2014 HSC Mathematics Extension 1 Marking Guidelines

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1 04 HSC Mathematics Etension Marking Guidelines Section I Multiple-choice Answer Key Question Answer D A 3 C 4 D 5 B 6 B 7 A 8 D 9 C 0 C

2 BOSTES 04 HSC Mathematics Etension Marking Guidelines Section II Question (a) Provides a correct solution 3 Obtains a quadratic in, or equivalent merit Attempts to solve quadratic in +, or equivalent merit Let n = + n 6n + 9 = 0 (n 3) = 0 n = 3 Now, + = = 0 ( )( ) = 0 = or = Question (b) Provides a correct solution Finds a correct epression for the probability of rain on or days, or equivalent merit P(rain ) = 0. P ( rain ) = 0. = 0.9 Probability that it rains on fewer than 3 days is 30 ) ( (0.) 0

3 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question (c) Draws a correct sketch Draws a graph showing the correct range or shape, or equivalent merit y y = 6tan < < 6 3 < y < 3 3

4 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question (d) Provides a correct solution 3 Obtains a correct primitive, or equivalent merit Attempts the given substitution, or equivalent merit 5 d = u + = u = d = udu = 5 u = u + u du u = u3 + u 3 = = = 3 4

5 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question (e) Provides a correct solution 3 Obtains one correct interval, or equivalent merit Obtains a quadratic inequality, or equivalent merit Multiply by : ( + 5) > 6 ie ( + 5) > 6 ( + 5 ) 6 > 0 ( ) > 0 ( )( 5 ) > 0 0 < < or > 5 5

6 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question (f) Finds the correct derivative Uses an appropriate differentiation rule, or equivalent merit e ln y = e. + e ln e ln. dy = d e + e ln e ln = Question (a) Question (a) (i) Provides a correct answer Distance travelled is ( + ) m, ie 4 m. 6

7 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question (a) (ii) Provides a correct solution Identifies when the particle is first at rest, or equivalent merit When at rest, = 0ie 6 cos 3t = 0 cos 3t = 0 3t = t = 6 First at rest when t = 6 Now, = sin 3t = 6cos 3t = 8 sin 3t when t = 6 = 8 sin = 8 acceleration = 8 m/s 7

8 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question (b) sin 8 8 = = Provides a correct solution 3 Obtains correct integral epression for volume in terms of cos 8, or equivalent merit Obtains integral epression for volume of the form k or equivalent merit y = cos 4 V = 8 y d 0 8 = cos 4d cos 8 = d 0 volume = unit cos 4 d, 8

9 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question (c) Provides a correct solution 3 Obtains epression for v possibly involving an undetermined constant Attempts to use a = d v = = e d v = e d = e + c = + e + c When = 0,v = 4 ( 6 ) = c c = 6 ( ) d v d v = + e + 6 v = 4 + 4e + or equivalent merit 9

10 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question (d) Provides a correct solution Attempts to use the binomial theorem. Consider ( + ) n. From the binomial theorem: ( + ) n = n n n n n n Let = n n n n n ( + ) = 0 = ( ) + ( ) + + ( ) 0 + n n n n n n 0 = + + ( ) 0 n n Question (e) Provides a valid eplanation y O a The tangent at = is drawn, and it intersects the ais at. As shown in the diagram, is further away from than. 0

11 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question (f) Provides a correct solution 3 Obtains A and B, or equivalent merit Shows A B =, or equivalent merit T = A Be 0.03t As t, T 3 A = 3 T = 3 Be 0.03t t = 0 T = = 3 B B = 0.03t T = 3 e Now, when T = t 0 = 3 e 0.03t e = t 3 e = t = ln 3 ln t = 0.03 = 5.985! t 6 minutes

12 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 3 (a) Provides a correct proof 3 Uses inductive assumption, or equivalent merit Establishes case for n =, or equivalent merit n + ( ) n+ n =, + ( ) = + = 3 which is divisible by 3 Assume true for n = k ie Assume k + ( ) k+ is divisible by 3 ie k + ( ) k+ = 3M where M is an integer. We need to prove true for n = k + ie That k+ + ( ) k+ is divisible by 3 k + k + k + + ( ) = ( k ) + ( ) k + k + = 3M + ( ) + ( ) = 3 M + ( ) k + k = 3M ( ) k + = 3M + ( ) k + Which is a multiple of 3 since M and k are integers. ie if k + ( ) k+ is divisible by 3, then k+ + ( ) k+ is divisible by 3 By the principle of mathematical induction n + ( ) n+ is divisible by 3 for all n.

13 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 3 (b) (i) Provides a correct solution Uses Pythagoras Theorem and attempts to differentiate, or equivalent merit L = 40 + dl d = = 40 + = L = cos. Question 3 (b) (ii) Provides a correct solution dl dt dl d = d dt dl = 3 d = 3cos (from part (i)). 3

14 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 3 (c) (i) Provides a correct solution Establishes one coordinate, or equivalent merit P(at, at ) S (0, a) t : 0 t + at a t + at Q =, t + t + at at Q = t +, t + Question 3 (c) (ii) Provides a correct solution at 0 m = t + OQ at 0 t + at t + = t + at = t 4

15 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 3 (c) (iii) Provides a correct solution 3 Obtains an equation in, y and a Attempts to use part (ii), or equivalent merit at at = t y = + t + m OQ = t = y (from part (ii)) a y = y + = ay y + ( + y ) = a y, for 0 + y = ay + y ay = 0 + y ay + a = a + (y a) = a which is a circle of centre (0,a) and radius a units and point Q lies on this circle. 5

16 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 3 (d) Question 3 (d) (i) Provides a correct eplanation BAC = CQP (ABQP is a cyclic quadrilateral; eterior angle equals interior opposite angle) 6

17 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 3 (d) (ii) Provides a correct proof Observes that OPA is isosceles, or equivalent merit From (i) OAP = PQC = OAP = OPA = (OA = OP radii; angles opposite equal sides of OAP are equal) OPA = TPC = (vertically opposite angles equal) TPC = PQC = OP is a tangent to circle through P, Q, C, since the angle between a tangent (OP) and a chord (CP) is equal to the angle in the alternate segment. Question 4 (a) (i) Provides a correct solution Attempts to eliminate t, or equivalent merit = Vt cos, y = gt + Vt sin t = V cos y = g + V..sin V cos V cos g = sec + tan V 7

18 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 4 (a) (ii) Provides a correct solution 3 Obtains the coordinate of P, or equivalent merit Obtains an equation in, or equivalent merit P is the point of intersection of gsec y = tan g sec and y = V = tan g sec V = g V ( + tan ) = g sec V ( + tan ) = 0 V = 0 or g sin = + V cos cos sec tan gsec = + tan V V V sin.cos g g cos = cos + (cos 0) = V cos (cos + sin) g Since D =, = D D V = cos( sin + cos ) g V D = cos( sin + cos ) g 8

19 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 4 (a) (iii) Provides a correct solution Uses double-angle formulae, or equivalent merit D = V [ cos + sin cos] g V + cos = + sin g V = [ + cos + sin ] g dd = [sin + cos ] d g V V = [cos sin ] g 9

20 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 4 (a) (iv) Provides a correct solution 3 Obtains =, or equivalent merit 8 Attempts to solve dd d = 0, or equivalent merit dd D has a maimum value when = 0 d ie sin = cos tan = = 4 = 8 d D V d = ( sin cos ) g V = 4 ( sin + cos ) g for =,sin + cos > 0 8 d D d < 0 D is maimum when = 8 0

21 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question 4 (b) (i) Provides a correct solution Shows probability that A wins on second turn is rq, or equivalent merit ( P A wins on st or nd ) = p + rq = p + r ( ( p + r)) = p + r pr r = p( r ) + r ( r ) = ( r )( p + r) Question 4 (b) (ii) Provides a correct solution 3 Observes limiting probability is a geometric series, or equivalent merit Finds probability that A wins on third turn, or equivalent merit 3 ( wins eventually ) = p + rq + r p + r q + P A = p( + r + r 4 + ) + rq ( + r + r 4 + ) = ( p + rq)( + r + r 4 + ) = ( p + rq) r = ( r) ( p + r) ( + r) ( r) (using part(i)) p + r = + r

22 BOSTES 04 HSC Mathematics Etension Marking Guidelines Mathematics Etension 04 HSC Eamination Mapping Grid Section I Question Content Syllabus outcomes.8 PE3 5.7 PE HE PE6, PE PE HE HE PE PE PE, P4 Section II Question Content Syllabus outcomes (a) PE3, P4 (b) 8. HE3 (c) 5.3 HE4 (d) 3.5 HE6 (e) 3.4E PE3 (f) 8.8,.5 P7, H3, PE5 (a) (i) 4.4 HE3 (a) (ii) 4.4 HE3 (b) 3.4, 3.6 H8 (c) 3 4.3E HE5 (d) 7.3E HE3, HE7 (e) 6.4 HE7, PE6 (f) 3 4.E HE3 3 (a) HE, PE6 3 (b) (i) 5., 8.8 P4, P7, PE6 3 (b) (ii) 4.E HE5, PE6 3 (c) (i) 6.7E, 9.6 PE3 3 (c) (ii) 9.6 PE3 3 (c) (iii) PE3 3 (d) (i).0 PE3, PE6 3 (d) (ii).9,.0 PE3, PE6 4 (a) (i) 4.3E HE3

23 BOSTES 04 HSC Mathematics Etension Marking Guidelines Question Content Syllabus outcomes 4 (a) (ii) 3 4.3E HE3 4 (a) (iii) 4.3, 5.7, 3.7 HE3, H5 4 (a) (iv) 3 5., 0.6 HE3, H5 4 (b) (i) 3.3 H5, HE7, PE6 4 (b) (ii) H5, HE7, PE6 3

2017 HSC Mathematics Extension 1 Marking Guidelines

2017 HSC Mathematics Extension 1 Marking Guidelines 07 HSC Mathematics Extension Marking Guidelines Section I Multiple-choice Answer Key Question Answer A B 3 B 4 C 5 D 6 D 7 A 8 C 9 C 0 B NESA 07 HSC Mathematics Extension Marking Guidelines Section II