Mathematics - aceh.b-cdn.net
Transcript of Mathematics - aceh.b-cdn.net
BAULKHAM HILLS
BAULKHAM HILLS HIGH SCHOOL
2016 YEAR 11 YEARLY
Mathematics
General Instructions bull Reading time - 5 minutes bull Working time - 2 hours bull Write using black or blue pen bull Board-approved calculators may be used bull All necessary working should be shown
in every question bull Marks may be deducted for careless or
badly arranged work bull Attempt all questions
Total marks - 70
This paper consists of TWO sections
Section 1 - Multiple Choice 10 marks
Section 2 - Extended Response 60 marks Attempt all questions Start a new page for each question
middot Section I-Multiple Choice (10 marks) Attempt all questions
Answer the following on the booklet provided
1 A yacht sailed from A to B on a bearing of 196degT To sail from B directly back to A the bearing it
should travel is
(C) 164degT
2 Simplify 1 r5-v3
(A) 5+3 (B) 5+3 (C) 5-3 (D) 5-V3 2 22 2 22
3 If a and ~ are the roots of 5x 2 + 15x - 3 = 0 then a 2 + 3aJ + (J2 is
48(A) 42 (B) _ 42 (C) 48 (D)5 5 5 5
4 Correct to the nearest degree the angle of inclination from the positive x-axis of a line with
x-intercept 2 and y-intercept 2[2 is
(B) 55deg CD) 155deg
5 For the function y = [(x) and the domain a ~ x ~ b which of the following is true
y
y = f(x)
o a b x
middot (A) [ex) gt 0 and [ex) lt 0 (B) [ ex) lt 0 and [ex) lt 0
middot (C) [ex) gt 0 and [(x) gt 0 (D) [(x) lt 0 and [(x) gt 0
middot 6 Wh h f h ( II h lf f cosec8sec8 7IC 0 t e 10 owmg expreSSIOns IS t e correct simp I IcatIOn 0 8 tan
2 2CD) cosec ()middot (A) sin2 () (B) cos ()
2
7 The solutions of v3tan x +1 = 0 for the domain 0deg 5 x 5 360deg are
(A) x = 30deg210deg (8) x = 150deg210deg
middot (C) x = 150deg330deg (D) x = 210deg330deg
8
w
A L-~a~________________ v C
In the diagram above ACD is a straight angle Which of the following statements is true
w2 2middot (A) = u 2 + v - 2uvcosfJ
2 2 + v(B) w = u 2 + 2uvcosfJ
2 2 + w 2(C) u = v - 2vwcosfJ
middot (D)~=~sina casp
119
B
L-~_______ D
+-______
2
~--~--+---r_--~------+x
-2
The diagram above shows the function y = [(x)
A student wishes to solve the equation [(x) = 2 - Ixl graphically The correct number of solutions
is
(A) (B) 2 (C) 3 (D) 4
10 Given [(x + 1) = x 2 - X + 1 then [(x) in its simplest form is
(A) x 2 + x + 1 (B) x 2 + x + 3
(C) x 2 - 3x + 3 (D) x 2
- 3x + 1
End of Section I
3
---
Section II - Extended Response All necessary working should be shown in every question
Marks middot Question 11 (15 marks) - Start on the appropriate page in your answer booklet
a) bull Simplify 2 x+3 x-2
5 4
2bull b) Solve 13x - 21 = 7
middot c) Factorise 6x 2 + llx -10 2
d) For the function [(x) = x2 - 16
(i) Find the domain and range 2
(ii) Is the function odd even or neither 2
(Justify your answer by showing working)
e) Write a set of inequalities which define the shaded region 2
x
f) Find the solutions of the equation 2 cos 2 e = 3 sin () + 3 for 0deg e 360deg 3
End of Question 11
4
Question 12 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) 3 Simplify
b) Differentiate with respect to x 4
(i) 2 i-x
(ii) 2xy3x + 1 2
2(iii) 4x-l
2x+l
c) Evaluate showing all working x 3 + 8
lim x-gt-2 2x + 4 2
x 2d) Find the equation of the tangent to y = - 4x at the point (1 -3) on it 2
e) Solve 12x-31=4x+1 2
End of Question 12
5
Question 13 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) Find the exact value of cos e given that tan e = 7 and sin e lt O 2
b) Find the value of x 2
Not to scale
12
c) If 6x 2 - 11 == A(x + 2)2 + Bx + C find the values of A Band C 3
d)
e)
For which values of k does the equation x 2 - 2x + 3 shy k = 0 have real roots
Consider the function [(x) = ~X4 - 2x3 + 2 2
(i)
(ii)
Find the coordinates of the stationary points of y = [(x)
Determine the nature of these stationary points
(iii) Sketch the graph of y = [(x) clearly showing the stationary points and
y-intercept (You need not find the x-intercepts)
2
2
2
2
End of Question 13
6
middot Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) The diagram shows the points A(1-2) B(-4-3) and C(-13) and the line Be
Not to scale
x
A(l - 2)
(i) Calculate the exact length of Be 1 (ii) Show that the equation of BC is 2x - y + 5 = o 2 (iii) Find the exact area of triangle ABe 3
b) Solve x 2 - 2x - 1 lt O 3
Question 14 continues on the next page
7
Question 14 (continued)
c) The diagram shows two right cones The small cone of radius PS=r em is enclosed within bull a larger cone of radius QR=5em as shown The larger cone has height 15cm and the small
cone has height hem
T
Not to scale
15 ern
hem
R
(i) Prove that L1TPSIIIL1TQR 2
(ii) Show that h = 15 shy 3r 1
(iii) Find the radius of the small cone such that its volume is a maximum 3
End of Examination
8
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middot Section I-Multiple Choice (10 marks) Attempt all questions
Answer the following on the booklet provided
1 A yacht sailed from A to B on a bearing of 196degT To sail from B directly back to A the bearing it
should travel is
(C) 164degT
2 Simplify 1 r5-v3
(A) 5+3 (B) 5+3 (C) 5-3 (D) 5-V3 2 22 2 22
3 If a and ~ are the roots of 5x 2 + 15x - 3 = 0 then a 2 + 3aJ + (J2 is
48(A) 42 (B) _ 42 (C) 48 (D)5 5 5 5
4 Correct to the nearest degree the angle of inclination from the positive x-axis of a line with
x-intercept 2 and y-intercept 2[2 is
(B) 55deg CD) 155deg
5 For the function y = [(x) and the domain a ~ x ~ b which of the following is true
y
y = f(x)
o a b x
middot (A) [ex) gt 0 and [ex) lt 0 (B) [ ex) lt 0 and [ex) lt 0
middot (C) [ex) gt 0 and [(x) gt 0 (D) [(x) lt 0 and [(x) gt 0
middot 6 Wh h f h ( II h lf f cosec8sec8 7IC 0 t e 10 owmg expreSSIOns IS t e correct simp I IcatIOn 0 8 tan
2 2CD) cosec ()middot (A) sin2 () (B) cos ()
2
7 The solutions of v3tan x +1 = 0 for the domain 0deg 5 x 5 360deg are
(A) x = 30deg210deg (8) x = 150deg210deg
middot (C) x = 150deg330deg (D) x = 210deg330deg
8
w
A L-~a~________________ v C
In the diagram above ACD is a straight angle Which of the following statements is true
w2 2middot (A) = u 2 + v - 2uvcosfJ
2 2 + v(B) w = u 2 + 2uvcosfJ
2 2 + w 2(C) u = v - 2vwcosfJ
middot (D)~=~sina casp
119
B
L-~_______ D
+-______
2
~--~--+---r_--~------+x
-2
The diagram above shows the function y = [(x)
A student wishes to solve the equation [(x) = 2 - Ixl graphically The correct number of solutions
is
(A) (B) 2 (C) 3 (D) 4
10 Given [(x + 1) = x 2 - X + 1 then [(x) in its simplest form is
(A) x 2 + x + 1 (B) x 2 + x + 3
(C) x 2 - 3x + 3 (D) x 2
- 3x + 1
End of Section I
3
---
Section II - Extended Response All necessary working should be shown in every question
Marks middot Question 11 (15 marks) - Start on the appropriate page in your answer booklet
a) bull Simplify 2 x+3 x-2
5 4
2bull b) Solve 13x - 21 = 7
middot c) Factorise 6x 2 + llx -10 2
d) For the function [(x) = x2 - 16
(i) Find the domain and range 2
(ii) Is the function odd even or neither 2
(Justify your answer by showing working)
e) Write a set of inequalities which define the shaded region 2
x
f) Find the solutions of the equation 2 cos 2 e = 3 sin () + 3 for 0deg e 360deg 3
End of Question 11
4
Question 12 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) 3 Simplify
b) Differentiate with respect to x 4
(i) 2 i-x
(ii) 2xy3x + 1 2
2(iii) 4x-l
2x+l
c) Evaluate showing all working x 3 + 8
lim x-gt-2 2x + 4 2
x 2d) Find the equation of the tangent to y = - 4x at the point (1 -3) on it 2
e) Solve 12x-31=4x+1 2
End of Question 12
5
Question 13 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) Find the exact value of cos e given that tan e = 7 and sin e lt O 2
b) Find the value of x 2
Not to scale
12
c) If 6x 2 - 11 == A(x + 2)2 + Bx + C find the values of A Band C 3
d)
e)
For which values of k does the equation x 2 - 2x + 3 shy k = 0 have real roots
Consider the function [(x) = ~X4 - 2x3 + 2 2
(i)
(ii)
Find the coordinates of the stationary points of y = [(x)
Determine the nature of these stationary points
(iii) Sketch the graph of y = [(x) clearly showing the stationary points and
y-intercept (You need not find the x-intercepts)
2
2
2
2
End of Question 13
6
middot Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) The diagram shows the points A(1-2) B(-4-3) and C(-13) and the line Be
Not to scale
x
A(l - 2)
(i) Calculate the exact length of Be 1 (ii) Show that the equation of BC is 2x - y + 5 = o 2 (iii) Find the exact area of triangle ABe 3
b) Solve x 2 - 2x - 1 lt O 3
Question 14 continues on the next page
7
Question 14 (continued)
c) The diagram shows two right cones The small cone of radius PS=r em is enclosed within bull a larger cone of radius QR=5em as shown The larger cone has height 15cm and the small
cone has height hem
T
Not to scale
15 ern
hem
R
(i) Prove that L1TPSIIIL1TQR 2
(ii) Show that h = 15 shy 3r 1
(iii) Find the radius of the small cone such that its volume is a maximum 3
End of Examination
8
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_ _
----
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---- --_- - ---shyolyen -3
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- - - ------ _- ------ - - -- ---- ___f _s _oluhrgts _ ____reg _2
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~ ~ X --middot- - I--------=- -- --- --------- --shy
-------l------~--~ --O- --- - - ---- --- --shy shy
---------------------shy
__f-J- _ liLt -s~ 2-$)-~_b___$ _0__~_ -- _b ~0 __ _~(I) _-
----t~ - 4 S~ 2 e - ~ S_~ __0 - G
c
- -- _ i
lt C I )
or
(I)
-shy Q-l2
( h __ ____oj 3~ -I - ~--(3 ~-X3--+--) CJ ~ chlta _ 1 ~ ~~1D
__ _ - l~_- 21_ ___ _ 2-~(3_ ~L) _ od 0 clvu-OrY ~Cl~ r
~ -
---- - 1---------- - - ------shy
3 h + ---- - - ---_ _-- ------ ---- --i-(--- -C-I ) middot middotCo r recf
2 --- - _ ----__shy
h -I ----~-)- - y--=== - I-G-=-fmiddot--shy
I -lt----- -~ --=--1 ---( ~ x) - --- - - -shy---_ - _ 1 - ---- --zr~- -
____ __ ___t~1 - x) __ _3)-
--- - - 1---- ----====------- -- shy----- - _ I_)__j- 2x J 3x-+-______ _ _1A_~ 2~_ IL --((
(--
2 - shy- - -
I -I~ ______ __ _________ _____ ___ __~ (3~-) 2 _ Y ~- 2(3A+I - 3 __bull _ Jlt _ _~ __ --- --- - - ~- --- ~t)(-X- 3 - - + ~ 3x +--1 - x 2shy
_____ ___Jt-____ AJ3~tJ_____ _ ____ _____ 2t3~+-I_____ _
_ _(J)_~ U k) __-------- ----- -- --- -- - ----- -- - _ -~-3x rroduC+ C 1c+
------_ -_- -------- -p--------------shy
~ -3X+- 1 _ ( I) - _lt-9 r_C~d- _ 95 --- - ---- - --- _-----shy
--7-- - J-Iz ~x l3X--ri)
----__-shy
------___ ----- ---- --_ - shy
- - - - --------1- ----_ _---shy
(Iii) ~ 1 r - k I T
L 2zt-l v Z I
I _ ( -vshy
1 - V 1shy
(Zx~i)4middot - (4-() L - (I)
lt vs~s 1v o+e-+- r-J(( - - -- - lZx Ir~
- g(+--If---E--x 1
_llz -)~ _
-=- -shy(Z~ 1-1) 2 _
I (ff1 (~Z)(pound~ -2x+-t )
x-2 x-) - 2 ___2-xyt 1shy
-- ---- -C~) --=-L(--=- -~)~- - --- shy
------ - - -
------ - -2--- --- shy
___ _ G__
--- ~~-j
--- l-J shy
- 2 __~~---_ J - corC ~ ct- 3c~d I~~
J j l _m tx-- ~C J - 2 (- -f )fT- 3
~ ~ ~
w ~ - 2x shy J -_ - - --- --- --I ~lt------- Ci ) CD rr ~- ~T
e) 2 x - 3 j - 4- J- L
2~ -3 4t-fc I
- _t _ ~K _ 2 ~ bx
~ ~ ~ 3middot shy
LI-l S =- -f - 2~f ~- 2 ~
__ _ _ gt~1 (J cJ Ib-ls - Z IS
_ middot _ f9+_~ _ ~J_Hcgt__ ______ cJ__-
( ) ~r ~ow Ct---J redegj rf -y -=- _
-- - -
-- - _ --~
Co~ 8 =- ~ f~6 - _L) ~r -~
_(1) _for ~d
-- b)shy
_ _ _ (x t-Lf)(t -7J-~ ~2_
- ---x2~-3~middotmiddot--2middot ~--~-o -j- -- -- J lt )
xt~middotx~ middot ~middot~f middot middot-0 middot- - -- ( -- - bullbull _P p - - shy
__kuL~gt 0 _ _
x - I lt raquo lt 2---_---------- -_ - - - -shy
- - - tJ- jJS =_ltix~+_t-X_Y-lf)--+-~b-t-S - -
_ _ __ __ _4x~middotLYA~ +-fA__+_Ex_-plusmn_C
-- - - - - =A(~ -+- -rf~ plusmnB)~J (~A- +lt) L ____ u_ J-tLS - x t- Ox ~l __ _
_ - - - A-2 - - - -- HiG)-J--B ~o_ f) f c- 7 -11 _- R -2 _ 3_~- -- - - ~ -- - _ -- b __ _ - - _ow _ __ C _shy~ ~ - - --- -- -C--) - _- - ---- - --f-) - - -- - -- - -- (1-)
- -
_d) pound9 r __ Ld- LC1~ 7 6 gt0 ___ _ _ _
--- - - A- b--Jtcu -- - - -_- - _ _ ___ __ __ _ =~2)~- t__-L13~_) ___ __lt____ __c )
- - - =--~~-rZ middot =Fccf1--middot --- - - -shy
----- ~ Cfl- g ~ 0-- VJ~ --k -2 --- -lt-- C_ ) --------- bull - ----- - _- - - - -- shy
x-7
u_____~l-
-- - -
--- --
-- - -- -- -------- --
- ~ I )I-or sampo~ltvj -1) _ 2~3 - 0~ _c- 0 _
- --- - - - - -- -2x-_ x -~-3)- - ~- - sj -
--------- - -- --~J - ~-2--- -- -
--- - --------- - ---- - -degr-~)~- --------til- --At-(~2=- - _ _ _ ____ U)
___-=- _ _8_ (9)1- - 1 _O~eC)~gtJ __o ___ _ 2 Co) = _ 4 - -~ J 4
---shy-----shy--shy
(3-~tpound)- --shy -shy___ ___ ___ ____ __j~ _-l3JL -=L2(3)
= _ _
AI
f Ii _
gt0 -- _ _
--- - - -- -- _- ------ shy -~0) ~ ll~)__ _ _ ____
sectJ_--0 __ ___ -- shy 7 -----------shy~------
(2) _
(3-~11 S -) shyV 1 J
---- - ---- ----- - - ----- ----
----
------
------- --- ---- --- - ----
(
- )
b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
___ __ __x ~ _b _-__ JnJ~ - LfCl_C_ __ ___
2co-shy _~ - ---bull--- -- - -- -__- - --- shy
2 - 1(-2)2 - lf~L 2lt I
2 -4- r-g ---- 2 J) - - ----=-~-~-- ---- --- I
2 ) 2 t 212
- ---shy2
____ __~----------~--l ~fi____ -----f-~~- -----shy -- -- - - shy
_ __ _~ _ 2 5) L
-__-- -- - -- - --------------=~ - -
_ ---- _ _ _ --i - f-2 --~ -x --~ --I-T ~-- shyS~JlI ~ Ii ()
-----1 -) -- -In- drER ---d--l _Qs --- - - - - - - - - -- L -TPfl- -- -L-+--Q-s- -C-9lVen - kgt-o-H-1Q ) -
_ ____ _ ___L_T s _CSJtntno~ _______
-- - -- - A _LJlaquojL4IQS_ -(yen-J-cJ -~J - LS -~iucJ)or (Alt)
_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
------- --- - --- --+------ lSi ~-SS~ -- - middot -shy
-shy--shy -- --shy -shyS
------i
cmiddot QILf
- -) - ~______ -
-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
__=-_--_-_-~__-=__~1_(_~~middotJL +_ 3__plusmn_~_)~_ ___
--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
_ -- - ------shy
- - -- -__-_ - -- _-------__---- -shy12 -~
_ __ M _ _ __ __ _ _
Xz-x - 3+3
-----shy-I+lt
____ --- ~U)_- ~ ------_shy
- - - -__---- -- --- _-- 2-__ _e~ _s_~- r()~5~ (~II~)
- - ~- ~1f -~ _~Lx--~ X ~
----- -- - ---- 13-2 -tCt 1) ~lt-----_ ( j )
j - - 2x L - - ------ -- --- - 5 - --
__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
_ _ _ __ ___ __ _ _~~_I l ~ [ )
____Imiddot__l~~_)-d = 2(1 )-(-2) S _ _= _ ~ ~ _ __ _ _ ___ _ __ ____J2~_tc~OL _ _ __ ~_ _
--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
- --- - 1- - ------ ----- - - -- -----
_ ___ _ middot_Vll =_3oTr -=-Q_1L(ZL)_ _ _____ ______ __ =--3_oTJ~--lgJJI------- ------
J~ shy
___---bTT~_IgJ[--fi-)- ------ --- -shy_ ~ 39T( ~6Qrr __ ___ _____ ____ _
lt 0 as 10 - ---- T--reYVTtmiddotdJ - 1- - --- -- r---3 Cfl1 middotmiddot 9vlZ J
- - _ - - --- - -- ~) --- - - -- -- shy
- - -- --- ---- -- - shy
-- ---I
-------_
- --_ -_ ------ --- shy
----__ - ------ -- -- shy
-- - _ - - __------- --
---_- ---- - --- - - -- - --- - - - - - --shy
_-------- shy
7 The solutions of v3tan x +1 = 0 for the domain 0deg 5 x 5 360deg are
(A) x = 30deg210deg (8) x = 150deg210deg
middot (C) x = 150deg330deg (D) x = 210deg330deg
8
w
A L-~a~________________ v C
In the diagram above ACD is a straight angle Which of the following statements is true
w2 2middot (A) = u 2 + v - 2uvcosfJ
2 2 + v(B) w = u 2 + 2uvcosfJ
2 2 + w 2(C) u = v - 2vwcosfJ
middot (D)~=~sina casp
119
B
L-~_______ D
+-______
2
~--~--+---r_--~------+x
-2
The diagram above shows the function y = [(x)
A student wishes to solve the equation [(x) = 2 - Ixl graphically The correct number of solutions
is
(A) (B) 2 (C) 3 (D) 4
10 Given [(x + 1) = x 2 - X + 1 then [(x) in its simplest form is
(A) x 2 + x + 1 (B) x 2 + x + 3
(C) x 2 - 3x + 3 (D) x 2
- 3x + 1
End of Section I
3
---
Section II - Extended Response All necessary working should be shown in every question
Marks middot Question 11 (15 marks) - Start on the appropriate page in your answer booklet
a) bull Simplify 2 x+3 x-2
5 4
2bull b) Solve 13x - 21 = 7
middot c) Factorise 6x 2 + llx -10 2
d) For the function [(x) = x2 - 16
(i) Find the domain and range 2
(ii) Is the function odd even or neither 2
(Justify your answer by showing working)
e) Write a set of inequalities which define the shaded region 2
x
f) Find the solutions of the equation 2 cos 2 e = 3 sin () + 3 for 0deg e 360deg 3
End of Question 11
4
Question 12 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) 3 Simplify
b) Differentiate with respect to x 4
(i) 2 i-x
(ii) 2xy3x + 1 2
2(iii) 4x-l
2x+l
c) Evaluate showing all working x 3 + 8
lim x-gt-2 2x + 4 2
x 2d) Find the equation of the tangent to y = - 4x at the point (1 -3) on it 2
e) Solve 12x-31=4x+1 2
End of Question 12
5
Question 13 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) Find the exact value of cos e given that tan e = 7 and sin e lt O 2
b) Find the value of x 2
Not to scale
12
c) If 6x 2 - 11 == A(x + 2)2 + Bx + C find the values of A Band C 3
d)
e)
For which values of k does the equation x 2 - 2x + 3 shy k = 0 have real roots
Consider the function [(x) = ~X4 - 2x3 + 2 2
(i)
(ii)
Find the coordinates of the stationary points of y = [(x)
Determine the nature of these stationary points
(iii) Sketch the graph of y = [(x) clearly showing the stationary points and
y-intercept (You need not find the x-intercepts)
2
2
2
2
End of Question 13
6
middot Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) The diagram shows the points A(1-2) B(-4-3) and C(-13) and the line Be
Not to scale
x
A(l - 2)
(i) Calculate the exact length of Be 1 (ii) Show that the equation of BC is 2x - y + 5 = o 2 (iii) Find the exact area of triangle ABe 3
b) Solve x 2 - 2x - 1 lt O 3
Question 14 continues on the next page
7
Question 14 (continued)
c) The diagram shows two right cones The small cone of radius PS=r em is enclosed within bull a larger cone of radius QR=5em as shown The larger cone has height 15cm and the small
cone has height hem
T
Not to scale
15 ern
hem
R
(i) Prove that L1TPSIIIL1TQR 2
(ii) Show that h = 15 shy 3r 1
(iii) Find the radius of the small cone such that its volume is a maximum 3
End of Examination
8
--
_ _
----
== - 15 _ 3 bull - -ci -~ -- ---7---- ~--3--
---- --_- - ---shyolyen -3
- - $= ------- --- -- _-- - - __ ~DpoundCvJ _ ~ 2 ~--tJ 2
~ --E~ ~f) --+ 0713 --- -- - _f_ - F -C1f _ 1fj~S~Jo~ shy~~
- - - ------ _- ------ - - -- ---- ___f _s _oluhrgts _ ____reg _2
_= (~3) plusmn ( =~ -)shy10 _ f(x4-t2~ X t-lrD-shy
-fl-x _) _ =___X_)2 - _(X -)+ I ~ --x1--_ -2X4-1 -X f 1- 4 -1
- -Xl- 3i + 3
----- - -~- =--E- -~~Ji
~ e-1-
---S-2 $6Ht __
12- 3
IS If S I
___-=-- ~ _ J2~~(~s-_~~_~__ _tJ~) lt0 11 i
____ -coc cto~_middot~ -fl((~) lt 0 -reg----- - shy
----_ - --- --- shy
--shy
~f3 2u~o Go gt~
Ict6 - -J - --- - _ ~r _ ~- = _ ~_ - ----- shy -- -- - ~G~LotJuL L _~ _3~ ) _
ISO 330 -- - - ) - - - --- - shy
-reg--- --shy- -g
---- -------5 ~~---(bs8
-- - - - I ~~
- - -=-S--S ~ltS--~ 6--shy
- - ------ -- shy- a-e
_ _ _ _
c amp11
lX-r3 J -5(7--~2) - )x-2 -_- -_ - -_ _ltlt--- -( -- --- ----middot-1 middot-shy4 20
~t 1- (2 _- 5_~+lQ
20
0( - 22- s _ -~lt______- 0 )
_ __ _ __ __3_~ _- __4 _~ ___ Qf__ __-=32C- 12 -= - _
__2t =- ~ __ ~ _ _ ___ _l-_- _ - 3xS__
)L ~ 3 (= _LI_) X ~-f i
--- -- ---- ---shy
__=-_
-shy --shy--shy __ -shy _ _
V lt _li -v ~ f ~~_ c )__ ___ - ---~~ ) -~q-- s shy-- ~ ---
-~-- -_ _--- ------ ---_- - shy_ ___--+-____ amp 1
--middot-------ilmiddot----~--========--
-------(1-)- pound (~)-=-J oX 2_ -_ _
_-poundr~ )~J-(~~~- shy--- --- - --------==- J- X-l - 1-(
-- -- --- --- --- -- --F(-~r)
--------- -- ---shy_______
J-4------( r )-- - -
---- ----- -shy
--~------ --------- -------- --shy
---middot=1l-1~ ---7-~ -f=-J 2- $---- -- 01- ---J -s--J~--t-~- - - - - ---- ----- - - - -- _ shy---------------=---~
~ ~ X --middot- - I--------=- -- --- --------- --shy
-------l------~--~ --O- --- - - ---- --- --shy shy
---------------------shy
__f-J- _ liLt -s~ 2-$)-~_b___$ _0__~_ -- _b ~0 __ _~(I) _-
----t~ - 4 S~ 2 e - ~ S_~ __0 - G
c
- -- _ i
lt C I )
or
(I)
-shy Q-l2
( h __ ____oj 3~ -I - ~--(3 ~-X3--+--) CJ ~ chlta _ 1 ~ ~~1D
__ _ - l~_- 21_ ___ _ 2-~(3_ ~L) _ od 0 clvu-OrY ~Cl~ r
~ -
---- - 1---------- - - ------shy
3 h + ---- - - ---_ _-- ------ ---- --i-(--- -C-I ) middot middotCo r recf
2 --- - _ ----__shy
h -I ----~-)- - y--=== - I-G-=-fmiddot--shy
I -lt----- -~ --=--1 ---( ~ x) - --- - - -shy---_ - _ 1 - ---- --zr~- -
____ __ ___t~1 - x) __ _3)-
--- - - 1---- ----====------- -- shy----- - _ I_)__j- 2x J 3x-+-______ _ _1A_~ 2~_ IL --((
(--
2 - shy- - -
I -I~ ______ __ _________ _____ ___ __~ (3~-) 2 _ Y ~- 2(3A+I - 3 __bull _ Jlt _ _~ __ --- --- - - ~- --- ~t)(-X- 3 - - + ~ 3x +--1 - x 2shy
_____ ___Jt-____ AJ3~tJ_____ _ ____ _____ 2t3~+-I_____ _
_ _(J)_~ U k) __-------- ----- -- --- -- - ----- -- - _ -~-3x rroduC+ C 1c+
------_ -_- -------- -p--------------shy
~ -3X+- 1 _ ( I) - _lt-9 r_C~d- _ 95 --- - ---- - --- _-----shy
--7-- - J-Iz ~x l3X--ri)
----__-shy
------___ ----- ---- --_ - shy
- - - - --------1- ----_ _---shy
(Iii) ~ 1 r - k I T
L 2zt-l v Z I
I _ ( -vshy
1 - V 1shy
(Zx~i)4middot - (4-() L - (I)
lt vs~s 1v o+e-+- r-J(( - - -- - lZx Ir~
- g(+--If---E--x 1
_llz -)~ _
-=- -shy(Z~ 1-1) 2 _
I (ff1 (~Z)(pound~ -2x+-t )
x-2 x-) - 2 ___2-xyt 1shy
-- ---- -C~) --=-L(--=- -~)~- - --- shy
------ - - -
------ - -2--- --- shy
___ _ G__
--- ~~-j
--- l-J shy
- 2 __~~---_ J - corC ~ ct- 3c~d I~~
J j l _m tx-- ~C J - 2 (- -f )fT- 3
~ ~ ~
w ~ - 2x shy J -_ - - --- --- --I ~lt------- Ci ) CD rr ~- ~T
e) 2 x - 3 j - 4- J- L
2~ -3 4t-fc I
- _t _ ~K _ 2 ~ bx
~ ~ ~ 3middot shy
LI-l S =- -f - 2~f ~- 2 ~
__ _ _ gt~1 (J cJ Ib-ls - Z IS
_ middot _ f9+_~ _ ~J_Hcgt__ ______ cJ__-
( ) ~r ~ow Ct---J redegj rf -y -=- _
-- - -
-- - _ --~
Co~ 8 =- ~ f~6 - _L) ~r -~
_(1) _for ~d
-- b)shy
_ _ _ (x t-Lf)(t -7J-~ ~2_
- ---x2~-3~middotmiddot--2middot ~--~-o -j- -- -- J lt )
xt~middotx~ middot ~middot~f middot middot-0 middot- - -- ( -- - bullbull _P p - - shy
__kuL~gt 0 _ _
x - I lt raquo lt 2---_---------- -_ - - - -shy
- - - tJ- jJS =_ltix~+_t-X_Y-lf)--+-~b-t-S - -
_ _ __ __ _4x~middotLYA~ +-fA__+_Ex_-plusmn_C
-- - - - - =A(~ -+- -rf~ plusmnB)~J (~A- +lt) L ____ u_ J-tLS - x t- Ox ~l __ _
_ - - - A-2 - - - -- HiG)-J--B ~o_ f) f c- 7 -11 _- R -2 _ 3_~- -- - - ~ -- - _ -- b __ _ - - _ow _ __ C _shy~ ~ - - --- -- -C--) - _- - ---- - --f-) - - -- - -- - -- (1-)
- -
_d) pound9 r __ Ld- LC1~ 7 6 gt0 ___ _ _ _
--- - - A- b--Jtcu -- - - -_- - _ _ ___ __ __ _ =~2)~- t__-L13~_) ___ __lt____ __c )
- - - =--~~-rZ middot =Fccf1--middot --- - - -shy
----- ~ Cfl- g ~ 0-- VJ~ --k -2 --- -lt-- C_ ) --------- bull - ----- - _- - - - -- shy
x-7
u_____~l-
-- - -
--- --
-- - -- -- -------- --
- ~ I )I-or sampo~ltvj -1) _ 2~3 - 0~ _c- 0 _
- --- - - - - -- -2x-_ x -~-3)- - ~- - sj -
--------- - -- --~J - ~-2--- -- -
--- - --------- - ---- - -degr-~)~- --------til- --At-(~2=- - _ _ _ ____ U)
___-=- _ _8_ (9)1- - 1 _O~eC)~gtJ __o ___ _ 2 Co) = _ 4 - -~ J 4
---shy-----shy--shy
(3-~tpound)- --shy -shy___ ___ ___ ____ __j~ _-l3JL -=L2(3)
= _ _
AI
f Ii _
gt0 -- _ _
--- - - -- -- _- ------ shy -~0) ~ ll~)__ _ _ ____
sectJ_--0 __ ___ -- shy 7 -----------shy~------
(2) _
(3-~11 S -) shyV 1 J
---- - ---- ----- - - ----- ----
----
------
------- --- ---- --- - ----
(
- )
b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
___ __ __x ~ _b _-__ JnJ~ - LfCl_C_ __ ___
2co-shy _~ - ---bull--- -- - -- -__- - --- shy
2 - 1(-2)2 - lf~L 2lt I
2 -4- r-g ---- 2 J) - - ----=-~-~-- ---- --- I
2 ) 2 t 212
- ---shy2
____ __~----------~--l ~fi____ -----f-~~- -----shy -- -- - - shy
_ __ _~ _ 2 5) L
-__-- -- - -- - --------------=~ - -
_ ---- _ _ _ --i - f-2 --~ -x --~ --I-T ~-- shyS~JlI ~ Ii ()
-----1 -) -- -In- drER ---d--l _Qs --- - - - - - - - - -- L -TPfl- -- -L-+--Q-s- -C-9lVen - kgt-o-H-1Q ) -
_ ____ _ ___L_T s _CSJtntno~ _______
-- - -- - A _LJlaquojL4IQS_ -(yen-J-cJ -~J - LS -~iucJ)or (Alt)
_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
------- --- - --- --+------ lSi ~-SS~ -- - middot -shy
-shy--shy -- --shy -shyS
------i
cmiddot QILf
- -) - ~______ -
-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
__=-_--_-_-~__-=__~1_(_~~middotJL +_ 3__plusmn_~_)~_ ___
--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
_ -- - ------shy
- - -- -__-_ - -- _-------__---- -shy12 -~
_ __ M _ _ __ __ _ _
Xz-x - 3+3
-----shy-I+lt
____ --- ~U)_- ~ ------_shy
- - - -__---- -- --- _-- 2-__ _e~ _s_~- r()~5~ (~II~)
- - ~- ~1f -~ _~Lx--~ X ~
----- -- - ---- 13-2 -tCt 1) ~lt-----_ ( j )
j - - 2x L - - ------ -- --- - 5 - --
__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
_ _ _ __ ___ __ _ _~~_I l ~ [ )
____Imiddot__l~~_)-d = 2(1 )-(-2) S _ _= _ ~ ~ _ __ _ _ ___ _ __ ____J2~_tc~OL _ _ __ ~_ _
--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
- --- - 1- - ------ ----- - - -- -----
_ ___ _ middot_Vll =_3oTr -=-Q_1L(ZL)_ _ _____ ______ __ =--3_oTJ~--lgJJI------- ------
J~ shy
___---bTT~_IgJ[--fi-)- ------ --- -shy_ ~ 39T( ~6Qrr __ ___ _____ ____ _
lt 0 as 10 - ---- T--reYVTtmiddotdJ - 1- - --- -- r---3 Cfl1 middotmiddot 9vlZ J
- - _ - - --- - -- ~) --- - - -- -- shy
- - -- --- ---- -- - shy
-- ---I
-------_
- --_ -_ ------ --- shy
----__ - ------ -- -- shy
-- - _ - - __------- --
---_- ---- - --- - - -- - --- - - - - - --shy
_-------- shy
---
Section II - Extended Response All necessary working should be shown in every question
Marks middot Question 11 (15 marks) - Start on the appropriate page in your answer booklet
a) bull Simplify 2 x+3 x-2
5 4
2bull b) Solve 13x - 21 = 7
middot c) Factorise 6x 2 + llx -10 2
d) For the function [(x) = x2 - 16
(i) Find the domain and range 2
(ii) Is the function odd even or neither 2
(Justify your answer by showing working)
e) Write a set of inequalities which define the shaded region 2
x
f) Find the solutions of the equation 2 cos 2 e = 3 sin () + 3 for 0deg e 360deg 3
End of Question 11
4
Question 12 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) 3 Simplify
b) Differentiate with respect to x 4
(i) 2 i-x
(ii) 2xy3x + 1 2
2(iii) 4x-l
2x+l
c) Evaluate showing all working x 3 + 8
lim x-gt-2 2x + 4 2
x 2d) Find the equation of the tangent to y = - 4x at the point (1 -3) on it 2
e) Solve 12x-31=4x+1 2
End of Question 12
5
Question 13 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) Find the exact value of cos e given that tan e = 7 and sin e lt O 2
b) Find the value of x 2
Not to scale
12
c) If 6x 2 - 11 == A(x + 2)2 + Bx + C find the values of A Band C 3
d)
e)
For which values of k does the equation x 2 - 2x + 3 shy k = 0 have real roots
Consider the function [(x) = ~X4 - 2x3 + 2 2
(i)
(ii)
Find the coordinates of the stationary points of y = [(x)
Determine the nature of these stationary points
(iii) Sketch the graph of y = [(x) clearly showing the stationary points and
y-intercept (You need not find the x-intercepts)
2
2
2
2
End of Question 13
6
middot Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) The diagram shows the points A(1-2) B(-4-3) and C(-13) and the line Be
Not to scale
x
A(l - 2)
(i) Calculate the exact length of Be 1 (ii) Show that the equation of BC is 2x - y + 5 = o 2 (iii) Find the exact area of triangle ABe 3
b) Solve x 2 - 2x - 1 lt O 3
Question 14 continues on the next page
7
Question 14 (continued)
c) The diagram shows two right cones The small cone of radius PS=r em is enclosed within bull a larger cone of radius QR=5em as shown The larger cone has height 15cm and the small
cone has height hem
T
Not to scale
15 ern
hem
R
(i) Prove that L1TPSIIIL1TQR 2
(ii) Show that h = 15 shy 3r 1
(iii) Find the radius of the small cone such that its volume is a maximum 3
End of Examination
8
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_ _
----
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- - - ------ _- ------ - - -- ---- ___f _s _oluhrgts _ ____reg _2
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_____ ___Jt-____ AJ3~tJ_____ _ ____ _____ 2t3~+-I_____ _
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LI-l S =- -f - 2~f ~- 2 ~
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_ middot _ f9+_~ _ ~J_Hcgt__ ______ cJ__-
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--- - -------
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Question 12 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) 3 Simplify
b) Differentiate with respect to x 4
(i) 2 i-x
(ii) 2xy3x + 1 2
2(iii) 4x-l
2x+l
c) Evaluate showing all working x 3 + 8
lim x-gt-2 2x + 4 2
x 2d) Find the equation of the tangent to y = - 4x at the point (1 -3) on it 2
e) Solve 12x-31=4x+1 2
End of Question 12
5
Question 13 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) Find the exact value of cos e given that tan e = 7 and sin e lt O 2
b) Find the value of x 2
Not to scale
12
c) If 6x 2 - 11 == A(x + 2)2 + Bx + C find the values of A Band C 3
d)
e)
For which values of k does the equation x 2 - 2x + 3 shy k = 0 have real roots
Consider the function [(x) = ~X4 - 2x3 + 2 2
(i)
(ii)
Find the coordinates of the stationary points of y = [(x)
Determine the nature of these stationary points
(iii) Sketch the graph of y = [(x) clearly showing the stationary points and
y-intercept (You need not find the x-intercepts)
2
2
2
2
End of Question 13
6
middot Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) The diagram shows the points A(1-2) B(-4-3) and C(-13) and the line Be
Not to scale
x
A(l - 2)
(i) Calculate the exact length of Be 1 (ii) Show that the equation of BC is 2x - y + 5 = o 2 (iii) Find the exact area of triangle ABe 3
b) Solve x 2 - 2x - 1 lt O 3
Question 14 continues on the next page
7
Question 14 (continued)
c) The diagram shows two right cones The small cone of radius PS=r em is enclosed within bull a larger cone of radius QR=5em as shown The larger cone has height 15cm and the small
cone has height hem
T
Not to scale
15 ern
hem
R
(i) Prove that L1TPSIIIL1TQR 2
(ii) Show that h = 15 shy 3r 1
(iii) Find the radius of the small cone such that its volume is a maximum 3
End of Examination
8
--
_ _
----
== - 15 _ 3 bull - -ci -~ -- ---7---- ~--3--
---- --_- - ---shyolyen -3
- - $= ------- --- -- _-- - - __ ~DpoundCvJ _ ~ 2 ~--tJ 2
~ --E~ ~f) --+ 0713 --- -- - _f_ - F -C1f _ 1fj~S~Jo~ shy~~
- - - ------ _- ------ - - -- ---- ___f _s _oluhrgts _ ____reg _2
_= (~3) plusmn ( =~ -)shy10 _ f(x4-t2~ X t-lrD-shy
-fl-x _) _ =___X_)2 - _(X -)+ I ~ --x1--_ -2X4-1 -X f 1- 4 -1
- -Xl- 3i + 3
----- - -~- =--E- -~~Ji
~ e-1-
---S-2 $6Ht __
12- 3
IS If S I
___-=-- ~ _ J2~~(~s-_~~_~__ _tJ~) lt0 11 i
____ -coc cto~_middot~ -fl((~) lt 0 -reg----- - shy
----_ - --- --- shy
--shy
~f3 2u~o Go gt~
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ISO 330 -- - - ) - - - --- - shy
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---- -------5 ~~---(bs8
-- - - - I ~~
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_ _ _ _
c amp11
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or
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- -
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= _ _
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b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
___ __ __x ~ _b _-__ JnJ~ - LfCl_C_ __ ___
2co-shy _~ - ---bull--- -- - -- -__- - --- shy
2 - 1(-2)2 - lf~L 2lt I
2 -4- r-g ---- 2 J) - - ----=-~-~-- ---- --- I
2 ) 2 t 212
- ---shy2
____ __~----------~--l ~fi____ -----f-~~- -----shy -- -- - - shy
_ __ _~ _ 2 5) L
-__-- -- - -- - --------------=~ - -
_ ---- _ _ _ --i - f-2 --~ -x --~ --I-T ~-- shyS~JlI ~ Ii ()
-----1 -) -- -In- drER ---d--l _Qs --- - - - - - - - - -- L -TPfl- -- -L-+--Q-s- -C-9lVen - kgt-o-H-1Q ) -
_ ____ _ ___L_T s _CSJtntno~ _______
-- - -- - A _LJlaquojL4IQS_ -(yen-J-cJ -~J - LS -~iucJ)or (Alt)
_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
------- --- - --- --+------ lSi ~-SS~ -- - middot -shy
-shy--shy -- --shy -shyS
------i
cmiddot QILf
- -) - ~______ -
-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
__=-_--_-_-~__-=__~1_(_~~middotJL +_ 3__plusmn_~_)~_ ___
--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
_ -- - ------shy
- - -- -__-_ - -- _-------__---- -shy12 -~
_ __ M _ _ __ __ _ _
Xz-x - 3+3
-----shy-I+lt
____ --- ~U)_- ~ ------_shy
- - - -__---- -- --- _-- 2-__ _e~ _s_~- r()~5~ (~II~)
- - ~- ~1f -~ _~Lx--~ X ~
----- -- - ---- 13-2 -tCt 1) ~lt-----_ ( j )
j - - 2x L - - ------ -- --- - 5 - --
__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
_ _ _ __ ___ __ _ _~~_I l ~ [ )
____Imiddot__l~~_)-d = 2(1 )-(-2) S _ _= _ ~ ~ _ __ _ _ ___ _ __ ____J2~_tc~OL _ _ __ ~_ _
--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
- --- - 1- - ------ ----- - - -- -----
_ ___ _ middot_Vll =_3oTr -=-Q_1L(ZL)_ _ _____ ______ __ =--3_oTJ~--lgJJI------- ------
J~ shy
___---bTT~_IgJ[--fi-)- ------ --- -shy_ ~ 39T( ~6Qrr __ ___ _____ ____ _
lt 0 as 10 - ---- T--reYVTtmiddotdJ - 1- - --- -- r---3 Cfl1 middotmiddot 9vlZ J
- - _ - - --- - -- ~) --- - - -- -- shy
- - -- --- ---- -- - shy
-- ---I
-------_
- --_ -_ ------ --- shy
----__ - ------ -- -- shy
-- - _ - - __------- --
---_- ---- - --- - - -- - --- - - - - - --shy
_-------- shy
Question 13 (15 marks) - Start on the appropriate page in your answer booklet Marks
a) Find the exact value of cos e given that tan e = 7 and sin e lt O 2
b) Find the value of x 2
Not to scale
12
c) If 6x 2 - 11 == A(x + 2)2 + Bx + C find the values of A Band C 3
d)
e)
For which values of k does the equation x 2 - 2x + 3 shy k = 0 have real roots
Consider the function [(x) = ~X4 - 2x3 + 2 2
(i)
(ii)
Find the coordinates of the stationary points of y = [(x)
Determine the nature of these stationary points
(iii) Sketch the graph of y = [(x) clearly showing the stationary points and
y-intercept (You need not find the x-intercepts)
2
2
2
2
End of Question 13
6
middot Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) The diagram shows the points A(1-2) B(-4-3) and C(-13) and the line Be
Not to scale
x
A(l - 2)
(i) Calculate the exact length of Be 1 (ii) Show that the equation of BC is 2x - y + 5 = o 2 (iii) Find the exact area of triangle ABe 3
b) Solve x 2 - 2x - 1 lt O 3
Question 14 continues on the next page
7
Question 14 (continued)
c) The diagram shows two right cones The small cone of radius PS=r em is enclosed within bull a larger cone of radius QR=5em as shown The larger cone has height 15cm and the small
cone has height hem
T
Not to scale
15 ern
hem
R
(i) Prove that L1TPSIIIL1TQR 2
(ii) Show that h = 15 shy 3r 1
(iii) Find the radius of the small cone such that its volume is a maximum 3
End of Examination
8
--
_ _
----
== - 15 _ 3 bull - -ci -~ -- ---7---- ~--3--
---- --_- - ---shyolyen -3
- - $= ------- --- -- _-- - - __ ~DpoundCvJ _ ~ 2 ~--tJ 2
~ --E~ ~f) --+ 0713 --- -- - _f_ - F -C1f _ 1fj~S~Jo~ shy~~
- - - ------ _- ------ - - -- ---- ___f _s _oluhrgts _ ____reg _2
_= (~3) plusmn ( =~ -)shy10 _ f(x4-t2~ X t-lrD-shy
-fl-x _) _ =___X_)2 - _(X -)+ I ~ --x1--_ -2X4-1 -X f 1- 4 -1
- -Xl- 3i + 3
----- - -~- =--E- -~~Ji
~ e-1-
---S-2 $6Ht __
12- 3
IS If S I
___-=-- ~ _ J2~~(~s-_~~_~__ _tJ~) lt0 11 i
____ -coc cto~_middot~ -fl((~) lt 0 -reg----- - shy
----_ - --- --- shy
--shy
~f3 2u~o Go gt~
Ict6 - -J - --- - _ ~r _ ~- = _ ~_ - ----- shy -- -- - ~G~LotJuL L _~ _3~ ) _
ISO 330 -- - - ) - - - --- - shy
-reg--- --shy- -g
---- -------5 ~~---(bs8
-- - - - I ~~
- - -=-S--S ~ltS--~ 6--shy
- - ------ -- shy- a-e
_ _ _ _
c amp11
lX-r3 J -5(7--~2) - )x-2 -_- -_ - -_ _ltlt--- -( -- --- ----middot-1 middot-shy4 20
~t 1- (2 _- 5_~+lQ
20
0( - 22- s _ -~lt______- 0 )
_ __ _ __ __3_~ _- __4 _~ ___ Qf__ __-=32C- 12 -= - _
__2t =- ~ __ ~ _ _ ___ _l-_- _ - 3xS__
)L ~ 3 (= _LI_) X ~-f i
--- -- ---- ---shy
__=-_
-shy --shy--shy __ -shy _ _
V lt _li -v ~ f ~~_ c )__ ___ - ---~~ ) -~q-- s shy-- ~ ---
-~-- -_ _--- ------ ---_- - shy_ ___--+-____ amp 1
--middot-------ilmiddot----~--========--
-------(1-)- pound (~)-=-J oX 2_ -_ _
_-poundr~ )~J-(~~~- shy--- --- - --------==- J- X-l - 1-(
-- -- --- --- --- -- --F(-~r)
--------- -- ---shy_______
J-4------( r )-- - -
---- ----- -shy
--~------ --------- -------- --shy
---middot=1l-1~ ---7-~ -f=-J 2- $---- -- 01- ---J -s--J~--t-~- - - - - ---- ----- - - - -- _ shy---------------=---~
~ ~ X --middot- - I--------=- -- --- --------- --shy
-------l------~--~ --O- --- - - ---- --- --shy shy
---------------------shy
__f-J- _ liLt -s~ 2-$)-~_b___$ _0__~_ -- _b ~0 __ _~(I) _-
----t~ - 4 S~ 2 e - ~ S_~ __0 - G
c
- -- _ i
lt C I )
or
(I)
-shy Q-l2
( h __ ____oj 3~ -I - ~--(3 ~-X3--+--) CJ ~ chlta _ 1 ~ ~~1D
__ _ - l~_- 21_ ___ _ 2-~(3_ ~L) _ od 0 clvu-OrY ~Cl~ r
~ -
---- - 1---------- - - ------shy
3 h + ---- - - ---_ _-- ------ ---- --i-(--- -C-I ) middot middotCo r recf
2 --- - _ ----__shy
h -I ----~-)- - y--=== - I-G-=-fmiddot--shy
I -lt----- -~ --=--1 ---( ~ x) - --- - - -shy---_ - _ 1 - ---- --zr~- -
____ __ ___t~1 - x) __ _3)-
--- - - 1---- ----====------- -- shy----- - _ I_)__j- 2x J 3x-+-______ _ _1A_~ 2~_ IL --((
(--
2 - shy- - -
I -I~ ______ __ _________ _____ ___ __~ (3~-) 2 _ Y ~- 2(3A+I - 3 __bull _ Jlt _ _~ __ --- --- - - ~- --- ~t)(-X- 3 - - + ~ 3x +--1 - x 2shy
_____ ___Jt-____ AJ3~tJ_____ _ ____ _____ 2t3~+-I_____ _
_ _(J)_~ U k) __-------- ----- -- --- -- - ----- -- - _ -~-3x rroduC+ C 1c+
------_ -_- -------- -p--------------shy
~ -3X+- 1 _ ( I) - _lt-9 r_C~d- _ 95 --- - ---- - --- _-----shy
--7-- - J-Iz ~x l3X--ri)
----__-shy
------___ ----- ---- --_ - shy
- - - - --------1- ----_ _---shy
(Iii) ~ 1 r - k I T
L 2zt-l v Z I
I _ ( -vshy
1 - V 1shy
(Zx~i)4middot - (4-() L - (I)
lt vs~s 1v o+e-+- r-J(( - - -- - lZx Ir~
- g(+--If---E--x 1
_llz -)~ _
-=- -shy(Z~ 1-1) 2 _
I (ff1 (~Z)(pound~ -2x+-t )
x-2 x-) - 2 ___2-xyt 1shy
-- ---- -C~) --=-L(--=- -~)~- - --- shy
------ - - -
------ - -2--- --- shy
___ _ G__
--- ~~-j
--- l-J shy
- 2 __~~---_ J - corC ~ ct- 3c~d I~~
J j l _m tx-- ~C J - 2 (- -f )fT- 3
~ ~ ~
w ~ - 2x shy J -_ - - --- --- --I ~lt------- Ci ) CD rr ~- ~T
e) 2 x - 3 j - 4- J- L
2~ -3 4t-fc I
- _t _ ~K _ 2 ~ bx
~ ~ ~ 3middot shy
LI-l S =- -f - 2~f ~- 2 ~
__ _ _ gt~1 (J cJ Ib-ls - Z IS
_ middot _ f9+_~ _ ~J_Hcgt__ ______ cJ__-
( ) ~r ~ow Ct---J redegj rf -y -=- _
-- - -
-- - _ --~
Co~ 8 =- ~ f~6 - _L) ~r -~
_(1) _for ~d
-- b)shy
_ _ _ (x t-Lf)(t -7J-~ ~2_
- ---x2~-3~middotmiddot--2middot ~--~-o -j- -- -- J lt )
xt~middotx~ middot ~middot~f middot middot-0 middot- - -- ( -- - bullbull _P p - - shy
__kuL~gt 0 _ _
x - I lt raquo lt 2---_---------- -_ - - - -shy
- - - tJ- jJS =_ltix~+_t-X_Y-lf)--+-~b-t-S - -
_ _ __ __ _4x~middotLYA~ +-fA__+_Ex_-plusmn_C
-- - - - - =A(~ -+- -rf~ plusmnB)~J (~A- +lt) L ____ u_ J-tLS - x t- Ox ~l __ _
_ - - - A-2 - - - -- HiG)-J--B ~o_ f) f c- 7 -11 _- R -2 _ 3_~- -- - - ~ -- - _ -- b __ _ - - _ow _ __ C _shy~ ~ - - --- -- -C--) - _- - ---- - --f-) - - -- - -- - -- (1-)
- -
_d) pound9 r __ Ld- LC1~ 7 6 gt0 ___ _ _ _
--- - - A- b--Jtcu -- - - -_- - _ _ ___ __ __ _ =~2)~- t__-L13~_) ___ __lt____ __c )
- - - =--~~-rZ middot =Fccf1--middot --- - - -shy
----- ~ Cfl- g ~ 0-- VJ~ --k -2 --- -lt-- C_ ) --------- bull - ----- - _- - - - -- shy
x-7
u_____~l-
-- - -
--- --
-- - -- -- -------- --
- ~ I )I-or sampo~ltvj -1) _ 2~3 - 0~ _c- 0 _
- --- - - - - -- -2x-_ x -~-3)- - ~- - sj -
--------- - -- --~J - ~-2--- -- -
--- - --------- - ---- - -degr-~)~- --------til- --At-(~2=- - _ _ _ ____ U)
___-=- _ _8_ (9)1- - 1 _O~eC)~gtJ __o ___ _ 2 Co) = _ 4 - -~ J 4
---shy-----shy--shy
(3-~tpound)- --shy -shy___ ___ ___ ____ __j~ _-l3JL -=L2(3)
= _ _
AI
f Ii _
gt0 -- _ _
--- - - -- -- _- ------ shy -~0) ~ ll~)__ _ _ ____
sectJ_--0 __ ___ -- shy 7 -----------shy~------
(2) _
(3-~11 S -) shyV 1 J
---- - ---- ----- - - ----- ----
----
------
------- --- ---- --- - ----
(
- )
b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
___ __ __x ~ _b _-__ JnJ~ - LfCl_C_ __ ___
2co-shy _~ - ---bull--- -- - -- -__- - --- shy
2 - 1(-2)2 - lf~L 2lt I
2 -4- r-g ---- 2 J) - - ----=-~-~-- ---- --- I
2 ) 2 t 212
- ---shy2
____ __~----------~--l ~fi____ -----f-~~- -----shy -- -- - - shy
_ __ _~ _ 2 5) L
-__-- -- - -- - --------------=~ - -
_ ---- _ _ _ --i - f-2 --~ -x --~ --I-T ~-- shyS~JlI ~ Ii ()
-----1 -) -- -In- drER ---d--l _Qs --- - - - - - - - - -- L -TPfl- -- -L-+--Q-s- -C-9lVen - kgt-o-H-1Q ) -
_ ____ _ ___L_T s _CSJtntno~ _______
-- - -- - A _LJlaquojL4IQS_ -(yen-J-cJ -~J - LS -~iucJ)or (Alt)
_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
------- --- - --- --+------ lSi ~-SS~ -- - middot -shy
-shy--shy -- --shy -shyS
------i
cmiddot QILf
- -) - ~______ -
-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
__=-_--_-_-~__-=__~1_(_~~middotJL +_ 3__plusmn_~_)~_ ___
--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
_ -- - ------shy
- - -- -__-_ - -- _-------__---- -shy12 -~
_ __ M _ _ __ __ _ _
Xz-x - 3+3
-----shy-I+lt
____ --- ~U)_- ~ ------_shy
- - - -__---- -- --- _-- 2-__ _e~ _s_~- r()~5~ (~II~)
- - ~- ~1f -~ _~Lx--~ X ~
----- -- - ---- 13-2 -tCt 1) ~lt-----_ ( j )
j - - 2x L - - ------ -- --- - 5 - --
__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
_ _ _ __ ___ __ _ _~~_I l ~ [ )
____Imiddot__l~~_)-d = 2(1 )-(-2) S _ _= _ ~ ~ _ __ _ _ ___ _ __ ____J2~_tc~OL _ _ __ ~_ _
--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
- --- - 1- - ------ ----- - - -- -----
_ ___ _ middot_Vll =_3oTr -=-Q_1L(ZL)_ _ _____ ______ __ =--3_oTJ~--lgJJI------- ------
J~ shy
___---bTT~_IgJ[--fi-)- ------ --- -shy_ ~ 39T( ~6Qrr __ ___ _____ ____ _
lt 0 as 10 - ---- T--reYVTtmiddotdJ - 1- - --- -- r---3 Cfl1 middotmiddot 9vlZ J
- - _ - - --- - -- ~) --- - - -- -- shy
- - -- --- ---- -- - shy
-- ---I
-------_
- --_ -_ ------ --- shy
----__ - ------ -- -- shy
-- - _ - - __------- --
---_- ---- - --- - - -- - --- - - - - - --shy
_-------- shy
middot Question 14 (15 marks) - Start on the appropriate page in your answer booklet
a) The diagram shows the points A(1-2) B(-4-3) and C(-13) and the line Be
Not to scale
x
A(l - 2)
(i) Calculate the exact length of Be 1 (ii) Show that the equation of BC is 2x - y + 5 = o 2 (iii) Find the exact area of triangle ABe 3
b) Solve x 2 - 2x - 1 lt O 3
Question 14 continues on the next page
7
Question 14 (continued)
c) The diagram shows two right cones The small cone of radius PS=r em is enclosed within bull a larger cone of radius QR=5em as shown The larger cone has height 15cm and the small
cone has height hem
T
Not to scale
15 ern
hem
R
(i) Prove that L1TPSIIIL1TQR 2
(ii) Show that h = 15 shy 3r 1
(iii) Find the radius of the small cone such that its volume is a maximum 3
End of Examination
8
--
_ _
----
== - 15 _ 3 bull - -ci -~ -- ---7---- ~--3--
---- --_- - ---shyolyen -3
- - $= ------- --- -- _-- - - __ ~DpoundCvJ _ ~ 2 ~--tJ 2
~ --E~ ~f) --+ 0713 --- -- - _f_ - F -C1f _ 1fj~S~Jo~ shy~~
- - - ------ _- ------ - - -- ---- ___f _s _oluhrgts _ ____reg _2
_= (~3) plusmn ( =~ -)shy10 _ f(x4-t2~ X t-lrD-shy
-fl-x _) _ =___X_)2 - _(X -)+ I ~ --x1--_ -2X4-1 -X f 1- 4 -1
- -Xl- 3i + 3
----- - -~- =--E- -~~Ji
~ e-1-
---S-2 $6Ht __
12- 3
IS If S I
___-=-- ~ _ J2~~(~s-_~~_~__ _tJ~) lt0 11 i
____ -coc cto~_middot~ -fl((~) lt 0 -reg----- - shy
----_ - --- --- shy
--shy
~f3 2u~o Go gt~
Ict6 - -J - --- - _ ~r _ ~- = _ ~_ - ----- shy -- -- - ~G~LotJuL L _~ _3~ ) _
ISO 330 -- - - ) - - - --- - shy
-reg--- --shy- -g
---- -------5 ~~---(bs8
-- - - - I ~~
- - -=-S--S ~ltS--~ 6--shy
- - ------ -- shy- a-e
_ _ _ _
c amp11
lX-r3 J -5(7--~2) - )x-2 -_- -_ - -_ _ltlt--- -( -- --- ----middot-1 middot-shy4 20
~t 1- (2 _- 5_~+lQ
20
0( - 22- s _ -~lt______- 0 )
_ __ _ __ __3_~ _- __4 _~ ___ Qf__ __-=32C- 12 -= - _
__2t =- ~ __ ~ _ _ ___ _l-_- _ - 3xS__
)L ~ 3 (= _LI_) X ~-f i
--- -- ---- ---shy
__=-_
-shy --shy--shy __ -shy _ _
V lt _li -v ~ f ~~_ c )__ ___ - ---~~ ) -~q-- s shy-- ~ ---
-~-- -_ _--- ------ ---_- - shy_ ___--+-____ amp 1
--middot-------ilmiddot----~--========--
-------(1-)- pound (~)-=-J oX 2_ -_ _
_-poundr~ )~J-(~~~- shy--- --- - --------==- J- X-l - 1-(
-- -- --- --- --- -- --F(-~r)
--------- -- ---shy_______
J-4------( r )-- - -
---- ----- -shy
--~------ --------- -------- --shy
---middot=1l-1~ ---7-~ -f=-J 2- $---- -- 01- ---J -s--J~--t-~- - - - - ---- ----- - - - -- _ shy---------------=---~
~ ~ X --middot- - I--------=- -- --- --------- --shy
-------l------~--~ --O- --- - - ---- --- --shy shy
---------------------shy
__f-J- _ liLt -s~ 2-$)-~_b___$ _0__~_ -- _b ~0 __ _~(I) _-
----t~ - 4 S~ 2 e - ~ S_~ __0 - G
c
- -- _ i
lt C I )
or
(I)
-shy Q-l2
( h __ ____oj 3~ -I - ~--(3 ~-X3--+--) CJ ~ chlta _ 1 ~ ~~1D
__ _ - l~_- 21_ ___ _ 2-~(3_ ~L) _ od 0 clvu-OrY ~Cl~ r
~ -
---- - 1---------- - - ------shy
3 h + ---- - - ---_ _-- ------ ---- --i-(--- -C-I ) middot middotCo r recf
2 --- - _ ----__shy
h -I ----~-)- - y--=== - I-G-=-fmiddot--shy
I -lt----- -~ --=--1 ---( ~ x) - --- - - -shy---_ - _ 1 - ---- --zr~- -
____ __ ___t~1 - x) __ _3)-
--- - - 1---- ----====------- -- shy----- - _ I_)__j- 2x J 3x-+-______ _ _1A_~ 2~_ IL --((
(--
2 - shy- - -
I -I~ ______ __ _________ _____ ___ __~ (3~-) 2 _ Y ~- 2(3A+I - 3 __bull _ Jlt _ _~ __ --- --- - - ~- --- ~t)(-X- 3 - - + ~ 3x +--1 - x 2shy
_____ ___Jt-____ AJ3~tJ_____ _ ____ _____ 2t3~+-I_____ _
_ _(J)_~ U k) __-------- ----- -- --- -- - ----- -- - _ -~-3x rroduC+ C 1c+
------_ -_- -------- -p--------------shy
~ -3X+- 1 _ ( I) - _lt-9 r_C~d- _ 95 --- - ---- - --- _-----shy
--7-- - J-Iz ~x l3X--ri)
----__-shy
------___ ----- ---- --_ - shy
- - - - --------1- ----_ _---shy
(Iii) ~ 1 r - k I T
L 2zt-l v Z I
I _ ( -vshy
1 - V 1shy
(Zx~i)4middot - (4-() L - (I)
lt vs~s 1v o+e-+- r-J(( - - -- - lZx Ir~
- g(+--If---E--x 1
_llz -)~ _
-=- -shy(Z~ 1-1) 2 _
I (ff1 (~Z)(pound~ -2x+-t )
x-2 x-) - 2 ___2-xyt 1shy
-- ---- -C~) --=-L(--=- -~)~- - --- shy
------ - - -
------ - -2--- --- shy
___ _ G__
--- ~~-j
--- l-J shy
- 2 __~~---_ J - corC ~ ct- 3c~d I~~
J j l _m tx-- ~C J - 2 (- -f )fT- 3
~ ~ ~
w ~ - 2x shy J -_ - - --- --- --I ~lt------- Ci ) CD rr ~- ~T
e) 2 x - 3 j - 4- J- L
2~ -3 4t-fc I
- _t _ ~K _ 2 ~ bx
~ ~ ~ 3middot shy
LI-l S =- -f - 2~f ~- 2 ~
__ _ _ gt~1 (J cJ Ib-ls - Z IS
_ middot _ f9+_~ _ ~J_Hcgt__ ______ cJ__-
( ) ~r ~ow Ct---J redegj rf -y -=- _
-- - -
-- - _ --~
Co~ 8 =- ~ f~6 - _L) ~r -~
_(1) _for ~d
-- b)shy
_ _ _ (x t-Lf)(t -7J-~ ~2_
- ---x2~-3~middotmiddot--2middot ~--~-o -j- -- -- J lt )
xt~middotx~ middot ~middot~f middot middot-0 middot- - -- ( -- - bullbull _P p - - shy
__kuL~gt 0 _ _
x - I lt raquo lt 2---_---------- -_ - - - -shy
- - - tJ- jJS =_ltix~+_t-X_Y-lf)--+-~b-t-S - -
_ _ __ __ _4x~middotLYA~ +-fA__+_Ex_-plusmn_C
-- - - - - =A(~ -+- -rf~ plusmnB)~J (~A- +lt) L ____ u_ J-tLS - x t- Ox ~l __ _
_ - - - A-2 - - - -- HiG)-J--B ~o_ f) f c- 7 -11 _- R -2 _ 3_~- -- - - ~ -- - _ -- b __ _ - - _ow _ __ C _shy~ ~ - - --- -- -C--) - _- - ---- - --f-) - - -- - -- - -- (1-)
- -
_d) pound9 r __ Ld- LC1~ 7 6 gt0 ___ _ _ _
--- - - A- b--Jtcu -- - - -_- - _ _ ___ __ __ _ =~2)~- t__-L13~_) ___ __lt____ __c )
- - - =--~~-rZ middot =Fccf1--middot --- - - -shy
----- ~ Cfl- g ~ 0-- VJ~ --k -2 --- -lt-- C_ ) --------- bull - ----- - _- - - - -- shy
x-7
u_____~l-
-- - -
--- --
-- - -- -- -------- --
- ~ I )I-or sampo~ltvj -1) _ 2~3 - 0~ _c- 0 _
- --- - - - - -- -2x-_ x -~-3)- - ~- - sj -
--------- - -- --~J - ~-2--- -- -
--- - --------- - ---- - -degr-~)~- --------til- --At-(~2=- - _ _ _ ____ U)
___-=- _ _8_ (9)1- - 1 _O~eC)~gtJ __o ___ _ 2 Co) = _ 4 - -~ J 4
---shy-----shy--shy
(3-~tpound)- --shy -shy___ ___ ___ ____ __j~ _-l3JL -=L2(3)
= _ _
AI
f Ii _
gt0 -- _ _
--- - - -- -- _- ------ shy -~0) ~ ll~)__ _ _ ____
sectJ_--0 __ ___ -- shy 7 -----------shy~------
(2) _
(3-~11 S -) shyV 1 J
---- - ---- ----- - - ----- ----
----
------
------- --- ---- --- - ----
(
- )
b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
___ __ __x ~ _b _-__ JnJ~ - LfCl_C_ __ ___
2co-shy _~ - ---bull--- -- - -- -__- - --- shy
2 - 1(-2)2 - lf~L 2lt I
2 -4- r-g ---- 2 J) - - ----=-~-~-- ---- --- I
2 ) 2 t 212
- ---shy2
____ __~----------~--l ~fi____ -----f-~~- -----shy -- -- - - shy
_ __ _~ _ 2 5) L
-__-- -- - -- - --------------=~ - -
_ ---- _ _ _ --i - f-2 --~ -x --~ --I-T ~-- shyS~JlI ~ Ii ()
-----1 -) -- -In- drER ---d--l _Qs --- - - - - - - - - -- L -TPfl- -- -L-+--Q-s- -C-9lVen - kgt-o-H-1Q ) -
_ ____ _ ___L_T s _CSJtntno~ _______
-- - -- - A _LJlaquojL4IQS_ -(yen-J-cJ -~J - LS -~iucJ)or (Alt)
_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
------- --- - --- --+------ lSi ~-SS~ -- - middot -shy
-shy--shy -- --shy -shyS
------i
cmiddot QILf
- -) - ~______ -
-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
__=-_--_-_-~__-=__~1_(_~~middotJL +_ 3__plusmn_~_)~_ ___
--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
_ -- - ------shy
- - -- -__-_ - -- _-------__---- -shy12 -~
_ __ M _ _ __ __ _ _
Xz-x - 3+3
-----shy-I+lt
____ --- ~U)_- ~ ------_shy
- - - -__---- -- --- _-- 2-__ _e~ _s_~- r()~5~ (~II~)
- - ~- ~1f -~ _~Lx--~ X ~
----- -- - ---- 13-2 -tCt 1) ~lt-----_ ( j )
j - - 2x L - - ------ -- --- - 5 - --
__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
_ _ _ __ ___ __ _ _~~_I l ~ [ )
____Imiddot__l~~_)-d = 2(1 )-(-2) S _ _= _ ~ ~ _ __ _ _ ___ _ __ ____J2~_tc~OL _ _ __ ~_ _
--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
- --- - 1- - ------ ----- - - -- -----
_ ___ _ middot_Vll =_3oTr -=-Q_1L(ZL)_ _ _____ ______ __ =--3_oTJ~--lgJJI------- ------
J~ shy
___---bTT~_IgJ[--fi-)- ------ --- -shy_ ~ 39T( ~6Qrr __ ___ _____ ____ _
lt 0 as 10 - ---- T--reYVTtmiddotdJ - 1- - --- -- r---3 Cfl1 middotmiddot 9vlZ J
- - _ - - --- - -- ~) --- - - -- -- shy
- - -- --- ---- -- - shy
-- ---I
-------_
- --_ -_ ------ --- shy
----__ - ------ -- -- shy
-- - _ - - __------- --
---_- ---- - --- - - -- - --- - - - - - --shy
_-------- shy
Question 14 (continued)
c) The diagram shows two right cones The small cone of radius PS=r em is enclosed within bull a larger cone of radius QR=5em as shown The larger cone has height 15cm and the small
cone has height hem
T
Not to scale
15 ern
hem
R
(i) Prove that L1TPSIIIL1TQR 2
(ii) Show that h = 15 shy 3r 1
(iii) Find the radius of the small cone such that its volume is a maximum 3
End of Examination
8
--
_ _
----
== - 15 _ 3 bull - -ci -~ -- ---7---- ~--3--
---- --_- - ---shyolyen -3
- - $= ------- --- -- _-- - - __ ~DpoundCvJ _ ~ 2 ~--tJ 2
~ --E~ ~f) --+ 0713 --- -- - _f_ - F -C1f _ 1fj~S~Jo~ shy~~
- - - ------ _- ------ - - -- ---- ___f _s _oluhrgts _ ____reg _2
_= (~3) plusmn ( =~ -)shy10 _ f(x4-t2~ X t-lrD-shy
-fl-x _) _ =___X_)2 - _(X -)+ I ~ --x1--_ -2X4-1 -X f 1- 4 -1
- -Xl- 3i + 3
----- - -~- =--E- -~~Ji
~ e-1-
---S-2 $6Ht __
12- 3
IS If S I
___-=-- ~ _ J2~~(~s-_~~_~__ _tJ~) lt0 11 i
____ -coc cto~_middot~ -fl((~) lt 0 -reg----- - shy
----_ - --- --- shy
--shy
~f3 2u~o Go gt~
Ict6 - -J - --- - _ ~r _ ~- = _ ~_ - ----- shy -- -- - ~G~LotJuL L _~ _3~ ) _
ISO 330 -- - - ) - - - --- - shy
-reg--- --shy- -g
---- -------5 ~~---(bs8
-- - - - I ~~
- - -=-S--S ~ltS--~ 6--shy
- - ------ -- shy- a-e
_ _ _ _
c amp11
lX-r3 J -5(7--~2) - )x-2 -_- -_ - -_ _ltlt--- -( -- --- ----middot-1 middot-shy4 20
~t 1- (2 _- 5_~+lQ
20
0( - 22- s _ -~lt______- 0 )
_ __ _ __ __3_~ _- __4 _~ ___ Qf__ __-=32C- 12 -= - _
__2t =- ~ __ ~ _ _ ___ _l-_- _ - 3xS__
)L ~ 3 (= _LI_) X ~-f i
--- -- ---- ---shy
__=-_
-shy --shy--shy __ -shy _ _
V lt _li -v ~ f ~~_ c )__ ___ - ---~~ ) -~q-- s shy-- ~ ---
-~-- -_ _--- ------ ---_- - shy_ ___--+-____ amp 1
--middot-------ilmiddot----~--========--
-------(1-)- pound (~)-=-J oX 2_ -_ _
_-poundr~ )~J-(~~~- shy--- --- - --------==- J- X-l - 1-(
-- -- --- --- --- -- --F(-~r)
--------- -- ---shy_______
J-4------( r )-- - -
---- ----- -shy
--~------ --------- -------- --shy
---middot=1l-1~ ---7-~ -f=-J 2- $---- -- 01- ---J -s--J~--t-~- - - - - ---- ----- - - - -- _ shy---------------=---~
~ ~ X --middot- - I--------=- -- --- --------- --shy
-------l------~--~ --O- --- - - ---- --- --shy shy
---------------------shy
__f-J- _ liLt -s~ 2-$)-~_b___$ _0__~_ -- _b ~0 __ _~(I) _-
----t~ - 4 S~ 2 e - ~ S_~ __0 - G
c
- -- _ i
lt C I )
or
(I)
-shy Q-l2
( h __ ____oj 3~ -I - ~--(3 ~-X3--+--) CJ ~ chlta _ 1 ~ ~~1D
__ _ - l~_- 21_ ___ _ 2-~(3_ ~L) _ od 0 clvu-OrY ~Cl~ r
~ -
---- - 1---------- - - ------shy
3 h + ---- - - ---_ _-- ------ ---- --i-(--- -C-I ) middot middotCo r recf
2 --- - _ ----__shy
h -I ----~-)- - y--=== - I-G-=-fmiddot--shy
I -lt----- -~ --=--1 ---( ~ x) - --- - - -shy---_ - _ 1 - ---- --zr~- -
____ __ ___t~1 - x) __ _3)-
--- - - 1---- ----====------- -- shy----- - _ I_)__j- 2x J 3x-+-______ _ _1A_~ 2~_ IL --((
(--
2 - shy- - -
I -I~ ______ __ _________ _____ ___ __~ (3~-) 2 _ Y ~- 2(3A+I - 3 __bull _ Jlt _ _~ __ --- --- - - ~- --- ~t)(-X- 3 - - + ~ 3x +--1 - x 2shy
_____ ___Jt-____ AJ3~tJ_____ _ ____ _____ 2t3~+-I_____ _
_ _(J)_~ U k) __-------- ----- -- --- -- - ----- -- - _ -~-3x rroduC+ C 1c+
------_ -_- -------- -p--------------shy
~ -3X+- 1 _ ( I) - _lt-9 r_C~d- _ 95 --- - ---- - --- _-----shy
--7-- - J-Iz ~x l3X--ri)
----__-shy
------___ ----- ---- --_ - shy
- - - - --------1- ----_ _---shy
(Iii) ~ 1 r - k I T
L 2zt-l v Z I
I _ ( -vshy
1 - V 1shy
(Zx~i)4middot - (4-() L - (I)
lt vs~s 1v o+e-+- r-J(( - - -- - lZx Ir~
- g(+--If---E--x 1
_llz -)~ _
-=- -shy(Z~ 1-1) 2 _
I (ff1 (~Z)(pound~ -2x+-t )
x-2 x-) - 2 ___2-xyt 1shy
-- ---- -C~) --=-L(--=- -~)~- - --- shy
------ - - -
------ - -2--- --- shy
___ _ G__
--- ~~-j
--- l-J shy
- 2 __~~---_ J - corC ~ ct- 3c~d I~~
J j l _m tx-- ~C J - 2 (- -f )fT- 3
~ ~ ~
w ~ - 2x shy J -_ - - --- --- --I ~lt------- Ci ) CD rr ~- ~T
e) 2 x - 3 j - 4- J- L
2~ -3 4t-fc I
- _t _ ~K _ 2 ~ bx
~ ~ ~ 3middot shy
LI-l S =- -f - 2~f ~- 2 ~
__ _ _ gt~1 (J cJ Ib-ls - Z IS
_ middot _ f9+_~ _ ~J_Hcgt__ ______ cJ__-
( ) ~r ~ow Ct---J redegj rf -y -=- _
-- - -
-- - _ --~
Co~ 8 =- ~ f~6 - _L) ~r -~
_(1) _for ~d
-- b)shy
_ _ _ (x t-Lf)(t -7J-~ ~2_
- ---x2~-3~middotmiddot--2middot ~--~-o -j- -- -- J lt )
xt~middotx~ middot ~middot~f middot middot-0 middot- - -- ( -- - bullbull _P p - - shy
__kuL~gt 0 _ _
x - I lt raquo lt 2---_---------- -_ - - - -shy
- - - tJ- jJS =_ltix~+_t-X_Y-lf)--+-~b-t-S - -
_ _ __ __ _4x~middotLYA~ +-fA__+_Ex_-plusmn_C
-- - - - - =A(~ -+- -rf~ plusmnB)~J (~A- +lt) L ____ u_ J-tLS - x t- Ox ~l __ _
_ - - - A-2 - - - -- HiG)-J--B ~o_ f) f c- 7 -11 _- R -2 _ 3_~- -- - - ~ -- - _ -- b __ _ - - _ow _ __ C _shy~ ~ - - --- -- -C--) - _- - ---- - --f-) - - -- - -- - -- (1-)
- -
_d) pound9 r __ Ld- LC1~ 7 6 gt0 ___ _ _ _
--- - - A- b--Jtcu -- - - -_- - _ _ ___ __ __ _ =~2)~- t__-L13~_) ___ __lt____ __c )
- - - =--~~-rZ middot =Fccf1--middot --- - - -shy
----- ~ Cfl- g ~ 0-- VJ~ --k -2 --- -lt-- C_ ) --------- bull - ----- - _- - - - -- shy
x-7
u_____~l-
-- - -
--- --
-- - -- -- -------- --
- ~ I )I-or sampo~ltvj -1) _ 2~3 - 0~ _c- 0 _
- --- - - - - -- -2x-_ x -~-3)- - ~- - sj -
--------- - -- --~J - ~-2--- -- -
--- - --------- - ---- - -degr-~)~- --------til- --At-(~2=- - _ _ _ ____ U)
___-=- _ _8_ (9)1- - 1 _O~eC)~gtJ __o ___ _ 2 Co) = _ 4 - -~ J 4
---shy-----shy--shy
(3-~tpound)- --shy -shy___ ___ ___ ____ __j~ _-l3JL -=L2(3)
= _ _
AI
f Ii _
gt0 -- _ _
--- - - -- -- _- ------ shy -~0) ~ ll~)__ _ _ ____
sectJ_--0 __ ___ -- shy 7 -----------shy~------
(2) _
(3-~11 S -) shyV 1 J
---- - ---- ----- - - ----- ----
----
------
------- --- ---- --- - ----
(
- )
b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
___ __ __x ~ _b _-__ JnJ~ - LfCl_C_ __ ___
2co-shy _~ - ---bull--- -- - -- -__- - --- shy
2 - 1(-2)2 - lf~L 2lt I
2 -4- r-g ---- 2 J) - - ----=-~-~-- ---- --- I
2 ) 2 t 212
- ---shy2
____ __~----------~--l ~fi____ -----f-~~- -----shy -- -- - - shy
_ __ _~ _ 2 5) L
-__-- -- - -- - --------------=~ - -
_ ---- _ _ _ --i - f-2 --~ -x --~ --I-T ~-- shyS~JlI ~ Ii ()
-----1 -) -- -In- drER ---d--l _Qs --- - - - - - - - - -- L -TPfl- -- -L-+--Q-s- -C-9lVen - kgt-o-H-1Q ) -
_ ____ _ ___L_T s _CSJtntno~ _______
-- - -- - A _LJlaquojL4IQS_ -(yen-J-cJ -~J - LS -~iucJ)or (Alt)
_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
------- --- - --- --+------ lSi ~-SS~ -- - middot -shy
-shy--shy -- --shy -shyS
------i
cmiddot QILf
- -) - ~______ -
-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
__=-_--_-_-~__-=__~1_(_~~middotJL +_ 3__plusmn_~_)~_ ___
--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
_ -- - ------shy
- - -- -__-_ - -- _-------__---- -shy12 -~
_ __ M _ _ __ __ _ _
Xz-x - 3+3
-----shy-I+lt
____ --- ~U)_- ~ ------_shy
- - - -__---- -- --- _-- 2-__ _e~ _s_~- r()~5~ (~II~)
- - ~- ~1f -~ _~Lx--~ X ~
----- -- - ---- 13-2 -tCt 1) ~lt-----_ ( j )
j - - 2x L - - ------ -- --- - 5 - --
__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
_ _ _ __ ___ __ _ _~~_I l ~ [ )
____Imiddot__l~~_)-d = 2(1 )-(-2) S _ _= _ ~ ~ _ __ _ _ ___ _ __ ____J2~_tc~OL _ _ __ ~_ _
--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
- --- - 1- - ------ ----- - - -- -----
_ ___ _ middot_Vll =_3oTr -=-Q_1L(ZL)_ _ _____ ______ __ =--3_oTJ~--lgJJI------- ------
J~ shy
___---bTT~_IgJ[--fi-)- ------ --- -shy_ ~ 39T( ~6Qrr __ ___ _____ ____ _
lt 0 as 10 - ---- T--reYVTtmiddotdJ - 1- - --- -- r---3 Cfl1 middotmiddot 9vlZ J
- - _ - - --- - -- ~) --- - - -- -- shy
- - -- --- ---- -- - shy
-- ---I
-------_
- --_ -_ ------ --- shy
----__ - ------ -- -- shy
-- - _ - - __------- --
---_- ---- - --- - - -- - --- - - - - - --shy
_-------- shy
--
_ _
----
== - 15 _ 3 bull - -ci -~ -- ---7---- ~--3--
---- --_- - ---shyolyen -3
- - $= ------- --- -- _-- - - __ ~DpoundCvJ _ ~ 2 ~--tJ 2
~ --E~ ~f) --+ 0713 --- -- - _f_ - F -C1f _ 1fj~S~Jo~ shy~~
- - - ------ _- ------ - - -- ---- ___f _s _oluhrgts _ ____reg _2
_= (~3) plusmn ( =~ -)shy10 _ f(x4-t2~ X t-lrD-shy
-fl-x _) _ =___X_)2 - _(X -)+ I ~ --x1--_ -2X4-1 -X f 1- 4 -1
- -Xl- 3i + 3
----- - -~- =--E- -~~Ji
~ e-1-
---S-2 $6Ht __
12- 3
IS If S I
___-=-- ~ _ J2~~(~s-_~~_~__ _tJ~) lt0 11 i
____ -coc cto~_middot~ -fl((~) lt 0 -reg----- - shy
----_ - --- --- shy
--shy
~f3 2u~o Go gt~
Ict6 - -J - --- - _ ~r _ ~- = _ ~_ - ----- shy -- -- - ~G~LotJuL L _~ _3~ ) _
ISO 330 -- - - ) - - - --- - shy
-reg--- --shy- -g
---- -------5 ~~---(bs8
-- - - - I ~~
- - -=-S--S ~ltS--~ 6--shy
- - ------ -- shy- a-e
_ _ _ _
c amp11
lX-r3 J -5(7--~2) - )x-2 -_- -_ - -_ _ltlt--- -( -- --- ----middot-1 middot-shy4 20
~t 1- (2 _- 5_~+lQ
20
0( - 22- s _ -~lt______- 0 )
_ __ _ __ __3_~ _- __4 _~ ___ Qf__ __-=32C- 12 -= - _
__2t =- ~ __ ~ _ _ ___ _l-_- _ - 3xS__
)L ~ 3 (= _LI_) X ~-f i
--- -- ---- ---shy
__=-_
-shy --shy--shy __ -shy _ _
V lt _li -v ~ f ~~_ c )__ ___ - ---~~ ) -~q-- s shy-- ~ ---
-~-- -_ _--- ------ ---_- - shy_ ___--+-____ amp 1
--middot-------ilmiddot----~--========--
-------(1-)- pound (~)-=-J oX 2_ -_ _
_-poundr~ )~J-(~~~- shy--- --- - --------==- J- X-l - 1-(
-- -- --- --- --- -- --F(-~r)
--------- -- ---shy_______
J-4------( r )-- - -
---- ----- -shy
--~------ --------- -------- --shy
---middot=1l-1~ ---7-~ -f=-J 2- $---- -- 01- ---J -s--J~--t-~- - - - - ---- ----- - - - -- _ shy---------------=---~
~ ~ X --middot- - I--------=- -- --- --------- --shy
-------l------~--~ --O- --- - - ---- --- --shy shy
---------------------shy
__f-J- _ liLt -s~ 2-$)-~_b___$ _0__~_ -- _b ~0 __ _~(I) _-
----t~ - 4 S~ 2 e - ~ S_~ __0 - G
c
- -- _ i
lt C I )
or
(I)
-shy Q-l2
( h __ ____oj 3~ -I - ~--(3 ~-X3--+--) CJ ~ chlta _ 1 ~ ~~1D
__ _ - l~_- 21_ ___ _ 2-~(3_ ~L) _ od 0 clvu-OrY ~Cl~ r
~ -
---- - 1---------- - - ------shy
3 h + ---- - - ---_ _-- ------ ---- --i-(--- -C-I ) middot middotCo r recf
2 --- - _ ----__shy
h -I ----~-)- - y--=== - I-G-=-fmiddot--shy
I -lt----- -~ --=--1 ---( ~ x) - --- - - -shy---_ - _ 1 - ---- --zr~- -
____ __ ___t~1 - x) __ _3)-
--- - - 1---- ----====------- -- shy----- - _ I_)__j- 2x J 3x-+-______ _ _1A_~ 2~_ IL --((
(--
2 - shy- - -
I -I~ ______ __ _________ _____ ___ __~ (3~-) 2 _ Y ~- 2(3A+I - 3 __bull _ Jlt _ _~ __ --- --- - - ~- --- ~t)(-X- 3 - - + ~ 3x +--1 - x 2shy
_____ ___Jt-____ AJ3~tJ_____ _ ____ _____ 2t3~+-I_____ _
_ _(J)_~ U k) __-------- ----- -- --- -- - ----- -- - _ -~-3x rroduC+ C 1c+
------_ -_- -------- -p--------------shy
~ -3X+- 1 _ ( I) - _lt-9 r_C~d- _ 95 --- - ---- - --- _-----shy
--7-- - J-Iz ~x l3X--ri)
----__-shy
------___ ----- ---- --_ - shy
- - - - --------1- ----_ _---shy
(Iii) ~ 1 r - k I T
L 2zt-l v Z I
I _ ( -vshy
1 - V 1shy
(Zx~i)4middot - (4-() L - (I)
lt vs~s 1v o+e-+- r-J(( - - -- - lZx Ir~
- g(+--If---E--x 1
_llz -)~ _
-=- -shy(Z~ 1-1) 2 _
I (ff1 (~Z)(pound~ -2x+-t )
x-2 x-) - 2 ___2-xyt 1shy
-- ---- -C~) --=-L(--=- -~)~- - --- shy
------ - - -
------ - -2--- --- shy
___ _ G__
--- ~~-j
--- l-J shy
- 2 __~~---_ J - corC ~ ct- 3c~d I~~
J j l _m tx-- ~C J - 2 (- -f )fT- 3
~ ~ ~
w ~ - 2x shy J -_ - - --- --- --I ~lt------- Ci ) CD rr ~- ~T
e) 2 x - 3 j - 4- J- L
2~ -3 4t-fc I
- _t _ ~K _ 2 ~ bx
~ ~ ~ 3middot shy
LI-l S =- -f - 2~f ~- 2 ~
__ _ _ gt~1 (J cJ Ib-ls - Z IS
_ middot _ f9+_~ _ ~J_Hcgt__ ______ cJ__-
( ) ~r ~ow Ct---J redegj rf -y -=- _
-- - -
-- - _ --~
Co~ 8 =- ~ f~6 - _L) ~r -~
_(1) _for ~d
-- b)shy
_ _ _ (x t-Lf)(t -7J-~ ~2_
- ---x2~-3~middotmiddot--2middot ~--~-o -j- -- -- J lt )
xt~middotx~ middot ~middot~f middot middot-0 middot- - -- ( -- - bullbull _P p - - shy
__kuL~gt 0 _ _
x - I lt raquo lt 2---_---------- -_ - - - -shy
- - - tJ- jJS =_ltix~+_t-X_Y-lf)--+-~b-t-S - -
_ _ __ __ _4x~middotLYA~ +-fA__+_Ex_-plusmn_C
-- - - - - =A(~ -+- -rf~ plusmnB)~J (~A- +lt) L ____ u_ J-tLS - x t- Ox ~l __ _
_ - - - A-2 - - - -- HiG)-J--B ~o_ f) f c- 7 -11 _- R -2 _ 3_~- -- - - ~ -- - _ -- b __ _ - - _ow _ __ C _shy~ ~ - - --- -- -C--) - _- - ---- - --f-) - - -- - -- - -- (1-)
- -
_d) pound9 r __ Ld- LC1~ 7 6 gt0 ___ _ _ _
--- - - A- b--Jtcu -- - - -_- - _ _ ___ __ __ _ =~2)~- t__-L13~_) ___ __lt____ __c )
- - - =--~~-rZ middot =Fccf1--middot --- - - -shy
----- ~ Cfl- g ~ 0-- VJ~ --k -2 --- -lt-- C_ ) --------- bull - ----- - _- - - - -- shy
x-7
u_____~l-
-- - -
--- --
-- - -- -- -------- --
- ~ I )I-or sampo~ltvj -1) _ 2~3 - 0~ _c- 0 _
- --- - - - - -- -2x-_ x -~-3)- - ~- - sj -
--------- - -- --~J - ~-2--- -- -
--- - --------- - ---- - -degr-~)~- --------til- --At-(~2=- - _ _ _ ____ U)
___-=- _ _8_ (9)1- - 1 _O~eC)~gtJ __o ___ _ 2 Co) = _ 4 - -~ J 4
---shy-----shy--shy
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--- - -------
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--- -
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--- - -------
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_____ ___Jt-____ AJ3~tJ_____ _ ____ _____ 2t3~+-I_____ _
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---- - ---- ----- - - ----- ----
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--- -
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--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
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--- - -------
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v-=- - 7lr2( 1~-3r)
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10 r ~(~--~- -()- 3
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__kuL~gt 0 _ _
x - I lt raquo lt 2---_---------- -_ - - - -shy
- - - tJ- jJS =_ltix~+_t-X_Y-lf)--+-~b-t-S - -
_ _ __ __ _4x~middotLYA~ +-fA__+_Ex_-plusmn_C
-- - - - - =A(~ -+- -rf~ plusmnB)~J (~A- +lt) L ____ u_ J-tLS - x t- Ox ~l __ _
_ - - - A-2 - - - -- HiG)-J--B ~o_ f) f c- 7 -11 _- R -2 _ 3_~- -- - - ~ -- - _ -- b __ _ - - _ow _ __ C _shy~ ~ - - --- -- -C--) - _- - ---- - --f-) - - -- - -- - -- (1-)
- -
_d) pound9 r __ Ld- LC1~ 7 6 gt0 ___ _ _ _
--- - - A- b--Jtcu -- - - -_- - _ _ ___ __ __ _ =~2)~- t__-L13~_) ___ __lt____ __c )
- - - =--~~-rZ middot =Fccf1--middot --- - - -shy
----- ~ Cfl- g ~ 0-- VJ~ --k -2 --- -lt-- C_ ) --------- bull - ----- - _- - - - -- shy
x-7
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--- - --------- - ---- - -degr-~)~- --------til- --At-(~2=- - _ _ _ ____ U)
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---shy-----shy--shy
(3-~tpound)- --shy -shy___ ___ ___ ____ __j~ _-l3JL -=L2(3)
= _ _
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f Ii _
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--- - - -- -- _- ------ shy -~0) ~ ll~)__ _ _ ____
sectJ_--0 __ ___ -- shy 7 -----------shy~------
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---- - ---- ----- - - ----- ----
----
------
------- --- ---- --- - ----
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b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
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_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
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cmiddot QILf
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-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
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--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
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_ __ M _ _ __ __ _ _
Xz-x - 3+3
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__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
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--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
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-- ---I
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-- - -
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Co~ 8 =- ~ f~6 - _L) ~r -~
_(1) _for ~d
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xt~middotx~ middot ~middot~f middot middot-0 middot- - -- ( -- - bullbull _P p - - shy
__kuL~gt 0 _ _
x - I lt raquo lt 2---_---------- -_ - - - -shy
- - - tJ- jJS =_ltix~+_t-X_Y-lf)--+-~b-t-S - -
_ _ __ __ _4x~middotLYA~ +-fA__+_Ex_-plusmn_C
-- - - - - =A(~ -+- -rf~ plusmnB)~J (~A- +lt) L ____ u_ J-tLS - x t- Ox ~l __ _
_ - - - A-2 - - - -- HiG)-J--B ~o_ f) f c- 7 -11 _- R -2 _ 3_~- -- - - ~ -- - _ -- b __ _ - - _ow _ __ C _shy~ ~ - - --- -- -C--) - _- - ---- - --f-) - - -- - -- - -- (1-)
- -
_d) pound9 r __ Ld- LC1~ 7 6 gt0 ___ _ _ _
--- - - A- b--Jtcu -- - - -_- - _ _ ___ __ __ _ =~2)~- t__-L13~_) ___ __lt____ __c )
- - - =--~~-rZ middot =Fccf1--middot --- - - -shy
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x-7
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-- - -
--- --
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--------- - -- --~J - ~-2--- -- -
--- - --------- - ---- - -degr-~)~- --------til- --At-(~2=- - _ _ _ ____ U)
___-=- _ _8_ (9)1- - 1 _O~eC)~gtJ __o ___ _ 2 Co) = _ 4 - -~ J 4
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(3-~tpound)- --shy -shy___ ___ ___ ____ __j~ _-l3JL -=L2(3)
= _ _
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f Ii _
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--- - - -- -- _- ------ shy -~0) ~ ll~)__ _ _ ____
sectJ_--0 __ ___ -- shy 7 -----------shy~------
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---- - ---- ----- - - ----- ----
----
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------- --- ---- --- - ----
(
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b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
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_ ____ _ ___L_T s _CSJtntno~ _______
-- - -- - A _LJlaquojL4IQS_ -(yen-J-cJ -~J - LS -~iucJ)or (Alt)
_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
------- --- - --- --+------ lSi ~-SS~ -- - middot -shy
-shy--shy -- --shy -shyS
------i
cmiddot QILf
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-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
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--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
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_ __ M _ _ __ __ _ _
Xz-x - 3+3
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- - ~- ~1f -~ _~Lx--~ X ~
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j - - 2x L - - ------ -- --- - 5 - --
__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
_ _ _ __ ___ __ _ _~~_I l ~ [ )
____Imiddot__l~~_)-d = 2(1 )-(-2) S _ _= _ ~ ~ _ __ _ _ ___ _ __ ____J2~_tc~OL _ _ __ ~_ _
--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
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J~ shy
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----__ - ------ -- -- shy
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---_- ---- - --- - - -- - --- - - - - - --shy
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--- --
-- - -- -- -------- --
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--------- - -- --~J - ~-2--- -- -
--- - --------- - ---- - -degr-~)~- --------til- --At-(~2=- - _ _ _ ____ U)
___-=- _ _8_ (9)1- - 1 _O~eC)~gtJ __o ___ _ 2 Co) = _ 4 - -~ J 4
---shy-----shy--shy
(3-~tpound)- --shy -shy___ ___ ___ ____ __j~ _-l3JL -=L2(3)
= _ _
AI
f Ii _
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--- - - -- -- _- ------ shy -~0) ~ ll~)__ _ _ ____
sectJ_--0 __ ___ -- shy 7 -----------shy~------
(2) _
(3-~11 S -) shyV 1 J
---- - ---- ----- - - ----- ----
----
------
------- --- ---- --- - ----
(
- )
b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
___ __ __x ~ _b _-__ JnJ~ - LfCl_C_ __ ___
2co-shy _~ - ---bull--- -- - -- -__- - --- shy
2 - 1(-2)2 - lf~L 2lt I
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2 ) 2 t 212
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_ __ _~ _ 2 5) L
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_ ---- _ _ _ --i - f-2 --~ -x --~ --I-T ~-- shyS~JlI ~ Ii ()
-----1 -) -- -In- drER ---d--l _Qs --- - - - - - - - - -- L -TPfl- -- -L-+--Q-s- -C-9lVen - kgt-o-H-1Q ) -
_ ____ _ ___L_T s _CSJtntno~ _______
-- - -- - A _LJlaquojL4IQS_ -(yen-J-cJ -~J - LS -~iucJ)or (Alt)
_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
------- --- - --- --+------ lSi ~-SS~ -- - middot -shy
-shy--shy -- --shy -shyS
------i
cmiddot QILf
- -) - ~______ -
-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
__=-_--_-_-~__-=__~1_(_~~middotJL +_ 3__plusmn_~_)~_ ___
--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
_ -- - ------shy
- - -- -__-_ - -- _-------__---- -shy12 -~
_ __ M _ _ __ __ _ _
Xz-x - 3+3
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__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
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____Imiddot__l~~_)-d = 2(1 )-(-2) S _ _= _ ~ ~ _ __ _ _ ___ _ __ ____J2~_tc~OL _ _ __ ~_ _
--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
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-- ---I
-------_
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----__ - ------ -- -- shy
-- - _ - - __------- --
---_- ---- - --- - - -- - --- - - - - - --shy
_-------- shy
---- - ---- ----- - - ----- ----
----
------
------- --- ---- --- - ----
(
- )
b) X 2 - 2~ - I lt 0
K 0 c--h _ X 2- -1L - ( = 0 _ _ _
___ __ __x ~ _b _-__ JnJ~ - LfCl_C_ __ ___
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2 - 1(-2)2 - lf~L 2lt I
2 -4- r-g ---- 2 J) - - ----=-~-~-- ---- --- I
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_ __ _~ _ 2 5) L
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_ ____ _ ___L_T s _CSJtntno~ _______
-- - -- - A _LJlaquojL4IQS_ -(yen-J-cJ -~J - LS -~iucJ)or (Alt)
_ ____ __ __(lL L _ --_ ___LS~_~ _ _r _ ~ _ ____ _ ___ IS h
--- -
-----_ - - - - - -----------I~ == -5 -
------- --- - --- --+------ lSi ~-SS~ -- - middot -shy
-shy--shy -- --shy -shyS
------i
cmiddot QILf
- -) - ~______ -
-shy - 2 L-shy=- 3Io--shy4(-2)
--shy shy
__=-_--_-_-~__-=__~1_(_~~middotJL +_ 3__plusmn_~_)~_ ___
--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
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_ __ M _ _ __ __ _ _
Xz-x - 3+3
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__ ____ _ _ _2~ -_j-d--_~_~-=-Q___ _
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--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
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-- ---I
-------_
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--- ~19 -t- 3~ --_ _ _ ----- ----- - ---- _ - ----- --~-
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_ __ M _ _ __ __ _ _
Xz-x - 3+3
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--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
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-- ---I
-------_
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----__ - ------ -- -- shy
-- - _ - - __------- --
---_- ---- - --- - - -- - --- - - - - - --shy
_-------- shy
--- - -------
__ - bullbull _- ---
_y -_____ TTr 2-~ _ bui_ -- -- -h~ IS-3r -CI)
v-=- - 7lr2( 1~-3r)
__ __ l~n 2- _ ___ _ _ = ___ I _ =- 3D_ r 3 __-t-~-- cI _
10 r ~(~--~- -()- 3
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J~ shy
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lt 0 as 10 - ---- T--reYVTtmiddotdJ - 1- - --- -- r---3 Cfl1 middotmiddot 9vlZ J
- - _ - - --- - -- ~) --- - - -- -- shy
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-- ---I
-------_
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----__ - ------ -- -- shy
-- - _ - - __------- --
---_- ---- - --- - - -- - --- - - - - - --shy
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