Extension 1 Mathematics 3U - Sydney...d) Find dy if y = e 2x cos-1 x dx e) Find dy at t = ::. if x = 2 cos t and y = 2 sin t dx 4 2x f) Solve - ~ x + 4 x-3 g) Use the substitution

Student Name Ft Le Teacher's Name:

Sydney Technical

igh ool

Extension 1 Mathematics TRIALHSC

August 2020

General · Working time - 120 minutes+ 10 minutes reading time Instructions · Write using black pen

• NESA approved calculators may be used • A reference sheet is provided at the back of this paper • In questions 11-14, show relevant mathematical reasoning and/or

calculations

Total marks: Section 1-10 marks 70 · Attempt Questions 1-10

• Allow about 15 minutes for this section

Section II - 60 marks • Attempt questions 11-14 • Allow about 1 hours and 45 minutes for this section

Section I

10 Marks Attempt Questions 1-10 Allow about 15 minutes for this section

Use the multiple choice answer sheet for Questions 1-10.

1. A coin is biased such that the probability of a head is 0.8. The probability that exactly four tails will be observed when the coin is flipped ten times is:

(A) 10 X 0.26 X 0.84

(B) 10C4 X 0.24 X 0.86

(C) 10C4 X 0.26 X 0.84

(D) 10 X 0.24 X 0.86

2. Which one of the following vectors is parallel to the vector OP= 12 i - 6j?

-(A) OA = 12i + 6j

-(B) 0B = -i + 2j

-(C) OC=2i+j

-(D) OD= -2i + j

f dx 3. Which of the following is the correct expression for r;::---z ?

✓9- X

(A) sin-1 3x + c

(B) cos-1 3x + c

(D) cos-1 :: + c 3

2

4. The region made between the curves y = x 2 + 1 and y = 4, and the y-axis is shown below. y

,.;:;-.i---+--+--+---~1----1--~--1--___,;i-::;,. X

-4 -2 2 4

Which of these expressions gives the area of the region shown?

(D) [ ~dy

5. The number of elephants, N, in a population at time t is given by N = Aekt + 750, with constants A > 0 and k > 0. Which of the following is the correct differential equation?

(A) dN = -k(N + 750) dt

(B) dN = k(N + 750) dt

dN (C) - = -k(N - 750)

dt

(D) dN = k(N - 750) dt

3

6. In the triangle below, 1~1 = 1~1 = 4.

What is the value of u · v?

(A) 16

(B) 8../2

(C) 8

(D) 8../3

7. Which polynomial has a multiple root at x = 1

(A) x 3 + 3x2 - 4

(B) x3 - 3x + 2

(C) x3 - 3x - 4

(D) x 3 + 3x2 + 2

8. When cos x + sin x is rewritten in the form R cos (x - a), then:

(A) R = ../2 and a = !: 4

1C (B) R = 2 and a = -

4

.M 3rr (C) R = v L. and a = -4

3rr (D) R = 2 and a = -

4

4

9. The slope field below is for a first-order differential equation.

J'

Which of the following is a possible differential equation?

(A) dy = sin 2x dx

(B) dy = sin 2y dx

(C) dy = cos 2x dx

(D) dy = cos 2y dx

5

10. The graph of the function y = f(x) is below. y

4

Which of the following is a graph of Y = lf~)i7

(A) (B)

y

4

2

X

I -4 2 I I

l-2 I I I I l-4 I I

(C) (D)

y

I I I 14 I I I I

I 2 I I I I

X

--4 -2 4

6

y

X

-4 -2 I 2 4 I I

l-2 I I I I l-4 I I !

y

4

2

X

-4

Section II

Total marks - 60 Attempt Question 11-14 Allow about 1 hour and 45 minutes for this section

Begin each question on a NEW page

In Questions 11-14, your responses should include relevant mathematical reasoning and/or calculations.

Question 11 (15 marks) Begin a NEW page.

a) Nine people are arranged around a circular table

1.

11.

How many ways can they be arranged?

If they are seated randomly, what is the probability that two individuals, Lois and Clark, are sitting next to each other?

1

1

b) The heights in a population are normally distributed with a mean of 173 cm and a 2 standard deviation of 7cm. Use the empirical rule to find the approximate probability that a randomly selected person has a height under 159 cm?

c) A standard die is rolled 5 times. What is the probability of rolling a four at least two times? 2

d) Find dy if y = e 2x cos-1 x dx

e) Find dy at t = ::. if x = 2 cos t and y = 2 sin t dx 4

2x f) Solve - ~ x + 4

x-3

g) Use the substitution u = 2 - x4, or otherwise, to find J 7x3(2 - x4)

5 dx

End of Question 11

7

2

2

3

2


a) Find the particular solution to dy = zx2

where y(0) = 1 dx 3y

3

b) In the trapezium below, 0A = u, 75c = v, and ID.Al = ZICBI 2

u (Not to scale)

Express the vector BA in terms of u and v. - -

c) An unbiased coin is flipped 6400 times. The random variable X is the number of heads recorded.

0

1.

2

3 ,,-:,..,"-..' ::-\::<r

1. Assuming that X can be accurately approximated by a normal distribution, 1 find the values ofµ and a such that X ~ N (µ, a 2

)

11. Find the z-scores of flipping 3260 heads and of flipping 3100 heads. 2

111. Hence, use the table below to estimate the probability of flipping between 2 3 100 and 3 260 heads.

.5

0.5000 0.5398 0.5793 0.6179 0.6554 0.6915 0.7257 0.7580 0.7881 0.8159

0.8413 0.8643 0.8849 0.9032 0.9192 0.9332 0.9452 0.9554 0.9641 0.9713

0.9772 0.9821 0.9861 0.9893 0.9918 0.9938 0.9953 0.9965 0.9974 0.9981

0.9987 0.9990 0.9993 0.9995 0.9997 0.9998 0.9998 0.9999 0.9999 1.0000

Question 12 is continued on the next page

8

e d) 1ft = tan-

2

6t-8 Show that 3 sin 0 - 4 cos 0 - 4 = --

2 l+t

11. Hence solve 3 sin 0 - 4 cos 0 = 4 for O s; 0 s; 2n

End of Question 12

9

2

3


a) The radius of the base of a cylinder is increasing at a rate of 5 cm/min. The height of 2 the cylinder is fixed at 30 cm and the formula for the volume of a cylinder is V = nr2 h. Find the exact rate of change of the volume of the cylinder at the instant where the radius is 10 cm.

b) Below, the region between y = 3 cos~ and the x-axis is shaded between x = -n and x = n. 2

y

-4

If the region is rotated around the x-axis, find the volume of the solid formed. Leave your answer to 2 decimal places.

3

c) Use mathematical induction to prove that 72n + 7n + 4 is divisible by 6 for integers n;?: 0. 4

10

d) Billy throws a pebble from a height of 2 metres at an angle of 0 to the horizontal, with a velocity of 30m/s.

y

T (10, 6)

e x2 m .. ·-·-···-·-·························-·-···········-············-····-·-·--····--·····-·······-►

<E"------+-----------------------____::;,.x

1. Show that the expressions for the horizontal and vertical displacement at t seconds 2 after projection are x = 30tcos 0 and y = -St2 + 30tsin 0 + 2 respectively. (Take the acceleration due to gravity as -10 m/s2 and take the origin to be the ground directly below Billy).

11. Show that the equation of the path of the particle is 2 -xz

y = - (1 + tan2 0) + x tan 0 + 2 180

111. If Billy manages to hit a target at point T, which is 10 away on the ground and 6 2 metres high, find two possible angles of projection, to the nearest degree.

End of Question 13

11


a) 25 quokkas are introduced to an island to promote the survival of the species. The growth of

their population, Q, overt years, can be modelled by dQ = 0.0006Q(15 000 - Q). dt

11.

111.

IV.

V.

Show that ----1--- = ~ (~ + --1

--) 0.0006Q(15 000-Q) 9 Q 15 000-Q

Hence, solve dQ = 0.0006Q(15 000 - Q), using integration, to show that dt

15 000 . Q = w where B IS some constant.

l+Be-

Find the value of B, and hence use the model to estimate the number of quokkas on the island after two months.

What is the maximum number of quokkas that the island will support?

After how many months is the number of quokkas to greater than half of the maximum amount?

Question 14 is continued on the next page

12

1

3

2

1

2

b) Adam walks in a straight line form the point A(-1, 4) to the point Q( 4, 16) with constant speed. His position vector can be expressed in the form p = a + tu, where t is the time after he - -starts walking. Adam arrives at the point Q at t = 3.

y

Q(4,16)

• B(ll,9)

~----+-------------------,:,;,- X

1. State the vectors a and u. 2

11. Bob is at point B(11,9). During Adam's walk from A to P, Bob wishes to throw a ball to Adam. Bob decides to throw the ball when Adam is at the closest point to B.

a) Write a vector w that is perpendicular to u. 1

{3) Hence or otherwise, find value oft when Bob throws the ball to Adam. 3

End of Exam

13

Official STHS Mathematics Paper

1 /VIL 2

3 l. e 4

5]_ ' D

6 . . .

7 s. L J_ '

8

'

4-. 0 9

10

11 S, D 12

13 6, D 14

.c.

15 1 ,. b-16

17 3- f\ 18

19 1. L 20

21 In. B 22

23

24

25

26 I

1

ll. Official STHS Mathematics Paper

1 ~ 'i/ t! ~ tos2a 2 1 x 2~11 I

·;-; I -3 1fo120 - t -4

5

6 b') f5~ = r~3 -2r'::/ . . . .

7

8 /tf,'/~ ') I

9 ~ ~llt7

10 2 ,.....,

P(X'-l5ql ~

J.·51· 11 -- 2 --- -./

12 ... ,. \ r r- ... .

13 ~J A: A-t l~f, ~o}:: 1- ( ~lCi)v+ l,~{i)~{¾) ') 14 - ~ -15

16 ~} '2.~

✓ -:: ~-1-x_ "'::e,, 17 \I\ I :_ 2e -Z ll ' v' - - ~'?..

V

18 '2-r I

19 . 1{: 2 :Z:l_ e -e ~- 1?L -- t/ 1- Jt .._ - .

20 I .J

21 0~ ~ ~t, ::. -2>~-<t ~ dt:; 2rAJ.-t

22 ') ~ ~ ..J. I

~ - ... ,~, -

23 ' - 2~+ ::: ~ ¾~-b - . ,I. -1. -24

A-t TT 1! -~ 25 -h :_ """Zf -:_- 1cz~q-

-26 ~-1

2


1 '2~ ....

2 +} -~-3 )Cc~ -s) r ( S( J 't) x-s J ~ -X-:/ 3 3 . J~ ( A-3) .. ? {;c 14 ){x-3) 2. 2 0 4 {x-1) /_~21- --- {1~4)(z-3) ~(:) 5 ( 'X-3J( 2t__ - ( -:r. 2 ri- ';(_ -ri) 20 6 ( 'X - 5) ( - X 2 + z_ f {2 ~I ? 0 .

7 - ( ;r_ - ] ) '( X 1- ~ X - }2 '7 0 8 - ( x-3 )( ~ -lf )( z +-3) 7 0

'-

9 f\

,~~

10 '\. f"\. lf ' -11 -1\ Vi\/~ 12

13 ' :l~ -3, 1~A.~Lf - . ~I

14

15 ~) . tA-:. 2- ~ lf JL\ 16 U JI\:. - lf'-x 3 e,{ A 4'.,,d drx.-:. --Lfz 1

17 I'

,

t'-

18 ' \ + ~ 1 ( 2- -,y L-f) ✓ d1-~ .

_J 19 ( d IA

+ \ :x_ :s s 20 - i x..3 ~

L,\ r "-' ,i

21 '::1 (

- , \ ~ J~ 22 ~ 0' -½ - -l-\6 23 -+ - ?; ~L 24 ..-- . I{ 'f -

25 - -:;-(_2~ ;tif)b

26 - . 24 +- C, -1


1 (A) o{ ~ ?-,-J - ~.'2 - -

2 J

,, 1·

3 j s~ '£ J~ ~ \ 2~ c;}7l J J ._

4 t,, 1 - -::t '2. ~c... "

5 f.tf; .J -;(--;.Q , .. ,-=- I I J

6 I·-:. 0-~ 7 c,·-:..- I 8 ~o "'~ ,- x,~.~1 -

u 9 ~ 1Jl'2~\ L.,

10 cJ ---3

11 b) &A ·- i - '2 v\ .... v \~ V'\ --.. -- - ' ,l 12 - "2. v\ - v ·--- -" -

13 - C

14 c.-) \/ JA = . l 6it00··r; -:).__

0- - 6' \ ' . Lt 00 "f 2 't- ·; -I

15 - 3200 &-~- (bVO -16 6- ~Lf-0 17 .

18 ,, __ / ~/ I l - s2Go ~ 510()

19 '3'2. 60 -3200 5100-3200: 20 2:-:. Lro ':- l"S ;?:--:: Lfo c:_ -2-5

.. 21

-22 ;;~ / P(31 oo ~ r =: 3160) ~

23 -=- r(-2·S ~?. <:. 1--s) /I~ /1/'lffr- -24 -.,__ P(i~r•s)- Plz z'2·SJ -2-.s 1 .. 5 2·S

25 . ~ P(l S(··S) .-( ( -- PC? ~2·5)) 26 ~ o-crn2-f +-O,q q 3'J


1 JJ) LJ1i :- ~ti'~e -Lt~e -~ 2 512+ Lf C 1--t ?_) - Lf ( I +-t-0

f ft"2- '1. "2... - ( +--l l ,t--b 3 --4 6-b -~ + Lfb 2

- t-r ·- 4-h <

5 - \.+{'1--6 6-b- g 7 -.

f ·+ b ?_ -8

9 t; I ~ ~~" e - Lf Ll,~ e-:: tt 10 s s:~B ~ Lr, e -~ - ir~ (J 11 6{-9 12 Hl,, (P_ \ ,}~ ~ ~o to~~- ( 13 ~ ·/-:- < '•· . •

14

15 ' fa_ ~ - H._ c1 ~e~.2/\ - . ) - l 16 ~ :.- 0- q2,=, - - . o~ fz =n-2.

17 &--- [r '8 S C2J11J -

\/

18

19 . l~ec~r i+. e~ ~ LMS:: s .Si<; 1T - 'f Wj'fr-

20 ~· '-r --. 21 , .f<MJ' 22

23 ' . e: r~ 8 s Ddo) . ,r-- -V ,

24

25

26

J..


1 d..r -2 0\) .·df -::S V =- TT r '2 r 30 3 dv 6 Tt - 011.r 4 dv dr A1/ "

µOv - dl ;c .::_..

5 ~t:. IA, 6 - Sr 60n-r - WOrir . . -7 Pri r- (0 8 dv 9 dt ....... 10C0rr -10 I'" CoS;r :: ~~ ~ - / ,....

11 6) v~ 2n (3~~J-dl ~ '? i" :. ~ (I ~ewtJ -u

12 (ii .,

13 ·- 1in- ~1-½ J~ n " "-JV

14 (it-·15 - qi\ I+ ~t_ d1-- f"'I -16 ,r-- .. .

'11r I

17 - '"t + ' - ~I"' ')t I ff'\ '-"' ._ V

18 "

19 - 1Tr( TT +s~"'" - { 0 + c;.;, 0 ) I -"- ,I

20 -

1,,.~ ,i

21 ~

l-\ 1 -22

23 - F25'· 83 C2tto f/\3 -24 ' 25

26

2


4

5

6

7

8

11

19

22

23 ~ ¾e ~~~{ ~ ~e. r vcO ;{ ,~ ~t~o 24 ~e. ~r- V\- \<l I -:'2 3 Lf ...

26 ~ l \ 2


1JJ~/ ;,:;Q 30 / ~:: .... , 0 "' -- / -5'= YJ.{n0

V 2 ~-c. :; :_ -( ()-t c+ ( "\

3 /r{ f;-::() x-:.~slJ / /+£ vt:O ~ ... 31Js' e '"I - ,.,

4 --·· c ::3o ~A i :3olo:>9 rJ _._ (.,,.: 305;"'1(3

-5 s:o x: so~e Sn ~-- -rot~?/) s~., 0 6 'A.. ::_ ~() { lot, e +-c-, . - L. ~ .. st 2

+ 30f )~,/J f c"' .

7 At -b--:-0_ ~:o A4 t!~o, ._.,~2 8 , C -o I

<1_ - . 2 - - . Ltf:::. '2

9 So :x:: ?-0-£ w.lJ ~ ~ .-: --S-t 2 +-30-&;~,0 +-2 10 v 11 --~ /

l ' '":J.- /ii

12 5o t=- - l.

30~0 a j ~ ~ -~2+ 3()-1: ~:~0-1-2 --(21 13 v

14 ~~b ( I) ~ ~1-o (i) -15

" '2.. x... 16 ~ :_ -5 ( '!>~r.~A) + 30 1~.,0 'r 30~0+2 17 u -$ - X '2. <;,a ?f) +z_~-9+2 18 .,..... >Oz -19 2-.

-)( r - .....

20 ~ 1 ~o l ( + 11{~/-& J 1 ~ ¼~9 + )_ -21

22

23

24

25

26

1


:· .>.ii .. ·

At ::L= b, .. ,,. .. .

1 ½~ tO 2 - /0 2

<./ _L " - { r + +a., ~eJ ...

3 l- 180 + l01tJ~ 0 "12 . -4 6 - -~ C t + 1a~ "0) + 1o~~e r1- 2 -5 S't-= - S- - s- 1n .. / A _ 1- r o-1-a"' & -I-/ f 6 s~~A - 10~~0 -I- '-f/:: 0 -

-

7 qO! J ro?.-. 11 ysY trl 8 ~-<-8:: 2~s · 9 10: J-::J280 .

. ,,

10 - 10 - .. ·•

11 -- -12 Cf ()+ 1) f 230 cro-J1120 ,·. ~e = 1:p:e ~. .· .·

13 lo 10 I

14 J,:_

15

( /Jo+e ~ e -16 ,~. 17 ,'. fj-:_ R'1°. 25° ~ltt.ie)

I 18

19

20

.,,; 21

22

,. 23

24 ..

25 ..

26 .. ..

. .. 2


1 ti\) ' I

) 1

2 RH~:: ~l a .+-- 1s000-Q 3 ' 15000 -Q +() \ 4 ~ q- l'- 0 ( i 5000--Q) J~ - ....

I ' ' 5 1sooo \ 6 ...--., ~ l Q(fsooo -6) )· -· 7 l 8 --- ~ (Q{\sooo -ci) - - ' ' 9

10 ..,..._ n-voo6 Q(lsooo-a) -11 ~ LH( 12

\ . f/

,; I H~le, c;.,, ... ---- O·OOOtl)(lsooO -{Q 13 J():. ..

14 d:l- ' \ l ' 15 d0 ~ -~', T 4-- . \ Soe() .-Q ) 16 1 I ( ' r

-I '/")

17 & -:_ 'f \._) Q otGl - I Sooa ~Q v1 l)( '-

18 t =- t {"' l G I ,_ l~ l 1sooo -0 I + l 19 1 ( -b- t' ~ r L-\ l , S~oo -& l

r'.l I qi-'f l 20 . ::. e ' . {sooo-N

21 (Q Aeqf , l.../i~e ft . -1l ISooo-a

.,._ - ,t - I --e 22 Q -=- \$000 Aeqi - ()Aeqf 23 Q( 1 + Aet1+) =- \ SOO() Ae q-f

' \St>OO/-le"t; 24 () - . I+ A-Pt11,.f . --

25 ,sooo 26 (9._ hi + I


1 ;, \ / Ai i-: 0 o:.2s '

2 lS()()O 3 1-s~ I+ ~ et'trO 4 1,,s+2s ~ =- Is o()O 5 '2sg ~ I L-f 11'5 6 R= ~qq 7

,

_·_Jl-f i=- 2... ( b yeo-,,--s) 8 - '~ 12. ("'

9 \5000 J -

10 G~ i +511 e-'ty Ti

11 _((\·3q5 -·--····

12 ' Es J-i ""t1te o{ I { l q f1 ok~a.;;. C\fter 2 i,-,-io,,,'¾r, ~ ,.

V 13

14

15 ~ JI ~rUj·;,..4 U1 l)C< ( : ~ a..~ +-,~ 16 u er I 0

ls on o 17 ' Q ➔ I +--0 - (Sovo ' . -18

·---·- -- ----- . -· .. ····-- -··· -··

19 ' 11,e,_ Ml",..,X; .-i"l"""' Me,tw-.L.er s'-'f r)/?Qf1~ ~ -

I I 20 ,~ l~oon 21

22

23

24

25

26


1

2 vl ~~UL ~ n-1--soo I

3 rsooo 4 1500 == f +-S cr1e~ 1~

5

6 l + S 11 e: 1+ = ) 7 I

~

8 -Cft -e - 5qq 9 . "'\

10 -1&= ( ~ (-ci-q ) 11

I ll\

12 t=- -*1V\(~1 13 - 0~1, '-~. '-1~ -14 --t·S21_._ J

~--~J, 15

16 - f._ Af +er- ( "" 0 'I rHi \ ¼e.. .-h~"""Ovi~+ o{

17 q '-1.0~k~ t½(-ce.eJ.J ~a. I+ ~f ~e . w. 11(.?C't J'l1!,7.....,'

18 V

19

20

21

22

23

24

25

26


1 hJ ·i/ AQ ~ - 4 - t-0 1 r--s ~

2 v I 6 -1t ~. L 12 3

4 a: V\J N,_qW'\ avh do Q r;f'O""" -A qf t;:_? ' -

J r---1, "- I

5

' 1-s I

6 . H~lL ~ - I L-(- ecvv} ~=- ~-r/ ( 2-'-"' ..,. .

r~' 7

8 r- ) ~ - -9

10

11

12 -

13 ,-,; d) A v<?{_ivr- {)fJ' (/2, .. ,i ;IA A 'llf t( ,

' I 14 -15 -Lf I

16 s w .-- ~ (S)(' er.,_~ - I

......, '- '-....

>Ctl I (J.r J~V\l~{)le, 17

18 I

19

.

20

21

22

23 - . ·-· -· -

24

25

26


1 r-n ) 1--lf "

2 1~ ) Ett 4 q Bo":\ a (- ·-Uie__ ·¼10-../ ~ ~ I I ·+(\I f5'

4 _ 1 I ----< ,

VA~ ~~ Ct11J J{e_~ ~~·Jo~ ,.__ I

3 tl.) " l.v- ~ C.f Q<Z,\

I "' (} I \../"

4 - - -

5 - -I I r ~ 7 r { I I r-t-r1 H~r~ 1t + -b I ~- I q + >-I ~ I

6 Lt -;, I - r - S-t- - ... '"" r f I ~ 1t A -l ....,_..J

7 -I+ 3 I

8 Lti t t ~ / 4+ .& J - <

'- -...I

9 C'\

rn_·_-r J st ~ n--1t1 crviJ lf.f Lt-b~ (+ f -11 -~+ s+ :__ s~ ~ 11-~ 12 t 2\ = sf;~ St l ·2+ 12+ ~ 1 =t-+S>- (2) 13 36~S-f

~-=-'

\l ~C() 14

15 -

16 ~1, C\ ·,\dr') (l) 17 36-Sf 18 \2 + l2i- ::: 21-+5 'f I~ 19 Sl36~-st-J 20 -ts + ir2t - \'")_ 21 ~(80 4 /Lflfb = \80 -2S-b 22 \6~-t--= 360 \ 23 f;-:._ 360 [©r t: 2- I 3 f__ 2.c)()) 1 Tt;q 24

25

26

Documents

Extension 1 Mathematics 3U - Sydney...d) Find dy if y = e 2x cos-1 x dx e) Find dy at t = ::. if x = 2 cos t and y = 2 sin t dx 4 2x f) Solve - ~ x + 4 x-3 g) Use the substitution