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NAME: TEACHER: MATHEMATICS AND STATISTICS 2013 (MATHEMATICS with CALCULUS) Standards: AS 91577 Apply the algebra of complex numbers in solving problems. Page 2 Page 13 Page 24 AS 91578 Apply differentiation methods in solving problems AS 91579 Apply integration methods in solving problems Produced by SINCOS Page 1 2013 EOY L3 M3 Mathematics Writer: David Fortune NAME: TEACHER: MATHEMATICS AND STATISTICS 2013 (MATHEMATICS) 91577 Apply the algebra of complex numbers in solving problems Credits: Five QUESTION AND ANSWER BOOKLET Make sure that you have a copy of the Formulae and Tables Booklet. You should answer ALL the questions in this booklet. Show ALL working for ALL questions. The questions in this booklet are NOT in order of difficulty. Attempt all questions otherwise you may not provide enough evidence to achieve the required standard. If you need any more room for any answer, use the extra space provided on page 12 at the back of this booklet and clearly number the question. YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE EXAMINATION. TOTAL Produced by SINCOS OVERALL LEVEL OF PERFORMANCE Page 2 t 2013 EOY L3 M3 Mathematics Writer: David Fortune You are advised to spend 60 minutes answering the questions in this booklet. Assessor’s use only QUESTION ONE (a) Solve the following equation to find an expression for x in terms of p: log 5 ( x p) 3 . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) Write 1 3 2 3 3 in the form a b 3 where a and b are rational numbers. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (c) Solve for x: x 3 3x . You must show all your working steps. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 3 2013 EOY L3 M3 Mathematics Writer: David Fortune (d) (i) Sketch the locus of the complex number z satisfying the condition z 1 1 z . Assessor’s use only _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Draw your answer on the grid. Produced by SINCOS Page 4 2013 EOY L3 M3 Mathematics Writer: David Fortune (ii) Find the complex number v that belongs to the locus in Q1d(i) above, with the greatest positive argument , ie where 0 . Assessor’s use only _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 5 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only QUESTION TWO (a) u and v are complex numbers where u 2 3i and v 3 2i Find: (i) 3u - expressing your answer in the form a b i . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) u v expressing your answer in polar form rcis . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) One root of the equation 3x 2 qx q 0 , q 0 , is three times the other root. Find the value of q. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 6 2013 EOY L3 M3 Mathematics Writer: David Fortune (c) Assessor’s use only Given that x and y are rational numbers solve the equation: 3 y 2i y i . 3x 2 y x i _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ __________________________________________________________________________ Produced by SINCOS Page 7 2013 EOY L3 M3 Mathematics Writer: David Fortune (d) Solve the equation for x : x 1 x m 1 x x 3 Assessor’s use only x 1, 0 . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 8 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only QUESTION THREE Write 2cis in the form a b i where a and b are rational numbers. 3 6 (a) _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) Solve for x: x 3 3 x . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 9 2013 EOY L3 M3 Mathematics Writer: David Fortune (c) Assessor’s use only Find all the solutions of z 3 1 3 i . Write your solutions in polar form. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (d) Solve the equation for x : 3 x 2 p 2 x 1 . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 10 2013 EOY L3 M3 Mathematics Writer: David Fortune (e) Assessor’s use only z 1 1 then x 0 . Let z x iy . Show that if z 1 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 11 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only Extra paper for continuing your answers, if required. Clearly number the question(s). Question Number _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 12 2013 EOY L3 M3 Mathematics Writer: David Fortune NAME: TEACHER: MATHEMATICS AND STATISTICS 2013 (MATHEMATICS) 91578 Apply differentiation methods in solving problems Credits: Six QUESTION AND ANSWER BOOKLET Make sure that you have a copy of the Formulae and Tables Booklet. You should answer ALL the questions in this booklet. Show ALL working for ALL questions. Show any derivative(s) that you need to find when solving the problems. The questions in this booklet are NOT in order of difficulty. Attempt all questions otherwise you may not provide enough evidence to achieve the required standard. If you need any more room for any answer, use the extra space provided on page 23 at the back of this booklet and clearly number the question. YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE EXAMINATION. TOTAL Produced by SINCOS OVERALL LEVEL OF PERFORMANCE Page 13 t 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only You are advised to spend 60 minutes answering the questions in this booklet. Show all working. QUESTION ONE (a) Differentiate y 3 2 x 4 . You do not need to simplify your answer. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) Find the equation of the tangent to the curve y 2 at the point 0, 2 . x 1 Show any derivatives that you need to find when solving this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (c) A meteor enters the earth’s atmosphere and burns up at a rate that is always proportional to its surface area. Assuming the meteor is always spherical, show that the radius decreases at a constant rate. Show any derivatives that you need to find when solving this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 14 2013 EOY L3 M3 Mathematics Writer: David Fortune (d) Assessor’s use only The graph below defines the function y f (x) . For the function f (x) find: (i) lim f ( x) . x 3 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) lim f ( x) . x 5 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (iii) all value(s) of x where f (x) is not differentiable. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ (iv) all value(s) of x where f ( x) 0 . ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ Produced by SINCOS Page 15 2013 EOY L3 M3 Mathematics Writer: David Fortune (e) Assessor’s use only A triangular prism has ends which are equilateral triangles. The total length of all the edges of the triangular prism is L cm. Show that the maximum possible surface area of the prism is L2 2(12 3 ) . Show any derivatives that you need to find when solving this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 16 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only QUESTION TWO (a) (a) Differentiate ex . y 2 4x 1 You do not need to simplify your answer. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) The graph shows the function y f (x) . On the axes below, sketch the graph of the derived function of f (x) . Produced by SINCOS Page 17 2013 EOY L3 M3 Mathematics Writer: David Fortune (c) Assessor’s use only An ellipse is defined by the parametric equations x 2 cos t and y 3 sin t . Find the gradient of the tangent to the curve at t 4 . Show any derivatives that you need to find when solving this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ 2 d 2 y dy 2 2 (d) Show that y ln( x 5) is a solution of the differential equation . 2 dx x 5 dx 2 Show any derivatives that you need to find when solving this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 18 2013 EOY L3 M3 Mathematics Writer: David Fortune (e) 3 -1 Coffee is poured at a uniform rate of 10 cm s into a cup whose inside shape is a truncated cone. If the upper and lower radii of the coffee cup are 4 cm and 2 cm, and the height of the cup is 6 cm, how fast will the level of the coffee be rising when the coffee is half way up? Assessor’s use only [It may be useful to extend the cup to form a cone.] Give any derivative(s) you need to find when solving this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 19 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only QUESTION THREE (a) Differentiate y e 4 x ln( 3x 4) . You do not need to simplify your answer. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) On Christmas Eve, in-store sales of Galaxy Tablets follow the model: S (20t 2t 2 )e 0.2t 0 t 10 where S is the number of sales (in hundreds) and t is the time in hours after opening the shop doors. Show any derivatives that you need to find when solving this problem. (i) Find the rate of change of the number of sales 2 hours after opening the doors. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) At what time is the maximum rate of sales? _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 20 2013 EOY L3 M3 Mathematics Writer: David Fortune (c) A liquid is draining through a conical filter that is 16 cm high and with a radius of 4 cm at the top. Assessor’s use only The liquid is coming out of the cone at a constant rate of 2 cm3/sec. At what rate is the depth of liquid in the cone changing when the liquid is 8 cm deep? Show any derivatives that you need to find when solving this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ ________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 21 2013 EOY L3 M3 Mathematics Writer: David Fortune (d) Assessor’s use only A trapezium is inscribed inside a semicircle of radius 4 cm so that one side is along the diameter. Find the maximum possible area for the trapezium. [Hint: it may be useful to find the area in terms of angle a.] Show any derivatives that you need to find when solving this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 22 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only Extra paper for continuing your answers, if required. Clearly number the question(s). Question Number _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 23 2013 EOY L3 M3 Mathematics Writer: David Fortune NAME: TEACHER: MATHEMATICS AND STATISTICS 2013 (MATHEMATICS) 91579 Apply integration methods in solving problems Credits: Six QUESTION AND ANSWER BOOKLET Make sure that you have a copy of the Formulae and Tables Booklet. You should answer ALL the questions in this booklet. Show ALL working for ALL questions. Show any integral(s) that you need to find when solving the problems. The questions in this booklet are NOT in order of difficulty. Attempt all questions otherwise you may not provide enough evidence to achieve the required standard. If you need any more room for any answer, use the extra space provided on page 34 at the back of this booklet and clearly number the question. YOU MUST HAND THIS BOOKLET TO YOUR TEACHER AT THE END OF THE EXAMINATION. TOTAL Produced by SINCOS OVERALL LEVEL OF PERFORMANCE Page 24 t 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only You are advised to spend 60 minutes answering the questions in this booklet. QUESTION ONE (a) Find the integrals: You do not need to simplify your answers. Do not forget the constant of integration. (i) (cosec 2 4 x) dx . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) 5 x 4 3x dx . x2 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 25 2013 EOY L3 M3 Mathematics Writer: David Fortune (b) Calculate the enclosed area between the graphs of y x 2 1 and y x 1 . Assessor’s use only Give the results of any integration needed to solve this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (c) A particle moves with velocity function v(t ) cos t m/s along a straight line where t is the time in seconds from the start of the motion. Assuming the distance after 1 second was 4 metres, find the distance function for the particle. Give the results of any integration needed to solve this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 26 2013 EOY L3 M3 Mathematics Writer: David Fortune (d) The temperature, T, of a cake cools at a rate proportional to the difference between its own temperature and the temperature Ts of the surroundings. (i) Assessor’s use only Write a differential equation that expresses the rate at which the cake cools. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) The hot cake had an initial temperature of 270 oC and cools down to 120 oC after 3 minutes. The surrounding room temperature is 20 oC. Find the time that the hot cake takes to cool to 40 oC. Give the results of any integration needed to solve this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 27 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only QUESTION TWO (a) Find the integral p ( x 3) 3 where p is a constant. dx You do not need to simplify your answer. Do not forget the constant of integration. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ 1 (b) Dierdre wanted to find ex 2 2 x dx . 0 She couldn’t integrate so decided to use Simpson’s Rule to find an approximation to the integral. She made up a table for the function as shown. x e x2 2 x 0 1 0.25 0.646 0.5 0.472 0.75 0.392 1 0.368 1 Use these values and Simpson’s Rule to find an approximate value for ex 2 2 x dx . 0 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 28 2013 EOY L3 M3 Mathematics Writer: David Fortune (c) Find the integral: 4x 1 dx . 2x 1 Assessor’s use only _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (d) The graph shown is y sin 2 x . Given that the shaded area between the curve, the x-axis and the line x = m is 0.2 find the value of m. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 29 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only (e) The curves cy 2 x 3 and y 2 ax (where a > 0 and c > 0) intersect at the origin and at the point P in the first quadrant. The areas of the regions enclosed by the arcs OP, the x-axis and the vertical line through P are A1 and A2 respectively. Prove that A1 3 . A2 5 Give the results of any integration needed to solve this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 30 2013 EOY L3 M3 Mathematics Writer: David Fortune Assessor’s use only QUESTION THREE (a) Find the integrals: You do not need to simplify your answers. Do not forget the constant of integration. (i) e 1 4x dx . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (ii) (16 cos 2x sin 2x) dx . _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (b) Find the general solution of the differential equation dy k ( x 2)( 2 y ) 0 x 2 and 0 y 2 . dx _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 31 2013 EOY L3 M3 Mathematics Writer: David Fortune b (c) Assessor’s use only (2 7h( x)) dx 2 . 1 b Find a simplified expression, in terms of b, for h( x) dx . 1 _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ (d) A water trough for stock is in the shape of an isosceles triangular prism. The top edges of the tank are parallel to the ground. The stock trough is 40 cm deep, 60 cm wide and 2 m long. Water evaporates from the trough in such a way that, at any time, the rate of change of depth of water in the tank is proportional to the top surface area of the water in the tank times half its depth. Initially the water trough is full. After 8 days the depth of water in the trough is 30 cm. After how many days is the water in the trough 10 cm deep? Give the results of any integration needed to solve this problem. _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 32 2013 EOY L3 M3 Mathematics Writer: David Fortune _________________________________________________________________________ Assessor’s use only _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 33 2013 EOY L3 M3 Mathematics Writer: David Fortune Extra paper for continuing your answers, if required. Clearly number the question(s). Assessor’s use only Question Number _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ _________________________________________________________________________ Produced by SINCOS Page 34 2013 EOY L3 M3 Mathematics Writer: David Fortune ASSESSMENT SCHEDULE 91577 Apply the algebra of complex numbers in solving problems Achievement Achievement with Merit Achievement with Excellence Apply the algebra of complex numbers in solving problems involves: Apply the algebra of complex numbers, using relational thinking, in solving problems involves one or more of: Apply the algebra of complex numbers, using extended abstract thinking, in solving problems involves one or more of: • selecting and using methods • demonstrating knowledge of concepts and terms • communicating using appropriate representations. • selecting and carrying out a logical sequence of steps • connecting different concepts or representations • demonstrating understanding of concepts • forming and using a model; • devising a strategy to investigate or solve a problem • identifying relevant concepts in context • developing a chain of logical reasoning, or proof • forming a generalisation; and also relating findings to a context, or communicating thinking using appropriate mathematical statements. and also using correct mathematical statements, or communicating mathematical insight. Sufficiency for each question: N0: No response, no relevant evidence. N1: Attempt at ONE question showing limited knowledge of algebra of complex numbers in solving problems. N2: 1 u A3: 2 u A4: 3 u M5: 1 r M6: 2 r E7: 1 t with minor errors ignored E8: 1 t with full excellence criteria Judgement Statement Score range Produced by SINCOS Not Achieved Achievement Achievement with Merit Achievement with Excellence 0-6 7 - 13 14 - 18 19 -24 Schedule Page 35 2013 EOY L3 M3 Mathematics Writer: David Fortune Question Evidence ONE Achievement (u) Apply the algebra of complex numbers in solving problems. 1a x p 53 1b 1 1c 1d(i) [ = p 125] 1 3 3 x 3 9x 2 9x 2 x 3 0 x 0.636 valid x 0.524 not valid Merit (r) Apply the algebra of complex numbers, using relational thinking, in solving problems. Excellence (t) Apply the algebra of complex numbers, using extended abstract thinking, in solving problems. Correct expression. Correct solution. 2 solutions given. Correct solution. (one only). Locus of z 1 1 is the shaded area inside circle centre (1,0) Locus of 1 z is the shaded area outside circle centre (0,0) Either correct locus shown. Correct solution (correct locus intersection shown – dark shading in right circle). 1d(ii) ( x 1) 2 y 2 x 2 y 2 2x 1 0 3 1 y ie x 2 2 y 3 Argument tan 1 tan 1 x 1 3 Meet at Produced by SINCOS Correct solution showing correct logical steps with correct mathematical statements. Schedule Page 36 2013 EOY L3 M3 Mathematics Writer: David Fortune Question TWO Evidence Achievement (u) Apply the algebra of complex numbers in solving problems. 2a(i) 2a(ii) 2b 3 7 i 26 cis 0.197 OR Apply the algebra of complex numbers, using extended abstract thinking, in solving problems. 26 cis11.3o If and 3 are the two roots then q q x ( x )( x 3 ) 3 3 q q 4 and 3 2 3 3 4 and thus q = 16 3 Correct line 2 or equivalent. Correct solution. (3 y 2i)( x i) (3x 2 y )( y i) 2 2 xi 2 y 2 (3x 2 y)i Equating real and imaginary parts y 2 1 and 2 x 3x 2 y y 1 x and Correct equating real and imaginary parts. 1 y 5 ie solutions are x 2d Apply the algebra of complex numbers, using relational thinking, in solving problems. Excellence (t) Correct expression. x2 2c Merit (r) 1 5 x y 1 1 y 1 5 Correct solution. x 1 x m2 2 1 x x 9 2 2 9 x 18x(1 x) 9(1 x) x(1 x)m 2 m2 x2 m2 x 9 0 Squaring Correct line 3. (or equivalent) m 2 m 4 36m 2 x 2m 2 1 m 2 36 = 2 2m Produced by SINCOS Schedule Page 37 Correct solution showing correct logical steps with correct mathematical statements. 2013 EOY L3 M3 Mathematics Writer: David Fortune Question Evidence THREE Achievement (u) Apply the algebra of complex numbers in solving problems. 3a 64 Correct expression. 3b x396 x x x 1 Correct solution. 3c z 3 2cis 2cis ( 2k ) 3 3 1 z1 2 3 cis ( ) 9 1 7 z 2 2 3 cis ( ) 9 1 1 13 5 z 3 2 3 cis ( ) or z 3 2 3 cis ( ) 9 9 3d 3e ( x 2 p) ln 3 ( x 1) ln 2 x(ln 3 ln 2) ln 2 2 p ln 3 ln 2 2 p ln 3 x ln 3 ln 2 x 1 yi x 1 yi ( x 1) 2 y 2 ( x 1) 2 y 2 Apply the algebra of complex numbers, using extended abstract thinking, in solving problems. Correct solution (all three) in either real or degree argument format. Correct expression without exponents. Correct solution. 1 Correct relationship for x 2 and y 2 without modulus signs. [line 2 or equivalent] positive 2x 2x 4x 0 x0 Produced by SINCOS Apply the algebra of complex numbers, using relational thinking, in solving problems. Excellence (t) Correct polar form for one solution. ( x 1) 2 y 2 ( x 1) 2 y 2 as Merit (r) Schedule Page 38 Correct solution showing correct logical steps with correct mathematical statements. 2013 EOY L3 M3 Mathematics Writer: David Fortune ASSESSMENT SCHEDULE 91578 Apply differentiation methods in solving problems Achievement Achievement with Merit Achievement with Excellence Apply differentiation methods in solving problems involves: Apply differentiation methods, using relational thinking, in solving problems involves one or more of: Apply differentiation methods, using extended abstract thinking, in solving problems involves one or more of: • selecting and using methods • demonstrating knowledge of concepts and terms • communicating using appropriate representations. • selecting and carrying out a logical sequence of steps • connecting different concepts or representations • demonstrating understanding of concepts • forming and using a model; • devising a strategy to investigate or solve a problem • identifying relevant concepts in context • developing a chain of logical reasoning, or proof • forming a generalisation; and also relating findings to a context, or communicating thinking using appropriate mathematical statements. and also using correct mathematical statements, or communicating mathematical insight. Sufficiency for each question: N0: No response, no relevant evidence. N1: Attempt at ONE question demonstrating limited knowledge of differentiation techniques N2: 1 u A3: 2 u A4: 3 u M5: 1 r M6: 2 r E7: 1 t with minor errors ignored E8: 1 t with full excellence criteria Judgement Statement Score range Produced by SINCOS Not Achieved Achievement Achievement with Merit Achievement with Excellence 0-6 7 - 13 14 - 18 19 -24 Schedule Page 39 2013 EOY L3 M3 Mathematics Writer: David Fortune Question Achievement (u) Merit (r) Excellence (t) Apply differentiation methods in solving problems. Apply differentiation methods, using relational thinking, in solving problems. Apply differentiation methods, using extended abstract thinking, in solving problems. Evidence ONE 2 1a 1b 1c 1d dy 1 (2 x 4) 3 (2) dx 3 Correct derivative. dy dy 2 2 . At x = 0 2 dx dx ( x 1) Equation of tangent y 2 2( x 0) ie y 2 x 2 4 dV V r3 4 r 2 A 4 r 2 3 dr dV kA 4k r 2 k a constant dt dr dr dV 4k r 2 dt dV dt 4 r 2 k a constant Correct derivative with correct solution. Correct dV . dt Correct derivative with correct solution. . (i) lim f ( x) 2 x 3 (ii) lim f ( x) does not exist x 4 Correct solution in at least 3 situations. (iii) f (x) is not differentiable at x 1, 3 and 5 (iv) x 5 1e Let x be side of triangle and y be length of prism. L 6x 3 2 Area = 3xy 0.5x sin 60 2 6x 3y L y xL 6 x 2 3 2 x 2 dA dA Correct . L 12 x 3 x dx dx At A 0 L 12 x 3x 0 L x 12 3 3 2 x Area xL 6 x 2 2 L2 6L2 3 L2 2 (12 3 ) 2 12 3 (12 3 ) 2 12L2 3L2 2 (12 3 ) 2 Produced by SINCOS L2 2(12 3 ) Schedule Page 40 Correct maximum for x. Correct derivative with correct solution. 2013 EOY L3 M3 Mathematics Writer: David Fortune Question Evidence TWO Achievement (u) Apply differentiation methods in solving problems. 2a dy (4 x 2 1)e x 8 xe x dx (4 x 2 1) 2 Apply differentiation methods, using relational thinking, in solving problems. Apply differentiation methods, using extended abstract thinking, in solving problems. Correct derivative. dx dy 2 sin t 3 cos t dt dt dy dy dt 3 cos t dx dt dx 2 sin t Correct 2d Excellence (t) Correct graph of derivative. [graph for x < 1 must be negative but could be different shape] 2b 2c Merit (r) 3 At t Grad of tangent = 4 2 dy 2x r 2 32 w 2 5 dx x 5 d 2 y d dy d 2 x dx 2 dx dx dx x 2 5 ( x 2 5) 2 2 x 2 x 2 x 2 10 ( x 2 5) 2 ( x 2 5) 2 2 Correct dy . dx Correct derivative with correct solution. dy . dx 4x dy 2 ( x 5) 2 dx 2 d 2 y dy 2 x 2 10 2 2 2 2 2 dx ( x 5) ( x 5) dx 12 h 1 dV 10 ie r h 4 r 3 dt 1 1 V r2 h h3 3 27 dV 1 h2 dh 9 dh dh dV 90 dt dV dt h2 dh 90 10 At h = 9 then dt 81 9 Correct derivative with correct solution. 2 2e Produced by SINCOS Correct dV . dh [or equivalent answer using diff variables] Schedule Page 41 Correct dh . dt Correct derivatives with correct solution. 2013 EOY L3 M3 Mathematics Writer: David Fortune Question Evidence THREE Achievement (u) Apply differentiation methods in solving problems. 3a 3b(i) 3b(ii) dy 3 4e 4 x dx 3x 4 dS dt (20 4t )e 0.2t (20t 2t 2 )( 0.2e 0.2t ) dS 3.75 At t = 2 dt t = 2.93 hours after shop opens 3d Excellence (t) Apply differentiation methods, using relational thinking, in solving problems. Apply differentiation methods, using extended abstract thinking, in solving problems. Correct derivative. Correct derivative with correct solution. [Units not required] From (i) max when (20 4t ) 0.2(20t 2t 2 ) 0 0.4t 2 8t 20 0 3c Merit (r) h 16 h ie r r 4 4 dV 2 and dt 1 1 V r 2 h h3 3 48 dV 1 dh dh dV 32 h2 dh 16 dt dV dt h 2 dh 1 At h = 8 [ = 0.159] dt 2 Two solution to quadratic equation given [2.93 and 17.07] Correct dV dh Correct solution. (Only one). Correct derivative with correct solution. 1 2 1 4 sin a 2 4 2 sin( 2a) 2 2 = 16 sin a 8 sin( 2a) Area= dA 16 cos a 16 cos( 2a) da = 0 when cos a cos( 2a) ie Correct 2 cos 2 a cos a 1 0 1 cos a or cos a 1 2 a Correct derivative with correct solution. 3 ie maximum area 20.78 cm2 Produced by SINCOS dA . da Schedule Page 42 2013 EOY L3 M3 Mathematics Writer: David Fortune ASSESSMENT SCHEDULE 91579 Apply integration methods in solving problems Achievement Achievement with Merit Achievement with Excellence Apply integration methods in solving problems involves: Apply integration methods, using relational thinking, in solving problems involves one or more of: Apply integration methods, using extended abstract thinking, in solving problems involves one or more of: • selecting and using methods • demonstrating knowledge of concepts and terms • communicating using appropriate representations. • selecting and carrying out a logical sequence of steps • connecting different concepts or representations • demonstrating understanding of concepts • forming and using a model; • devising a strategy to investigate or solve a problem • identifying relevant concepts in context • developing a chain of logical reasoning, or proof • forming a generalisation; and also relating findings to a context, or communicating thinking using appropriate mathematical statements. and also using correct mathematical statements, or communicating mathematical insight. Sufficiency for each question: N0: No response, no relevant evidence. N1: Attempt at ONE question showing limited knowledge of integration techniques. N2: 1 u A3: 2 u A4: 3 u M5: 1 r M6: 2 r E7: 1 t with minor errors ignored E8: 1 t with full excellence criteria Judgement Statement Score range Produced by SINCOS Not Achieved Achievement Achievement with Merit Achievement with Excellence 0-6 7 - 13 14 - 18 19 -24 Schedule Page 43 2013 EOY L3 M3 Mathematics Writer: David Fortune Question Evidence ONE Achievement (u) Apply integration methods in solving problems. 1a(i) 1a(ii) 1b 1 cot 4 x C 4 Correct integration. 5 3 x 3 ln x C 3 Correct integration. not essential. x 1 (x 1) dx ( x x ) dx 2 0 1 (ii) Apply integration methods, using extended abstract thinking, in solving problems. 1 2 0 1d(i) Apply integration methods, using relational thinking, in solving problems. Excellence (t) Meet at x 2 1 x 1 ie x = 0 and x = 1 Area required = 1 1c Merit (r) 1 1 1 x 2 x3 = 6 3 0 2 sin t C Distance = At t = 1 4 C sin t 4 Distance = dT dT k (Ts T ) OR k (T Ts ) dt dt dT k (T 20) dt dT k dt T 20 ln( T 20) kt C Correct integration. Correct integration. Correct integration with correct solution. Correct DE. T 20 Ae kt At t = 0, T = 270 so T 20 250e kt When t = 3 T = 120 100 250e 3k ln 0.4 k 0.3054 3 When T = 40 then 40 20 250e 0.3054t 0.3054t ln 0.08 t 8.28 ie temperature is 40 oC after 8.3 minutes Produced by SINCOS Correct integration with correct solution. Correct integration with correct general solution for T. Correct solution showing correct integrations and correct mathematical logic and statements. Schedule Page 44 2013 EOY L3 M3 Mathematics Writer: David Fortune Question Evidence TWO Achievement (u) Apply integration methods in solving problems. 2a Merit (r) Excellence (t) Apply integration methods, using relational thinking, in solving problems. Apply integration methods, using extended abstract thinking, in solving problems. Correct integration. p ( x 3) 4 C 4 Integral = 2b 0.25 1 0.368 4(0.646 0.392) 2(0.472) 3 Correct integration. = 0.539 2c 4x 1 3 dx (2 )dx 2x 1 2x 1 = 2 x 1.5 ln 2 x 1 C Correct integration with minor error in division or substitution. Correct integration with correct solution. not essential. m 2d sin 2x. dx 0.2 Area 0 m 1 cos 2 x 0.2 2 0 1 1 cos 2m 0.2 2 2 cos 2m 0.6 m 0.464 2e Correct integration. Correct integration with correct solution. Meet at cax x 3 ie x = 0 and x ac ac A1 ac ydx 0 c 1 3 x 2 dx 0 ac 5 3 2 4 4 2 x = a c 5 5 c 0 5 2 ac A2 ac ydx a 0 Set up one correct definitive integral for area. One correct integration for an area completed. 1 x 2 dx 0 ac 5 3 32 2 4 4 a c = x 3 0 2 A 3 ie 1 5 A2 2 5 3 2 a 3 Produced by SINCOS Correct solution showing correct integrations and correct mathematical statements. Schedule Page 45 2013 EOY L3 M3 Mathematics Writer: David Fortune Question Evidence THREE Achievement (u) Apply integration methods in solving problems. 3a(i) 3a(ii) 1 e 4 x C 4 Merit (r) Apply integration methods, using relational thinking, in solving problems. Excellence (t) Apply integration methods, using extended abstract thinking, in solving problems. Correct integration. (8 sin 4x) dx Correct integration. 2 cos 4x C 3b 1 dy k ( x 2)dx 2 y 1 dy k (2 x)dx y2 1 ln( y 2) 2kx kx 2 C 2 2 kx 0.5 kx 2 OR y 2 Ae y 2 e 2kx0.5kx 2 c Correct integration. Correct integration with correct general solution. OR equivalent b 3c (2 7h( x)) dx 2 1 b ie 2 x 7 h( x) dx 2 b 1 Correct definite integration. 1 b ie 2b 2 7 h( x) dx 2 1 b 7 h( x) dx 2b ie 1 Produced by SINCOS Correct integration with correct solution. b 2b h( x) dx 7 1 Schedule Page 46 2013 EOY L3 M3 Mathematics Writer: David Fortune Question Evidence THREE 3d Achievement (u) Merit (r) Excellence (t) Let width of water at height h be w. 3h h 40 ie w 2 0.5 w 30 dh h h kA k 200 w 150kh2 dt 2 2 1 dh 150k 1 dt h2 1 150kt C h 1 At t = 0 h = 40 C 40 At t = 8 h = 30 1 1 150k 8 30 40 1 k 144000 When h = 10 1 150t 1 10 144000 40 Correct integration. Correct integration with correct logical steps and correct solution. t 72 ie 10 cm deep after 72 days Produced by SINCOS Schedule Page 47 2013 EOY L3 M3 Mathematics Writer: David Fortune