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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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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---

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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- -Xl- 3i + 3

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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

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20

0( - 22- s _ -~lt______- 0 )

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--- -- ---- ---shy

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-~-- -_ _--- ------ ---_- - shy_ ___--+-____ amp 1

--middot-------ilmiddot----~--========--

-------(1-)- pound (~)-=-J oX 2_ -_ _

_-poundr~ )~J-(~~~- shy--- --- - --------==- J- X-l - 1-(

-- -- --- --- --- -- --F(-~r)

--------- -- ---shy_______

J-4------( r )-- - -

---- ----- -shy

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---------------------shy

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c

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or

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( h __ ____oj 3~ -I - ~--(3 ~-X3--+--) CJ ~ chlta _ 1 ~ ~~1D

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~ -

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2 --- - _ ----__shy

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(--

2 - shy- - -

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_____ ___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

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(Iii) ~ 1 r - k I T

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------ - - -

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~ ~ ~

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__ _ _ gt~1 (J cJ Ib-ls - Z IS

_ middot _ f9+_~ _ ~J_Hcgt__ ______ cJ__-

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-- - -

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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 - -

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-- - - - - =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-

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

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~ --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

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----_ - --- --- shy

--shy

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ISO 330 -- - - ) - - - --- - shy

-reg--- --shy- -g

---- -------5 ~~---(bs8

-- - - - I ~~

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- - ------ -- 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---------------=---~

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-------l------~--~ --O- --- - - ---- --- --shy shy

---------------------shy

__f-J- _ liLt -s~ 2-$)-~_b___$ _0__~_ -- _b ~0 __ _~(I) _-

----t~ - 4 S~ 2 e - ~ S_~ __0 - G

c

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lt C I )

or

(I)

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( h __ ____oj 3~ -I - ~--(3 ~-X3--+--) CJ ~ chlta _ 1 ~ ~~1D

__ _ - l~_- 21_ ___ _ 2-~(3_ ~L) _ od 0 clvu-OrY ~Cl~ r

~ -

---- - 1---------- - - ------shy

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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- - -

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_____ ___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

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--- ~~-j

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J j l _m tx-- ~C J - 2 (- -f )fT- 3

~ ~ ~

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e) 2 x - 3 j - 4- J- L

2~ -3 4t-fc I

- _t _ ~K _ 2 ~ bx

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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

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- ---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-

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)

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( 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

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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-

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

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___-=-- ~ _ J2~~(~s-_~~_~__ _tJ~) lt0 11 i

____ -coc cto~_middot~ -fl((~) lt 0 -reg----- - shy

----_ - --- --- shy

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