Mathematics Extension 2

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1 A C E EXAM PAPER Student name: 08 YEAR YEARLY EXAMINATION Mathematics Extension General Instructions Working time - 80 minutes Write using black pen NESA approved calculators may be used A reference sheet is provided at the back of this paper In Questions -6, show relevant mathematical reasoning and/or calculations Total marks: 00 Section I 0 marks Attempt Questions -0 Allow about 5 minutes for this section Section II 90 marks Attempt Questions -6 Allow about 65 minutes for this section

2 Year Mathematics Extension Section I 0 marks Attempt questions - 0 Allow about 5 minutes for this section Use the multiple-choice answer sheet for questions -0. Let arg(z) ( ) for a certain complex number z. What is arg(z* )? (A) (B) (C) (D) 7π 5 3π 5 π 5 3π 5 <. If ln(tan56 x) + x : dx, which of the following integrals uses the correct substitution? (A) (B) (C) (D) < lnu du ( C ( < lnu + tan : u du lnu du ( < lnu + tan : u du (x ) 3. Ellipse: : + y: 9 Hyperbola: x : y : How many points do the graphs of the above equations have in common? (A) 0 (B) (C) (D) 3. A force of magnitude N acts in the north-east direction and another force of 3 N acts in the easterly direction. What is the resultant magnitude (in N) of these two forces? (A) (B) (C) (D) G5 G

3 Year Mathematics Extension 5. What is the eccentricity of the ellipse 3x : + 5y : x + 30y + 0? (A) (B) I 5 I 3 5 (C) (D) I 5 I The curve y x : x J and the x-axis between x 0 and x is rotated π radians about the y-axis. Which of the following is an expression for the volume V of the solid formed? (A) (B) (C) (D) J π K y J dy π K y dy J π K y dy J 8π K y dy 7. A particle of mass kg moves in a circular motion on a smooth frictionless table at a speed of 3 m/s. It is attached to a fixed point in the middle of the table by a light, inelastic string of length metres. What is the tension in the string? (A) (B) (C) (D) 6 N N 8 N 36 N 8. The equation x J + px + q 0 where p 0 and q 0 has roots α, β, γ and δ. What is α + β + γ + δ? (A) q (B) (C) (D) p p q p q 3

4 Year Mathematics Extension 9. The point P Vcp, X Z lies on the rectangular hyperbola xy Y c:. What is the equation of the normal to the hyperbola at P? (A) py c p < (x cp) (B) (C) (D) px p y cp: [ p :\ x + pqy c(p + q) x + p : y cp 0. A particle is projected with a speed of 0 m/s and passes through a point P whose horizontal distance from the point of projection is 30 m and whose vertical height above the point of projection is 8.75 m. What is the angle of projection? (A) tan [ 3 \ (B) tan [ 3 \ (C) tan [ 3 \ (D) tan [ 3 \

5 Year Mathematics Extension Section II 90 marks Attempt questions - 6 Allow about 65 minutes for this section Answer each question in the appropriate writing booklet. Your responses should include relevant mathematical reasoning and/or calculations. Question (5 marks) Marks (a) For z i, w 3 i, find: z + w z : w : (b) Find the exact value of: < x + x : + x + 5 dx : : K x : dx 3 (c) Find the equation of the tangent to the curve x xy + y 3 at the point P (, ) to the curve. 3 dx (d) Find 9x : + 6x (e) The equation x < 3x : 5x 0 has roots α, β and γ. Find the value of α < β < γ < 5

6 Year Mathematics Extension Question (5 marks) Marks (a) The region under the curve y e 5bc and above the x-axis for a x a is rotated about the y-axis to form a solid. Calculate the volume of the solid using the method of cylindrical shells. 3 What is the limiting value of the volume of the solid as a approaches infinity? (b) z + i and z 3 i Find z z in the form a + ib where a and b are real. Write z 6 and z : in modulus-argument form. Write cos )( 6: as a surd by equating equivalent expressions for z z. (c) Find the values of A, B, C and D such that: 5x 3 3x + x x + x A x + B Cx + D + x x + Hence evaluate 5x< 3x : + x x J + x : dx (d) Solve the polynomial equation x 6x 3 + 9x + x 0, given that the equation has a double root. 3 6

7 Year Mathematics Extension Question 3 (5 marks) Marks (a) In the Argand diagram, ABCD is a square, and OE and OF are parallel and equal in length to AB and AD respectively. The vertices A and B correspond to the complex numbers z and z respectively. Explain why the point E corresponds to z : z 6. What complex number corresponds to point F? (iii) What complex number corresponds to vertex D? (b) Evaluate the integral lnx x : dx (c) A motor bike travels around a circular bend that is banked at an angle of α to the horizontal. The bike travels at a line on the road where the radius of the curve is r metres, at a constant speed, so there is no sideways frictional force acting on the bike. Find expressions for Nsinα and Ncosα by resolving forces. Derive an expression for the velocity (v) of the bike. Find the value of α if the radius of the curve is 0 metres and the road is banked to allow vehicles to travel at 90 km/h. (Use g 0 m/s) (d) A solid is formed by rotating about the line x the region bounded by the parabola y x, the x-axis, x 0 and x. Find the volume of this solid using the method of slicing. 3 (e) A motel has four vacant rooms. Each room can accommodate a maximum of four people. In how many different ways can six people be accommodated in the four rooms? 7

8 Year Mathematics Extension Question (5 marks) Marks (a) Consider the polynomial equation: ax < + bx : + cx + d 0. What are the relations between the roots a, b, γof the equation, in terms of the coefficients a, b, c and d? The equation 36x < x : x + 0 has roots a, b and c. Find a if α β + γ. (iii) The equation x < + bx : + cx + d 0 has roots a, b and c. Show that b < bc + 8d 0 if α β + γ. (b) The graph of f(x) e 5b is shown above. Draw separate one-third page sketches of these functions. Indicate clearly any asymptotes and intercepts with the axes. y f(x) y {f(x)} : y f(x) y lnf(x) (c) A light inextensible string OP is fixed at the end O and is attached at the other end P to a particle of mass m (in kg) which is moving uniformly in a horizontal circle whose centre is vertically below and distant x (in metres) from O. Let g be the acceleration due to gravity. Show that the period of motion is given by the formula: 3 T πi x g What is the effect on the motion of the particle if the mass is doubled? If the number of revolutions per second is increased from to 3, find the change in x. Answer correct to three decimal places and use 0 ms - for gravity. 8

9 Year Mathematics Extension Question 5 (5 marks) Marks (a) The circles XPYS and XYRQ intersect at the points X and Y. PXQ, PYR, QSY, PST and QTR are straight lines. Explain why STQ YRQ + YPS. Show that YRQ + YPS + SXQ π Prove that STQX is a cyclic quadrilateral. Let QPY α and PQY β. Show that STQ α + β 3 (b) P(acosθ, bsinθ) and P(acosφ, bsinφ) are the end points of a diameter of the ellipse shown below. Tangents to the ellipse at P, Q cut the x-axis at X, U respectively, and the y-axis at Y, V respectively. Show that the tangent to the ellipse at P is the following equation. xcos θ ysin θ + a b Show that φ θ ± π What are the coordinates of X, Y, U, V in terms of a, b and θ? ab Show that the area of XYUV is sinθ 9

10 Year Mathematics Extension Question 6 (5 marks) Marks (a) I 6 ( + x : ) dx n,,3,... Show that I n+ n n I n + Hence evaluate 6 n+ n,,3, 3 n ( + x : ) < dx (b) Prove the identity: cos 3 A 3 cosa cos3a Show that x cosa satisfies the cubic equation x < 6x + 0 given that cos3a 6 : : What are the three roots of the equation x < 6x + 0? Answer correct to four decimal places. (c) Given y x< x : Find the coordinates of all the stationary points. What are the equations of the asymptotes of the curve? Hence sketch the curve y x< x :. End of paper 0

11 Year Mathematics Extension

12 Year Mathematics Extension

13 3 Year Mathematics Extension

14 Year Mathematics Extension ACE Examination 08 Year Mathematics Extension Yearly Examination Worked solutions and marking guidelines Section I Solution Arg(% & ) 7Arg(%) Let / tan 3 5 6/ When x 0, u 0 and 5 3, / < < > ln(tan3 5) > ln/6/ Criteria Mark: B Mark: C 3 Mark: D There are 3 points in common. Force E cos35 K5 + Mark: B L L L L 3( ) + 5(L 9 + 6L + 9) (5 ) 9 + 5(L + 3) 9 5 (5 ) 9 (L + 3) \ P 5, Q 3 R 9 Q9 P 9 S 3T 9 S 5T 9 5 or R U 5 Area of the slice is an annulus V L 5 V L (5 9 ) 9 L 5 9 ±E L 5 9 ± E L XY +(Z 9 [ 9 )XL +\S + E LT S E LT ]XL +E L XL V Y lim c + `a ad V E L XL + > E L 6L Mark: A Mark: B

15 7 8 9 e fg9 [ 39 8 N Sum of the roots (j + k + l + X) Q P 0 0 j V + mj + n 0 k V + mk + n 0 l V + ml + n 0 X V + e + n 0 j V + k V + l V + X V + m(j + k + l + X) + n 0 j V + k V + l V + X V + m 0 + n 0 j V + k V + l V + X V n 5L p 9 L + 5 6L L 65 L 5 6L 65 p m pm p pm 9 m 9 \Gradient of the normal is m 9 L p m m9 (5 pm) ml p m (5 pm) Equations of projectile motion 5 Yrcosj 30 0rcosj r 3 cosj L tr9 + Yrsinj Year Mathematics Extension Mark: C Mark: A Mark: A r9 + 0rsinj 35 0r rsinj Substituting equation () into equation () w cosj x w cosj x sinj 35 5sec 9 j + 0tanj 7 9(tan 9 j + ) + tanj 9tan 9 j tanj (3tanj ) 9 0 3tanj tanj 3 j tan 3 w 3 x Mark: D

16 Section II (a) % + z 3{ 5 Solution Year Mathematics Extension Criteria mark: Correct (a) (b) (b) % 9 z 9 ( {) 9 (3 {) 9 { (9 { + { 9 ) { 5 + { 5 + 0{ Use the substitution / / / (5 + )65 When x then u 3 and when x 3 then u / > 65 > / 9} 0 3 Use the substitution 5 sin~ 65 6~ cos~ 65 cos~6~ When x 0 then ~ 0 and when 5 then ~ < V < V / 9 > E > cos~ cos~6~ 9 < V > w + cos~x 6~ ~ + sin~} w x + + < V mark: Shows some understanding. mark: Finds the primitive function or sets up the integration using substitution. 3 marks: Finds the primitive function. mark: Correctly expresses the integral in terms of q (c) 5 9 5L + L 5 wl + 5 6L 6L x + 3L (3L 9 5) 6L 65 L 5 At P (, ) 6L 65 L 5 3L 9 5 \Equation of the tangent L (5 ) 5 + L marks: Evaluates the derivative at P to find the gradient of the tangent. mark: Differentiates implicitly. 3

17 (d) (e) (a) (a) (b) Complete the square w x w x w5 + 3 x 9 + (35 + ) Therefore 65 > > 65 (35 + ) tan3 w 35 + x + jkl 6 P j k l (jkl) Cylindrical shells with radius of x and height R 3ÄÅ Ç Y lim c +5R 3ÄÅ X5 `Ä Äd Ç + > 5R 3ÄÅ 65 +ÉR 3ÄÅ Ç Ñ +S R 3ÇÅ T lim +S Ç Ö R3ÇÅ T + % + { % 9 3 { 3 + { 3 + { 3 + { 3 + Year Mathematics Extension 3 marks: Completes the square and sets up integration. mark: Shows some understanding of the problem. mark: correct answer mark: Finds the radius and height of the cylindrical shell mark: Correct mark: Correct (b) % (cos + + {sin + ) % 9 Ücos á + 6 à + {sin á + 6 àâ mark: Finds one of the points in modulusargument form.

18 Year Mathematics Extension (b) (iii) (c) (c) % % 9 Ücos á+ + 6 à + {sin á+ + 6 àâ 3 + { 3 + Equating the real parts cos w5+ x 3 \cos w 5+ 6 x ã5(5 9 + ) + å(5 9 + ) + (5 + ç) 5 9 ã5 + ã5 + å5 9 + å ç5 9 Therefore (ã + ) 5 55 (å + ç) ã5 5 3 å Hence A, B, C 3 and D > V > w x 65 > w x 65 ln ln(59 + ) tan (d) ê(5) 5 V ê (5) By trial and error (x ) is a double root (ê() ê () 0) 3(a) Dividing 5 V by gives Therefore ê(5) 5 V (5 ) 9 (5 3)(5 + ) \ x, and 3 mark: Correct mark: Finds two of the pronumerals or shows some understanding. mark: Correctly finds one of the integrals 3 marks: Finds the double root or makes significant progress. mark: Uses the derivative of the function. mark: Correct OE//AB and OE AB \OABE is a parallelogram Let w be the vector that correspond to point E. z + % % 9 \z % 9 % 5

19 3(a) 3(a) (iii) 3(b) 3(c) íe íî and OF OE \F corresponds to {(% 9 % ) (multiplying a complex number by i corresponds to an anticlockwise rotation about the origin through 90 ) Since AD//OF and AD OF Point D corresponds to the complex number: % + {(% 9 % ) % ( {) + {% 9 Integration by parts > ln > ln w 5 x 65 ln5 5 > 6 65 ln ln ln Resolving forces ecosj fg9 ïsinj [ 0 fg9 ïsinj [ \ïsinj fg9 [ Also esinj ïcosj mg \ïcosj mg Year Mathematics Extension mark: Correct mark: Correct mark: Sets up integration by parts. mark: Finds either ïcosj or ïsinj 3(c) ïsinj ïcosj fg 9 [ ft tanj g9 [t g 9 [ttanj g E[ttanj mark: Correct 3(c) (iii) Frictional force is 0 g 90 km h m/s tanj g9 [t j 7 3 mark: Uses the results for tanj with one correct value. 6

20 Year Mathematics Extension 3(d) 3(e) (a) Same volume as L (5 ) 9 rotated about the y-axis Area of the slice is a circle with a radius of x and height y. XY +(5 ) 9 XL +SEL T 9 XL V Y lim c + SEL T 9 XL `a ad V Y + > L EL + 6L + L9 8 3 L 9 + L} + w x V 8+ cubic units 3 6 people in rooms with no restrictions: 6 ways V 6 people in rooms with a room of 6: C V ù 6 people in rooms with a room of 5: C C C ú ways \6 people in rooms with a room of : ù V V ù C C C C ú 00 j + k + l Q P jk + kl + lj p P 3 marks: Correct integral for the volume of the solid. mark: Sets up the area of the slice mark: Shows some understanding mark: Correct (a) jkl 6 P j + k + l 36 j + j 3 mark: Correct (a) (iii) (b) j 6 j + k + l Q j + j Q j Q w Q x + Q w Q 9 x + p w Q x Q 8 + Q Qp Q + Q Qp Q Qp mark: Finds a or shows some understanding. mark: Correct 7

21 Year Mathematics Extension (b) mark: Correct (b) (iii) mark: Shows one asymptote only. (b) (iv) mark: Correct (c) Let w be the angular velocity of the particle P about C. The forces acting on the particle are its weight mg and the tension T in the string. Resolving the forces at P: Horizontally f[ü 9 cos á + ~à sin~ Vertically ft cos~ Dividing the above two equations f[ü 9 ft sin~ cos~ [ü 9 t tan~ but tan~ [ 5 \ [ü9 t [ 5 or ü Kt 5 3 marks: Deriving w or making significant progress. mark: Resolves the forces vertically and horizontally 8

22 The time for one complete revolution (or the period of motion) + ü Year Mathematics Extension + K t 5 +U 5 t (c) (c) (iii) 5(a) 5(a) 5(a) (iii) In the division of the two equations of motion the mass is cancelled out and hence there is no effect on the motion if the mass is doubled. Making x the subject of the above equation 5 t ü 9 Let w and w be the angular velocities of P in two situations in the problem. ü revolutions per sec + radians per sec ü 9 3 revolutions per sec 6+ radians per sec Using the above equation for x 5 t (ü ) 9 t t (ü 9 ) 9 t t + 9 w 6 36 x 5t m \The particle rises by about metres In Z Z + (Exterior angle of a triangle is equal to the sum of the two interior opposite angles) (Angles in the same segment are equal) Z + + (Opposite angles of a cyclic quadrilateral are supplementary) + + (Straight line measures 80 ) \ Z \ Z YXS + + (Straight line measures 80 ) \ Z Z + + ã + (from and ) \ STQX is a cyclic quadrilateral as the opposite angles are supplementary. mark: Correct mark: Showing some understanding of the problem. mark: Correct mark: Makes some progress towards the solution. mark: Makes some progress towards the solution. 9

23 Year Mathematics Extension 5(a) (iv) 5(b) 5(b) 5(b) (iii) and (Angles in the same segment are equal) (Angles in the same segment are equal) j + + (Exterior angle of a triangle is equal to the sum of the two interior opposite angles) + (Opposite angles of cyclic quadrilateral STQX are supplementary) (PXQ is a straight line) + + k + j (Exterior angle of a triangle is equal to the sum of the two interior opposite angles) 5 acos~and L Qsin~ 6L 65 6L 6~ 6~ 65 Qcos~ asin~ Qcos~ asin~ Equation of the tangent L Qsin~ Qcos~ (5 acos~) asin~ LPsin~ PQs{Ø 9 ~ 5Qcos~ + PQcos 9 ~ 5Qcos~ + LPsin~ PQ(s{Ø 9 ~ + cos 9 ~) \ 5cos ~ P + Lsin ~ Q Gradients of OP. f L 9 L Qsin~ acos~ Q P tan~ Similarly, the gradient of OQ is Ç tan± Hence O, P, Q are collinear (gradients are equal). tan~ tan± since ~ ± then ± ~ ± + 5cos ~ + Lsin ~ P Q Tangent cuts the x-axis at X á Ç, 0à and y-axis at Y á0, µ Similarly, since cos ± cos ~ and sin ± sin ~ then U á Ç µ, 0à and Y á0, µ à µ à 3 marks: Makes significant progress towards the solution mark: Applies a relevant circle theorem. mark: Finds the gradient of the tangent. mark: Finds the gradient of the OP and OQ mark: Finds the coordinates of one point. 0

24 Year Mathematics Extension 5(b) (iv) XYUV is a rhombus since the diagonals XU and YV bisect each other at right angles at O. ã 5L πy Y P cos~ Q sin~ PQ sin ~ 6(a) ª º > ( ) º 65 Ø,,3,... 6(a) [5( ) 3º ] > 5( Ø)( ) 3º3 (5)65 3º + Ø > [( ) ]( ) 3º3 65 3º + Ø > É( ) 3º ( ) 3(ºø) Ñ 65 3º + ت º ت ºø ت ºø (Ø )ª º + 3º ª ºø Ø Ø ª º + Ø ºø ª 3 ª w ª + x ª + ª > [tan 3 5] mark: Deduces that XYUV is a rhombus. 3 marks: Makes significant progress towards the solution mark: Correctly applies integration by parts. mark: Applies the recurrence relation to find an expression for I 3 6(b) 6(b) \ > ( ) 3 cos ã cos(ã + ã) cos 9 ãcosã sin 9 ãsinã (cos 9 ã )cosã cosãsinãsinã cos ã cosã cosãsin 9 ã cos ã cosã cosã( cos 9 ã) cos ã 3cosã cos ã cos ã 3 cosa Substituting 5 cosa into S cosãt + 6 S cosãt cos ã cosã cos ã 3 cosã \ cos3ã 8 cos3ã 6 8 mark: Makes some progress towards the solution. mark: Makes some progress towards the solution.

25 Year Mathematics Extension 6(a) ª º > 6(a) ( ) º 65 Ø,,3,... [5( ) 3º ] > 5( Ø)( ) 3º3 (5)65 3º + Ø > [( ) ]( ) 3º3 65 3º + Ø > É( ) 3º ( ) 3(ºø) Ñ 65 3º + ت º ت ºø ت ºø (Ø )ª º + 3º ª ºø Ø Ø ª º + Ø ºø ª 3 ª w ª + x ª + ª > [tan 3 5] 3 marks: Makes significant progress towards the solution mark: Correctly applies integration by parts. mark: Applies the recurrence relation to find an expression for I 3 6(b) 6(b) 6(b) (iii) \ > ( ) 3 cos 3 ã cos(ã + ã) cos 9 ãcosã sin 9 ãsinã (cos 9 ã )cosã cosãsinãsinã cos ã cosã cosãsin 9 ã cos ã cosã cosã( cos 9 ã) cos ã 3cosã cos3 ã cos 3 ã 3 cosa Substituting 5 cosa into S cosãt + 6 S cosãt cos ã cosã cos ã 3 cosã \ cos3ã 8 cos3ã 6 8 cos3ã 3ã ±cos 3 w x + Ø+ where n is an integer Taking n 0,, and the positive branch to obtain the three roots 5 cosa.68,.607, mark: Makes some progress towards the solution. mark: Makes some progress towards the solution. mark: Correct

26 6(c) ê (5) (59 )(35 9 ) (5 )(5) (5 9 ) 9 59 ( ) (5 9 ) 9 59 (5 9 ) (5 9 ) 9 Stationary points f (x) (5 9 ) 0 \5 0 or 5 ± ± 3 When 5 3 then When 5 3 then \Stationary points are (0, 0), S 3, 3 3T, S 3, 3 3T Year Mathematics Extension mark: Finds one stationary point 6(c) ê(5) (5 + )(5 ) \Vertical asymptotes are 5 ± ê(5) (5 + )(5 ) lim ê(5) 5 Ä Ö mark: Finds the vertical or oblique asymptotes. 6(c) (iii) \Oblique asymptote y x When x 0 then y 0 \Point of intersection with coordinate axes is (0, 0) S 3, 3 3T is a minima (ê (5) changes sign from to +) S 3, 3 3T is a maxima (ê (5) changes sign from + to ) L ê(5) is an odd function mark: Shows most of the features of the curve. 3