2001 HSC Mathematics Marking Guidelines

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Phoebe Elaine Glenn
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1 00 HSC Mathematics Marking Guidelines Question (a) ( marks) Outcomes assessed: P Gives correct answer =. Ê 5 ˆ Correct simplification Á or better Ë 87 Correct rounding of incorrect simplification Question (b) ( marks) Outcomes assessed: P4 Gives correct answer 5< x < and plots graph Gives correct answer 5< x < Gives correct graph of incorrect answer

2 00 HSC Mathematics Marking Guidelines Question (c) ( marks) Outcomes assessed: P, P4 Finds roots x = 4 and x = showing relevant working Correctly substitutes into quadratic formula Correctly factors LHS Finds roots from incorrect factorisation Question (d) ( marks) Outcomes assessed: P8, H5 Gives answer x + ln x (+ C) Finds primitive of Finds primitive of x Question (e) ( marks) Outcomes assessed: P Gives answer in the form Recognises factorisation of x 4 Correctly expands and factors numerator x + 4 x 4

3 00 HSC Mathematics Marking Guidelines Question (f) ( marks) Outcomes assessed: P Calculates original price = $890 Provides correct equation for x Question (a) ( marks) Gives the equation as y 4 5 x = ( ) equivalent Shows that dy = 5 at (, 4) dx Shows that dy = x + dx Gives equation of a line through (, 4) with incorrect slope Question (b) (i) ( marks) Outcomes assessed: P, P, P4, H5 Shows that y = ( x 4) Shows that A and B lie on the line given by the equation 5 Shows that mab = mab = 4 +

4 00 HSC Mathematics Marking Guidelines Question (b) (ii) ( mark) Outcomes assessed: P, P, P4, H5 Correctly substitutes into the formula to give correct length 40 ( ) Question (b) (iii) ( marks) Outcomes assessed: P, P4, H5 Calculates the distance 0 showing relevant working Shows that the distance is ( )+ ( ) Question (b) (iv) ( marks) Outcomes assessed: P4, H5 Calculates the area as 6 sq units Shows that the area = 0. 0 Question (b) (v) ( marks) Calculates the distance Ê Ë 6 ˆ 9 Shows that 6 = h OA Shows equation of BC is 5x + y = 6 4

5 00 HSC Mathematics Marking Guidelines Question (a) ( marks) Outcomes assessed: H, H5 Provides answer ln 5 ln 4 equivalent Gives correct indefinite integral Correctly substitutes limits into incorrect indefinite integral Question (b) ( marks) Outcomes assessed: H, H4 Gives correct numerical expression for the surface area Gives correct value for k Gives a value for the surface area consistent with incorrect value for k Question (c) (i) ( marks) Outcomes assessed: P7, H Gives answer x x 9 Gives denominator as x 9 Gives numerator as x 5

6 00 HSC Mathematics Marking Guidelines Question (c) (ii) ( marks) Outcomes assessed: P7, H Gives correct algebraic expression for the answer Applies quotient/product rule Question (d) (4 marks) Outcomes assessed: P, H5 Gives answer as Gives answer as 7± Shows = x + 7 7x Finds correct numerical approximation to x via sine rule Two of the mark criteria below Shows = x x cos60 Discards negative value for answer Correct numerical expression for x from quadratic formula 6

7 00 HSC Mathematics Marking Guidelines Question 4 (a) ( marks) Outcomes assessed: P4 Shows that there are no real roots when k > Finds the discriminant 4 k Applies another valid method Question 4 (b) (i) ( marks) Outcomes assessed: P, H, H5, H9 Shows that ACB = ABC and ACB = 80 x with reason Shows that CML = x Question 4 (b) (ii) ( marks) Outcomes assessed: P, H, H5, H9 Shows that TNL = NBM + BMN = x with reasons ( ) Shows that CBN = x straight Shows that BMN = x ( vertically opposite) Both above without reasons 7

8 00 HSC Mathematics Marking Guidelines Question 4 (c) (i) ( marks) Outcomes assessed: P5, H5, H9 Correctly sketches the curve Sketches a sine curve but with incorrect (or missing) amplitude or period Question 4 (c) (ii) ( marks) Outcomes assessed: P5, H8 Correctly sketches the line AND the area bounded by the line and the p curve drawn in 4 (c) (i) between x = 0 and x = 4 Correctly sketches the line the area bounded by their line and the p curve drawn in 4 (c) (i) between x = 0 and x = 4 Question 4 (c) (iii) ( marks) Outcomes assessed: H8 Gives correct answer Shows È cosx x Í Î 8 Ê p ˆ 8Ë 4 p 4 0 Correctly evaluates their expression for the indefinite integral of similar complexity 8

9 00 HSC Mathematics Marking Guidelines Question 5 (a) ( marks) Outcomes assessed: P5 { } { 0 y 0} States domain = x 5 x 5 and range = y States correct domain Shows that x 5 AND states correct range Shows that x 5 Recognises correct range Question 5 (b) (i) ( marks) Outcomes assessed: H Evaluates answer correct to at least one decimal place Uses the logarithm rule loga p = plog a Question 5 (b) (ii) ( mark) Outcomes assessed: H Deduces answer from their response to (b)(i) 9

10 00 HSC Mathematics Marking Guidelines Question 5 (c) ( marks) Outcomes assessed: P, H4, H5 Gives correct radius 5 mm Gives correct numerical expression for radius = 8 p 6 Gives the correct answer apart from an error in converting to radian measure Question 5 (d) (i) ( marks) Outcomes assessed: H4, H8 Gives correct numerical answer () Correctly applies trapezoidal rule Question 5 (d) (ii) ( marks) Outcomes assessed: H4 Give correct numerical expression for volume/hour based on answer in 5 (d) (i) Shows the volume/sec = area velocity Question 6 (a) (i) ( marks) Gives T 60 = 94 Shows that d = 5 0

11 00 HSC Mathematics Marking Guidelines Question 6 (a) (ii) ( marks) Gives correct sum (8790) Gives correct sum based on values in 6(a)(i) Substitutes into correct formula for S 60 Question 6 (b) ( marks) Outcomes assessed: H Gives correct decimal equivalent of ln(.) Shows a = ln(. ) t ln 00e (. ) = P Question 6 (c) (i) ( marks) Outcomes assessed: P6, H5, H6 49 Correctly determines the coordinates of A(, ), and B Ê, ˆ Ë 7 with appropriate working Correctly determines the coordinates of A and B apart from one computational error Shows that dy dx = 0

12 00 HSC Mathematics Marking Guidelines Question 6 (c) (ii) ( marks) Outcomes assessed: H6, H7 d y Shows that when x >, > 0 dx Shows concavity changes when x = Indicates concave up when d y > 0 dx Question 6 (c) (iii) ( mark) Outcomes assessed: H9 Gives correct answer 49 < k < 7 Answer consistent with response to (c)(i) Question 7 (a) ( marks) Outcomes assessed: H8 Gives numerical expression equivalent to 64 p Ê Provides expression p 8 x ˆ Á dx Ë as volume of elemental disk AND Provides correct limits of integration Ê Provides expression p 8 x ˆ Á dx Ë as volume of elemental disk Provides correct limits of integration

13 00 HSC Mathematics Marking Guidelines Question 7 (b) (i) ( marks) Provides correct answer Provides probability of failing to connect on first attempt (0.5) Recognises that probability of failing to connect on first attempt has to be multiplied by 0.75 Question 7 (b) (ii) ( mark) Gives correct answer ( 05. ) Consistent with (b) (i) Question 7 (c) (i) ( mark) Outcomes assessed: P Gives correct answer =

14 00 HSC Mathematics Marking Guidelines Question 7 (c) (ii) ( marks) Outcomes assessed: P, H4, H5 Gives all expressions from below Gives any expressions from below 4 Shows that x = - t + Correctly differentiates to find velocity Correctly differentiates velocity to find acceleration Question 7 (c) (iii) ( mark) Outcomes assessed: H9 States that since ẋ π 0, the particle is never at rest Gives answer consistent with (c)(ii) Question 7 (c) (iv) ( mark) Outcomes assessed: H9 Gives correct answer to limiting velocity = 0 Gives answer consistent with (c)(ii) 4

15 00 HSC Mathematics Marking Guidelines Question 8 (a) (5 marks) Outcomes assessed: H, H4, H9 Gives answers N 0 = 8, k = Ê ˆ 5 n 50 or equivalent Ë N = 95 or equivalent numerical expression Carries wrong k through correctly above with wrong N Shows N e k 70k ( 00)= 8 and 5000 = 8e above with wrong N 0 Shows = e k above with wrong N 0 Gives N 0 = 8 above with wrong N 0 5 Question 8 (b) (i) ( mark) Gives correct answer Ê ˆ Ë 5 Question 8 (b) (ii) ( marks) Gives correct probability Ê ˆ Ë0 Shows 5, 4,,, 5

16 00 HSC Mathematics Marking Guidelines Question 8 (c) (i) ( marks) Outcomes assessed: H7, H9 Gives y approximately =, and y approximately = 7 Gives either y or y Question 8 (c) (ii) ( marks) Outcomes assessed: H6, H7, H9 Correctly graphs a curve joining (0,0) and (5,0) AND indicates concavity changes at y = and y = 7 or at values consistent with answer in (c)(i) Graphs a curve joining (0,0) and (5,0) Shows correct concavity changes on wrong graph Question 9 (a) (i) ( marks) Shows that ADC = p 5 AND Shows that BDA = BAC Ê = Ë p ˆ 5 AND Correctly shows these results demonstrate similarity Shows two from above Shows one from above 6

17 00 HSC Mathematics Marking Guidelines Question 9 (a) (ii) ( mark) Uses properties of similar triangles to deduce answer Question 9 (a) (iii) ( marks) Outcomes assessed: H, H5, H9 Uses cosine rule to find x = cos 5 and uses the result to deduce answer Uses cosine rule to find x = cos p 5 Question 9 (b) (i) ( mark) Outcomes assessed: H Gives correct answer (4 litres/hour) Question 9 (b) (ii) ( marks) t t Gives correct expression for V e e = ( ) Finds correct primitive for V Evaluates constant for incorrect primitive 7

18 00 HSC Mathematics Marking Guidelines Question 9 (b) (iii) ( mark) Outcomes assessed: H Correctly shows required result Question 9 (b) (iv) ( marks) Outcomes assessed: H Finds t = ln or equivalent numerical answer t t Factorises e e correctly Question 0 (a) (i) ( mark) Outcomes assessed: H4, H9 Gives correct numerical expression for balance Question 0 (a) (ii) ( marks) Outcomes assessed: H4, H5, H9 [ ]= ( ) [ ] n 00 n Shows that the sum + (. 06)+... +( 06. ) equivalent Correct intermediate numerical expression for B n involving a GP Correctly finds the sum of a GP or 8

19 00 HSC Mathematics Marking Guidelines Question 0 (a) (iii) ( marks) Outcomes assessed: H4, H5 Gives a correct expression for m Deduces that B 0 ( 06. ) m Provides a correct expression for A m module evaluating the sum of a GP Question 0 (b) (i) ( marks) Outcomes assessed: H4, H5, H9 50secq Shows that T = tanq and T = 5 Correctly expresses T T Question 0 (b) (ii) (4 marks) Outcomes assessed: H4, H5, H9 Shows that the latest time is (or equivalent numerical expression) seconds after 8 am 5 6 Recognises that sinq = 4 5 gives maximum value of T dt Gives correct numerical expression for dq Recognises that T = T T has to be a maximum 9

2010 HSC NOTES FROM THE MARKING CENTRE MATHEMATICS

2010 HSC NOTES FROM THE MARKING CENTRE MATHEMATICS Contents 00 HSC NOTES FROM THE MARKING CENTRE MATHEMATICS Introduction... Question... Question... Question 3... 3 Question 4... 4 Question 5... 5 Question 6... 6 Question 7... 7 Question 8... 7 Question