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Final Examinations2006/2007 AdditionalMathematics
E 5EF SectionA (62 marks) I.
Eva l uatef_-I .r(2x+ 3),,o-. (3 rnarksi
2'
Let f(x) =
Irind f'(x) by thefirstprinciple.
*;.
(4 marks) 3.
( a ) I ti sg i ve ntl ta t2 }co s0 *2l sin0= r cos( 0+o) , Findthevalueof r andcrto thenearest 0.1u.
wher er >0ancloisanacut eangl e.
(\ -b/)l 2 [email protected] e ry:si n g tl re re su l ti n (a) ,findthefangeofthevaluesofy. (5 marks) 4.
Solvethe following equations:
( a ) l 2x -l l =i t; ,,) (b) lr l"r--5xl_tl=t,
5'
Show;by mathematicar induction,that,for
(5 marks) ail positiveintegersn,
e:;c).@#)+ ..'i&l : #1 I
(5 marks) 6'
(a) Expand(2x - r)6 compretery in descending orclerof powersof x. (b) If theconstant termin theexpansion of (2x -1161zx* is r7, find thevalue(s)of a. 31, X (5 marks)
7.
(a) Showthatsin(4x+ I )cos(4.r _ 3n . - I ' ' .1 ' 7 1:-cos 4 /-v e \
8x.
(b) Hencefind the generalsolution of the equation
sin(4x +-1)cos(4; -I 4
Fi nal E x am . 06/ A 7F .s A rl d i ti o n a l
4'
Ma th e ma ti c s
f= 1 4
(6 marks) 2/8
Final Examinations2A0612007 AdditionalMathematics
F.sEF
: y : -2x2+ 10x+ 10and regionABC is formedby L: 5x + y - 38 A' C1: In Figure1, theshaded (12,-22) of A'. B aud C are (4,18),(8,-38) and c,..y = x2 *16* + 26.The coordinates Findthe areaof the shadedregion' respectively.
8.
A(4,18
8,-38 Q:!=l x Tfi Qv +10 (5 rnarks)
Figure1
g.
In Figure 2, the coordinatesof I and B are (0, 4) and thatAC : CB : I : r, where r > 0'
(8, 0) respectively'c is a point on lB such
A(0,
Figure 2 (a) Expressthe coordinatesof C in terms of r' (b) Find the value(s)of r if r: :0; unitsfrom the straightline 2x +y - 8 (i) C is9 L
1?
(ii) theareaof LCPQ= +
units'whereP: (2' 4) and0: (5' 5)' square (6 rnarks)
=
1 0 . L e t A = (a ,-1 ),B = (l '!) and C 7-2,3)' -2 -> a, b, i andj. (a) Find AB and BC in termsof --> .-> + -> ll AB and AC BC I andbif O C a of values the (b) F'ind
Final Exam. 06/0? F. 5 Additional Mathematics
(6 marks) 3/8
Final Examinations200612007 AdditionalMathematics
N 5EF ll.
?^
y=tan-Jx. Let
(a) (i) Find $.
dx
(ii) Hence ,tto*'thut S
18(3seca3x -2s..2 3*1. T
(b) Find the slopeof the normalto thecurve y = tan23x at x = :
4
(6 marks)
12. Let a and B be therootsof theequation*
crB Express il + I
(a) (i)
'P c'
2 - ,r-t"- 1= 0 wherem is a realnurnber' ,
in tet'msof m.
(ii) Findtherangeof valuesof m suchthat I + I Bcx
(b) rf t * I Bct
< 3(a + B) .
< k(cr+ B) for all valuesof m, find therangeof valuesof k. (6 marks)
End of SectionA SectionB (48 marks) Answer any FOUR questionsonly from this section.
of with OA: OB :2 and/.AOB: 120o.D is themid-point 13. In Figure3, OACBis a paralleogram OA. CD rneetsAB at E. ->+ and OB :b, Le t OA :a (a)
Find
(a)
(i)
C
Find a . b.
(ii) l{encc, rind1OEl. \-4
marks)
(b) (i) Let BE : EA: ), : (1 - l,) *-> ExpressOE in termsof 1",a andb. ( ii ) L e tC E : E D : p : (1 - p) . ExpressOE in termsof p, a andb. ( i i i ) H e n ce ,sh o u ,th aOi: t
?"*!A. 33
Figure3 (5 marks)
(c)
Using the aboveresultsand dot productof vectors,find ZOEB.
(4 marks) !'inal Exam. 06/07 b-.5 Additional Mathematics
4/8
F.sEF
Final Examinations200612007 AdditionalMathematics
= familyof circlesF:x2+y2 + (7k- 4)x* 61,- Q3k+ 20) 0, u'hereftis real' 14. Consiclerthe Cr is thecircle,' * y'* 18x- 6y + 26 : 0, asshownin Figure4'
* t * )' t -l8 x -6 )' + 2 6 : 0
L: 4x+3y- 36:0 Figure4
(a)
Showthat Cr is a circle in F (1 mark)
(b)
= Cz is a circle centredat (4,75)andtouchesL: 4x a-3y - 36 0' (i) F'indthe equationof Cz. (ii) Show that Cz touchesCr.
(4 marks) (cj
The locus of the centresof the circles in F is a straightline Ir" Firrdthe equationof Lr-.
(2 marks) (d)
Cl is anothercircle in F. The line joining the centresof C: and C2 makesan angle 0 with Zp, wheretan 0 :
I{ . f in.f the possibleequation(s)of C3'
(5 rnarks)
Final Exam. 06/07 F. 5 Additional Mathematics
5/8
Final Examinations2AA6?0A7 AdditionalMathematics
5EF t<
Figure5 showsa tetrahedronOPQR with .ROperpendicularto the plane OPQ. IPRQ = 0, ZPOQ: $, /.RPO: a andIRQO = b . fi
/ o,+, q- -* ----l
\
a
Figure 5 Show that cos $ =
(a) (i)
(PR cosa)2 + (QR cosh)z- PQ2_ 2(PR)(QR)cosacos&
(ii) Henceor otherwise,show that cos O:
tot0
r - tana tanb. cos4 coso (5 marks)
(b) In Figure6 (a),ABCD is a rectangularplate.AB ll EF ll GH. AE - 3 cm,EG:4 cm, GD :5 cm and CD = h cm.The plate is folded along EG andG.Flsuchthat a triangular prism is formed as shown in Figure 6 (b),
a
E
n
3cm
G 4cm
D 5cm /
hcm //./
B
H
C
Figure 6 (a)
5 crn
/?
B cm
,4 cm
Figure6 (b) (i)
0.1o lf h = [2, using the result in (a) or otherwise,find IHAF correctto thenearest
(ii) If ZHA|-:
30o,lind the exactvalue of lz.
(7 marks)
FinalExam.06/07n 5 AdditionalMathematics
6t8
200612007 Final Examinations AdditionalMathematics
F. sEF
16. (a) (i) Evaluate irir,'"dt. (il) snow tnat
f .4
xdx: 1"- 1sin2x* Jsln 8432
1
sin4x+C. (5 marks)
(b)
Figure 7 (a) shows a region boundecl by the y-axis and the curve x : sin2/ * ko
.
where0
4r
by revolvingtheregionabout the y-axis. andfr> 0. Avesselis generated
J
The cross-sectionof the vesselis shown also in Figure 7 (a). The opening of the vesselis a circlewith radiusr units.
4 units Figure7 (b) Figure7 (a) Find the capacityof the vesselin terms of fr. (ii) Find the capacityof the vesselwhen r : 7.
(i)
(iii) The vesselis placedcompletelyinsidea tall rectangularbox as shownin Figure7 (b). 'fhe baseof the box is a squareof side 4 units. Find, if ftvaries, the capacityof the largestvesselthat can be placedinsidethe box. (Note:ignorethe thicknessof the vessel.) (7 marks)
Final Exam. 06/07 F. 5 Additional Mathematics
'il8
Final Examinations200612007 AdditionalMathematics
F. 5EF /
suchthatAB : 7^11cm,AP = 3 cm,BQ = 4 cm 17. In Figure8,AB. AP andBQ arelinesegments andIBAP: IABQ : 90".R is a variablepointonAB suchthatIARP: 0 and/BRQ = $.
3 cm
A
l<-
-t; /vJ cm
F'igure8 (a) lf R is moving towards B at a speedof I cm/s, by consideringRB in terms of Q,frnd the rate of changeof $ when R is rF cm from A.
(3 marks) (b) (i)
Showthat 3cot0+ 4cot$= 7.,r5,
(ii) Hencerrntl 99 in termsof o andg,
de
(3 marks)
(c) Let L=P R+QR. (i) ExpressL in terms of 0 and {. (ii) Using (c) (i) and (b) (ii). or otherwise,show that dI,
de
3 (co s$ -cos0) sin20
(iii) Find the positionof R so that L is mtntmum. ('l'estingfor minimum is not required.) (6 marks)
End of Paper
Final Exam. 06/07 R 5 Additional Mathematics
8/8
E 5EF SectionA (62 marks) I.
Eva l uatef_-I .r(2x+ 3),,o-. (3 rnarksi
2'
Let f(x) =
Irind f'(x) by thefirstprinciple.
*;.
(4 marks) 3.
( a ) I ti sg i ve ntl ta t2 }co s0 *2l sin0= r cos( 0+o) , Findthevalueof r andcrto thenearest 0.1u.
wher er >0ancloisanacut eangl e.
(\ -b/)l 2 [email protected] e ry:si n g tl re re su l ti n (a) ,findthefangeofthevaluesofy. (5 marks) 4.
Solvethe following equations:
( a ) l 2x -l l =i t; ,,) (b) lr l"r--5xl_tl=t,
5'
Show;by mathematicar induction,that,for
(5 marks) ail positiveintegersn,
e:;c).@#)+ ..'i&l : #1 I
(5 marks) 6'
(a) Expand(2x - r)6 compretery in descending orclerof powersof x. (b) If theconstant termin theexpansion of (2x -1161zx* is r7, find thevalue(s)of a. 31, X (5 marks)
7.
(a) Showthatsin(4x+ I )cos(4.r _ 3n . - I ' ' .1 ' 7 1:-cos 4 /-v e \
8x.
(b) Hencefind the generalsolution of the equation
sin(4x +-1)cos(4; -I 4
Fi nal E x am . 06/ A 7F .s A rl d i ti o n a l
4'
Ma th e ma ti c s
f= 1 4
(6 marks) 2/8
Final Examinations2A0612007 AdditionalMathematics
F.sEF
: y : -2x2+ 10x+ 10and regionABC is formedby L: 5x + y - 38 A' C1: In Figure1, theshaded (12,-22) of A'. B aud C are (4,18),(8,-38) and c,..y = x2 *16* + 26.The coordinates Findthe areaof the shadedregion' respectively.
8.
A(4,18
8,-38 Q:!=l x Tfi Qv +10 (5 rnarks)
Figure1
g.
In Figure 2, the coordinatesof I and B are (0, 4) and thatAC : CB : I : r, where r > 0'
(8, 0) respectively'c is a point on lB such
A(0,
Figure 2 (a) Expressthe coordinatesof C in terms of r' (b) Find the value(s)of r if r: :0; unitsfrom the straightline 2x +y - 8 (i) C is9 L
1?
(ii) theareaof LCPQ= +
units'whereP: (2' 4) and0: (5' 5)' square (6 rnarks)
=
1 0 . L e t A = (a ,-1 ),B = (l '!) and C 7-2,3)' -2 -> a, b, i andj. (a) Find AB and BC in termsof --> .-> + -> ll AB and AC BC I andbif O C a of values the (b) F'ind
Final Exam. 06/0? F. 5 Additional Mathematics
(6 marks) 3/8
Final Examinations200612007 AdditionalMathematics
N 5EF ll.
?^
y=tan-Jx. Let
(a) (i) Find $.
dx
(ii) Hence ,tto*'thut S
18(3seca3x -2s..2 3*1. T
(b) Find the slopeof the normalto thecurve y = tan23x at x = :
4
(6 marks)
12. Let a and B be therootsof theequation*
crB Express il + I
(a) (i)
'P c'
2 - ,r-t"- 1= 0 wherem is a realnurnber' ,
in tet'msof m.
(ii) Findtherangeof valuesof m suchthat I + I Bcx
(b) rf t * I Bct
< 3(a + B) .
< k(cr+ B) for all valuesof m, find therangeof valuesof k. (6 marks)
End of SectionA SectionB (48 marks) Answer any FOUR questionsonly from this section.
of with OA: OB :2 and/.AOB: 120o.D is themid-point 13. In Figure3, OACBis a paralleogram OA. CD rneetsAB at E. ->+ and OB :b, Le t OA :a (a)
Find
(a)
(i)
C
Find a . b.
(ii) l{encc, rind1OEl. \-4
marks)
(b) (i) Let BE : EA: ), : (1 - l,) *-> ExpressOE in termsof 1",a andb. ( ii ) L e tC E : E D : p : (1 - p) . ExpressOE in termsof p, a andb. ( i i i ) H e n ce ,sh o u ,th aOi: t
?"*!A. 33
Figure3 (5 marks)
(c)
Using the aboveresultsand dot productof vectors,find ZOEB.
(4 marks) !'inal Exam. 06/07 b-.5 Additional Mathematics
4/8
F.sEF
Final Examinations200612007 AdditionalMathematics
= familyof circlesF:x2+y2 + (7k- 4)x* 61,- Q3k+ 20) 0, u'hereftis real' 14. Consiclerthe Cr is thecircle,' * y'* 18x- 6y + 26 : 0, asshownin Figure4'
* t * )' t -l8 x -6 )' + 2 6 : 0
L: 4x+3y- 36:0 Figure4
(a)
Showthat Cr is a circle in F (1 mark)
(b)
= Cz is a circle centredat (4,75)andtouchesL: 4x a-3y - 36 0' (i) F'indthe equationof Cz. (ii) Show that Cz touchesCr.
(4 marks) (cj
The locus of the centresof the circles in F is a straightline Ir" Firrdthe equationof Lr-.
(2 marks) (d)
Cl is anothercircle in F. The line joining the centresof C: and C2 makesan angle 0 with Zp, wheretan 0 :
I{ . f in.f the possibleequation(s)of C3'
(5 rnarks)
Final Exam. 06/07 F. 5 Additional Mathematics
5/8
Final Examinations2AA6?0A7 AdditionalMathematics
5EF t<
Figure5 showsa tetrahedronOPQR with .ROperpendicularto the plane OPQ. IPRQ = 0, ZPOQ: $, /.RPO: a andIRQO = b . fi
/ o,+, q- -* ----l
\
a
Figure 5 Show that cos $ =
(a) (i)
(PR cosa)2 + (QR cosh)z- PQ2_ 2(PR)(QR)cosacos&
(ii) Henceor otherwise,show that cos O:
tot0
r - tana tanb. cos4 coso (5 marks)
(b) In Figure6 (a),ABCD is a rectangularplate.AB ll EF ll GH. AE - 3 cm,EG:4 cm, GD :5 cm and CD = h cm.The plate is folded along EG andG.Flsuchthat a triangular prism is formed as shown in Figure 6 (b),
a
E
n
3cm
G 4cm
D 5cm /
hcm //./
B
H
C
Figure 6 (a)
5 crn
/?
B cm
,4 cm
Figure6 (b) (i)
0.1o lf h = [2, using the result in (a) or otherwise,find IHAF correctto thenearest
(ii) If ZHA|-:
30o,lind the exactvalue of lz.
(7 marks)
FinalExam.06/07n 5 AdditionalMathematics
6t8
200612007 Final Examinations AdditionalMathematics
F. sEF
16. (a) (i) Evaluate irir,'"dt. (il) snow tnat
f .4
xdx: 1"- 1sin2x* Jsln 8432
1
sin4x+C. (5 marks)
(b)
Figure 7 (a) shows a region boundecl by the y-axis and the curve x : sin2/ * ko
.
where0
4r
by revolvingtheregionabout the y-axis. andfr> 0. Avesselis generated
J
The cross-sectionof the vesselis shown also in Figure 7 (a). The opening of the vesselis a circlewith radiusr units.
4 units Figure7 (b) Figure7 (a) Find the capacityof the vesselin terms of fr. (ii) Find the capacityof the vesselwhen r : 7.
(i)
(iii) The vesselis placedcompletelyinsidea tall rectangularbox as shownin Figure7 (b). 'fhe baseof the box is a squareof side 4 units. Find, if ftvaries, the capacityof the largestvesselthat can be placedinsidethe box. (Note:ignorethe thicknessof the vessel.) (7 marks)
Final Exam. 06/07 F. 5 Additional Mathematics
'il8
Final Examinations200612007 AdditionalMathematics
F. 5EF /
suchthatAB : 7^11cm,AP = 3 cm,BQ = 4 cm 17. In Figure8,AB. AP andBQ arelinesegments andIBAP: IABQ : 90".R is a variablepointonAB suchthatIARP: 0 and/BRQ = $.
3 cm
A
l<-
-t; /vJ cm
F'igure8 (a) lf R is moving towards B at a speedof I cm/s, by consideringRB in terms of Q,frnd the rate of changeof $ when R is rF cm from A.
(3 marks) (b) (i)
Showthat 3cot0+ 4cot$= 7.,r5,
(ii) Hencerrntl 99 in termsof o andg,
de
(3 marks)
(c) Let L=P R+QR. (i) ExpressL in terms of 0 and {. (ii) Using (c) (i) and (b) (ii). or otherwise,show that dI,
de
3 (co s$ -cos0) sin20
(iii) Find the positionof R so that L is mtntmum. ('l'estingfor minimum is not required.) (6 marks)
End of Paper
Final Exam. 06/07 R 5 Additional Mathematics
8/8
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