dx`
	`v`	`=sin x`
	`dv`	`=cos x\ dx`

`A_n`	`= int_0^(pi/2) cos x cos^(2n − 1)x\ dx`
	`= [sin x cos^(2n − 1)x]_0^(pi/2) − int_0^(pi/2) sin x*-(2n-1) sinx cos^(2n − 2) x\ dx`
	`= 0 + (2n − 1) int_0^(pi/2) sin^2 x cos^(2n − 2) x\ dx`
	`= (2n − 1) int_0^(pi/2) (1 − cos^2x)cos^(2n − 2)x\ dx`
	`= (2n − 1)(int_0^(pi/2) cos^(2n − 2)x\ dx − int_0^(pi/2) cos^(2n)x\ dx)`
	`= (2n − 1) A_(n − 1) − (2n − 1) A_n`

`A_n + (2n − 1) A_n`	`= (2n − 1) A_(n − 1)`
`2nA_n`	`= (2n − 1) A_(n − 1)`
`:.nA_n`	`= (2n − 1)/2 A_(n − 1)\ \ text(for)\ \ n ≥ 1.`

♦♦ Mean mark part (ii) 32%.

(ii)	`u`	`=cos^(2n) x`
	`du`	`=-2n\ sinx cos^(2n − 1) x\ dx`
	`v`	`= x`
	`dv`	`= dx`

`A_n`	`= int_0^(pi/2) 1 xx cos^(2n)x\ dx`
	`= [x cos^(2n) x]_0^(pi/2) − int_0^(pi/2) x * -2n\ sinx cos^(2n − 1) x\ dx`
	`= 0 + 2n int_0^(pi/2) x sin x cos^(2n − 1) x\ dx`
	`= 2n int_0^(pi/2) x sin x cos^(2n − 1)x\ dx\ \ \ text(for)\ \ n ≥ 1`

♦♦♦ Mean mark part (iii) 8%.

(iii)	`u`	`=sin x cos^(2n-1) x`
	`du`	`=cosx cos^(2n-1)x -(2n-1) cos^(2n-2)x *sin^2x\ dx`
		`=cos^(2n)x – (2n-1)cos^(2n-2) x* (1-cos^2 x)\ dx`
		`=cos^(2n)x – (2n-1)cos^(2n-2)x +(2n-1)cos^(2n)\ dx`
		`=2n cos^(2n)x-(2n-1)cos^(2n-2)x`
	`v`	`= x^2/2`
	`dv`	`= x\ dx`

`A_n`	`= 2n int_0^(pi/2) x sin x cos^(2n − 1)x\ dx`
	`= 2n[x^2/2 sin x cos^(2n − 1) x]_0^(pi/2)`
	`− 2n int_0^(pi/2) x^2/2 (2n cos^(2n)x-(2n-1)cos^(2n-2)x)\ dx`
	`= 0 −2n[n int_0^(pi/2) x^2 cos^(2n) x\ dx − (2n − 1)/2 int_0^(pi/2) x^2 cos^(2n − 2) x \ dx]`
	`= -2n(nB_n-(2n-1)/2 B_(n-1))`
	`= -2n^2B_n+n(2n-1)B_(n-1)`
`:.(A_n)/(n^2)`	`= (2n − 1)/n B_(n − 1) − 2B_n\ \ \ text(for)\ \ n ≥ 1`

♦♦♦ Mean mark part (iv) 13%.

(iv)	`(A_n)/(n^2)`	`= ((2n −1)B_(n − 1))/n − 2B_n\ \ \ text(for)\ \ n ≥ 1`
	`:.1/(n^2)`	`= ((2n − 1)B_(n − 1))/(nA_n) − (2B_n)/(A_n)`
		`= ((2n − 1)B_(n − 1))/((2n − 1)/2 A_(n − 1)) − (2B_n)/(A_n)\ \ \ \ \ text{(using part (i))}`
		`= (2B_(n − 1))/(A_(n − 1)) − (2B_n)/(A_n)`
		`= 2((B_(n − 1))/(A_(n − 1)) − (B_n)/(A_n))\ \ \ text(for)\ \ n ≥ 1`

♦♦♦ Mean mark part (v) 9%.

(v)	`sum_(k = 1)^n 1/(k^2)`	`= 1/(1^2) + 1/(2^2) + 1/(3^2) + … + 1/(n^2)`
	`sum_(k = 1)^n 1/(k^2)`	`= 2((B_0)/(A_0) − (B_1)/(A_1)) + 2((B_1)/(A_1) − (B_2)/(A_2)) + … + 2((B_(n − 1))/(A_(n − 1)) − (B_n)/(A_n))`
		`= 2(B_0)/(A_0) − 2(B_n)/(A_n)`

`A_0 = int_0^(pi/2) dx = [x]_0^(pi/2) = pi/2`

`B_0 = int_0^(pi/2)x^2\ dx = [(x^3)/3]_0^(pi/2) = (pi^3)/24`

`sum_(k = 1)^n 1/(k^2)`	`= 2 xx ((pi^3)/24)/(pi/2) − 2(B_n)/(A_n)`
	`= (pi^2)/6 − 2(B_n)/(A_n)`

♦♦♦ Mean mark part (vi) 8%.

(vi)	`B_n`	`= int_0^(pi/2) x^2 cos^(2n)x\ dx`
	`B_n`	`= int_0^(pi/2) x^2 (1 − sin^2 x)^n\ dx`

`text(S)text(ince)\ \ sin x ≥ 2/pi x,\ \ 0 ≤ x ≤ pi/2`

`:.B_n`	`≤ int_0^(pi/2) x^2 (1 − (2/pi x)^2)^n\ dx`
	`≤ int_0^(pi/2) x^2 (1 − (4x^2)/(pi^2))^n\ dx`

(vii)	`u`	`=x,\ \ \ du=dx`
	`dv`	`=x(1-(4x^2)/(pi^2))^n\ dx`
	`v`	`=(- pi^2)/(8(n+1)) (1-(4x^2)/(pi^2))^(n+1)`

♦♦♦ Mean mark part (vii) 2%.

`int_0^(pi/2) x * x(1 − (4x^2)/(pi^2))^n dx`

`= [(-pi^2)/(8(n + 1))(1 − (4x^2)/(pi^2))^(n + 1) * x]_0^(pi/2) – int_0^(pi/2) (-pi^2)/(8(n + 1))(1 − (4x^2)/(pi^2))^(n + 1)\ dx`

`= (-pi^2)/(8(n + 1)) (0 − 0) + (pi^2)/(8(n + 1)) int_0^(pi/2) (1 − (4x^2)/(pi^2))^(n + 1) dx`

`= (pi^2)/(8(n + 1)) int_0^(pi/2) (1 − (4x^2)/(pi^2))^(n + 1) dx`

♦♦♦ Mean mark part (viii) 4%.

(viii)	`text(Using)\ \ x`	`=pi/2 sin t\ \ \ \ dx=pi/2 cos t\ dt`
	`text(When)\ \ x`	`=0,\ \ \ t=0`
	`text(When)\ \ x`	`=pi/2,\ \ \ t=pi/2`

`B_n`	`≤ (pi^2)/(8(n + 1)) int_0^(pi/2) (1 − (4x^2)/(pi^2))^(n + 1)\ dx`
	`≤ (pi^2)/(8(n + 1)) int_0^(pi/2) (1 − sin^2 t)^(n + 1) *pi/2 cos t\ dt`
	`≤ (pi^3)/(16(n + 1)) int_0^(pi/2) (cos^2 t)^(n + 1) cos t\ dt`
	`≤ (pi^3)/(16(n + 1)) int_0^(pi/2) cos^(2n + 3) t\ dt`

`text(Consider the RHS of the inequality)`

`text(RHS)`	`=(pi^3)/(16(n + 1)) int_0^(pi/2) cos^(2n + 3) t\ dt`
	`≤(pi^3)/(16(n + 1)) int_0^(pi/2) cos^(2n) t\ dt,\ \ \ \ text{(as cos t ≤ 1)}`
	`≤(pi^3)/(16(n + 1)) A_n\ \ \ \ text(… as required)`

♦♦♦ Mean mark part (ix) 3%.

(ix) `sum_(k = 1)^n 1/(k^2) = (pi^2)/6 − 2(B_n)/(A_n)\ \ \ \ text{(from part (v))}`

`:.sum_(k = 1)^n 1/(k^2) < (pi^2)/6\ \ \ \ (A_n > 0`, `B_n > 0)`

`text{Using part (viii)}`

`B_n`	`≤(pi^3)/(16(n + 1)) A_n`
`(2B_n)/A_n`	`≤(pi^3)/(8(n + 1)) `

`:.sum_(k = 1)^n 1/(k^2) ≥ (pi^2)/6 − (pi^3)/(8(n + 1))`

`:. (pi^2)/6 − (pi^3)/(8(n + 1)) ≤ sum_(k = 1)^n 1/(k^2) < (pi^2)/6`

♦♦♦ Mean mark part (x) 19%.

(x) `text(S)text(ince)\ (pi^3)/(8(n + 1)) → 0\ text(as)\ n → ∞`

`lim_(n → ∞) sum_(k = 1)^n 1/(k^2) = (pi^2)/6`

Polynomials, EXT2 2010 HSC 7c

Let `P(x) = (n − 1)x^n − nx^(n − 1) + 1`, where `n` is an odd integer, `n ≥ 3`.

Show that `P(x)` has exactly two stationary points. (1 mark)
Show that `P(x)` has a double zero at `x = 1`. (1 mark)
Use the graph `y = P(x)` to explain why `P(x)` has exactly one real zero other than `1`. (2 marks)
Let `α` be the real zero of `P(x)` other than `1`.
Given that `2^x>=3x-1` for `x>=3`, or otherwise, show that `-1 < α ≤ -1/2`. (2 marks)
Deduce that each of the zeros of `4x^5 − 5x^4 + 1` has modulus less than or equal to `1`. (2 marks)

Show Answers Only

`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`

Show Worked Solution

(i)	`P′(x)`	`= n(n − 1)x^(n − 1) − n(n − 1)x^(n − 2)`
		`= n(n − 1)x^(n − 2)(x − 1)`

`P′(x) = 0\ \ text(when)\ \ x = 0\ \ text(or)\ \ x = 1.`

`:.P(x)\ text(has exactly two stationary points.)`

(ii)	`P(1)`	`= (n − 1) xx 1 − n xx 1 + 1 = 0, and`
	`P′(1) `	`= 0`

`:. P(x)\ text(has a double zero at)\ \ x=1`

(iii) `P(x)\ text{has exactly two stationary points at}\ \ x=0 and 1`

♦♦ Mean mark part (iii) 26%.

`text(S)text(ince)\ \ P(0) = 1 and P(1)=0`

`=>\ text{A local maximum occurs at (0,1)}`

`text(Noting that)\ \ P(x)`	`-> oo\ \ text(as)\ \ x-> oo, and`
`P(x)`	`-> -oo\ \ text(as)\ \ x-> -oo`

`:.\ text(The double zero at)\ \ x=1,\ text{means that}\ \ P(x)`

`text(has exactly one real zero other than)\ \ x=1.`

♦♦♦ Mean mark part (iv) 3%.

(iv)	`P(-1)`	`=(n-1)(-1)^n-n xx (-1)^(n-1) + 1`
		`=(n-1)(-1) – nxx1 +1\ \ \ \ text{(given}\ n\ text{is odd)}`
		`=-2n+2`
		`<0\ \ \ text{(for odd}\ n>=3text{)}`

`P(-1/2)`	`= (n − 1)(-1/2)^n − n(-1/2)^(n − 1)+1`
	`= -(n − 1) xx 1/(2^n) − n/(2^(n − 1)) + 1\ \ \ \ text{(given}\ n\ text{is odd)}`
	`= (-n + 1 − 2n + 2^n)/(2^n)`
	`= (2^n − (3n − 1))/(2^n)`
	`>=0\ \ \ \ \ text{(given}\ \ 2^n>=3n-1\ \ text{for}\ \ n>=3, and 2^n > 0)`

`text(S)text(ince)\ \ P(x)\ \ text(is continuous, when it changes sign, it cuts the)\ x text(-axis)`

`:. -1 < α ≤ -1/2`

(v) `P(x) = 4x^5 − 5x^4 + 1\ \ text(is of the form)`

♦♦♦ Mean mark part (v) 2%.

`P(x) = (n − 1)x^n − nx^(n − 1) + 1`

`:.P(x)\ \ text(has 5 zeros)\ \=> 1, 1, alpha, beta, bar beta\ \ \ \ (text(where)\ beta\ text{is not real})`

`text(The zeros 1 and α have a modulus ≤ 1.)`

`text(Consider the product of the roots)`

`11alphabetabar beta`	`=- 1/4`
`alphabetabar beta`	`=- 1/4`
`\|\ alphabetabar beta\ \|`	`=\|\ – 1/4\ \|`
`\|\ alpha\ \|*\|\ beta\ \|^2`	`= 1/4`

`text(S)text(ince)\ \ |\ α\ | > 1/2,\ \ |\ beta\ |<1`

`:.text(Each of the zeros of)\ x^5 − 5x^4 + 1\ text(has a modulus)\ <=1.`

Harder Ext1 Topics, EXT2 2010 HSC 7a

In the diagram `ABCD` is a cyclic quadrilateral. The point `K` is on `AC` such that `∠ADK = ∠CDB`, and hence `ΔADK` is similar to `ΔBDC`.

Copy or trace the diagram into your writing booklet.

Show that `ΔADB` is similar to `ΔKDC`. (2 marks)
Using the fact that `AC = AK + KC`,
show that `BD xx AC = AD xx BC + AB xx DC`. (2 marks)
A regular pentagon of side length `1` is inscribed in a circle, as shown in the diagram.

Let `x` be the length of a chord in the pentagon.
Use the result in part (ii) to show that `x = (1 + sqrt5)/2`. (2 marks)

Show Answers Only

`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`

Show Worked Solution

(i)

Harder Ext1 Topics, EXT2 2010 HSC 7a Answer

`text(S)text(ince)\ \ ∠ADB`	`=∠ADK +∠KDB, and`
`∠KDC`	`=∠CDB +∠KDB`
`=>∠ADB`	`= ∠KDC`
`∠ABD`	`= ∠ACD\ \ \ text{(angles in the same segment on arc}\ ADtext{)}`

`∠ADK`	`= ∠CDB\ \ \ `	`text{(corresponding angles of similar}`
		`text{triangles,}\ ΔADK\ text(\|\|\|)\ ΔBDC text{)}`

`:.ΔADB\ text(|||)\ ΔKDC\ \ text{(equiangular)}`

(ii) `text(S)text(ince)\ ΔADB\ text(|||)\ ΔKDC`

`(BD)/(DC) = (AD)/(DK) = (AB)/(KC)\ \ text{(corresponding sides of similar triangles)}`

`:.AB xx DC = KC xx BD\ \ \ \ …\ (1)`

♦♦ Mean mark part (ii) 31%.

`text(Similarly, using)\ \ ΔADK\ text(|||)\ ΔBDC\ \ text{(given)}`

`(AD)/(BD)=(AK)/(BC)`

`:. BC xx AD = AK xx BD\ \ \ \ …\ (2)`

`text{Add (1) + (2)}`

`AB xx DC + AD xx BC`	`=KC xx BD+AK xx BD`
	`=BD(AK+KC)`
`:.BD xx AC`	`=AD xx BC+AB xx DC`

(iii) `text(Each diagonal of the pentagon is)\ x.`

♦♦ Mean mark part (iii) 24%.

`text{Hence in applying the result in (i) you have}`

`BD = AC = x, AD = DC = CB = 1, BA = x`

`x xx x`	`= 1 xx 1 + x xx 1`
`x^2 − x − 1`	`= 0`
`x`	`= (1 ± sqrt(1 + 4))/2`
	`= (1 ± sqrt(5))/2`
`:.x`	`= (1 + sqrt(5))/2,\ \ \ (x>0)`

Harder Ext1 Topics, EXT2 2010 HSC 5c

A TV channel has estimated that if it spends `$x` on advertising a particular program it will attract a proportion `y(x)` of the potential audience for the program, where

`(dy)/(dx) = ay(1 − y)`

and `a > 0` is a given constant.

Explain why
`(dy)/(dx)` has its maximum value when `y = 1/2`. (1 mark)
Using
`int (dy)/(y(1 − y)) = ln (y/(1 − y)) + c`, or otherwise, deduce that
`y(x) = 1/(ke^(-ax) + 1)` for some constant `k > 0`. (3 marks)
The TV channel knows that if it spends no money on advertising the program then the audience will be one-tenth of the potential audience.
Find the value of the constant `k` referred to in part (c)(ii). (1 mark)
What feature of the graph
`y = 1/(ke^(-ax) + 1)` is determined by the result in part (c)(i)? (1 mark)
Sketch the graph
`y(x) = 1/(ke^(-ax) + 1)` (1 mark)

Show Answers Only

`text(See Worked Solutions.)`
`text{Proof (See Worked Solutions)}`
`9`
`y = 1/2`
`text(See Worked Solutions.)`

Show Worked Solution

(i) `(dy)/(dx) = ay(1 − y)`

♦♦ Mean mark part (i) 32%.

`=>text(Given)\ \ a>0,\ \ ay(1 − y)\ text(is an inverted parabola)`

`text(with zeros at)\ \ y = 0\ \ text(and)\ \ y = 1`

`=>\ text(Parabola symmetry means that a maximum occurs when)`

`y = (0+1)/2=1/2`

`:.(dy)/(dx)\ \ text(has its maximum value when)\ y = 1/2.`

♦ Mean mark part (ii) 41%.

(ii) `dy/dx=ay(1 − y)`

`int (dy)/(y(1 − y))`	`=int a\ dx`
`ax`	`= ln (y/(1 − y))+c`
`e^(ax − c)`	`= y/(1 − y)`
`e^(ax − c) − ye^(ax − c)`	`= y`
`y(1 + e^(ax − c))`	`= e^(ax − c)`
`y`	`= (e^(ax − c))/(1 + e^(ax − c))`
`y`	`= 1/(e^c e^(− ax) + 1)`

`text(Let)\ k = e^c`

`:.y(x) = 1/(ke^(-ax) + 1)\ text(for some constant)\ \ k > 0.`

(iii) `text(When)\ \ x = 0, y = 0.1`

`0.1`	`= 1/(ke^0 + 1)`
`k + 1`	`= 10`
`k`	`= 9`

♦♦ Mean mark part (iv) 25%.

(iv) `text(The gradient the curve is greatest at)\ y = 1/2\ \ \ text{from part (i)}`

`:. text(A point of inflection occurs at)\ y =1/2.`

(v) `text(As)\ \ x->oo,\ \ y->1/(0+1)=1`

♦♦ Mean mark part (v) 12%.

Harder Ext1 Topics, EXT2 2010 HSC 4c

Let `k` be a real number, `k ≥ 4`.

Show that, for every positive real number `b`, there is a positive real number `a` such that `1/a + 1/b = k/(a + b)`. (3 marks)

Show Answers Only

`text{Proof (See Worked Solutions)}`

Show Worked Solution

`1/a + 1/b`	` = k/(a+b)`
`text(Multiply b.s. by)\ \ ab(a+b)`
`b(a+b)+a(a+b)`	`=kab`
`(a + b)^2`	`=kab`
`a^2 + 2ab + b^2`	`=kab`
`a^2+(2-k)ba+b^2`	`=0`

`text(Given)\ \ b>0,\ \ a\ \ text(is real when)\ \ Delta>=0`

♦♦♦ Mean mark 13%.

`(2-k)^2 b^2-4b^2`	`>=0`
`(4-4k+k^2)b^2-4b^2`	`>=0`
`b^2k(k-4)`	`>=0`
`k`	`>=4`

`:. a \ \ text(is real)`

`text(Using the quadratic formula to solve for)\ \ a`

`a`	`= (-(2-k)b +- sqrt Delta)/2`
	`=1/2 (2-k)b +- sqrt Delta`

`text(S)text(ince)\ \ b>0,\ \ (k-2)>0,\ \ and sqrt Delta>0`

`=>a=1/2 (2-k)b + sqrt Delta>=0`

`text(i.e. there exists a positive real number)\ \ a\ \ text(that satisfies.)`

Conics, EXT2 2010 HSC 3d

The diagram shows the rectangular hyperbola `xy = c^2`, with `c > 0`.

The points `A(c, c)`, `R(ct, c/t)` and `Q(-ct, -c/t)` are points on the hyperbola,
with `t ≠ ±1`.

The line `l_1` is the line through `R` perpendicular to `QA`.
Show that the equation of `l_1` is
1. `y = -tx + c(t^2 + 1/t)`. (2 marks)
The line `l_2` is the line through `Q` perpendicular to `RA`.
Write down the equation of `l_2`. (1 mark)
Let `P` be the point of intersection of the lines `l_1` and `l_2`.
Show that `P` is the point `(c/(t^2), ct^2)`. (2 marks)
Give a geometric description of the locus of `P`. (1 mark)

Show Answers Only

`text{Proof (Show Worked Solutions)}`
`y = tx + c(t^2 − 1/t)`
`text{Proof (Show Worked Solutions)}`
`text{Proof (Show Worked Solutions)}`

Show Worked Solution

(i)

`m_(QA)`	`= (c + c/t)/(c + ct)`
	`= (1 + 1/t)/(1 + t)`
	`= (t + 1)/(t(1 + t))`
	`= 1/t`

`:.\ text(Gradient of perpendicular to)\ QA\ \ (l_1) = -t`

`:.text(Equation of)\ \ l_1`

`y − c/t`	`= -t(x − ct)`
`y`	`= -tx + ct^2 + c/t`
	`= -tx + c(t^2 + 1/t)\ \ \ …\ (1)`

(ii)	`m_(RA)`	`= (c − c/t)/(c − ct)`
		`= (1 − 1/t)/(1 − t)`
		`= (t − 1)/(t(1 − t))`
		`= -1/t`

`:.m\ text(of)\ l_2 = t`

`:.\ text(Equation of)\ l_2`

`y + c/t`	`= t(x + ct)`
`y`	`= tx + ct^2 – c/t`
	`= tx + c(t^2 – 1/t)\ \ \ …\ (2)`

(iii) `text{Solving (1) and (2) simultaneously}`

`-tx + c(t^2 + 1/t)`	`= tx + c(t^2 − 1/t)`
`ct^2+c/t-ct^2+c/t`	`=2tx`
`(2c)/t`	`= 2tx`
`x`	`= c/(t^2)`

`text{Substitute into (2)}`

`y`	`= (ct)/(t^2) + ct^2 − c/t`
	` = ct^2`

`:.P\ text(has coordinates)\ (c/(t^2), ct^2)`

(iv) `text(Using the coordinates of)\ P`

♦♦ Mean mark part (iv) 1%.
MARKER’S COMMENT: Most students found the correct locus, although very few indicated that it was only the branch in the 1st quadrant.

`=>t^2 = c/x,\ \ t^2 = y/c`

`y/c`	`= c/x`
`:.xy`	`=c^2`

`text(S)text(ince the parameter of)\ P\ text(is)\ t^2`

`=>\ text(the locus is in the first quadrant.)`

`:.\ text(The locus of)\ P\ text(is the branch of the rectangular hyperbola)`

`xy = c^2\ text{in the first quadrant (excluding}\ A, t≠±1 text{)}`

Harder Ext1 Topics, EXT2 2010 HSC 3c

Two identical biased coins are each more likely to land showing heads than showing tails.

The two coins are tossed together, and the outcome is recorded. After a large number of trials it is observed that the probability that the two coins land showing a head and a tail is `0.48`.

What is the probability that both coins land showing heads? (2 marks)

Show Answers Only

`0.36`

Show Worked Solution

`text(Let)\ \ p=P(H),\ \ =>(1-p)=P(T)`

♦♦♦ Mean mark 19%.

`text(S)text(ince)\ \ P(H,T)+P(T,H)=0.48`

`p(1-p)+(1-p)p`	`=0.48`
`2p(1-p)`	`=0.48`
`p^2-p+0.24`	`=0`
`(p-0.6)(p-0.4)`	`=0`

`:. p=0.6\ \ \ \ \ text{(given}\ P(H)>P(T)text{)}`

`:.P(H,H)=0.6 xx 0.6 = 0.36`

Polynomials, EXT2 2011 HSC 8c

Let `beta` be a root of the complex monic polynomial

`P(z) = z^n + a_(n - 1) z^(n - 1) + … + a_1z + a_0.`

Let `M` be the maximum value of `|\ a_(n - 1)\ |,\ |\ a_(n - 2)\ |,\ …\ ,\ |\ a_0\ |.`

Show that
`|\ beta\ |^n <= M(|\ beta\ |^(n - 1) + |\ beta\ |^(n - 2) + … + |\ beta\ | + 1).` (2 marks)
Hence, show that for any root `beta` of `P(z)`
1. `|\ beta\ | < 1 + M.` (3 marks)
Let `S(x) = sum_(k = 0)^n c_k(x + 1/x)^k`, where the real numbers `c_k` satisfy `|\ c_k\ | <= |\ c_n\ |` for all `k < n`, and `c_n != 0.`
Using parts (i) and (ii), or otherwise, show that `S(x) = 0` has no real solutions. (3 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i) `P(z) = z^n + a_(n – 1) z^(n – 1) + … + a_1z + a_0.`

♦♦♦ Mean mark part (i) 7%.

`M = text(Max)\ (\ |\ a_(n – 1)\ |, |\ a_(n – 2)\ |, … , |\ a_0|\ )`

`P(beta) = 0`

`0`	`=beta^n + a_(n – 1) beta^(n – 1) + … + a_1 beta + a_0`
`beta^n`	`= -(a_(n – 1) beta^(n – 1) + … + a_1 beta + a_0)`
`\|\ beta^n\ \|`	`=\|\ a_(n – 1) beta^(n – 1) + … + a_1 beta + a_0\ \|`
	`<= \|\ a_(n – 1) beta^(n – 1)\ \| + … + \|\ a_1 beta\ \| + \|\ a_0\ \|`
`\|\ beta\ \|^n`	`<= \|\ a_(n – 1)\ \|\|\ beta\ \|^(n – 1) + … + \|\ a_1\ \|\|\ beta\ \| + \|\ a_0\ \|`
	`<= M (\|\ beta\ \|^(n – 1) + … + \|\ beta\ \| + 1)`

(ii) `(1 + |\ beta\ | + … + |\ beta\ |^(n – 1))`

♦♦♦ Mean mark part (ii) 3%.

`=>text(GP where)\ \ a = 1,\ \ r = |\ beta\ |, and n\ text(terms)`

`text(Consider)\ |\ beta\ | > 1`

`1 + |\ beta\ | + … + |\ beta\ |^(n – 1) = (1(|\ beta\ |^n – 1))/(|\ beta\ | – 1)`

`text(Using)\ \ |\ beta\ |^n <= M(|\ beta\ |^(n – 1) + … + |\ beta\ | + 1)\ \ \ text{(part (i))}`

`\|\ beta\ \|^n`	`<= M((\|\ beta\ \|^n – 1))/(\|\ beta\ \| – 1)`
`\|\ beta\ \|^n (\|\ beta\ \| – 1)`	`<= M(\|\ beta\ \|^n – 1)`
`\|\ beta\ \| – 1`	`<= M ((\|\ beta\ \|^n – 1))/\|\ beta\ \|^n`
`\|\ beta\ \|`	`<= 1 + M (1 – 1/\|\ beta\ \|^n)`
`\|\ beta\ \|`	`< 1 + M`

`text(Consider)\ |\ beta\ | <= 1`

`|\ beta\ | < 1 + M,\ \ text(as)\ \ M > 0.`

`:.|\ beta\ | < 1 + M\ \ text(for any root)\ \ beta`

(iii) `S(x)`	`=sum_(k=0)^n c_k(x + 1/x)^k`
	`=c_n(x + 1/x)^n + c_(n – 1)(x + 1/x)^(n – 1) + … + c_1(x + 1/x) + c_0`
`(S(x))/c_n`	`=(x + 1/x)^n + (c_(n-1))/c_n (x + 1/x)^(n – 1) + … + c_1/c_n(x + 1/x) + c_0/c_n`

`text(This is a monic polynomial of the form in part (i))`

♦♦♦ Mean mark part (iii) 2%.

`text(Let)\ \ z = x + 1/x`

`P(z) = z^n + a_(n – 1) z^(n – 1) + … + a_1 z + a_0`

`text(Consider the coefficients)\ \ a_k = c_k/c_n`

`text(S)text(ince)\ \ |\ c_k\ |<=|\ c_n\ |,\ \ \ text{(given)}`

`=>\|\ a_k\ \| <=`	`(\|\ c_k\ \|)/\|\ c_n\ \| <= 1\ \ text(from above)`
`=>M <=`	`1\ \ \ \ text{(since}\ M\ text(is the max of)\ \|\ a_k\ \| text{)}`

`text(If)\ \ beta\ \ text(is a real root of)\ \ P(z) and |\ beta\ | < 1 + M,\ \ text(then)`

`=>|\ beta\ | < 2`

`text(However,)\ \ beta`	`= x + 1/x`
`:. \|\ beta\ \|`	`= \|\ x + 1/x\ \|`
	`= \|\ x\ \| + 1/\|\ x\ \|\ \ \ \ text{(for}\ x\ text{real)}`
	`>= 2`

`:.text(We have)\ \ |\ beta\ | < 2 and |\ beta\ | = |\ x + 1/x\ | >= 2`

`text(This is a contradiction and hence our assumption that)`

`beta\ \ text(is a real root of)\ \ S(x)\ \ text(is incorrect.)`

`text(Alternative strategy to show a contradiction)`

`text(Graphing)\ \ y = x + 1/x`

`text(S)text(ince the turning points occur at)\ \ y=+- 2`

`=>|\ x + 1/x\ | >= 2.`

`:.S(x)\ \ text(has no real solutions.)`

Harder Ext1 Topics, EXT2 2011 HSC 8b

A bag contains seven balls numbered from `1` to `7`. A ball is chosen at random and its number is noted. The ball is then returned to the bag. This is done a total of seven times.

What is the probability that each ball is selected exactly once? (1 mark)
What is the probability that at least one ball is not selected? (1 mark)
What is the probability that exactly one of the balls is not selected? (2 marks)

Show Answers Only

`(6!)/7^6=720/(117\ 649)`
`1 – (6!)/7^6=(116\ 929)/(117\ 649)`
`(3 xx 6!)/7^5=2160/(16\ 807)`

Show Worked Solution

(i) `P text{(each ball is selected once)}`

♦♦ Mean mark part (i) 32%.

`=7/7 xx 6/7 xx 5/7 xx 4/7 xx 3/7 xx 2/7 xx 1/7`

`=(6!)/7^6`

`=720/(117\ 649)`

(ii) `P text{(at least one ball is not selected)}`

`=1 – P text{(each ball once)}`

`=1 – (6!)/7^6`

`=(116\ 929)/(117\ 649)`

(iii) `text{Consider when 1 of the balls is not selected (say #1).}`

♦♦♦ Mean mark part (iii) 2%.

`=>\ text(One of the other 6 balls is selected twice.)`

`text(e.g. 2, 2, 3, 4, 5, 6, 7).`

`text(This can be done in)\ \ (7!)/(2!)\ \ text(ways.)`

`text(With 6 differently numbered pairs possible,)`

`text(This can be done in)\ \ 6 xx (7!)/(2!)\ \ text(ways)`

`text(S)text(ince there are 7 numbers that can be left out,)`

`text(Total number of ways to leave 1 number out)`

`=7 xx 6 xx (7!)/(2!)`

`=21 xx 7!`

`:.P text{(exactly one ball not selected)}`	`=(21 xx 7!)/7^7`
	`=(3 xx 6!)/7^5`
	`=2160/(16\ 807)`

Calculus, EXT2 C1 2011 HSC 8a

For every integer `m >= 0` let

`I_m = int_0^1 x^m (x^2 - 1)^5\ dx.`

Prove that for `m >= 2`

`I_m = (m - 1)/(m + 11) \ I_(m - 2).` (3 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

`I_m = int_0^1 x^m(x^2 – 1)^5 dx`

`u`	`=x^(m – 1)\ \ \ \ \ \ `	`\ \ \ \ u′`	`=(m – 1)x^(m – 2)`
`v′`	`=x(x^2 – 1)^5`	`v`	`=1/12 (x^2 – 1)^6`

♦♦♦ Mean mark 24%.

`I_m`	`=[x^(m-1) xx 1/12(x^2 – 1)^6]_0^1 – int_0^1 1/12(x^2 – 1)^6 xx (m – 1)x^(m – 2) dx`
	`=[0 – 0] – (m – 1)/12 int_0^1 x^(m – 2) (x^2 – 1)^6 dx`
	`=-(m – 1)/12 int_0^1 x^(m – 2) (x^2 – 1)^5 (x^2 – 1) dx`
	`=-(m – 1)/12 int_0^1 x^m (x^2 – 1)^5 dx + (m – 1)/12 int _0^1 x^(m – 2) (x^2 – 1)^5 dx`
	`=-(m – 1)/12 I_m + (m – 1)/12 I_(m – 2)`

`12I_m + mI_m – I_m`	`= (m – 1)I_(m – 2)`
`(m + 11)I_m`	` = (m – 1)I_(m-2)`
`:.I_m`	`= (m – 1)/(m + 11) I_(m – 2)`

Complex Numbers, EXT2 N2 2011 HSC 6c

On an Argand diagram, sketch the region described by the inequality

`|\ 1 + 1/z\ | <= 1.` (2 marks)

Show Answers Only

Show Worked Solution

♦♦♦ Mean mark 18%.
MARKER’S COMMENT: Substituting `z=x+iy` immediately was common and caused major algebraic problems.

`\|\ 1 + 1/z\ \|<= `	`1`
`\|\ (z + 1)/z\ \|<= `	`1`
`\|\ z + 1\ \|/\|\ z\ \|<= `	`1`
`\|\ z + 1\ \|<= `	`\|\ z\ \|`
`sqrt ((x + 1)^2 + y^2) <=`	`sqrt (x^2 + y^2)`
`(x + 1)^2 + y^2 <=`	`x^2 + y^2`
`x^2 + 2x + 1 <=`	` x^2`
`:.x <=`	`-1/2`

Graphs, EXT2 2011 HSC 6b

Let `f (x)` be a function with a continuous derivative.

Prove that `y = (f(x))^3` has a stationary point at `x = a` if `f(a) = 0` or `f prime(a) = 0.` (2 marks)
Without finding `f″(x)`, explain why `y = (f(x))^3` has a horizontal point of inflection at `x = a` if `f(a) = 0` and `f prime (a) != 0.` (1 mark)
The diagram shows the graph `y = f(x).`
Copy or trace the diagram into your writing booklet.
On the diagram in your writing booklet, sketch the graph `y = (f(x))^3`, clearly distinguishing it from the graph `y = f(x).` (3 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(See Worked Solutions)`

Show Worked Solution

(i) `text(If)\ \ f(x)\ \ text(is a function with a continuous derivative,)`

`=> f prime (x)\ \ text(exists for all)\ \ x.`

`y`	`= (f(x))^3`
`(dy)/(dx)`	`= 3 xx (f(x))^2 xx f prime(x)`

`:.\ (dy)/(dx) = 0\ \ text(when)\ \ f(x) = 0 or f prime (x) = 0`

`:.\ x = a\ \ text(is a stationary point if)\ \ f(a) = 0 or f prime(a) = 0`

♦♦♦ Mean mark part (ii) 0%!
A BEAST!

(ii) `text(If)\ \ f prime (a) != 0\ \ text(then either)\ \ f prime (a) > 0 or f prime (a) < 0,`

`and f prime (x)\ \ text(keeps that sign either side of)\ \ x = a.`

`:. text(If)\ \ f(a) = 0 and f prime(a) != 0, text(there is a stationary point at)\ \ x = a`

`text(where the slope of the curve does not change either side of)\ \ x = a.`

`text(i.e. a horizontal point of inflection occurs at)\ \ x=a`

(iii) `y = (f(x))^3`

`text(When)\ \ f(x) = 0,\ \ (f(x))^3 = 0\ \ text{(horizontal P.I. from part (ii))}`

`text(When)\ \ f(x) = 1,\ \ (f(x))^3 = 1`

`text(If)\ \ 0 < f(x) < 1,\ \ 0 < (f(x))^3 < f(x)`

`text(If)\ \ f(x) > 1,\ \ (f(x))^3 > f(x)`

`text(When)\ f prime (a) = 0,\ \ (f(x))^3\ \ text(has a maximum turning point)`

`f(x) < 0,\ \ (f(x))^3 < 0`

Mechanics, EXT2 M1 2011 HSC 6a

Jac jumps out of an aeroplane and falls vertically. His velocity at time `t` after his parachute is opened is given by `v(t)`, where `v(0) = v_0` and `v(t)` is positive in the downwards direction. The magnitude of the resistive force provided by the parachute is `kv^2`, where `k` is a positive constant. Let `m` be Jac’s mass and `g` the acceleration due to gravity. Jac’s terminal velocity with the parachute open is `v_T.`

Jac’s equation of motion with the parachute open is

`m (dv)/(dt) = mg - kv^2.` (Do NOT prove this.)

Explain why Jac’s terminal velocity `v_T` is given by

`sqrt ((mg)/k).` (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
By integrating the equation of motion, show that `t` and `v` are related by the equation

`t = (v_T)/(2g) ln[((v_T + v)(v_T - v_0))/((v_T - v)(v_T + v_0))].` (3 marks)

--- 6 WORK AREA LINES (style=lined) ---
Jac’s friend Gil also jumps out of the aeroplane and falls vertically. Jac and Gil have the same mass and identical parachutes.

Jac opens his parachute when his speed is `1/3 v_T.` Gil opens her parachute when her speed is `3v_T.` Jac’s speed increases and Gil’s speed decreases, both towards `v_T.`

Show that in the time taken for Jac's speed to double, Gil's speed has halved. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(See Worked Solutions)`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i. `m (dv)/(dt) = mg – kv^2`

`text(As)\ \ v ->text(terminal velocity,)\ \ (dv)/(dt) -> 0`

`mg – kv_T^2`	`= 0`
`v_T^2`	`= (mg)/k`
`v_T`	`= sqrt ((mg)/k)`

ii. `m (dv)/(dt) = mg – kv^2`

`int_0^t dt`	`=int_(v_0)^v m/(mg – kv^2)\ dv`
`t`	`= m/k int_(v_0)^v (dv)/((mg)/k – v^2)\ \ \ \ text{(using}\ \ v_T^2= (mg)/k text{)}`
	`= (v_T^2)/g int_(v_0)^v (dv)/(v_T^2 – v^2)`

♦ Mean mark part (ii) 39%.

`text(If)\ \ v_0 < v_T,\ \ text(then)\ \ v < v_T\ \ text(at all times.)`

`:. t`	`=(v_T^2)/g int_(v_0)^v (dv)/(v_T^2 – v^2)`
	`=(v_T^2)/g int_(v_0)^v (dv)/((v_T – v)(v_T + v))`
	`=(v_T^2)/(2 g v_T) int_(v_0)^v (1/((v_T – v)) + 1/((v_T + v))) dv`
	`=v_T/(2g)[-ln (v_T – v) + ln (v_T + v)]_(v_0)^v`
	`=v_T/(2g)[ln(v_T + v)-ln(v_T – v)-ln(v_T + v_0)+ln(v_T – v_0)]`
	`=v_T/(2g) ln[((v_T + v)(v_T – v_0))/((v_T – v)(v_T + v_0))]`

`text(If)\ \ v_0 > v_T,\ \ text(then)\ \ v > v_T\ \ text(at all times.)`

`text(Replace)\ \ v_T – v\ \ text(by) – (v – v_T)\ \ text(in the preceeding)`

`text(calculation, leading to the same result.)`

iii. `text{From (ii) we have}:\ \ t = v_T/(2g) ln [((v_T + v)(v_T – v_0))/((v_T – v)(v_T + v_0))]`

`text(For Jac,)\ \ v_0 = V_T/3\ \ text(and we have to find the time)`

`text(taken for his speed to be)\ \ v = (2v_T)/3.`

`t`	`=v_T/(2g) ln[((v_T + (2v_T)/3)(v_T – v_T/3))/((v_T – (2v_T)/3)(v_T + v_T/3))]`
	`=v_T/(2g) ln [(((5v_T)/3)((2v_T)/3))/((v_T/3)((4v_T)/3))]`
	`=v_T/(2g) ln[(10/9)/(4/9)]`
	`=v_T/(2g) ln (5/2)`

`text(For Gil),\ \ v_0 = 3v_T\ \ text(and we have to find)`

♦♦♦ Mean mark part (iii) 16%.

`text(the time taken for her speed to be)\ \ v = (3v_T)/2.`

`t`	`=v_T/(2g) ln [((v_T + (3v_T)/2)(v_T – 3v_T))/((v_T – (3v_T)/2)(v_T + 3v_T))]`
	`=v_T/(2g) ln [(((5v_T)/2)(-2v_T))/((-v_T/2)(4v_T))]`
	`=v_T/(2g) ln [(-5)/-2]`
	`=v_T/(2g) ln (5/2)`

`:.\ text(The time taken for Jac’s speed to double is)`

`text(the same as it takes for Gil’s speed to halve.)`

Conics, EXT2 2011 HSC 5c

The diagram shows the ellipse `x^2/a^2 + y^2/b^2 = 1`, where `a > b`. The line `l` is the tangent to the ellipse at the point `P`. The foci of the ellipse are `S` and `S prime`. The perpendicular to `l` through `S` meets `l` at the point `Q`. The lines `SQ` and `S prime P` meet at the point `R`.

Copy or trace the diagram into your writing booklet.

Use the reflection property of the ellipse at `P` to prove that `SQ = RQ.` (2 marks)
Explain why `S prime R = 2a.` (1 mark)
Hence, or otherwise, prove that `Q` lies on the circle `x^2 + y^2 = a^2.` (3 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(See Worked Solutions)`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i)

`text(Let)\ \ l\ \ text(cut the)\ \ y text(-axis at)\ \ M`

`/_ MPS prime`	`= /_ QPS\ \ \ \ \ text{(reflection property of an ellipse)}`
`/_ RPQ`	`=/_ MPS prime\ \ \ \ \ text{(vertically opposite angles)}`

♦♦♦ Mean mark 21%.

`text(In)\ \ Delta SPQ and Delta RPQ`

`/_ SPQ = /_ RPQ\ \ \ text{(both equal}\ \ /_ MPS prime text{)}`

`/_ SQP = /_ RQP=90^@\ \ \ text{(given that}\ \ PQ⊥SRtext{)}`

`PQ\ \ text(is a common side)`

`:.\ Delta SPQ -= Delta RPQ\ \ \ \ text{(AAS)}`

`:. SQ = RQ\ \ \ text{(corresponding sides of congruent triangles)}`

♦ Mean mark part (ii) 46%.

(ii) `SP = PR\ \ \ text{(corresponding sides of congruent triangles)}`

`S prime P+ PS`	`= 2a\ \ \ \ text{(locus property of an ellipse)`
`:.S prime P+PR`	`= 2a`
`:.S prime R`	`= 2a`

(iii) `text(Join)\ \ QO`

♦♦♦ Mean mark 5%.

`text(Consider)\ \ Delta SS prime R and Delta SOQ`

`(SO)/(SS prime) = 1/2\ \ \ \ text{(}O\ \ text(is the midpoint of)\ \ SS prime text{)}`

`(SQ)/(SR) = 1/2\ \ \ \ text{(}Q\ \ text(is the midpoint of)\ \ SR prime text{)}`

`/_ S prime SR = /_ OSQ\ \ text{(common angle)}`

`:.\ Delta SS prime R\ text(|||)\ Delta SOQ`

`\ \ \ text{(AAS)}`

`(OQ)/(S prime R)`	`= 1/2`	`\ \ \ text{(corresponding sides of}`
		`\ \ \ text{similar triangles)}`
`(OQ)/(2a)`	`=1/2`
`:.\ OQ`	`=a`

`:.\ Q\ \ text(lies on the circle)\ \ x^2 + y^2 = a^2`

Harder Ext1 Topics, EXT2 2011 HSC 4c

A mass is attached to a spring and moves in a resistive medium. The motion of the mass satisfies the differential equation

`(d^2y)/(dt^2) + 3 (dy)/(dt) + 2y = 0,`

where `y` is the displacement of the mass at time `t.`

Show that, if `y = f(t)` and `y = g(t)` are both solutions to the differential equation and `A` and `B` are constants, then
1. `y = A f (t) + Bg (t)`
is also a solution. (2 marks)
A solution of the differential equation is given by `y = e^(kt)` for some values of `k`, where `k` is a constant.
Show that the only possible values of `k` are `k = -1` and `k = -2.` (2 marks)
A solution of the differential equation is
1. `y = Ae^(-2t) + Be^-t.`
When `t = 0`, it is given that `y = 0` and `(dy)/(dt) = 1`.
Find the values of `A` and `B.` (3 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`A = -1,\ \ B = 1`

Show Worked Solution

(i) `(d^2y)/(dt^2) + 3(dy)/(dt) + 2y = 0`

♦♦♦ Mean mark 18%.

`text(Given)\ \ y`	`=Af(t) + Bg (t)\ \ text(then)`
`y′`	`=Af′(t)+ Bg′(t)`
`y″`	`=Af″(t)+Bg″(t)`

`text(LHS)`	`=d^2/(dt^2) (Af (t) + Bg (t)) + 3 d/(dt) (Af (t) + Bg (t)) + 2 (Af (t) + Bg(t))`
	`=Af″ (t) + Bg″ (t) + 3 Af prime (t) + 3 Bg prime (t) + 2 Af (t) + 2 Bf (t)`
	`=A (f″(t) + 3 f prime (t) + 2 f (t)) + B(g″ (t) + 3g prime (t) + 2 f(t))`
	`=A(0) + B(0)`
	`=0`

`:.y = Af(t) + Bg(t)\ \ text(is a solution of)\ \ (d^2y)/(dt^2) + 3 (dy)/(dt) + 2y = 0`

(ii) `y = e^(kt),\ \ (dy)/(dt) = ke^(kt),\ \ (d^2y)/(dt^2) = k^2 e^(kt)`

`text(Substituting into)`

`(d^2y)/(dt^2) + 3 (dy)/(dt) + 2y`	`= 0`
`k^2 e^(kt) + 3 ke^(kt) + 2e^(kt)`	`=0`
`e^(kt) (k^2 + 3k + 2)`	`=0`
`k^2 + 3k + 2`	`=0\ \ \ \ (e^(kt) != 0)`
`(k + 1) (k + 2)`	`=0`

`:.k = -1 or -2`

(iii)	`y`	`= Ae^(-2t) + Be^-t`
	`(dy)/(dt)`	`= -2 Ae^(-2t) – Be^-t`

`text(When)\ \ t = 0,\ \ y = 0`

`=>A + B = 0\ \ \ …\ text{(i)}`

`text(When)\ \ t = 0,\ \ (dy)/(dt) = 1`

`=>-2A – B = 1\ \ \ …\ text{(ii)}`

`text{Add (i) + (ii)}`

`-A = 1 `

`:.A = -1 and B=1`

Conics, EXT2 2011 HSC 3d

The equation `x^2/16 - y^2/9 = 1` represents a hyperbola.

Find the eccentricity `e.` (1 mark)
Find the coordinates of the foci. (1 mark)
State the equations of the asymptotes. (1 mark)
Sketch the hyperbola. (1 mark)
For the general hyperbola
`x^2/a^2 - y^2/b^2 = 1`,
describe the effect on the hyperbola as `e -> oo.` (1 mark)

Show Answers Only

`e = 5/4`
`text(Foci) (+-5, 0)`
`y = +- 3/4 x`
`text(The hyperbola approaches the)\ \ y text(-axis from both sides.)`

Show Worked Solution

(i) `x^2/16 – y^2/9 = 1,\ \ \ =>a^2 = 16,\ \ \ b^2 = 9`

`b^2`	`= a^2 (e^2 – 1)`
`9`	`= 16 (e^2 – 1)`
`e^2`	`= 1 + 9/16 `
	`= 25/16`
`:. e`	`= 5/4`

(ii) `S (ae, 0),\ \ S prime (-ae, 0)`

`S(5, 0),\ \ S prime (–5, 0)`

(iii) `y = +- b/a x`

`y = +- 3/4 x`

(iv)

♦♦♦ Mean mark part (v) 24%.

(v) `text(As)\ \ e -> oo,\ \ b -> oo\ \ text(and the asymptotes approach the)`

`y text{-axis from both sides (i.e. the gradients of the asymptotes →∞)}`

` text(Thus the hyperbola approaches the)\ \ y text(-axis from both sides.)`

Proof, EXT2 P1 2012 HSC 16c

Let `n` be an integer where `n > 1`. Integers from `1` to `n` inclusive are selected randomly one by one with repetition being possible. Let `P(k)` be the probability that exactly `k` different integers are selected before one of them is selected for the second time, where `1 ≤ k ≤ n`.

Explain why `P(k) = ((n − 1)!k)/(n^k(n − k)!)`. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
Suppose `P(k) ≥ P(k − 1)`. Show that `k^2- k- n ≤ 0`. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---
Show that if `sqrt(n + 1/4) > k − 1/2` then the integers `n` and `k` satisfy `sqrtn > k − 1/2`. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
Hence show that if `4n + 1` is not a perfect square, then `P(k)` is greatest when `k` is the closest integer to `sqrtn`.

You may use part (iii) and also that `k^2 − k − n >0` if `P(k)< P(k − 1)`. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`

Show Worked Solution

i. `text(Let)\ P(1)=text{(After 1st number chosen, the second draw matches)}`

`P(1) =n/n xx 1/n=1/n`

`text(Let)\ P(2)=text{(After 1st two numbers chosen, the third draw matches)}`

`P(2) = n/n xx (n − 1)/n xx 2/n`

♦♦♦ Mean mark part (i) 7%.

`P(3)=n/n xx (n-1)/n xx (n-2)/n xx 3/n`

`vdots`

`=>text{On the}\ (k+1)text(th draw)`

`P(k)`	`= n/n xx (n − 1)/n xx (n − 2)/n xx …\ xx (n − k+1)/n xx k/n`
	`=((n-1)!)/n^k xx k/(1 xx 2 xx … xx (n-k))`
	`=((n-1)!\ k)/(n^k (n-k)!)`

♦ Mean mark part (ii) 38%.

ii.	`P(k)`	`≥ P(k − 1)`
	`((n − 1)!\ k)/(n^k(n − k)!)`	`≥ ((n − 1)! (k − 1))/(n^(k − 1)(n − k + 1)!)`
	`k(n − k + 1)`	`≥ n(k − 1)`
	`kn − k^2 + k`	`≥ nk − n`
	`:.k^2 − k − n`	`≤ 0`

♦♦♦ Mean mark part (iii) 10%.

iii.	`sqrt(n + 1/4)`	`> k − 1/2`
	`n + 1/4`	`> (k − 1/2)^2`
	`n + 1/4`	`> k^2 − k + 1/4`
	`n`	`>k^2 − k`
	`n`	`> k(k − 1)`

`text(S)text(ince)\ n\ text(and)\ k\ text(are integers such that)\ 1 ≤ k ≤ n.`

`=>n\ text(is at least the next integer after)\ k.`

`=>(k −1)\ text(is the integer before)\ k.`

`:.n`	`>k^2 − k +1/4`
`n`	`>(k-1/2)^2`
`:.sqrt n`	`>k-1/2`

iv. `text(Find the largest integer)\ \ k\ \ text(for which)\ \ P(k) ≥ P(k − 1)`

`P(k) ≥ P(k − 1)\ \ text(when)\ \ k^2 − k − n ≤ 0\ \ \ text{(part (ii))}`

`text(Solving)\ \ k^2 − k − n ≤ 0`

♦♦♦ Mean mark part (iii) 1%.

`(1 – sqrt(4n + 1))/2 ≤ k ≤ (1 + sqrt(4n+1))/2`

`text(Given)\ \ n>0,\ \ (1 – sqrt(4n + 1))/2<0`

`:.1 ≤ k ≤ (1 + sqrt(4n+1))/2`

`text(S)text(ince)\ \ 4n+1\ \ text(is not a perfect square and)\ \ k\ \ text(is an integer)`

`k<`	` (1 + sqrt(1 + 4n))/2`
`k<`	` 1/2 + sqrt(n + 1/4)`
`k-1/2<`	`sqrt(n + 1/4)`
`k-1/2<`	`sqrt n\ \ \ \ \ text{(from part (iii))}`
`k<`	`1/2+sqrt n`

`:. P(k)\ text(is greatest when)\ \ k\ \ text(is the closest integer to)\ n.`

Harder Ext1 Topics, EXT2 2012 HSC 16a

In how many ways can `m` identical yellow discs and `n` identical black discs be arranged in a row? (1 mark)
In how many ways can 10 identical coins be allocated to 4 different boxes? (1 mark)

Show Answers Only

`((m + n)!)/(m!n!)`
`(13!)/(10!3!)`

Show Worked Solution

(i) `((m + n)!)/(m!n!)\ text{ways (By definition)}`

♦ Mean mark part (i) 43%.

(ii) `text{Consider this as being “arrange 10 coins in 4 boxes}`

♦♦♦ Mean mark part (ii) 1%!

`text{with 3 separators between the boxes, making a total}`

`text{of 13 items” (as per the diagram).}`

`:.\ text(10 identical coins and 3 identical separators to arrange.)`

`:.\ text(Number of ways) = (13!)/(10!3!)\ \ \ \ text{(from part (i))}`

Polynomials, EXT2 2012 HSC 15b

Let `P(z) = z^4 − 2kz^3 + 2k^2z^2 − 2kz + 1`, where `k` is real.

Let `α = x + iy`, where `x` and `y` are real.

Suppose that `α` and `iα` are zeros of `P(z)`, where `bar α ≠ iα`.

Explain why `bar α` and `-i bar α` are zeros of `P(z)`. (1 mark)
Show that `P(z) = z^2(z − k)^2 + (kz − 1)^2`. (1 mark)
Hence show that if `P(z)` has a real zero then
1. `P(z) = (z^2 + 1)(z+ 1)^2` or `P(z) = (z^2 + 1)(z − 1)^2.` (2 marks)
Show that all zeros of `P(z)` have modulus `1`. (2 marks)
Show that `k = x − y`. (1 mark)
Hence show that `-sqrt2 ≤ k ≤ sqrt2`. (2 marks)

Show Answers Only

`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`

Show Worked Solution

(i) `P(z) = z^4 − 2kz^3 + 2k^2z^2 − 2kz +1`

`P(α) = P(iα) = 0, \ \ bar α ≠ iα.`

`text(S)text(ince)\ \ P(z)\ text(has real coefficients,)`

`=>\ text(Its zeros occur in conjugate pairs.)`

`text(i.e.)\ \ P(α) = 0\ text(and)\ P(bar α) = 0`

`bar α`	`= x − iy`
`bar(i α)`	`=bar (i(x+iy))`
	`=bar (ix-y)`
	`=-y-ix`
	`=-i(x-iy)`
	`=-i barα`

`:. -i bar α\ text(is a zero of)\ P(z)\ text(as it is the conjugate of)\ iα.`

(ii)	`H(z)`	`= z^2(z − k)^2 + (kz − 1)^2`
		`= z^2 (z^2 − 2kz + k^2) + (k^2z^2 − 2kz + 1)`
		`= z^4 − 2kz^3 + k^2z^2 + k^2z^2 − 2kz +1`
		`= z^4 − 2kz^3 + 2k^2z^2 − 2kz + 1`
		`= P(z)`

(iii) `text(If)\ z\ text(is real then)\ z^2(z − k)^2\ text(and)\ (kz −1)^2\ text(are real and)`

`text(either positive or zero.)`

♦♦♦ Mean mark part (iii) 11%.

`:.text(If)\ \ P(z) = z^2(z − k)^2 + (kz − 1)^2 = 0`

`=>(z − k)^2 = 0\ text(and)\ (kz − 1)^2 = 0`

`text(When)\ \ z = k,\ \ \ k^2 − 1 = 0\ \ text(or)\ k = ±1.`

`text(If)\ k = 1`

`P(z)`	`= z^2(z − 1)^2 + (z − 1)^2`
	`= (z^2 + 1)(z − 1)^2.`

`text(If)\ k = -1`

`P(z)`	`= z^2(z + 1)^2 + (-z − 1)^2`
	`= (z^2 + 1)(z + 1)^2`

(iv) `text(Product of the roots) = e/a=1`

♦♦♦ Mean mark part (iv) 20%.

`:. α bar α iα *(-i barα)`	`=1`
`(α bar α)^2`	`=1`
`(\|α\|)^4`	`=1`
`\|α\|`	`=1`

`:.\ text(All zeros have modulus 1.)`

(v) `text(Sum of zeroes)\ = -b/a=-(-2k)/1 = 2k`

♦♦♦ Mean mark part (v) 20%.

`:. 2k`	`=α + bar α + iα + (-i bar α )`
	`=x + iy + x − iy + (-y + ix) − i(x − iy)`
	`=2x − y + ix − ix − y`
	`=2x − 2y`
`:. k`	`=x-y`

(vi) `text(S)text(ince)\ \ |α| = 1\ \ \ text{(part (iv))}`

♦♦♦ Mean mark part (vi) 2%.

`=>x^2 + y^2`	`=1`
`text(Substitute)\ \ y=x-k\ \ \ text{(part (v))}`
`x^2 + (x − k)^2`	`=1`
`2x^2 − 2kx + k^2 − 1`	`=0`

`text(For a real solution to exist), Δ ≥ 0`

`4k^2 − 8(k^2 − 1) `	`≥ 0`
`-4k^2 + 8`	`≥ 0`
`k^2`	`≥ 2`

`:. -sqrt2 ≤ k ≤ sqrt2`

Proof, EXT2 P1 2012 HSC 15a

Prove that `sqrt(ab) ≤ (a + b)/2`, where `a ≥ 0` and `b ≥ 0`. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
If `1 ≤ x ≤ y`, show that `x(y − x + 1) ≥ y`. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---
Let `n` and `j` be positive integers with `1 ≤ j ≤ n`.

Prove that `sqrtn ≤ sqrt(j(n − j + 1)) ≤ (n + 1)/2.` (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
For integers `n ≥ 1`, prove that

`(sqrtn)^n ≤ n! ≤ ((n + 1)/2)^n`. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`

Show Worked Solution

i.	`(sqrta − sqrtb)^2`	`≥ 0`
	`a − 2sqrt(ab) + b`	`≥ 0`
	`a + b`	`≥ 2sqrt(ab)`
	`sqrt(ab)`	`≤ (a + b)/2`

ii. `text(Solution 1)`

♦♦ Mean mark part (ii) 28%.

`text(S)text(ince)\ \ 1 ≤ x ≤ y`

`y-x`	`>=0`
`y(x-1)-x(x-1)`	`>=0,\ \ \ \ (x-1>=0)`
`xy-x^2+x-y`	`>=0`
`:.xy-x^2+x`	`>=y`

`text(Solution 2)`

`x( y − x + 1)`	`= xy − x^2 + x`
	`= -y + xy − x^2 + x + y`
	`= y(x − 1) − x(x − 1)+ y`
	`= (x − 1)( y − x) + y`

`text(S)text(ince)\ \ x − 1 ≥ 0\ \ text(and)\ \ y − x ≥ 0`

`=>(x − 1)( y − x) + y`	`>=y`
`:.x(y − x + 1)`	`>=y`

iii. `text(Let)\ c = n − j + 1, \ c > 0\ text(as)\ n ≥ j.`

♦ Mean mark part (iii) 39%.

`=> sqrt(j(n − j + 1))\ \ text(can be written as)\ \ sqrt(jc).`

`sqrt(jc)`	`≤ (j + c)/2\ \ \ \ text{(part (i))}`
`sqrt(j(n − j + 1))`	`≤ (j + n-j+1)/2`
`=>sqrt(j(n − j + 1))`	`≤ (n+1)/2`
`j(n − j +1)`	`≥ n\ \ \ \ text{(part (ii))}`
`=>sqrt(j(n − j + 1))`	`≥ sqrt n`

`:.sqrtn ≤ sqrt(j(n − j + 1)) ≤ (n + 1)/2`

iv. `text{From (iii)}\ n ≤ j(n− j +1) ≤ (n + 1)/2`

♦♦♦ Mean mark part (iv) just 2%!

`text(Let)\ j\ text(take on the values from 1 to)\ n.`

`j = 1:`	`sqrtn ≤ sqrt(1(n)) ≤ (n + 1)/2`
`j = 2:`	`sqrtn ≤ sqrt(2(n−1)) ≤ (n + 1)/2`
	`vdots`
`j = n:`	`sqrtn ≤ sqrt(n(1)) ≤ (n + 1)/2`

`text{Multiply the corresponding parts of each line}`

`(sqrtn)^n ≤ sqrt(n! xx n!) ≤ ((n + 1)/2)^n`

`(sqrtn)^n ≤ n! ≤ ((n + 1)/2)^n`

Volumes, EXT2 2012 HSC 14c

The solid `ABCD` is cut from a quarter cylinder of radius `r` as shown. Its base is an isosceles triangle `ABC` with `AB = AC`. The length of `BC` is `a` and the midpoint of `BC` is `X`.

The cross-sections perpendicular to `AX` are rectangles. A typical cross-section is shown shaded in the diagram.

Find the volume of the solid `ABCD`. (4 marks)

Show Answers Only

`(ar^2)/3\ \ text(u³)`

Show Worked Solution

`text(Need to find the width of the rectangular cross-section)`

`text(Using similar triangles)`

`(AF)/(AX)`	`= (DF)/(BX)`
`h/r`	`= (DF)/(a/2)`
`DF`	`= (ah)/(2r)`
`:.DE`	`= (ah)/r`

`text(Need to find the height of the cross-section)`

`JF`	`=sqrt((AJ)^2-(AF)^2)`
	` = sqrt(r^2 − h^2)`

`=>text(Rectangle has base length)\ (ah)/r\ text(and height)\ sqrt(r^2 − h^2)`

`text(Volume of solid)`	`=lim_(δh→0) ∑_(h=0)^r (ah)/r sqrt(r^2 − h^2)\ δh`
	`= int_0^r (ah)/r sqrt(r^2 − h^2)\ dh`
	`= a/r int_0^r h sqrt(r^2 − h^2)\ dh`

`text(Let)\ \ u^2`	`=r^2-h^2`
`2u\ du`	`=-2h\ dh`
`-u\ du`	`=h\ dh`

`text(If)\ \ h=r,\ \ u^2=0,\ \ u=0`

`text(If)\ \ h=0,\ \ u^2=r^2,\ \ u=r\ \ (r>0)`

`:.\ text(Volume of solid)`	`=a/r int_r^0 u xx -u\ du`
	`=-a/r [u^3/3]_r^0`
	`=-a/r((-r^3)/3)`
	`=(ar^2)/3\ \ text(u³)`

Mechanics, EXT2 2013 HSC 16b

A small bead `P` of mass `m` can freely move along a string. The ends of the string are attached to fixed points `S` and `S′`, where `S′` lies vertically above `S`. The bead undergoes uniform circular motion with radius `r` and constant angular velocity `omega` in a horizontal plane.

The forces acting on the bead are the gravitational force and the tension forces along the string. The tension forces along `PS` and `PS′` have the same magnitude `T`.

The length of the string is `2a` and `SS′= 2ae`, where `0 < e < 1`. The horizontal plane through `P` meets `SS′` at `Q`. The midpoint of `SS′` is `O` and `beta = /_S′PQ`. The parameter `theta` is chosen so that `OQ =a cos theta.`

What information indicates that `P` lies on an ellipse with foci `S` and `S′`, and with eccentricity `e`? (1 mark)
Using the focus–directrix definition of an ellipse, or otherwise, show that
`SP = a(1 − e cos theta ).` (1 mark)
Show that
`sin beta = (e + cos theta)/(1 + e cos theta).` (2 marks)
By considering the forces acting on `P` in the vertical direction, show that
1. `mg = (2T(1 - e^2) cos theta)/(1 - e^2 cos^2 theta).` (2 marks)
Show that the force acting on `P` in the horizontal direction is
1. `mr omega^2 = (2T sqrt(1 - e^2) sin theta)/(1 - e^2 cos^2 theta).` (3 marks)
Show that
1. `tan theta = (r omega^2)/g sqrt(1 - e^2).` (1 mark)

Show Answers Only

`text(See Worked Solutions)`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i) `SP + S′P = 2a and SS prime = 2ae,\ \ 0 < e < 1`

♦♦ Mean mark 22%. 

`text(This is the condition for an ellipse with foci)`

`text(at)\ \S and S′ text(and eccentricity)\ e.`

♦ Mean mark 40%.

(ii)	`SP`	`= ePM\ \ \ \ \ text{(where}\ M\ text{is on the directrix)}`
		`=e(OM-OQ)`
		`= e (a/e – a cos theta)`
		`= a – a e cos theta`
		`= a(1 – e cos theta)`

♦ Mean mark 39%. 

(iii) `S prime P`	`= 2a – (a – a e cos theta)`
	`= a + a e cos theta`

`sin beta`	`= (QS prime)/(S prime P)`
	`= (ae + a cos theta)/(a + a e cos theta)`
	`= (e + cos theta)/(1 + e cos theta)`

♦♦ Mean mark 43%. 

(iv)

`T sin beta = T ((e + cos theta)/(1 + e cos theta))`

`sin /_ QPS`	`= (QS)/(SP)`
	`= (ae – a cos theta)/(a (1 – e cos theta)`
	`= (e – cos theta)/(1 – e cos theta)`

`text(Resolving the forces vertically)`

`mg`	`= T sin beta – T sin /_ QPS`
`mg`	`= T ((e + cos theta)/(1 + e cos theta) – (e – cos theta)/(1 – e cos theta))`
`mg`	`= T ((e-e^2 cos theta + cos theta – e cos^2 theta-(e + e^2 cos theta – cos theta – e cos^2 theta))/(1 – e^2 cos^2 theta))`
`mg`	`= T((-2e^2 cos theta + 2 cos theta)/(1 – e^2 cos^2 theta))`
`mg`	`= (2T(1 – e^2) cos theta)/(1 – e^2 cos^2 theta)`

(v) `text(Resolving the forces horizontally)`

♦♦♦ Mean mark 8%. 

`mr omega^2= T cos beta + T cos /_ QPS`

`cos beta`	`= r/(S prime P)`
	`= r/(2a – SP)`
	`= r/(2a – (a – a e cos theta))`
	`= r/(a (1 + e cos theta))`

`cos /_ QPS`	`= r/(SP)`
	`= r/(a (1 – e cos theta))`

`r`	`= sqrt (SP^2 – SQ^2)`
	`= sqrt (a^2 (1 – e cos theta)^2 – a^2 (e – cos theta)^2)`
	`= a sqrt (1 – 2e cos theta + e^2 cos ^2 theta – (e^2 – 2 e cos theta + cos^2 theta))`
	`= a sqrt (1 – e^2 – (1 – e^2) cos^2 theta)`
	`= a sqrt ((1 – e^2) (1 – cos^2 theta))`
	`= a sin theta sqrt (1 – e^2)`

`:. mr omega^2`	`= T ((a sin theta sqrt(1 – e^2))/(a(1 + e cos theta)) + (a sin theta sqrt(1 – e^2))/(a(1 – e cos theta)))`
	`= T sin theta sqrt (1 – e^2) ((1 – e cos theta + 1 + e cos theta)/(1 – e^2 cos^2 theta))`
	`= (2T sqrt (1 – e^2) sin theta)/(1 – e^2 cos^2 theta)`

♦ Mean mark 38%.

(vi) `cos theta`	`= (mg (1 – e^2 cos^2 theta))/(2T (1 – e^2))`
`sin theta`	`= (mr omega^2(1 – e^2 cos^2 theta))/(2T sqrt (1 – e^2))`
`tan theta`	`= (mr omega^2(1 – e^2 cos^2 theta))/(2T sqrt(1 – e^2)) xx (2T(1 – e^2))/(mg(1 – e^2 cos^2 theta))`
	`= (r omega^2 sqrt (1 – e^2))/g`

Proof, EXT2 P1 2013 HSC 16a

Find the minimum value of `P(x) = 2x^3 - 15x^2 + 24x + 16`, for `x >= 0.` (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
Hence, or otherwise, show that for `x >= 0`,

`(x + 1) (x^2 + (x + 4)^2) >= 25x^2.` (1 mark)

--- 5 WORK AREA LINES (style=lined) ---
Hence, or otherwise, show that for `m >= 0` and `n >= 0`,

`(m + n)^2 + (m + n + 4)^2 >= (100mn)/(m + n + 1).` (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`0`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

♦ Mean mark 43%. 
STRATEGY: The best responses carefully considered the domain restrictions, notably `P(0).`

i.	`P(x)`	`= 2x^3 – 15x^2 + 24x + 16,\ \ \ x >= 0`
	`P prime(x)`	`= 6x^2 – 30x +24`
		`= 6 (x^2 – 5x + 4)`
		`= 6 (x – 1) (x – 4)`
	`P″(x)`	` = 12x – 30`

`text(MAX or MIN when)\ \ P prime (x)=0`

`text(i.e. when)\ \ x=1 or 4`

`P″(1) = -6 < 0\ \ \ \ P″(4) = 18 > 0`

`:.text(Minimum turning point of)\ \ P(x)\ \ text(at)\ \ x = 4,`

`P(4) = 128 – 240 + 96 + 16 = 0`

`text(Checking limits:)`

`P(0) = 16`

`text(As)\ \ x->oo,\ \ y->oo`

`:.text(Minimum value of)\ \ P(x)\ \ text(is)\ \ 0\ \ text(when)\ \ x = 4.`

♦ Mean mark 47%. 

ii. `text(LHS)`	`= (x + 1) (x^2 + (x + 4)^2)`
	`= (x + 1) (2x^2 + 8x + 16)`
	`= 2x^3 + 8x^2 + 16x + 2x^2 + 8x + 16`
	`= 2x^3 + 10x^2 + 24x + 16`
	`= (2x^3 – 15x^2 + 24x + 16) + 25x^2`
	`= P(x) + 25x^2`

`text(The minimum value of)\ \ P(x)\ \ text(for)\ \ x>= 0\ \ text(is)\ \ 0,`

`=>P(x) + 25x^2`	`>= 25x^2`
`:.(x + 1) (x^2 + (x + 4)^2)`	`>= 25x^2,\ \ \ text(for)\ x >= 0`

♦♦ Mean mark 19%. 
STRATEGY: The best responses carefully considered the domain restrictions, notably `P(0).`

iii.	`(m + n)^2 + (m + n + 4)^2`
	`= (m + n)^2 + (m + n)^2 + 8(m + n) + 16`
	`= 2 (m + n)^2 + 8 (m + n) + 16`

`text(Let)\ \ x = m + n`

`(m + n)^2 + (m + n + 4)^2 = 2x^2 + 8x + 16`

`text{Using part (ii)}`

`(x + 1) (2x^2 + 8x + 16)`	`>= 25x^2`
`2x^2 + 8x + 16`	`>= (25x^2)/(x + 1),\ \ \ x >= 0`
`(m + n)^2 + (m + n + 4)^2`	`>= (25 (m + n)^2)/(m + n + 1)`

`text(Consider)\ \ (m – n)^2`

`text(S)text(ince)\ \ (m – n)^2`	`>= 0`
`m^2 + n^2`	`>= 2mn`
`(m+n)^2-2mn`	`>= 2mn`
`(m + n)^2`	`>= 4mn`
`:.(m + n)^2 + (m + n + 4)^2`	`>= (100mn)/(m + n + 1)`

Mechanics, EXT2 M1 2013 HSC 15d

A ball of mass `m` is projected vertically into the air from the ground with initial velocity `u`. After reaching the maximum height `H` it falls back to the ground. While in the air, the ball experiences a resistive force `kv^2`, where `v` is the velocity of the ball and `k` is a constant.

The equation of motion when the ball falls can be written as

`m dot v = mg-kv^2.` (Do NOT prove this.)

Show that the terminal velocity `v_T` of the ball when it falls is
`sqrt ((mg)/k).` (1 mark)

--- 4 WORK AREA LINES (style=lined) ---
Show that when the ball goes up, the maximum height `H` is
`H = (v_T^2)/(2g) ln (1 + u^2/(v_T^2)).` (3 marks)

--- 8 WORK AREA LINES (style=lined) ---
When the ball falls from height `H` it hits the ground with velocity `w`.
Show that `1/w^2 = 1/u^2 + 1/(v_T^2).` (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

i. `m dot v = mg-kv^2`

`t = 0,\ \ \ v = 0,\ \ \ x = 0\ \ \ text(Ball falling)`

`text{For terminal velocity}\ \(v_T),\ \ \ dot v = 0`

`v_T^2`	`= (mg)/k`
`:.v_T`	`= sqrt ((mg)/k)`

ii. `text(When the ball rises),\ \ m dot v = -mg-kv^2`

♦ Mean mark 43%.
 MARKER’S COMMENT: More than half of students incorrectly wrote the equation to solve as `m dot v=mg-kv^2!`

`text(Using)\ \ dot v`	`= v (dv)/(dx)`
`mv (dv)/(dx)`	`= -mg-kv^2`
`dx`	`=(-mv)/(mg + kv^2) dv`

`int_0^H dx`	`= -int_u^0 (mv)/(mg + kv^2) dv`
`[x]_0^H`	`= -m/(2k) [log_e (mg + kv^2)]_u^0`
`H`	`= -m/(2k) (log_e (mg)-log_e (mg + ku^2))`
	`= m/(2k) log_e ((mg + ku^2)/(mg))`
	`= m/(2k) log_e (1 + (ku^2)/(mg))\ \ \ \ text{(using}\ \ k = (mg)/v_T^2 text{)}`
`:.H`	`=(v_T^2)/(2g) log_e (1 + u^2/(v_T^2))`

iii. `text(When the ball falls),\ \ m dot v = mg-kv^2`

♦♦ Mean mark 14%.

`mv (dv)/(dx)`	`= mg-kv^2`
`dx`	`=(mv)/(mg-kv^2) dv`

`int_0^H dx`	`= int_0^w (mv)/(mg-kv^2)dv`
`[x]_0^H`	` =-m/(2k)[log_e (mg-kv^2)]_0^w`
`H`	`= -m/(2k) (log_e (mg-kw^2)-log_e (mg))`
	`= -m/(2k) log_e ((mg-kw^2)/(mg))`
	`= -m/(2k) log_e (1-(kw^2)/(mg))\ \ \ \ text{(using}\ \ k = (mg)/v_T^2 text{)}`
`H`	`=-(v_T^2)/(2g) log_e (1-w^2/(v_T^2))`
	`=-(v_T^2)/(2g) log_e ((v_T^2-w^2)/(v_T^2))`
	`=(v_T^2)/(2g) log_e ((v_T^2)/(v_T^2-w^2))`

`text{Using part (ii):}`

`(v_T^2)/(2g) log_e ((v_T^2)/(v_T^2-w^2))`	`=(v_T^2)/(2g) log_e (1 + u^2/(v_T^2))`
`(v_T^2)/(v_T^2-w^2)`	`=(v_T^2 + u^2)/(v_T^2)`
`(v_T^2 + u^2)(v_T^2-w^2)`	`=v_T^4`
`v_T^4-v_T^2 w^2+v_T^2 u^2-w^2 u^2`	`=v_T^4`
`v_T^2 w^2+w^2 u^2`	`=v_T^2 u^2`
`(v_T^2 w^2)/(v_T^2w^2u^2)+(w^2 u^2)/(v_T^2w^2u^2)`	`=(v_T^2 u^2)/(v_T^2w^2u^2)`
`:.1/u^2 + 1/(v_T^2)`	`=1/w^2`

Harder Ext1 Topics, EXT2 2013 HSC 14d

A triangle has vertices `A, B` and `C`. The point `D` lies on the interval `AB` such that `AD = 3` and `DB = 5`. The point `E` lies on the interval `AC` such that `AE = 4`, `DE = 3` and `EC = 2`.

Prove that `Delta ABC` and `Delta AED` are similar. (1 mark)
Prove that `BCED` is a cyclic quadrilateral. (1 mark)
Show that `CD = sqrt 21`. (2 marks)
Find the exact value of the radius of the circle passing through the points `B, C, E and D`. (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`(3 sqrt 105)/10`

Show Worked Solution

(i) `text(In)\ \ Delta ABC and Delta AED`

`/_ BAC = /_ EAD\ \ \ \ text{(common angle)}`

`(AB)/(AE)`	`= 8/4 = 2/1`
`(AC)/(AD)`	`= 6/3 = 2/1`
`(AB)/(AE)`	`= (AC)/(AD) = 2/1`

`:.\ Delta ABC\ \ text(\|\|\|)\ \ Delta AED`	`\ \ \ \ \ text{(sides about equal angles}`
	`\ \ \ \ \ text{are in the same ratio)}`

(ii) `/_ ABC = /_ AED\ \ \ \ \ text{(corresponding angles of similar triangles)}`

`/_ AED\ \ text(is an exterior angle of quadrilateral)\ \ BCED`

`:.\ BCED\ \ text(is a cyclic quadrilateral as an exterior)`

`text(angle equals the interior opposite angle.)`

(iii) `cos /_ AED = (3^2 + 4^2 – 3^2)/(2 xx 3 xx 4) = 2/3`

♦♦ Mean mark 33%. 

`cos /_ CED`	`= cos(pi – /_AED)`
	`=-cos/_AED`
	`=-2/3`

`text(In)\ \ Delta CDE`

`CD^2`	`= 2^2 + 3^2 – 2 xx 2 xx 3 xx (-2/3)`
	`= 13 + 8`
	`=21`
`:.CD`	`= sqrt 21`

(iv)

`text(Mark the centre)\ \ O,\ \ text(draw the radii)\ \ OC, OD`

`text(Let)\ \ /_ CBD`	`= alpha`
`:. /_ COD`	`= 2 alpha`	` \ \ \ \ text{(angles at circumference and}`
		`\ \ \ \ text{centre on arc}\ \ CD text{)}`
`/_ DEC`	`= pi – alpha`	` \ \ \ \ text{(opposite angles of cyclic}`
		`\ \ \ \ text{quadrilateral}\ \ BCED text{)}`

♦♦♦ Mean mark 7%. 
COMMENT: This part had the lowest mean mark in the entire 2013 exam.

`cos alpha = cos /_ AED = 2/3\ \ \ \ text{(part (iii))}`

`text(Let)\ \ r= text(circle radius)`

`text(In)\ \ Delta DOC`

`CD^2`	`=r^2 + r^2 – 2r xx r xx cos 2 alpha`
`21`	`=2r^2 – 2r^2 (2 cos^2 alpha – 1)`
	`=2r^2 – 2r^2 (8/9 – 1)`
	`=2r^2 + (2r^2)/9`
	`=(20r^2)/9`
`r^2`	`=(9 xx 21)/20`
`:.r`	`=(3 sqrt 21)/(2 sqrt 5)`
	`=(3 sqrt 105)/10`

Proof, EXT2 P1 2014 HSC 16b

Suppose `n` is a positive integer.

Show that

`-x^(2n) ≤ 1/(1 + x^2) − (1 − x^2 + x^4 − x^6 + … + (-1)^(n − 1) x^(2n − 2)) ≤ x^(2n)`. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---
Use integration to deduce that

`-1/(2n + 1) ≤ pi/4 − (1 − 1/3 + 1/5 − … + (-1)^(n − 1) 1/(2n − 1)) ≤ 1/(2n + 1)`. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
Explain why `pi/4 = 1 − 1/3 + 1/5 − 1/7 + …`. (1 mark)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`

Show Worked Solution

i. `text(Let)\ \ S_n=1 − x^2 + x^4 − x^6 + … + (-1)^(n − 1) x^(2n − 2)`

`=>text(GP where)\ \ a=1, r=-x^2`

♦♦ Mean mark 12%.
STRATEGY: Applying the GP formula and simplifying the middle term is worth 2 full marks in this question.

`:.S_n`	`=(1(1-(-x^2)^n))/(1-(-x^2))`
	`= (1 − (-x^2)^n)/(1 + x^2)`

`:.1/(1 + x^2) − (1 − x^2 + x^4 − x^6 + … + (-1)^(n − 1) x^(2n − 2))`

`=1/(1 + x^2)-(1 − (-x^2)^n)/(1 + x^2)`

`=((-x^2)^n)/(1 + x^2)`

`=((-1)^nx^(2n))/(1 + x^2)`

`text(S)text(ince)\ \ (1 + x^2)>=1,\ \ \ -x^(2n) ≤ ((-1)^nx^(2n))/(1 + x^2) ≤ x^(2n)`

`:.\ text(We can conclude)`

`-x^(2n) ≤ 1/(1 + x^2) − (1 − x^2 + x^4 − x^6 + … + (-1)^(n − 1) x^(2n − 2))\ ≤ x^(2n)`

ii. `text{Integrating part (i) between 0 and 1}`

♦ Mean mark 43%.

`int_0^1 -x^(2n)\ dx`	`=(-1)/(2n + 1)[x^(2n + 1)]_0^1`
	`=(-1)/(2n + 1)`

`int_0^1 1/(1 + x^2)`	`=[tan^(-1) x]_0^1`
	`=pi/4`

`int_0^1 (1 − x^2 + x^4 − x^6 + … + (-1)^(n − 1) x^(2n − 2))\ dx`

`=[x − (x^3)/3 + (x^5)/5 −… + ((-1)^(n − 1)x^(2n − 1))/(2n − 1)]_0^1`

`=1 − 1/3 + 1/5 − … + ((-1)^(n − 1))/(2n − 1)`

`int_0^1 x^(2n)\ dx`	`=1/(2n + 1)[x^(2n + 1)]_0^1`
	`=1/(2n + 1)`

`:.\ text(We can conclude)`

`(-1)/(2n + 1) ≤ pi/4 − (1 − 1/3 + 1/5 − … + (-1)^(n − 1) 1/(2n − 1)) ≤ 1/(2n + 1)`

iii. `(-1)/(2n + 1) ≤ pi/4 − (1 − 1/3 + 1/5 − … + ((-1)^(n − 1))/(2n − 1)) ≤ 1/(2n + 1)`

`text(As)\ n → ∞,`

`=>(-1)/(2n + 1) → 0^-\ \ text(and)\ \ 1/(2n + 1) → 0^+`

`=> pi/4 − (1 − 1/3 + 1/5 − … + 1/(2n − 1)) → 0`

♦♦ Mean mark 28%.

`:. pi/4 = 1 -1/3 + 1/5 − 1/7 + …`

Mechanics, EXT2 2014 HSC 15c

A toy aeroplane `P` of mass `m` is attached to a fixed point `O` by a string of length `l`. The string makes an angle `ø` with the horizontal. The aeroplane moves in uniform circular motion with velocity `v` in a circle of radius `r` in a horizontal plane.

The forces acting on the aeroplane are the gravitational force `mg`, the tension force `T` in the string and a vertical lifting force `kv^2`, where `k` is a positive constant.

By resolving the forces on the aeroplane in the horizontal and the vertical directions, show that
`(sin\ ø)/(cos^2\ ø) = (lk)/m − (lg)/(v^2)`. (3 marks)
Part (i) implies that
`(sin\ ø)/(cos^2\ ø) < (lk)/m`. (Do NOT prove this.)
Use this to show that
1. `sin\ ø < (sqrt(m^2 + 4l^2k^2) − m)/(2lk).` (2 marks)
Show that
`(sin\ ø)/(cos^2\ ø)` is an increasing function of `ø` for `-pi/2 < ø < pi/2`. (2 marks)
Explain why `ø` increases as `v` increases. (1 mark)

Show Answers Only

`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`

Show Worked Solution

(i) `text(Vertically)`

`kv^2 = mg + T\ sin\ ø\ \ \ \ \ …\ (1)`

`text(Horizontally)`

`T cos\ ø`	`=(mv^2)/r`
	`=(mv^2)/(l cos\ ø)\ \ \ \ \ \ (r=lcos\ ø)`
`T`	`=(mv^2)/(l cos^2\ ø)\ \ \ \ …\ (2)`

`text{Substitute (2) into (1)}`

`mg + (mv^2)/(l cos^2\ ø)\ sin\ ø`	`=kv^2`
`lmg + (mv^2)/(cos^2\ ø)\ sin\ ø`	`=lkv^2`
`(mv^2)/(cos^2\ ø)\ sin\ ø`	`=lkv^2 – lmg`
`(sin\ ø)/(cos^2\ ø)`	`=(lk)/m − (lg)/(v^2)\ \ \ text(… as required)`

♦♦♦ Mean mark 14%.
STRATEGY: The form of the required proof strongly hints that the quadratic formula is relevant – a clue that was ignored by many students.

(ii)	`(sin\ ø)/(cos^2\ ø)`	`< (lk)/m`
	`sin\ ø`	`< (lk)/m(1 − sin^2\ ø)`
	`m sin\ ø`	`< lk – lksin^2\ ø`
	`lk\ sin^2\ ø + m\ sin\ ø − lk< 0`

`text(Using the quadratic formula)`

`sin\ ø = (-m ± sqrt(m^2 + 4l^2k^2))/(2lk)`

`text(S)text(ince)\ \ 0<ø<pi/2\ \ \ \ =>\ 0 < sin\ ø < 1`

`:. sin\ ø = (sqrt(m^2 + 4l^2k^2)-m)/(2lk)`

`=> lk\ sin^2\ ø + m\ sin\ ø − lk< 0\ \ text(is true)`

`text(when)\ \ sin\ ø = 0`

`=>sqrt(m^2 + 4l^2k^2)>m`

`:.sin\ ø < (sqrt(m^2 + 4l^2k^2) − m)/(2lk)\ \ \ \ text(… as required)`

(iii)	`f(x)`	`= (sin\ ø)/(cos^2\ ø)`
	`f′(x)`	`= (cos\ ø\ cos^2\ ø − sin\ ø xx 2\ cos\ ø(-sin\ ø))/(cos^4\ ø)`
		`= (cos^2\ ø + 2sin^2\ ø)/(cos^3\ ø)`

♦♦ Mean mark 25%.

`text(Consider)\ \ -pi/2 < ø < pi/2`

`cos\ ø>0, \ \ cos^3\ ø > 0, \ \ 2sin^2\ ø>=0`

`=> f′(x) > 0`

`:. f(x)=(sin\ ø)/(cos^2\ ø)\ text(is an increasing function for)\ \ \ -pi/2 < ø < pi/2.`

(iv) `text(Consider)\ \ (sin\ ø)/(cos^2\ ø) = (lk)/m − (lg)/(v^2)`

♦♦♦ Mean mark 8%.
COMMENT: This question had the lowest mean mark in the entire 2014 exam.

`text(As)\ v\ text(increases)`	`=>(lg)/(v^2)\ text(decreases)`
	`=>(lk)/m – (lg)/(v^2)\ text(increases)`
	`=>(sin\ ø)/(cos^2\ ø)\ text(increases.)`

`text{From (iii)},\ (sin\ ø)/(cos^2\ ø)\ text(is an increasing function as)\ ø\ text(increases.)`

`:.ø\ text(increases as)\ \ v\ \ text(increases.)`

Complex Numbers, EXT2 N2 2014 HSC 15b

Using de Moivre’s theorem, or otherwise, show that for every positive integer `n`,

`(1 + i)^n + (1 − i)^n = 2(sqrt2)^n\ cos\ (npi)/4`. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
Hence, or otherwise, show that for every positive integer `n` divisible by 4,

`((n),(0)) − ((n),(2)) + ((n),(4)) − ((n),(6)) + … + ((n),(n)) = (-1)^(n/4)(sqrt2)^n` (3 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`

Show Worked Solution

i.	`1+i`	`=sqrt2(1/sqrt2 + 1/sqrt2 i)`
		`=sqrt2(cos\ pi/4 + i sin\ pi/4)`
	`(1+i)^n`	`=(sqrt2)^n (cos\ (n pi)/4 + i sin\ (n pi)/4)`

	`text(Similarly,)`
	`1-i`	`=sqrt2(cos (-pi/4) + i sin (-pi/4))`
		`=sqrt2(cos\ pi/4 – i sin\ pi/4)`
	`(1-i)^n`	`=(sqrt2)^n (cos\ (n pi)/4 – i sin\ (n pi)/4)`

`:.(1 + i)^n + (1 − i)^n`

`= (sqrt2)^n (cos\ (n pi)/4 + i sin\ (n pi)/4) +(sqrt2)^n (cos\ (n pi)/4 – i sin\ (n pi)/4)`

`= (sqrt2)^n(cos\ (npi)/4 + i\ sin\ (npi)/4 + cos\ (npi)/4 − i\ sin\ (npi)/4) `

`= 2(sqrt2)^n\ cos\ (npi)/4\ \ \ text(… as required)`

ii. `(1 + i)^n= ((n), (0)) + i((n),(1)) − ((n),(2)) − i((n), (3)) + ((n),(4)) + … + i^n((n),(n))`

`(1 − i)^n=((n),(0)) − i((n),(1)) − ((n),(2)) + i((n),(3)) + … + (-1)^ni^n((n),(n))`

`(1 + i)^n + (1 − i)^n`

♦♦ Mean mark 22%.
MARKER’S COMMENT: Few students could link part (i) to solving part (ii).

`= 2((n),(0)) − 2((n),(2)) + 2((n),(4)) + … + 2((n),(n))`

`text(Given)\ n\ text(is divisible by 4, the last term)=+((n),(n))`

`text{Using part (i):}`

`2((n),(0)) − 2((n),(2)) + 2((n),(4)) + … + 2((n),(n))`	`=2(sqrt2)^n cos\ (npi)/4`
`((n),(0)) − ((n),(2)) + ((n),(4)) + … + ((n),(n))`	`=(sqrt2)^n cos\ (npi)/4`

`text(If)\ n\ text(is a positive integer divisible by 4 then)\ cos\ (npi)/4 = ±1,`

`text(depending on whether)\ n\ text(is an even or odd multiple of 4.)`

`:. ((n),(0)) − ((n),(2)) + ((n),(4)) + … + ((n),(n)) = (-1)^(n/4)(sqrt2)^n`

Proof, EXT2 P1 2014 HSC 15a

Three positive real numbers `a`, `b` and `c` are such that `a + b + c = 1` and `a ≤ b ≤ c`.

By considering the expansion of `(a + b + c)^2`, or otherwise, show that

`qquad 5a^2 + 3b^2 +c^2 ≤ 1`. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{Proof (See Worked Solutions)}`

Show Worked Solution

`a + b+ c = 1,\ \ a ≤ b ≤ c`

♦♦ Mean mark 13%.

`(a + b+ c)^2 = 1`

`a^2 + b^2 + c^2 + 2ab+ 2bc + 2ac = 1`

`text(S)text(ince)\ a ≤ b\ \ =>a^2 ≤ ab`

`text(S)text(ince)\ b ≤ c\ \ =>b^2 ≤ bc`

`text(S)text(ince)\ a ≤ c\ \ =>a^2 ≤ ac`

`=> a^2 + b^2 + c^2 + 2a^2 + 2b^2 + 2a^2`	`≤ 1`
`:.5a^2 + 3b^2 + c^2`	`≤ 1\ \ \ text(… as required)`

Mechanics, EXT2* M1 2004 HSC 7a

The rise and fall of the tide is assumed to be simple harmonic, with the time between successive high tides being 12.5 hours. A ship is to sail from a wharf to the harbour entrance and then out to sea. On the morning the ship is to sail, high tide at the wharf occurs at 2 am. The water depths at the wharf at high tide and low tide are 10 metres and 4 metres respectively.

Show that the water depth, `y` metres, at the wharf is given by

`y = 7 + 3 cos\ ((4pit)/(25))`, where `t` is the number of hours after high tide. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---
An overhead power cable obstructs the ship’s exit from the wharf. The ship can only leave if the water depth at the wharf is 8.5 metres or less.

Show that the earliest possible time that the ship can leave the wharf is 4:05 am. (2 marks)

--- 8 WORK AREA LINES (style=lined) ---
At the harbour entrance, the difference between the water level at high tide and low tide is also 6 metres. However, tides at the harbour entrance occur 1 hour earlier than at the wharf. In order for the ship to be able to sail through the shallow harbour entrance, the water level must be at least 2 metres above the low tide level.

The ship takes 20 minutes to sail from the wharf to the harbour entrance and it must be out to sea by 7 am. What is the latest time the ship can leave the wharf? (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`4:28\ text(am)`

Show Worked Solution

i. `text(Period)`

`(2pi)/n =`	` 12.5`
`12.5n =`	` 2pi`
`25n =`	` 4pi`
`n =`	` (4pi)/25`

`text(Amplitude)`

`a`	`= 1/2(10 − 4)`
	`= 3`

`=>\ text(Motion centres around)\ \ x = 7\ \ text(with)`

`y = 10\ \ text(when)\ \ t= 0`

`:.\ text(Water depth is given by)`

`y`	`= 7+a cos nt`
	`= 7 + 3 cos\ ((4pit)/(25))\ \ …\ text(as required.)`

ii.

`text(Find)\ \ t\ \ text(when)\ \ y = 8.5:`

`8.5`	`= 7 + 3\ cos\ ((4pit)/25)`
`3\ cos\ ((4pit)/25)`	`= 1.5`
`cos\ ((4pit)/25)`	`= 1/2`
`(4pit)/25`	`=pi/3`
`t`	`=(25pi)/(3 xx 4pi)`
	`=2 1/12`
	`= 2\ text(hrs 5 mins)`

`:.\ text(The earliest time the ship can leave)`

`text(is 4:05 am … as required.)`

iii. `text(2 metres above low tide = 6 m)`

`text(Find)\ t\ text(when)\ y = 6:`

`6 = 7 + 3\ cos\ ((4pit)/(25))`

`3\ cos\ ((4pit)/(25))`	`= -1`
`cos\ ((4pit)/(25))`	`= -1/3`
`(4pit)/25`	`= 1.9106…`
`:.t`	`= (25 xx 1.9106…)/(4pi)`
	`= 3.801…`
	`= 3\ text{hr 48 min (nearest min)}`

`:.\ text(At the harbour entrance, water depth)`

`text(of 6 m occurs when)\ \ t = 2\ text(hr 48 min.)`

`:.\ text(Given 20 mins sailing time, the latest the ship can)`

`text(leave the wharf is at)\ \ t = 2\ text(hr 28 min, or 4:28 am.)`

Mechanics, EXT2* M1 2007 HSC 7b

A small paintball is fired from the origin with initial velocity `14` metres per second towards an eight-metre high barrier. The origin is at ground level, `10` metres from the base of the barrier.

The equations of motion are

`x = 14t\ cos\ theta`

`y = 14t\ sin\ theta – 4.9t^2`

where `theta` is the angle to the horizontal at which the paintball is fired and `t` is the time in seconds. (Do NOT prove these equations of motion)

Show that the equation of trajectory of the paintball is

`y = mx − ((1 + m^2)/40)x^2`, where `m = tan\ theta`. (2 mark)

--- 6 WORK AREA LINES (style=lined) ---
Show that the paintball hits the barrier at height `h` metres when

`m = 2 ± sqrt(3 − 0.4h)`.

Hence determine the maximum value of `h`. (2 marks)

--- 8 WORK AREA LINES (style=lined) ---
There is a large hole in the barrier. The bottom of the hole is `3.9` metres above the ground and the top of the hole is `5.9` metres above the ground. The paintball passes through the hole if `m` is in one of two intervals. One interval is `2.8 ≤ m ≤ 3.2`.

Find the other interval. (2 marks)

--- 8 WORK AREA LINES (style=lined) ---
Show that, if the paintball passes through the hole, the range is

`(40m)/(1 + m^2)\ \ text(metres.)`

Hence find the widths of the two intervals in which the paintball can land at ground level on the other side of the barrier. (3 marks)

--- 10 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{Proof (See Worked Solutions)}`
`text{Proof (See Worked Solutions)}`
`0.8 ≤ m ≤ 1.2`
`text{1.3 m (to 1 d.p.) and 0.5 m (to 1 d.p.)}`

Show Worked Solution

i.	`x`	`= 14t\ cos\ theta`	`\ \ …\ (1)`
	`y`	`= 14t\ sin\ theta-4.9t^2`	`\ \ …\ (2)`

`text(Substitute)\ \ t = x/(14\ cos\ theta)\ \ text{from (1) into (2)}`

`y`	`= 14(x/(14\ cos\ theta))\ sin\ theta − 4.9(x/(14\ cos\ theta))^2`
	`= x\ tan\ theta − (4.9/(14^2))((x^2)/(cos^2\ theta))`
	`= x\ tan\ theta − (x^2)/40\ sec^2\ theta`
	`= x\ tan\ theta − (x^2)/40(1 + tan^2\ theta)`
	`= mx − ((1 + m^2)/40)x^2\ \ \ \ \ (text(Given)\ \ m = tan\ theta)`

ii. `text(Show paintball hits at)\ \ h\ \ text(when)`

`m = 2 ± sqrt(3 − 0.4h)`

`text(i.e.)\ \ y = h\ \ text(when)\ \ x = 10`

`10m − ((1 + m^2)/40) · 10^2`	`= h`
`10m − 5/2(1 + m^2)`	`= h`
`20m − 5 − 5m^2`	`= 2h`
`5m^2 − 20m + 2h + 5`	`= 0`

`text(Using the quadratic formula)`

`m`	`=(20 ± sqrt((-20)^2 − 4 · 5 · (2h + 5)))/(2 · 5)`
	`= (20 ± sqrt(400 − 40h −100))/10`
	`= (20 ± sqrt(300 − 40h))/10`
	`= (20 ± 10sqrt(3 − 0.4h))/10`
	`= 2 ± sqrt(3 − 0.4h)\ \ \ text(… as required)`

`text(Find maximum)\ \ h`

`sqrt(3 − 0.4h)`	`≥ 0`
`3 − 0.4h`	`≥ 0`
`0.4h`	`≤ 3`
`h`	`≤ 7.5`

`:.\ text(Maximum)\ \ h = 7.5\ text(m)`

iii.

`text{Using part (ii)}`

`text(When)\ \ h = 3.9`

`m`	`= 2 ± sqrt(3 − 0.4(3.9))`
	`= 2 ± sqrt(1.44)`
	`= 2 ± 1.2`
	`= 3.2\ \ text(or)\ \ 0.8`

`text(When)\ \ h = 5.9`

`m`	`= 2 ± sqrt(3 − 0.4(5.9))`
	`= 2 ± sqrt(0.64)`
	`= 2 ± 0.8`
	`= 2.8\ \ text(or)\ \ 1.2`

`:.\ text(The other interval is)\ \ \ 0.8 ≤ m ≤ 1.2`

iv. `text(Find)\ \ x\ \ text(when)\ \ y = 0`

`mx − ((1 + m^2)/40)x^2`	`= 0`
`x[m − ((1 + m^2)/40)x]`	`= 0`
`((1 + m^2)/40)x`	`= m,\ \ \ \ x ≠ 0`
`:. x`	`= (40m)/(1 + m^2)\ \ …\ text(as required)`

`text(Consider the interval)\ \ \ 2.8 ≤ m ≤ 3.2`

`text(When)\ \ m = 2.8`

`=> x = (40(2.8))/(1 + 2.8^2) = 12.669…\ text(m)`

`text(When)\ \ m = 3.2`

`=>x = (40(3.2))/(1 + 3.2^2) = 11.387…\ text(m)`

`:.\ text(Landing width interval)`

`= 12.669… − 11.387…`

`= 1.281…`

`= 1.3\ text(m)\ \ text{to 1 d.p.}`

`text(Consider the interval)\ \ \ 0.8 ≤ m ≤ 1.2`

`text(When)\ \ m = 0.8`

`=>x = (40(0.8))/(1 + 0.8^2) = 19.512…\ text(m)`

`text(When)\ \ m = 1.2`

`=>x = (40(1.2))/(1 + 1.2^2) = 19.672…`

`text(S)text(ince interval includes)\ \ m = 1\ \ text(where the)`

`text(paintball has maximum range.)`

`x_(text(max)) = (40(1))/(1 + 1^2) = 20\ text(m)`

`:.\ text(Landing width interval)`

`= 20 − 19.512…`

`= 0.487…`

`= 0.5\ text(m)\ \ \ text{(to 1 d.p.)}`

L&E, EXT1 2007 HSC 7a

The graphs of the functions `y = kx^n` and `y = log_e x` have a common tangent at `x = a`, as shown in the diagram.

By considering gradients, show that
1. `a^n = 1/(nk)`. (1 mark)
Express `k` as a function of `n` by eliminating `a`. (2 marks)

Show Answers Only

`text{Proof (See Worked Solutions)}`
`1/(en)`

Show Worked Solution

(i)	`y_1`	`= kx^n`
	`(dy_1)/(dx)`	`= nkx^(n − 1)`

`text(At)\ \ x = a,\ \ (dy_1)/(dx) = nka^(n − 1)`

`y_2`	`= log_e x`
`(dy_2)/(dx)`	`= 1/x`

`text(At)\ \ x = a,\ \ (dy_2)/dx = 1/a`

`text(S)text(ince tangents have the same gradient at)\ \ x=a,`

`:. nka^(n − 1)`	`= 1/a`
`a^n nk`	`= 1`
`a^n`	`= 1/(nk)\ \ …\ text(as required)`

(ii) `text(At)\ \ x = a`

`y_1`	`= ka^n`
`y_2`	`= log_e a`

`text(Given a common tangent)`

♦♦♦ “Vast majority” could not eliminate `a` in part (ii).

`ka^n`	`= log_e a`
`k(1/(nk))`	`= log_e a\ \ \ text{(from (i))}`
`1/n`	`= log_e a`
`:.a`	`= e^(1/n)`

`text(Substitute)\ \ a = e^(1/n)\ \ text(into)\ \ a^n = 1/(nk)`

`(e^(1/n))^n`	`= 1/(nk)`
`e`	`= 1/(nk)`
`:.k`	`= 1/(en)`

Functions, EXT1 F1 2007 HSC 6b

Consider the function `f(x) = e^x − e^(-x)`.

Show that `f(x)` is increasing for all values of `x`. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
Show that the inverse function is given by

`qquad qquad f^(-1)(x) = log_e((x + sqrt(x^2 + 4))/2)` (3 marks)

--- 7 WORK AREA LINES (style=lined) ---
Hence, or otherwise, solve `e^x - e^(-x) = 5`. Give your answer correct to two decimal places. (1 mark)

--- 4 WORK AREA LINES (style=lined) ---

Show Answers Only

`text{Proof (See Worked Solutions.)}`
`text{Proof (See Worked Solutions.)}`
`1.65\ \ text{(to 2 d.p.)}`

Show Worked Solution

i.	`f(x)`	`= e^x − e^(-x)`
	`f′(x)`	`= e^x + e^(-x)`

`text(S)text(ince)\ \ `	`e^x`	`> 0\ \ text(for all)\ x`
	`e^(-x)`	`> 0\ \ text(for all)\ x`
	`f′(x)`	`> 0\ \ text(for all)\ x`

`:.f(x)\ \ text(is an increasing function for all)\ x.`

ii. `y = e^x − e^(-x)`

`text(Inverse function)`

`x`	`= e^y − 1/(e^y)`
`xe^y`	`= e^(2y) − 1`
`e^(2y) − xe^y − 1`	`= 0`

`text(Let)\ \ A = e^y`

`:.A^2 − xA − 1 = 0`

`text(Using the quadratic formula)`

`A`	`=(x ± sqrt((-x)^2 − 4 · 1 · (-1)))/(2 · 1)`
	`=(x ± sqrt(x^2 + 4))/2`

`text(S)text(ince)\ \ (x – sqrt(x^2 + 4))/2<0\ \ text(and)\ \ e^y>0,`

`:.e^y`	`= (x + sqrt(x^2 + 4))/2`
`log_e e^y`	` = log_e((x + sqrt(x^2 + 4))/2)`
`y`	`= log_e((x + sqrt(x^2 + 4))/2)`
`:.f^(-1)(x)`	`= log_e((x + sqrt(x^2 + 4))/2)\ \ …\ text(as required)`

iii.	`e^x − e^(-x)`	`= 5`
	`f(x)`	`= 5`
	`f^(-1)(5)`	`= x`

`f^(-1)(5)`	`= log_e((5 + sqrt(5^2 + 4))/2)`
	`= log_e((5 + sqrt29)/2)`
	`= 1.647…`
	`= 1.65\ \ text{(to 2 d.p.)}`

Trig Calculus, EXT1 2006 HSC 7

A gutter is to be formed by bending a long rectangular metal strip of width `w` so that the cross-section is an arc of a circle.

Let `r` be the radius of the arc and `2 theta` the angle at the centre, `O`, so that the cross-sectional area, `A`, of the gutter is the area of the shaded region in the diagram on the right.

Show that, when `0 < theta <= pi/2`, the cross-sectional area is
2. `A = r^2 (theta - sin theta cos theta).` (2 marks)
The formula in part (i) for `A` is true for `0 < theta < pi.` (Do NOT prove this.)
By first expressing `r` in terms of `w` and `theta`, and then differentiating, show that
2. `(dA)/(d theta) = (w^2 cos theta (sin theta - theta cos theta))/(2 theta^3).`
for `0 < theta < pi.` (3 marks)
Let `g(theta) = sin theta - theta cos theta.`
By considering `g prime(theta)`, show that `g(theta) > 0` for `0 < theta < pi.` (3 marks)
Show that there is exactly one value of `theta` in the interval `0 < theta < pi` for which
1. `(dA)/(d theta) = 0.` (2 marks)
Show that the value of `theta` for which `(dA)/(d theta) = 0` gives the maximum cross-sectional area. Find this area in terms of `w.` (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`w^2/(2 pi)\ \ text(u²)`

Show Worked Solution

(i) `text(Show)\ \ A = r^2(theta – sin theta cos theta)`

`text(Area of segment)\ \ OBC`

`= (2 theta)/(2 pi) xx pi r^2`

`= r^2 theta`

`text(Area of)\ \ Delta OBC`	`= 1/2 ab sin C`
	`= 1/2 r * r * sin 2 theta`
	`= 1/2 r^2 * 2 sin theta cos theta`
	`= r^2 sin theta cos theta`

`:.\ text(Shaded Area (A))`

`= text(Area of segment)\ \ OBC – text(Area of)\ \ Delta OBC`

`= r^2 theta – r^2 sin theta cos theta`

`= r^2 (theta – sin theta cos theta)\ \ text(… as required.)`

(ii) `text(Consider Arc length)\ \ BC`

`w`	`= (2 theta)/(2 pi) xx 2 pi r`
	`= 2 theta r`
`:.\ r`	`= w/(2 theta)`

`:. A`	`= (w/(2 theta))^2 (theta – sin theta cos theta)`
	`= w^2/(4 theta) – (w^2 sin theta cos theta)/(4 theta^2)`

`:. (dA)/(d theta)`	`= (-w^2)/(4 theta^2) – (w^2/4) [((sin theta xx -sin theta + cos theta * cos theta) theta^2 – 2 theta sin theta cos theta)/theta^4]`
	`= (-w^2)/(4 theta^2) – (w^2/(4 theta^4))[(cos^2 theta – sin^2 theta) theta^2 – 2 theta sin theta cos theta]`
	`= (-w^2)/(4 theta^2) – (w^2/(4 theta^3))[(2 cos^2 theta – 1) theta – 2 sin theta cos theta]`
	`= (-w^2)/(4 theta^2) – (w^2/(4 theta^3))[(2 cos^2 theta – 1) theta – 2 sin theta cos theta]`
	`= w^2/(4 theta^3) [-theta – 2 theta cos^2 theta + theta + 2 sin theta cos theta]`
	`= w^2/(4 theta^3) [2 sin theta cos theta – 2 theta cos^2 theta]`
	`= w^2/(2 theta^3) (sin theta cos theta – theta cos^2 theta)`
	`= (w^2 cos theta\ (sin theta – theta cos theta))/(2 theta^3)\ \ text(… as required)`

(iii)	`g(theta)`	`= sin theta- theta cos theta`
	`g prime (theta)`	`= cos theta – (theta xx -sin theta + cos theta * 1)`
		`= cos theta + theta sin theta – cos theta`
		`= theta sin theta`

`text(S)text(ince)\ \ 0 < theta < pi,`

`=> sin theta > 0`

`:.\ g prime (theta) > 0`

`g(0) = sin 0 – 0 * cos 0 = 0`

`:.\ text(S)text(ince)\ \ g(0) = 0 and g(theta)\ \ text(is an increasing function)`

`(g prime (theta) > 0)\ \ text(for)\ \ 0 < theta < pi , text(then)\ \ g(theta) > 0.`

(iv) `text(If)\ \ (dA)/(d theta) = 0\ ,\ \ \ 0 < theta < pi`

`(w^2 cos theta\ (sin theta – theta cos theta))/(2 theta^3) = 0`

`text(Consider)`

`(w^2 cos theta)/(2 theta^3) = 0\ ,\ \ theta != 0`

`=>theta = pi/2`

`text(Consider)`

`sin theta – theta cos theta=`	` 0`
`text(i.e.)\ \ g(theta)=`	` 0`

`=>\ text(No solution for)\ \ 0 < theta < pi\ \ text{(using part (iii))}`

`:.\ text(There is only one value of)\ \ theta.`

(v) `(dA)/(d theta)`	`= (w^2 cos theta\ (sin theta – theta cos theta))/(2 theta^3)`
	`= (w^2 cos theta * g(theta))/(2 theta^3)`

`text(If)\ \ theta = pi/4\ ,\ \ g(pi/4) > 0\ ,\ \ cos (pi/4) > 0`

`:.\ (dA)/(d theta) > 0`

`text(If)\ \ theta = (3 pi)/4\ ,\ \ g((3 pi)/4) > 0\ ,\ \ cos ((3 pi)/4) < 0`

`:.\ (dA)/(d theta) < 0`

`:.\ text(Maximum when)\ \ theta = pi/2`

`text(When)\ \ theta = pi/2`

`A`	`= r^2 (theta – sin theta cos theta)`
	`= (w/(2 theta))^2 (theta – sin theta cos theta)`
	`= w^2/(2^2 xx (pi/2)^2) (pi/2 – sin\ pi/2 cos\ pi/2)`
	`= w^2/pi^2 (pi/2)`
	`= w^2/(2 pi)\ \ text(u²)`

Mechanics, EXT2* M1 2006 HSC 6a

Two particles are fired simultaneously from the ground at time `t = 0.`

Particle 1 is projected from the origin at an angle `theta, \ \ 0 < theta < pi/2`, with an initial velocity `V.`

Particle 2 is projected vertically upward from the point `A`, at a distance `a` to the right of the origin, also with an initial velocity of `V.`

It can be shown that while both particles are in flight, Particle 1 has equations of motion:

`x = Vt cos theta`

`y = Vt sin theta -1/2 g t^2,`

and Particle `2` has equations of motion:

`x = a`

`y = Vt -1/2 g t^2.` Do NOT prove these equations of motion.

Let `L` be the distance between the particles at time `t.`

Show that, while both particles are in flight,

`L^2 = 2V^2t^2 (1 - sin theta) - 2aVt cos theta + a^2.` (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
An observer notices that the distance between the particles in flight first decreases, then increases.

Show that the distance between the particles in flight is smallest when

`t = (a cos theta)/(2V(1 - sin theta))` and that this smallest distance is `a sqrt ((1 - sin theta)/2).` (3 marks)

--- 12 WORK AREA LINES (style=lined) ---
Show that the smallest distance between the two particles in flight occurs while Particle 1 is ascending if

`V > sqrt((a g cos theta)/(2 sin theta \ (1 - sin theta))).` (1 mark)

--- 6 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

`text(Show)\ \ L^2 = 2V^2t^2 (1 – sin theta) – 2aVt cos theta + a^2`

`text(Consider)\ \ P_1`

`x_1 = Vt cos theta`

`y_1 = Vt sin theta – 1/2 g t^2`

`text(Consider)\ \ P_2`

`x_2 = a`

`y_2 = Vt -1/2 g t^2`

`text(Let)\ \ d=\ text(Vertical distance between particles)`

`d= Vt -1/2 g t^2 – (Vt sin theta – 1/2 g t^2)`

`d= Vt (1 – sin theta)`

`text(Using Pythagoras:)`

`L^2`	`= (a – x_1)^2 + d^2`
	`= (a – Vt cos theta)^2 + V^2t^2 (1 – sin theta)^2`
	`= a^2 – 2aVt cos theta + V^2t^2 cos ^2 theta + V^2t^2 (1 – 2 sin theta + sin^2 theta)`
	`= a^2 – 2aVt cos theta + V^2 t^2 (cos^2 theta + sin^2 theta + 1 – 2 sin theta)`
	`= a^2 – 2aVt cos theta + V^2 t^2 (2 – 2 sin theta)`
	`= 2 V^2 t^2 (1 – sin theta) – 2aVt cos theta + a^2\ \ text(… as required.)`

ii. `L^2 = 2V^2 t^2 (1 – sin theta) – 2a Vt cos theta + a^2`

`(d(L^2))/(dt) = 4 V^2 t\ (1 – sin theta) – 2aV cos theta`

`text(Max or min when)\ \ (d(L^2))/(dt) = 0`

`4V^2t\ (1 – sin theta)`	`= 2aV cos theta`
`t`	`= (2a V cos theta)/(4V^2 (1 – sin theta))`
	`= (a cos theta)/(2V(1 – sin theta)`

`(d^2(L^2))/(dt^2) = 4V^2(1 – sin theta) > 0\ \ text(for)\ \ V > 0,\ \ 0 < theta < pi/2`

`:.\ L^2\ \ text(is a minimum)`

`:.\ L\ \ text(is a minimum when)\ \ t = (a cos theta)/(2V (1 – sin theta)`

`text(Show minimum distance is)\ \ a sqrt {(1 – sin theta)/2}`

`text(When)\ \ t = (a cos theta)/(2V(1 – sin theta))`

`L^2`	`= 2V^2 ((a^2 cos ^2 theta)/(4V^2 (1 – sin theta)^2)) (1 – sin theta)`
	`\ \ – 2aV ((a cos theta)/(2V (1 – sin theta))) cos theta + a^2`
	`= (a^2 cos^2 theta)/(2 (1 – sin theta)) – (a^2 cos^2 theta)/((1 – sin theta)) + a^2`
	`= a^2[(cos^2 theta – 2 cos^2 theta + 2 (1 – sin theta))/(2(1 – sin theta))]`
	`= a^2[(-cos^2 theta + 2 – 2 sin theta)/(2(1 – sin theta))]`
	`= a^2[(-(1 – sin^2 theta) + 2 – 2 sin theta)/(2 (1 – sin theta))]`
	`= a^2 [(sin^2 theta – 2 sin theta + 1)/(2(1 – sin theta))]`
	`= a^2 [(1 – sin theta)^2/(2 (1 – sin theta))]`
	`= a^2 [((1 – sin theta))/2]`
`:.\ L`	`= sqrt ((a^2(1 – sin theta))/2)`
	`= a sqrt ((1 – sin theta)/2)\ \ text(… as required.)`

iii. `text(Smallest distance occurs when)`

`t = (a cos theta)/(2V (1 – sin theta)`

`text(If)\ \ P_1\ \ text(is ascending,)\ \ dot y_1 > 0`

`y_1 = Vt sin theta – 1/2 g t^2`

`dot y_1 = V sin theta – g t`

`:.\ V sin theta – g ((a cos theta)/(2V (1 – sin theta)))`	`> 0`
`2V^2 sin theta\ (1 – sin theta) – a g cos theta`	`> 0`
`2V^2 sin theta\ (1 – sin theta)`	`> ag cos theta`
`V^2`	`> (a g cos theta)/(2 sin theta\ (1 – sin theta))`
`V`	`> sqrt ((a g cos theta)/(2 sin theta\ (1 – sin theta)))\ \ text(… as required.)`

Mechanics, EXT2* M1 2004 HSC 6b

A fire hose is at ground level on a horizontal plane. Water is projected from the hose. The angle of projection, `theta`, is allowed to vary. The speed of the water as it leaves the hose, `v` metres per second, remains constant. You may assume that if the origin is taken to be the point of projection, the path of the water is given by the parametric equations

`x = vt\ cos\ theta`

`y = vt\ sin\ theta − 1/2 g t^2`

where `g\ text(ms)^(−2)` is the acceleration due to gravity. (Do NOT prove this.)

Show that the water returns to ground level at a distance`(v^2\ sin\ 2theta)/g` metres from the point of projection. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---

This fire hose is now aimed at a 20 metre high thin wall from a point of projection at ground level 40 metres from the base of the wall. It is known that when the angle `theta` is 15°, the water just reaches the base of the wall.

Show that `v^2 = 80g`. (1 mark)

--- 3 WORK AREA LINES (style=lined) ---
Show that the cartesian equation of the path of the water is given by

`y = x\ tan\ theta − (x^2\ sec^2\ theta)/160`. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
Show that the water just clears the top of the wall if

`tan^2\ theta − 4\ tan\ theta + 3 = 0`. (2 marks)

--- 6 WORK AREA LINES (style=lined) ---
Find all values of `theta` for which the water hits the front of the wall. (2 marks)

--- 10 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(See Worked Solutions)`
`text(See Worked Solutions)`
`text(See Worked Solutions)`
`text(See Worked Solutions)`
`15^@ ≤ theta ≤ 45^@ \ \ text(and)\ \ 71.6^@ ≤ theta ≤ 75^@`

Show Worked Solution

i.	`x`	`= vt\ cos\ theta`
	`y`	`= vt\ sin\ theta − 1/2 g t^2`

`text(Find)\ \ t\ \ text(when)\ \ y = 0`

`vt\ sin\ theta − 1/2 g t^2`	`= 0`
`t(v\ sin\ theta − 1/2 g t)`	`= 0`
`v\ sin\ theta − 1/2 g t`	`= 0, \ \ t ≠ 0`
`1/2 g t`	`= v\ sin\ theta`
`t`	`= (2v\ sin\ theta)/g`

`text(Find)\ \ x\ \ text(when)\ \ t = (2v\ sin\ theta)/g`

`x`	`= v · (2v\ sin\ theta)/g\ cos\ theta`
	`= (v^2 · \ 2\ sin\ theta\ cos\ theta)/g`
	`= (v^2\ sin\ 2theta)/g`

`:.\ text(When)\ \ x = (v^2\ sin\ 2theta)/g,\ \ \text(the water returns)`

`text(to ground level … as required.)`

ii. `text(Show)\ \ v^2 = 80\ text(g)`

`text(When)\ \ theta = 15^@, \ x = 40`

`text{From part (i)}`

`40`	`= (v^2\ sin\ 30^@)/g`
`v^2 xx 1/2`	`= 40g`
`v^2`	`= 80g\ \ \ …text(as required)`

iii. `text(Show)\ \ y = x\ tan\ theta − (x^2\ sec^2\ theta)/160`

`x`	`= vt\ cos\ theta`
`:.t`	`= x/(v\ cos\ theta)`

`text(Substitute into)`

`y`	`= vt\ sin\ theta − 1/2 g t^2`
	`= v · x/(v\ cos\ theta) · sin\ theta −1/2 g (x/(v\ cos\ theta))^2`
	`= x\ tan\ theta − 1/2 g (x^2/(v^2\ cos^2\ theta))`
	`= x\ tan\ theta − 1/2 g · x^2/(80g\ cos^2\ theta)`
	`= x\ tan\ theta − (x^2\ sec^2\ theta)/160\ \ \ …text(as required.)`

iv. `text(Water clears the top if)\ \ y = 20\ \ text(when)\ \ x = 40`

`text{Substitute into equation from (iii)}`

`40\ tan\ theta − (40^2\ sec^2\ theta)/160`	`= 20`
`40\ tan\ theta − 10\ sec^2\ theta`	`= 20`
`40\ tan\ theta − 10(1 + tan^2\ theta)`	`= 20`
`40\ tan\ theta − 10 − 10\ tan^2\ theta`	`= 20`
`10\ tan^2\ theta − 40\ tan\ theta\ + 30`	`= 0`
`tan^2\ theta − 4\ tan\ theta + 3`	`= 0\ \ \ text(… as required)`

`text(Water hits the bottom of the wall when)`

`x = 40\ \ text(and)\ \ y = 0`

`40\ tan\ theta − (40^2\ sec^2\ theta)/160`	`= 0`
`40\ tan\ theta − 10\ sec^2\ theta`	`= 0`
`4\ tan\ theta − (1 + tan^2\ theta)`	`= 0`
`tan^2\ theta − 4\ tan\ theta + 1`	`= 0`

`text(Using the quadratic formula)`

`tan\ theta`	`= (+4 ± sqrt(16 − 4 · 1 · 1))/ (2 xx 1)`
	`= (4 ± sqrt12)/2`
	`= 2 ± sqrt3`
`theta`	`= 15^@\ \ text(or)\ \ 75^@`

`text(Water hits the top of the wall when)`

`x = 40\ text(and)\ \ y = 20`

`tan^2\ theta − 4\ tan\ theta + 3`	`= 0\ \ \ text{from (iv)}`
`(tan\ theta − 1)(tan\ theta − 3)`	`= 0`

`tan\ theta`	`= 1`	`text(or)`	`tan\ theta`	`= 3`
`theta`	`= 45^@`		`theta`	`= tan^(−1)\ 3`
				`= 71.565…`
				`= 71.6^@\ \ \text{(to 1 d.p.)}`

`:.\ text(Following the motion of the water as)\ \ theta\ \ text(increases,)`

`text(water hits the wall when)`

`15^@ ≤ theta ≤ 45^@`	`\ text(and)`
`71.6^@ ≤ theta ≤ 75^@`

Mechanics, EXT2* M1 2006 HSC 4b

A particle is undergoing simple harmonic motion on the `x`-axis about the origin. It is initially at its extreme positive position. The amplitude of the motion is 18 and the particle returns to its initial position every 5 seconds.

Write down an equation for the position of the particle at time `t` seconds. (2 marks)

--- 4 WORK AREA LINES (style=lined) ---
How long does the particle take to move from a rest position to the point halfway between that rest position and the equilibrium position? (3 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`x = 18 cos ((2 pi)/5 t)`
`5/6\ \ text(seconds)`

Show Worked Solution

i. `text{Amplitude (A)} = 18`

♦♦♦ “Very poorly” answered (exact data unavailable).

`text(Period) = (2 pi)/n = 5`

`5n`	`= 2 pi`
`n`	`= (2 pi)/5`

`text(Using)\ \ x`	`= A cos n t`
`x`	`= 18 cos ((2 pi)/5 t)`

ii. `text(When)\ \ t= 0,\ \ \ x = 18`

`text(Find)\ \ t\ \ text(when)\ \ x = 9`

`9`	`= 18 cos ((2 pi)/5 t)`
`cos ((2 pi)/5 t)`	`= 1/2`
`(2 pi)/5 t`	`= pi/3`
`t`	`= (5 pi)/(3 xx 2 pi)`
	`= 5/6\ \ text(seconds)`

`:.\ text(It takes the particle)\ \ 5/6\ \ text(seconds to move from)`

`text(rest position and half way to equilibrium.)`

Plane Geometry, EXT1 2006 HSC 3d

The points `P, Q` and `T` lie on a circle. The line `MN` is tangent to the circle at `T` with `M` chosen so that `QM` is perpendicular to `MN`. The point `K` on `PQ` is chosen so that `TK` is perpendicular to `PQ` as shown in the diagram.

Show that `QKTM` is a cyclic quadrilateral. (1 mark)
Show that `/_KMT = /_KQT.` (1 mark)
Hence, or otherwise, show that `MK` is parallel to `TP.` (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`
`text(Proof)\ \ text{(See Worked Solutions)}`

Show Worked Solution

(i)

`/_ QMT = 90^@\ \ \ (QM _|_ MN)`

`/_ QKT = 90^@\ \ \ (PQ _|_ TK)`

`:.\ /_ QMT + /_ QKT = 180^@`

`:.\ QKTM\ \ text(is cyclic)\ \ text{(opposite angles are supplementary)}`

`text(… as required.)`

(ii) `text(Show)\ \ /_ KMT = /_ KQT`

`/_ KMQ = /_ KTQ = theta`

`text{(angles in the same segment on arc}\ \ KQ text{)}`

`/_ KQT`	`= 90 – theta\ \ text{(angle sum of}\ \ Delta KQT text{)}`
`/_ KMT`	`= /_ QMT – /_ KMQ`
	`= 90 – theta`

`:.\ /_ KMT = /_ KQT\ \ text(… as required.)`

(iii) `text(Show)\ \ MK\ text(||)\ TP`

`/_ NTP`	`= /_ KQT\ \ text{(angle in alternate segment)`
	`= 90 – theta`

`:.\ /_ NTP = /_ KMT\ \ text{(from part (ii))}`

`:.\ MK\ text(||)\ TP\ \ text{(corresponding angles are equal)}`

Mechanics, EXT2* M1 2005 6b

An experimental rocket is at a height of 5000 m, ascending with a velocity of ` 200 sqrt 2\ text(m s)^-1` at an angle of 45° to the horizontal, when its engine stops.

After this time, the equations of motion of the rocket are:

`x = 200t`

`y = -4.9t^2 + 200t + 5000,`

where `t` is measured in seconds after the engine stops. (Do NOT show this.)

What is the maximum height the rocket will reach, and when will it reach this height? (2 marks)

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The pilot can only operate the ejection seat when the rocket is descending at an angle between 45° and 60° to the horizontal. What are the earliest and latest times that the pilot can operate the ejection seat? (3 marks)

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For the parachute to open safely, the pilot must eject when the speed of the rocket is no more than `350\ text(m s)^-1`. What is the latest time at which the pilot can eject safely? (2 marks)

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Show Answers Only

`7041\ text(metres)`

`20.4\ text(seconds.)`
`40.8\ text(seconds)\ ;\ 55.8\ text(seconds)`
`49.7\ text(seconds)`

Show Worked Solution

`y = -4.9t^2 + 200t + 5000`

`dot y = -9.8t + 200`

`text(Max height occurs when)\ \ dot y = 0`

`9.8t`	`= 200`
`t`	`= 20.408…`
	`= 20.4\ text{seconds (to 1 d.p.)}`

`text(When)\ t = 20.408…`

`y`	`= -4.9 (20.408…)^2 + 200 (20.408…) + 5000`
	`= 7040.816…`
	`= 7041\ text{m (to nearest metre)}`

`:.\ text(The rocket will reach a maximum height of)`

`text(7041 metres when)\ \ t = 20.4\ text(seconds.)`

ii. `text(The rocket is descending at 45° at point)\ A\ text(on the graph.)`

`text(The symmetry of the parabolic motion means that)\ \ A`

`text(occurs when)\ \ t = 2 xx text(time to reach max height, or)`

`40.8\ text(seconds.)`

`text(Point)\ B\ text(occurs when the rocket is descending at 60°)`

`text(to the horizontal.)`

`text(At)\ \ B,`

`-tan 60° = (dot y)/(dot x)`

`text{(The gradient of the projectile becomes}`

`text{negative after reaching its max height.)}`

`dot y = -9.8t + 200`

`dot x = d/(dt) (200t) = 200`

`:.\ – sqrt 3`	`= (-9.8t + 200)/200`
`-200 sqrt 3`	`= -9.8t + 200`
`9.8t`	`= 200 + 200 sqrt 3`
`t`	`= (200 + 200 sqrt 3)/9.8`
	`= (546.410…)/9.8`
	`= 55.756…`
	`= 55.8\ text{seconds (nearest second)}`

`:.\ text(The pilot can operate the ejection seat)`

`text(between)\ \ t = 40.8 and t = 55.8\ text(seconds.)`

iii.

`v^2 = (dot x)^2 + (dot y)^2`

`text(When)\ \ v = 350`

`350^2`	`= 200^2 + (-9.8t + 200)^2`
`122\ 500`	`= 40\ 000 + (-9.8t + 200)^2`
`(-9.8t + 200)^2`	`= 82\ 500`
`-9.8t + 200`	`= +- sqrt (82\ 500)`
`9.8t`	`= 200 +- sqrt (82\ 500)`
`t`	`= (200 +- sqrt (82\ 500))/9.8`
	`= 49.717…\ \ \ (t > 0)`
	`= 49.7\ text{seconds (to 1 d.p.)}`

`:.\ text(The latest time the pilot can eject safely)`

`text(is when)\ \ t = 49.7\ text(seconds.)`

Polynomials, EXT1 2005 HSC 3a

Show that the function `g(x) = x^2 - log_e (x + 1)` has a zero between `0.7` and `0.9.` (1 mark)
Use the method of halving the interval to find an approximation to this zero of `g(x)`, correct to one decimal place. (2 marks)

Show Answers Only

`text(Proof)\ \ text{(See Worked Solutions)}`
`0.7\ \ text{(to 1 d.p.)}`

Show Worked Solution

(i) `g(x)`	`= x^2 – log_e (x + 1)`
`g(0.7)`	`= 0.7^2 – log_e (0.7 + 1)`
	`= 0.49 – log_e 1.7`
	`= -0.04…`
`g(0.9)`	`= 0.9^2 – log_e (0.9 + 1)`
	`= 0.81 – log_e 1.9`
	`= 0.168…`

`:.\ text(S)text(ince the sign changes, a zero exists)`

`text(between 0.7 and 0.9.)`

(ii) `text(Halving the interval)`

`g(0.8)`	`= 0.8^2 – log_e (0.8 + 1)`
	`= 0.64 – log_e 1.8`
	`= 0.0522…`

`:.\ text(S)text(ince)\ \ g(0.8) > 0,\ \ text(the zero exists)`

`text(between 0.7 and 0.8.)`

♦♦♦ Part (ii) was “very poorly done”.

`g(0.75)`	`= 0.75^2 – log_e 1.75`
	`= 0.00288…`

`=>text(S)text(ince)\ \ g(0.75) > 0,\ \ text(the zero exists)`

`text(between 0.7 and 0.75.)`

`:.\ text{The zero will be 0.7 (to 1 d.p.)}`

Calculus, 2ADV C3 2007 HSC 10b

The noise level, `N`, at a distance `d` metres from a single sound source of loudness `L` is given by the formula

`N = L/d^2.`

Two sound sources, of loudness `L_1` and `L_2` are placed `m` metres apart.

The point `P` lies on the line between the sound sources and is `x` metres from the sound source with loudness `L_1.`

Write down a formula for the sum of the noise levels at `P` in terms of `x`. (1 mark)

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There is a point on the line between the sound sources where the sum of the noise levels is a minimum.

Find an expression for `x` in terms of `m`, `L_1` and `L_2` if `P` is chosen to be this point. (4 marks)

--- 8 WORK AREA LINES (style=lined) ---

Show Answers Only

`N = L_1/x^2 + L_2/(m-x)^2`
`x = (root 3 L_1\ m)/{(root 3 L_2 + root 3 L_1)}`

Show Worked Solution

`N = L/d^2`

`text(Noise from)\ L_1`	`= L_1/x^2`
`text(Noise from)\ L_2`	`= L_2/(m-x)^2`
`:. N`	`= L_1/x^2 + L_2/(m-x)^2`

ii. `N = L_1\ x^-2 + L_2 (m – x)^-2`

`(dN)/(dx)`	`= -2 L_1 x^-3 + -2 L_2 (m – x)^-3 xx d/(dx) (m – x)`
	`= (-2 L_1)/x^3 + (2 L_2)/(m – x)^3`

`text(Max or min when)\ (dN)/(dx) = 0`

`(2 L_1)/x^3`	`= (2 L_2)/(m – x)^3`
`2 L_1 (m – x)^3`	`= 2 L_2\ x^3`
`L_1 (m – x)^3`	`= L_2\ x^3`
`root 3 L_1 (m – x)`	`= root 3 L_2\ x`

`root 3 L_1\ m – root 3 L_1\ x`	`= root 3 L_2\ x`
`root 3 L_2\ x + root 3 L_1\ x`	`= root 3 L_1\ m`
`x (root 3 L_2 + root 3 L_1)`	`= root 3 L_1\ m`
`x`	`= (root 3 L_1\ m)/{(root 3 L_2 + root 3 L_1)}`

`(dN)/(dx)`	`= -2 L_1\ x^-3 + 2 L_2 (m – x)^-3`
`(d^2N)/(dx^2)`	`= 6 L_1\ x^-4 – 6 L_2 (m – x)^-4 xx -1`
	`= (6 L_1)/x^4 + (6 L_2)/(m – x)^4 > 0`

`:.\ text(A minimum occurs when)`

`x = (root 3 L_1\ m)/{(root 3 L_2 + root 3 L_1)}`

Calculus in the Physical World, 2UA 2007 HSC 10a

An object is moving on the `x`-axis. The graph shows the velocity, `(dx)/(dt)`, of the object, as a function of time, `t`. The coordinates of the points shown on the graph are `A (2, 1), B (4, 5), C (5, 0) and D (6, –5)`. The velocity is constant for `t >= 6`.

Using Simpson’s rule, estimate the distance travelled between `t = 0` and `t = 4`. (2 marks)
The object is initially at the origin. During which time(s) is the displacement of the object decreasing? (1 mark)
Estimate the time at which the object returns to the origin. Justify your answer. (2 marks)
Sketch the displacement, `x`, as a function of time. (2 marks)

Show Answers Only

`~~ 6\ \ text(units)`
`t > 5\ \ text(seconds)`
`7.2\ \ text(seconds)`

Show Worked Solution

(i)

`text(Distance travelled)`

`= int_0^4 (dx)/(dt)\ dt`

`~~ h/3 [y_0 + 4y_1 + y_2]`

`~~ 2/3 [0 + 4 (1) + 5]`

`~~ 2/3 [9]`

`~~ 6\ \ text(units)`

(ii) `text(Displacement is reducing when the velocity is negative.)`

`:. t > 5\ \ text(seconds)`

(iii) `text(At)\ B,\ text(the displacement) = 6\ text(units)`

`text(Considering displacement from)\ B\ text(to)\ D.`

`text(S)text(ince the area below the graph from)`

`B\ text(to)\ C\ text(equals the area above the)`

`text(graph from)\ C\ text(to)\ D,\ text(there is no change)`

`text(in displacement from)\ B\ text(to)\ D.`

`text(Considering)\ t >= 6`

`text(Time required to return to origin)`

`t`	`= d/v`
	`= 6/5`
	`= 1.2\ \ text(seconds)`

`:.\ text(The particle returns to the origin after 7.2 seconds.)`

(iv)