2017 CSSA HSC MATHEMATICS 2 UNIT MARKING CRITERIA

x(J, CSSA CATHOLIC SECONDARY SCHOOLS ASSOCIATION 2017 TRIAL HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS ~ MARKING GUIDELINES These marking guidelines show the criteria to be applied to responses along with the marks to be awarded in line with the quality of responses. These guidelines are suggested and not prescriptive. This is not intended to be an exhaustive list but rather an indication of the considerations that students could include in their responses. Section I 10 Marks Questions 1-10 (1 mark each) . Targeted Question Answer Outcomes Assessed Performance Bands 1 C P4, HS 3 2 D P4 3 3 C H8 3 4 A PS5 3-4 5 D P3, P4, H5 4 6 B P4, H5 4 7 B P4, H5 4 8 C P3,H5 4-5 9 A H6 5 10 D H38 6 -1- DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers. 6200-2

Section II 90 Marks Question 11 (15 marks) (a) (1 mark) Qutcomes Assessed: P3, H1 Targeted Performance Bands: 2 Criteria Mark o Correctly determines the value to 2 significant figures 1 Sample answer: 37 (b) (1 mark) Qutcomes Assessed: P4 Targeted Performance Bands: 3 Criteria Mark Correctly factorises the expression 1 Sample answer: (5x+3)(x-1) (c) (3 marks) Outcomes Assessed: P4 Targeted Performance Bands: 4 Criteria Marks Correctly solves the inequality and correct sketching of inequality on number line 3 Correctly solves the inequality 2 Some progress towards solution 1 Sample answer: —8=<3x-4=<8 -4=3x<12 L - 4 -——=sx=4 3 3 2- DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff, It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

) (2 marks) Outcomes Assessed: P3 Targeted Performance Bands: 3-4 Criteria Marks o Correct answer 2 s Progress towards answer 1 Sample answer: lim 3(x+1) x—>=1 (x+1)}{x=1) _ lim 3 x=>-1 (x-1) =3 2 (g) (2 marks) Outcomes Assessed: P3 Targeted Performance Bands: 3-4 Criteria Marks o Findsbd 2 o Progress towards a solution 1 Sample answer: —2+4=l DISCLAIMER HSC papers. The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Triat

(h) (2 marks) Outcomes Assessed: P3 Targeted Performance Bands: 3-4 Criteria Marks e Correct answer 2 e Progress towards answer using index laws 1 Sample answer: 5x+Z = _1_ = 5—1 5 x+2=-1 Sx==3 DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranfy is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

(b) (3 marks) Outcomes Assessed: H2 Targeted Performance Bands: 3-4 Criteria Marks o Correct reasoning is used to establish AMPO = AOSN 3 o Significant progress is made towards proving AMPO = AOSN 2 o Correctly establishes that APSM = AOSN 1 Sample answer: In A's PSM and OSN £PMS = ZONS(given) PS = 08§ (given) LPSM = LOSN (vertically opposite angles) - APSM = AOSN (AAS) In A's MPO and NOP PM = ON (corresponding sides of congruent triangles) PQ is common OM = PN (since SM = SN and P§S = 0S5) . AMPO = AOSN (SSS) (c) (3 marks) Outcomes Assessed: P3 Targeted Performance Bands: 4 Criteria Marks e Obtains the correct values of 3 o Significant progress is made towards correct solution 2 e Correctly establishes that cos8 = —% 1 Sample answer: 25in6.cose +1=0 sin - DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

@ (@) (1 mark) Outcomes Assessed: P4 Targeted Performance Bands: 3 Criteria Mark e Correctly shows that 48 is parallel to DC Sample answer: My =—— AB 2___2 (d) (i) (1 mark) Outcomes Assessed: P4 Targeted Performance Bands: 3 Criteria Mark e Correctly shows the equation of ABis x -2y +10=0 Sample answer: 1 = —(x =2 y 5 (x--2) 2y -8=x+2 SLx-2y+10=0 (d) (iii) (2 marks) Outcomes Assessed: P4 Targeted Performance Bands: 3 Criteria Marks e Correctly determines the perpendicular distance from 45 to point C e Significant progress is made towards correct answer Sample answer: w20 +10 DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in refation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

Question 13 (15 marks) (@ @) (1 mark) Qutcomes Assessed: P7, H3 Targeted Performance Bands: 3-4 Criteria Mark » Find the correct derivative 1 Sample answer: dx (2)(i1) (2 marks) Outcomes Assessed: P7, H5 Targeted Performance Bands: 4 Criteria Marks ® Obtains the correct expression for the derivative 2 e Progress towards the correct derjvative 1 Sample answer: 2 (nx)(2x) - (2 +5)% = X ax (Inx) -10- DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA. assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

(b) (2 marks) Qutcomes Assessed: HS Targeted Performance Bands: 4 Criteria Marks s Correct solution is obtained 2 o Progress is made towards correct solution 1 Sample answer: [ 1 =| 4x-7)? dx L -2 DS e —x4 2 =_f£l +C 2 DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in refation to any specific trial exam. question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

(c)(ii) (2 marks) Outcomes Assessed: H5, H6 Targeted Performance Bands: 3-4 Criteria Marks e Correctly determines point of inflexion with concavity change 2 e Correctly determines the point of inflexion 1 Sample answer: Possible Point of inflexion at Z;}Z) =0 12x+6=0 x=— 0.5, at this value of x the y value is 13.5 x -1 -0.5 0 2 % -6 0 6 A change of concavity has occurred. .'. a point of inflexion occurs at (0.5 , 13.5) (c)(iil) (3 marks) Outcomes Assessed: H5, H6 Targeted Performance Bands: 4 Criteria Marks o Correctly draws graph, including all stationary points, points of inflexion & end points 3 e Significant progress is made towards correct graph 2 ¢ Some attempt, but does not include all points 1 Sample answer: 227y 7 (-3,16) (-0.5,13.5) (3,52) -3 1,0) 13- DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

(c) (iv) (1 mark) Outcomes Assessed: H7 Targeted Performance Bands: 3 Criteria Mark o Correct answer obtained 1 Sample answer: Maximum value of the function in the domain -3 = x = 3 is 52 (¢) (v) (1 mark) Outcomes Assessed: H7 Targeted Performance Bands: 5 Criteria Mark s Correct domain and inequality symbols given 1 Sample answer: —l <x<l 2 -14- DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possibie Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

(b)(@) (1 mark) Outcomes Assessed: P3, P4 Targeted Performance Bands: 3 Criteria Mark o Gives the correct coordinate for T 1 Sample answer: y=2- xz, cuts the x-axis when y=0. 0=2-x" x=442 ,x>0 So T is the coordinate (\/-2_,0). (b)(ii) (3 marks) Outcomes Assessed: H5, HS, H9 Targeted Performance Bands: 4-5 Criteria Marks o Evaluates the volume correctly 3 e Correctly integrates the expression 2 ¢ Some progress towards the solution 1 Sample answer: Let y =x* andy,=2-x L V2 y =2 dren s 1 Jz = 7z"|.x6 dx+flj(2—x2)zdx 0 1 i = flj.x6 abc+7z'_[(4—4x2 +x4)dx 1] 1 i T 4% x° 0 1 — %Jr 445_§*3/_5+i@ _[4_i+1j 3242 286 15 105 =0.921 (3decimal places ) -16- DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer, The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

(©)() (1 mark) Qutcomes Assessed: H4, HS Targeted Performance Bands: 3-4 Criteria Mark o Gives the correct expression for the annuity after one year 1 Sample answer: Let the amount in her account at the start of Helen’s annuity be $500 000; let the rate » = 0.06 and let Ay be the amount remaining in the annuity after n years. A =500 000(1 ‘06) -M (c) (iD)(3 marks) Outcomes Assessed: H4, H5, H9 Targeted Performance Bands: 4-5 Criteria Marks e Correctly determines the value of $M 3 e Progress towards finding $M 2 o Forms an equation involving $M 1 Sample answer: 4, = 4,(1.06)- M =(500 000(1.06)~ M )(1.06)~ M =500 000(1.06)° - M (1+1.06) so continuing the pattern Agy =500 000(1.06)" - M (1+1.06 +...+1.06” ) But 4, =0and (1+1.06+...+1.06”) is a GP where =30, a=1 and 7 = 1.06 (1 06%° - 1) So, 0= 500 000(1.06)" - M x 1.06-1 _0.06x500 000 (1.06)" (1.06% -1) =$36 32446 (2 decimal places) -17- DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or wartanty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completencss or usefulness of any Marking Guidelines provided for the Trial HSC papers.

Question 15 (15 marks) (a2)(1) (2 marks) Outcomes Assessed: H5, H9 Targeted Performance Bands: 4 Criteria Marks o Correctly finds the coordinate P 2 e Progress towards an answer 1 Sample answer: sin2x = cos 2x tan2x =1 S 4 Sz 8 2x=£, 4 z X=—, 8 b4 When x = E’ y= —1- So P is the coordinate z 1 8°J2) V2 DISCLAIMER . 4 , xis in first quadrant, so x =—. The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the intention of CSSA to provide specitic marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.

(a)(ii) (3 marks) Outcomes Assessed: H5, H8 Targeted Performance Bands: 3-5 Criteria Marks e Evaluates shaded area correctly 3 e Significant progress towards correct solution 2 » Correctly determines part integral of sin2x or cos2x correctly 1 Sample answer T 3 A= ZJ (1—cos2x)dx 0 K] -20- DISCLAIMER The information contained in this document is intended for the professional assistance of teaching staff. It does not constitute advice to students. Further it is not the inteation of CSSA to provide specific marking outcomes for all possible Trial HSC answers. Rather the purpose is to provide teachers with information so that they can better explore, understand and apply HSC marking requirements, as established by the NESA. No guarantee or warranty is made or implied with respect to the application or use of CSSA Marking Guidelines in relation to any specific trial exam question or answer. The CSSA assumes no liability or responsibility for the accuracy, completeness or usefulness of any Marking Guidelines provided for the Trial HSC papers.