11 2016MATHMETHEXAM2

SECTION B – Question 1–continuedTURN OVER

Question 1 (11marks)

Let f :[0,8π]→ R, f xx( ) cos=

+2

2π .

a. Findtheperiodandrangeof f. 2marks

b. Statetheruleforthederivativefunction f ′. 1mark

c. Findtheequationofthetangenttothegraphof f at x=π. 1mark

SECTION B

Instructions for Section BAnswerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

2016MATHMETHEXAM2 12

SECTION B – continued

d. Findtheequationsofthetangentstothegraphof f :[0,8π]→ R, f x x( ) cos=

+2

2π that

haveagradientof1. 2marks

e. Theruleof f ′canbeobtainedfromtheruleof f underatransformationT,suchthat

T R R Txy a

xy b

: ,2 2 1 00

→

=

+

−

π

Findthevalueofaandthevalueofb. 3marks

f. Findthevaluesofx,0≤x≤8π,suchthat f (x)=2f ′(x)+π. 2marks

2013MATHMETH(CAS)EXAM2 12

SECTION 2 – Question 1–continued

Question 1 (12marks)Triggthegardenerisworkinginatemperature-controlledgreenhouse.Duringaparticular24-hour

timeinterval,thetemperature(T°C)isgivenbyT(t)=25+2cos π t8

,0≤t≤24,wheretisthe

timeinhoursfromthebeginningofthe24-hourtimeinterval.

a. Statethemaximumtemperatureinthegreenhouseandthevaluesoftwhenthisoccurs. 2marks

b. StatetheperiodofthefunctionT. 1mark

c. FindthesmallestvalueoftforwhichT=26. 2marks

SECTION 2

Instructions for Section 2Answerallquestionsinthespacesprovided.Inallquestionswhereanumericalanswerisrequired,anexactvaluemustbegivenunlessotherwisespecified.Inquestionswheremorethanonemarkisavailable,appropriateworkingmust beshown.Unlessotherwiseindicated,thediagramsinthisbookarenotdrawntoscale.

13 2013MATHMETH(CAS)EXAM2

SECTION 2 – Question 1–continuedTURN OVER

d. Forhowmanyhoursduringthe24-hourtimeintervalisT≥26? 2marks

2013MATHMETH(CAS)EXAM2 14

SECTION 2 – Question 1–continued

Triggisdesigningagardenthatistobebuiltonflatground.Inhisinitialplans,hedrawsthegraphof y=sin(x)for0≤x≤2πanddecidesthatthegardenbedswillhavetheshapeoftheshadedregions showninthediagrambelow.Heincludesagardenpath,whichisshownaslinesegmentPC.

ThelinethroughpointsP 23

32

π ,

andC(c,0)isatangenttothegraphofy=sin(x)atpointP.

XO

y

xC(c, 0)

1

–1

P 2π3

32

,

2π

e. i. Finddydx

whenx = 23π. 1mark

ii. Showthatthevalueofcis 3 23

+π . 1mark

15 2013MATHMETH(CAS)EXAM2

SECTION 2–continuedTURN OVER

Infurtherplanningforthegarden,Triggusesatransformationoftheplanedefinedasadilationoffactorkfromthex-axisandadilationoffactormfromthey-axis,wherekandmarepositiverealnumbers.f. LetX′,P′andC′betheimage,underthistransformation,ofthepointsX,PandCrespectively. i. FindthevaluesofkandmifX′P′=10andX′C′=30. 2marks

ii. FindthecoordinatesofthepointP′. 1mark

2010 MATHMETH(CAS) EXAM 2 16

Question 3An ancient civilisation buried its kings and queens in tombs in the shape of a square-based pyramid, WABCD.The kings and queens were each buried in a pyramid with WA = WB = WC = WD = 10 m.Each of the isosceles triangle faces is congruent to each of the other triangular faces.

The base angle of each of these triangles is x, where 4 2< < .

Pyramid WABCD and a face of the pyramid, WAB, are shown here.

Axx

B A B

D C

Z Z

Y

WW

Z is the midpoint of AB.a. i. Find AB in terms of x.

ii. Find WZ in terms of x.

1 + 1 = 2 marks

b. Show that the total surface area (including the base), S m2, of the pyramid, WABCD, is given by S = 400(cos2 (x) + cos (x) sin (x)).

2 marks

SECTION 2 – Question 3 – continued

17 2010 MATHMETH(CAS) EXAM 2

c. Find WY, the height of the pyramid WABCD, in terms of x.

2 marks

d. The volume of any pyramid is given by the formula Volume = 13

× area of base × vertical height.

Show that the volume, T m3, of the pyramid WABCD is 4000

324 6cos cosx x .

1 mark

Queen Hepzabah’s pyramid was designed so that it had the maximum possible volume.

e. Find dTdx

and hence find the exact volume of Queen Hepzabah’s pyramid and the corresponding value of x.

4 marks

SECTION 2 – Question 3 – continuedTURN OVER

2010 MATHMETH(CAS) EXAM 2 18

SECTION 2 – continued

Queen Hepzabah’s daughter, Queen Jepzibah, was also buried in a pyramid. It also had

WA = WB = WC = WD = 10 m.

The volume of Jepzibah’s pyramid is exactly one half of the volume of Queen Hepzabah’s pyramid. The volume of Queen Jepzibah’s pyramid is also given by the formula for T obtained in part d.f. Find the possible values of x, for Jepzibah’s pyramid, correct to two decimal places.

2 marks

Total 13 marks

2008 MATHMETH(CAS) EXAM 2 22

Question 4The graph of f : (–π, π) ∪ (π, 3π) → R, f (x) = tan

x2

⎛⎝⎜

⎞⎠⎟ is shown below.

x

y

O– 32

a. i. Find f ' π2

⎛⎝⎜

⎞⎠⎟

.

ii. Find the equation of the normal to the graph of y = f (x) at the point where x = π2

.

iii. Sketch the graph of this normal on the axes above. Give the exact axis intercepts.

1 + 2 + 3 = 6 marks

SECTION 2 – Question 4 – continued

23 2008 MATHMETH(CAS) EXAM 2

b. Find the exact values of x∈ − ∪( , ) ( , )π π π π3 such that f ' (x) = f ' π2

⎛⎝⎜

⎞⎠⎟

.

2 marks

Let g(x) = f (x – a).c. Find the exact value of a ∈ ( –1, 1) such that g(1) = 1.

2 marks

Let h : ( , ) ( , )− ∪π π π π3 → R, h(x) = sin tanx x2 2

2⎛⎝⎜

⎞⎠⎟

+ ⎛⎝⎜

⎞⎠⎟

+ . d. i. Find h' (x).

ii. Solve the equation h' (x) = 0 for x∈ − ∪( , ) ( , )π π π π3 . (Give exact values.)

1 + 2 = 3 marks

SECTION 2 – Question 4 – continuedTURN OVER

2008 MATHMETH(CAS) EXAM 2 24

e. Sketch the graph of y = h(x) on the axes below.• Give the exact coordinates of any stationary points.• Label each asymptote with its equation.• Give the exact value of the y-intercept.

y

xO

2 marks

Total 15 marks

END OF QUESTION AND ANSWER BOOK