Mathematics (Project Maths Phase 2) Higher Level

Transcription

1 L.7/0 Pre-Leaving Certificate Examination, 03 Mathematics (Project Maths Phase ) Higher Level Marking Scheme Paper Pg. Paper Pg. 5 Page of 48

2 exams Pre-Leaving Certificate Examination, 03 Mathematics (Project Maths Phase ) Higher Level Paper Marking Scheme (300 marks) General Instructions Instructions There are three sections in this examination paper: Section A Concepts and Skills 00 marks 4 questions Section B Contexts and Applications 00 marks questions Section C Functions and Calculus (old syllabus) 00 marks questions Answer all eight questions. Marks will be lost if all necessary work is not clearly shown. Answers should include the appropriate units of measurement, where relevant. Answers should be given in simplest form, where relevant. 03 L.7/0_MS /48 Page of 48 exams

3 Summary of Marks Q. a 5C (0, 3, 4, 5) + Q.4 a 0B (0, 7, 0) 5B (0, 3, 5) b i 5B (0, 3, 5) b 5C (0, 3, 4, 5) ii 5B (0, 3, 5) c 5B (0, 3, 5) ii 5B (0, 3, 5) d 5C (0, 3, 4, 5) 5 5 Q.5 a 5B (0, 3, 5) Q. a 0C* (0, 5, 8, 0) b 0C (0, 5, 8, 0) b 5B* (0, 3, 5) c 5B (0, 3, 5) c 0B (0, 7, 0) d 0C (0, 5, 8, 0) 5 e 0C (0, 5, 8, 0) f 5B (0, 3, 5) Q.3 a 0C (0, 5, 8, 0) g 5B (0, 3, 5) b i 5B (0, 3, 5) 50 ii 0C (0, 5, 8, 0) 5 Q.6 a 0C (0, 5, 8, 0) b 0C (0, 5, 8, 0) c 0C (0, 5, 8, 0) d 0C (0, 5, 8, 0) e 0C (0, 5, 8, 0) Q.7 a 0, Att. 3 b i 0, Att. 3 ii 0, Att. 3 c i 0, Att. 3 ii 0, Att. 3 Q.8 a 0, Att. 3 b i 0, Att. 3 ii 0, Att. 3 c 5, Att. + 5, Att. + 5, Att. + 5, Att L.7/0_MS 3/48 Page 3 of 48 exams

4 Marking Scheme Sections A and B Structure of the marking scheme Students responses are marked according to different scales, depending on the types of response anticipated. Scales labelled A divide students responses into two categories (correct and incorrect). Scales labelled B divide responses into three categories (correct, partially correct, and incorrect), and so on. These scales and the marks that they generate are summarised in the following table: Scale label A B C D E No of categories mark scale 0, 3, 5 0, 3, 4, 5 0 mark scale 0, 7, 0 0, 5, 8, 0 0, 3, 5, 8, 0 5 mark scale 0,, 5 0, 8, 4,5 0, 5, 0, 4, 5 0 mark scale 0, 4, 8, 0 0, 6,, 8, 0 A general descriptor of each point on each scale is given below. More specific directions in relation to interpreting the scales in the context of each question are given in the scheme, where necessary. Marking scales level descriptors A-scales (two categories) incorrect response (no credit) correct response (full credit) B-scales (three categories) response of no substantial merit (no credit) partially correct response (partial credit) correct response (full credit) C-scales (four categories) response of no substantial merit (no credit) response with some merit (low partial credit) almost correct response (high partial credit) correct response (full credit) D-scales (five categories) response of no substantial merit (no credit) response with some merit (low partial credit) response about half-right (middle partial credit) almost correct response (high partial credit) correct response (full credit) E-scales (six categories) response of no substantial merit (no credit) response with some merit (low partial credit) response almost half-right (lower middle partial credit) response more than half-right (upper middle partial credit) almost correct response (high partial credit) correct response (full credit) In certain cases, typically involving incorrect rounding or omission of units, a mark that is one mark below the full-credit mark may also be awarded. Such cases are flagged with an asterisk. Thus, for example, scale 0C* indicates that 9 marks may be awarded. 03 L.7/0_MS 4/48 Page 4 of 48 exams

5 Section A Concepts and Skills 00 marks Answer all four questions from this section. Question (5 marks) Let z i and z i, where i. (a) Find z.z and hence plot z, z and z.z on an Argand diagram. 4 3 Im ( + i)( + i) ( + i) + i( + i) + i i + i 4 + 0i (, ) z.z (0, 4) 4 3 z z (, ) 3 4 Re 3 4 Calculating z, z (5C) Scale 5C (0, 3, 4, 5) Low partial credit: (3 marks) Attempt at multiplication. High partial credit: (4 marks) Not in form a + bi. Diagram (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) At least two points plotted. Axes fully filled in. (b) Express z, z and z.z in polar form. (5C) z (cos 45º + i sin 45º) z 8(cos 35º + i sin 35º) z. z 4(cos 80º +i sin 80º) Scale 5C (0, 3, 4, 5) Low partial credit: (3 marks) One correct solution. High partial credit: (4 marks) Two correct solutions. 03 L.7/0_MS 5/48 Page 5 of 48 exams

6 (c) Using your answers to parts (a) and (b), explain what happens when you multiply two complex numbers. Multiply argument (5B) Add angles 45º +35º 80 Scale 5B (0, 3, 5) Partial credit: (3 marks) Multiplying argument or adding angles. (d) Use De Moivre s theorem to evaluate (z ) 6, giving your answer(s) in rectangular form. (5C) ( (cos 45º + i sin 45º) 6 8(cos 70º + i sin 70º) 8(0 ) 8 + 0i Scale 5C (0, 3, 4, 5) Low partial credit: (3 marks) 8 or 70º calculated. High partial credit: (4 marks) 8, but not in form 8 + 0i. 03 L.7/0_MS 6/48 Page 6 of 48 exams

7 Question (5 marks) Future population size can be described using the exponential equation P(t) Ae bt, where A and b are constants. The size of population size P(t) can be determined at various points in time t. The population of a certain village was 500 in 000 and 3560 in 00. (a) Find the value of a. Find the value of b, correct to three decimal places. (0C*) P(t) Ae bt P(0) Ae 0 P(0) A 500 [5 marks, low partial credit] P(0) Ae 0t 500e 0t t 3560 e 500 e 0t.3733 lne 0b ln.3733 [8 marks, high partial credit] 0b b b Scale 0C* (0, 5, 8, 9, 0) Low partial credit: (5 marks) A calculated. Partial credit: (8 marks) An equation with b isolated. High partial credit: (9 marks) Not to 3 decimal places. (b) Determine the population size of the village in 00, correct to three significant figures. (5B*) P(t) Ae bt P(0) 500e 0.086(0) 500e.7 500(5.5845) Scale 5B* (0, 3, 4, 5) Low partial credit: (3 marks) P(0) fully substituted. High partial credit: (4 marks) Not to 3 significant figures. (c) During what year will the population of the village reach 5 000? (0B) P(t) 500e 0.086t 5000 e 0.086t e 0.086t 0 [6 marks, partial credit] ln e 0.086t ln t.305 t population will reach by 06 Scale 0B (0, 7, 0) Partial credit: (7 marks) e 0.086t 0 or similar calculation. 03 L.7/0_MS 7/48 Page 7 of 48 exams

8 Question 3 (5 marks) (a) Solve the simultaneous equations: (0C) x y z x 3y z x y 0. x + y z x 3y z y 0 x z 0 x z 0 x 0 z 0 Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks) Reduced to two equations. High partial credit: (8 marks) At least one value fully evaluated. (b) (i) Write the following as a single fraction: (5B) +. x y y x xy xy x y xy Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct common denominator identified. (ii) Hence, or otherwise, show that (x y) 4, x y given that x, y 0 and x, y. x y (x + y) xy 4 [6 marks, low partial credit] x x x 4xy 4y xy 4 4xy 4 y 8xy 8xy 4 y 0 [8 marks, high partial credit] (x y) 0 (0C) Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks) x y xy fully substituted. High partial credit: (8 marks) Quadratic equation found. 03 L.7/0_MS 8/48 Page 8 of 48 exams

9 Question 4 (5 marks) (a) Write a polynomial function for the following graph in its simplest form. (0B) 60 y x 80 ( x 6)( x 4)( x )( x 3) 0 ( x 0x 4)( x 4x 3) x 4x 3x 0x 40x 30x 4x 96x x 6x 3x 66x 7 0 Scale 0B (0, 7, 0) Partial credit: (7 marks) ( x 6)( x 4)( x )( x 3) or ( x 6 )( x 4 )( x )( x 3). 03 L.7/0_MS 9/48 Page 9 of 48 exams

10 (b) (i) Using the same axis and scales, sketch graphs of the functions f : x x 6 and g : x x. (5B) 6 y f(x) (g)x x Scale 5B (0, 3, 5) Partial credit: (3 marks) One correct point or line identified. (ii) Use your graph to solve the inequality x 6 x. (5B) x > 6 4 y f(x) x 6 < x (g)x x Scale 5B (0, 3, 5) Partial credit: (3 marks) Incorrect area shaded, i.e. x 6 > x. (iii) Verify your answer algebraically. (5B) (x 6) < (x + ) x x +36 < x + 4x + 4 [3 marks, low partial credit] 6x < 3 x < x > [5 marks, full credit] Scale 5B (0, 3, 5) Partial credit: (3 marks) (x 6) < (x + ) or x <. 03 L.7/0_MS 0/48 Page 0 of 48 exams

11 Section B Contexts and Applications 00 marks Answer both Question 5 and Question 6. Question 5 (50 marks) A clothing company produces one type of shirt. Market research has found that if the company prices the shirts at 30 each, they will sell 500 units per week. It was also found that if the price was set at 55 each, the company will sell none. The clothing company prices the shirts at x each, where 30 x 55. (a) Draw a straight line graph to represent possible sales per week. (5B) 500 Quantity Price () Scale 5B (0, 3, 5) Partial credit: (3 marks) One correct point on graph. (b) Find an expression for sales per week, in terms of x. (0C) (P, Q): (30, 500) (55, 0) Q 0 00 (x 55) 7 [8 marks, high partial credit] 7Q 00x Q 00x Q 0(x 55) Q 0x + 00 Q 00 0x Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks) Slope identified. Point identified. High partial credit: (8 marks) Equation fully filled-in. 03 L.7/0_MS /48 Page of 48 exams

12 5(c) Write an expression for the value in euro of weekly sales, in terms of x. (5B) sales price (00 0x)x 00x 0x Scale 5B (0, 3, 5) Partial credit: (3 marks) sales price. Answer above x. (d) Given that the clothing company s fixed costs are 000 per week and production costs are 0 for each shirt, find an expression for costs per week. (0C) Costs (00 0x) x x Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks),000 written. 0(00 0x) or similar. High partial credit: (8 marks) (00 0x) or similar. (e) Show that the weekly profit is 0x 500x (0C) 00x 0x ( x) 0x + 00x + 400x x + 500x 4000 Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks) sales cost. High partial credit: (8 marks) (sales cost) filled in. 03 L.7/0_MS /48 Page of 48 exams

13 5(f) A graph of the clothing company s weekly profits as a function of x is shown below. Use the graph to determine the price that the company should charge in order to maximise profits (5B) 6,000 Profit ( ) 4,000, x 50 60,000 Scale 5B (0, 3, 5) Partial credit: (3 marks) Maximum value indicated on graph. (g) Hence, calculate the number of shirts that will sell per week at this price. (5B) Q 00 0(37.5) shirts Scale 5B (0, 3, 5) Partial credit: (3 marks) 37.5 put into equation. 03 L.7/0_MS 37/48 Page 3 of 48 exams

14 Question 6 (50 marks) Lisa has won a major prize in a lottery game. When she goes to collect her prize, she is offered one of the following options: Option A: Option B: Receive a payment of 500, at the beginning of each month for 5 years. Receive a single payment lump sum immediately. Lisa is unsure of which option to take. Initial exploration: (a) Lisa feels that if she takes option A, it may give her a regular income in the future. She plans to put the monthly payment in a bank while she decides what to do. The bank is offering a rate of interest which corresponds to an annual equivalent rate (AER) of 3 5%. Find the rate of interest per month that would, if paid and compounded monthly, correspond to an annual equivalent rate (AER) of 3 5%. ( + i) (.035) ( + i) (.035) + i.0087 i Rate per month 0.87% or 0.9% Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks) ( + i) given. (.035) given. High partial credit: (8 marks) ( + i) (.035). (0C) (b) After three months, Lisa decides what she wants to do. She plans to continue saving all of the money as part of a pension for the future. Find the present values of the first three monthly payments lodged in her bank account. First payment: 500(.0087) Second payment: 500(.0087) Third payment: 500(.0087) ** Accept answer from appropriate value in (a). (0C) Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks) One correct value implied or calculated. High partial credit: (8 marks) Two correct values implied or calculated. (c) Show that the total value of Lisa s pension, assuming an annual equivalent rate (AER) of 3 5% over the period of the payments, can be represented by a geometric series. (0C) 500(.0087) + 500(.0087) (.0087) 300 a 500(.0087) r.0087 n 300 ( 5) geometric series ** Accept answer from appropriate value in (a). Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks) Geometric series expressed. High partial credit: (8 marks) At least two of a, r, n stated. 03 L.7/0_MS 38/48 Page 4 of 48 exams

15 6(d) Alternatively, Lisa could have accepted a single payment lump sum (option B). How much would this payment need to be to match the future value of Lisa s pension plan? S n n a( r ) r (.0087 ) ( ) (.36336) ,33.95 ** Accept answer from appropriate value in (a). Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks) S n with substitution. High partial credit: (8 marks) S n fully substituted. (0C) (e) Lisa was worried that if she received a large sum of money, she would spend it carelessly and then it would be gone. Assuming that she would spend no more than 750 every month, how much better off would Lisa be if she accepted the single payment lump sum (option B) and save the remainder as a pension under the same conditions? 74,33.95(.035) 5,688, (.0087) (.0087 ) ( ) (.36336) , , , ,68.85 ** Accept answer from appropriate value in (a). Scale 0C (0, 5, 8, 0) Low partial credit: (5 marks) New series fully substituted. High partial credit: (8 marks) 357,64.0 or S n fully evaluated. (0C) 03 L.7/0_MS 39/48 Page 5 of 48 exams

16 Marking Scheme Section C. Penalties of three types are applied to student s work, as follows: Blunders - mathematical errors / omissions B (3) Slips - numerical errors S ( ) Misreadings (provided task is not oversimplified) M ( ) Frequently occurring errors to which these penalties must be applied are listed in this marking scheme. They are labelled B, B, B3,..., S, S, S3,..., M, M, M3,..., etc. Note that these lists are not exhaustive.. When awarding attempt marks, e.g. Att. (3), it is essential to note that: any correct relevant step in a part of a question merits at least the attempt mark for that part if deductions result in a mark which is lower than the attempt mark, then the attempt mark must be awarded a mark between zero and the attempt mark is never awarded. 3. Worthless work must be awarded zero marks. 4. The phrase hit or miss means that partial marks are not awarded - the student receives all of the relevant marks or none. 5. The phrase and stops means that no more work of merit is shown by the student. 6. Special notes relating to the marking of a particular part of a question are indicated by a double asterisk. These notes immediately follow the relevant solution. 7. The sample solutions for each question are not intended to be exhaustive - there may be other correct solutions. 8. Unless otherwise indicated, accept the best of two or more attempts - even when attempts have been cancelled. 9. The same error in the same section of a question is penalised once only. 0. Particular cases, verification and answers derived from diagrams (unless requested) qualify for attempt marks at most.. A serious blunder, omission or misreading merits the attempt mark at most.. Allow comma for decimal point, e.g may be written as 3, L.7/0_MS 40/48 Page 6 of 48 exams

17 Section C Functions and Calculus (old syllabus) 00 marks Answer both Question 7 and Question 8. Question 7 (50 marks) (a) Let f (x) x 3 6x k, where k and x. Taking x as the first approximation of a root of the function f (x) 0 and x as the second approximation of this root, use the Newton-Raphson method to find the value of k. (0, Att. 3) 3 f (x) x 6x k f '( x) 3x 6 f () 6 k k 5 f '() x n f ( xn) xn f '( xn) k 5 6 k k 6 k 8 k 8 Blunders ( 3) B: Differentiation. B: f(). B3: f '(). B4: Algebraic. Slips ( ) S: Numerical. Attempts (3) A: f '( x) or f() or f '() evaluated. 03 L.7/0_MS 4/48 Page 7 of 48 exams

18 7(b) (i) Differentiate sin x with respect to x from first principles. (0, Att. 3) f (x) sin x f ( x h) sin (x + h) f ( x h) f ( x) sin(x+ h) sin x f ( x h) f ( x) h f ( x h) f ( x) h limit h 0 f ( x h) f ( x) h sin (A) sin (B) cos x h x x cos sin x h h cos sin h h sin x h cos h limit h 0 cos cos x + cos x x h A B sin h x limit h 0 A B h sin h Blunders ( 3) B: Trig formula. B:. B3: Limits. Slips ( ) S: Numerical. S: Trig values Attempts (3) A: f(x + h). 03 L.7/0_MS 4/48 Page 8 of 48 exams

19 7(b)(ii) sin x Given that y (cos x sin x) dy, find the value of when x 0. (0, Att. 3) dx Let u sin x dy dx cos x v cos x + sin x dv dx sin x + cos x dy (cos x sin x)(cos x) (sin x)( sin x cos x) dx (cos x sin x) x (cos0 sin 0)(cos0) (sin 0)( sin 0 cos0) ( 0)() (0)(0 ) () (cos0 sin 0) Blunders ( 3) B: Differentiation. B: Trig values. B3: Substitution. Slips ( ) S: Numerical. S: Trig values. Attempts (3) A: Error in differentiation formula. A: Some correct differentiation. 03 L.7/0_MS 43/48 Page 9 of 48 exams

20 7(c) A curve is defined by the equation x y xy 6. (i) Find dy in terms of x and y. (0, Att. 3) dx x y xy 6 u x v y u x v y dy dx x dv dy dy dx dx dx dy dy x xy + xy y dx dx 0 dy ( x xy) dx y xy dy dx y x xy xy dv dx y dy dx Blunders ( 3) B: Differentiation. B: uv formula. B3: Indices. B4: Transposition. Attempts (3) A: Correct differentiation. A: Error in differentiation formula. (ii) Determine whether the tangents are parallel at the point x. (0, Att. 3) x x y dy dx 4y + y 6 y + 4y 6 0 y + y 3 0 (y )(y +3) y + y 3 x y 3 dy dx tangents are not parallel Blunders ( 3) B: Factors. dy B:. dx Slips ( ) S: Numerical. Attempts (3) A: Quadratic equation. A: dy evaluated. dx 03 L.7/0_MS 44/48 Page 0 of 48 exams

21 Question 8 (50 marks) (a) Find x 4 x 3 dx. (0, Att. 3) x x 3x dx x 3 x 3x 3 + C 3 3 x 3 x C x 3 Blunders ( 3) B: Integration. B: Indices. B3: No + C. Attempts (3) A: Only C correct. Worthless (0) W: Differentiation instead of integration. (b) (i) Evaluate dx 5 4x x. (0, Att. 3) 5 4x x + 4 4x x () + ( x) dx sin ( ) ( x) sin sin 0 0 x Blunders ( 3) B: Integration. B: Completing the square. B3: In correct order of applying limits. Slips ( ) S: Numerical. S: Trig value. Attempts (3) A: Square completed. 03 L.7/0_MS 45/48 Page of 48 exams

22 8(b)(ii) Evaluate 3 0 x dx. (0, Att. 3) x Let u x + du dx du dx u x + u x + u du u u du u u u 3 4u 3 4 / 3/ 3/ u / / u u / du 3 x x (8) Blunders ( 3) B: Integration. B: Differentiation. B3: Indices. B4: Limits. B5: Incorrect order of applying limits. B6: Not changing limits Slips ( ) S: Numerical. Attempts (3) A: Differentiation unless required. 03 L.7/0_MS 46/48 Page of 48 exams

23 8(c) The diagram shows the curve y x, y and the line 4x 8y Calculate the area of the shaded region enclosed by the curve and the line. (5, 5, 5, 5, Att.,,, ) x 4x + 8y y 33 4x y 33 4x 8 x 8 33x 4x 4x 33x x 33x x +8 4x(x 8) (x 8) (4x ) (x 8) x 4 x 8 (5, Att. ) x 8 dx x 33 x dx 8 33 x x 4 33 [33 6] ln x 8 4 ln 8 ln 4 (5, Att. ) 8 ln 4 ln 3 (5, Att. ) Area 03 ln3 64 (5, Att. ) Blunders ( 3) B: Error in area formula. B: Error in integration. B3: Error or incorrect integration limits. B4: value. B5: Fails to substitute limits. Slips ( ) S: Numerical slip in calculations (max. of 3). Attempts (,,, ) A: Some attempt at integration. Worthless (0) W: Differentiation instead of integration except where other work merits attempt mark. 03 L.7/0_MS 47/48 Page 3 of 48 exams

24 Notes: 03 L.7/0_MS 48/48 Page 4 of 48 exams

25 . exams Pre-Leaving Certificate Examination, 03 Project Maths (Phase ) Higher Level Paper Marking Scheme (300 marks) Instructions There are two sections in this examination paper. Section A Concepts and Skills 50 marks 6 questions Section B Contexts and Applications 50 marks questions Students must answer all eight questions, as follows: In Section A, answer: Questions to 5 and either Question 6A or Question 6B. In Section B, answer Questions 7 and 8. Marks will be lost if all necessary work is not clearly shown. Answers should include the appropriate units of measurement, where relevant. Answers should be given in simplest form, where relevant. 03 L.7/0_MS 3/48 Page 5 of 48 exams

26 Summary of Marks Q. a i 5B (0, 3, 5) Q.6A a 5B (0, 3, 5) i 5B (0, 3, 5) b 5B (0, 3, 5) + b i 5B (0, 3, 5) 5B (0, 3, 5) ii 5B (0, 3, 5) 0C (0, 4, 8, 0) iii 5B (0, 3, 5) 5 5 OR Q. a 5B (0, 3, 5) Q.6B a 5C (0, 8,, 5) b 0C (0, 4, 8, 0) b 0B (0, 5, 0) c 5A (0, 5) 5 d 5B (0, 3, 5) 5 Q.7 a 5C (0, 8,, 5) b 5C (0, 8,, 5) Q.3 a 5B (0, 3, 5) c 5C (0, 8,, 5) b 0C (0, 4, 8, 0) d i 0C (0, 4, 8, 0) c 0C (0, 4, 8, 0) ii 5B (0, 3, 5) 5 iii 5B (0, 3, 5) e 0C (0, 4, 8, 0) Q.4 a 0C (0, 4, 8, 0) 75 b 5C (0, 3, 4, 5) c 0C (0, 4, 8, 0) Q.8 a 5C (0, 8,, 5) 5 b 0C (0, 4, 8, 0) + 0C (0, 4, 8, 0) + Q.5 a 5B (0, 3, 5) 0B* (0, 5, 0) b 0C (0, 4, 8, 0) c 0C (0, 4, 8, 0) c 0C (0, 4, 8, 0) d i 0C (0, 4, 8, 0) 5 ii 0C (0, 4, 8, 0) L.7/0_MS 4/48 Page 6 of 48 exams

27 Structure of the marking scheme Students responses are marked according to different scales, depending on the types of response anticipated. Scales labelled A divide students responses into two categories (correct and incorrect). Scales labelled B divide responses into three categories (correct, partially correct, and incorrect), and so on. These scales and the marks that they generate are summarised in the following table: Scale label A B C D E No of categories mark scale 0, 3, 5 0, 3, 4, 5 0 mark scale 0, 5, 0 0, 4, 8, 0 5 mark scale 0, 8,, 5 0 mark scale 0, 7, 8, 0 0, 7, 0, 8, 0 5 mark scale 0, 5, 0,, 5 0, 5, 0, 5, 0, 5 A general descriptor of each point on each scale is given below. More specific directions in relation to interpreting the scales in the context of each question are given in the scheme, where necessary. Marking scales level descriptors A-scales (two categories) incorrect response (no credit) correct response (full credit) B-scales (three categories) response of no substantial merit (no credit) partially correct response (partial credit) correct response (full credit) C-scales (four categories) response of no substantial merit (no credit) response with some merit (low partial credit) almost correct response (high partial credit) correct response (full credit) D-scales (five categories) response of no substantial merit (no credit) response with some merit (low partial credit) response about half-right (middle partial credit) almost correct response (high partial credit) correct response (full credit) E-scales (six categories) response of no substantial merit (no credit) response with some merit (low partial credit) response almost half-right (lower middle partial credit) response more than half-right (upper middle partial credit) almost correct response (high partial credit) correct response (full credit) In certain cases, typically involving incorrect rounding or omission of units, a mark that is one mark below the full-credit mark may also be awarded. Such cases are flagged with an asterisk. Thus, for example, scale 0C* indicates that 9 marks may be awarded. 03 L.7/0_MS 5/48 Page 7 of 48 exams

28 exams Pre-Leaving Certificate Examination, 03 Project Maths (Phase ) Higher Level Paper Marking Scheme (300 marks) Section A Concepts and Skills 50 marks Answer all six questions from this section. (5 marks each). The Venn diagram shows the probability of two events A and B occurring. (a) (i) Find the value of P(A B). (5B) S A B ( ) (0.8) 0. Scale 5B (0, 3, 5) Partial credit: (3 marks) or states total. (ii) Verify that P(A B) P(A) P(B) P(A B) and hence, state what this tells us about A and B events mutually exclusive (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) Shows equal, but no conclusion. 03 L.7/0_MS 6/48 Page 8 of 48 exams

29 (b) A fair coin is tossed four times giving either heads or tails each time. (i) Complete the sample space below to show all the possible outcomes. (5B) H H H H H H H T H H H H H T H H T H H H T T H H H H H T H T H T T H H T T T H T H H T T H T T T T H T T T T T T H H T H H T T H T H T H T T T H Scale 5B (0, 3, 5) Partial credit: (3 marks) At least 3 more combinations. (ii) E is the probability that the first three results are heads and F is the probability that the fourth result is heads. Find P(E) and P(F).. P(E) P(F) (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) P(E) correct only. P(F) correct only. (iii) Show that P(E and F) P(E) P(F) and hence, state what this tells us about E and F. (5B) events independent Scale 5B (0, 3, 5) Partial credit: (3 marks) or, but no conclusion L.7/0_MS 7/48 Page 9 of 48 exams

30 . One of the highlights of the Olympic Games is the marathon. It is a long-distance running event with an official distance of 4 95 kilometres that is usually run as a road race. It is one of the original events of the modern Olympic Games revived in 896 although it did not feature in the original Games in ancient Greece. A sports researcher, wants to investigate the connection, if any, between athletes performances in the marathon and the athletes ages. He takes a random sample of 5 athletes who completed the race in the 0 London Olympic Games and compares the race times of these athletes to each of their ages. The results are shown in the table below. Race Time (minutes) Age of Athlete (years) (a) Explain the following terms in relation to the above information: (5B) (i) Sample; a subset of the population (ii) Population. all the possible subjects or cases of interest Scale 5B (0, 3, 5) Partial credit: (3 marks) One correct item. 03 L.7/0_MS 8/48 Page 30 of 48 exams

31 (b) Create a suitable graphical representation to illustrate the data. (0C) 50 Race Time (minutes) Athlete s Age (years) Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) Scatter plot. High partial credit: (8 marks) Axes not labelled properly. (c) What kind of relationship, if any, does the observed data suggest exists between athletes performances and the athletes ages in the marathon? older runners are faster Scale 5A (0, 5) Hit / Miss. (5A) (d) Can you make the same hypothesis for all athletes that compete in the marathon? no (5B) Explain your answer. as sample may be too small for the population Scale 5B (0, 3, 5) Partial credit (3 marks) Correct answer, but reason not given. 03 L.7/0_MS 9/48 Page 3 of 48 exams

32 3. l is the line x 3y 0 and m is the line x y 0. (a) Write down the slope of l and the slope of m. Slope of l 3 y l m (5B) Slope of m x x Scale 5B (0, 3, 5) Partial credit: (3 marks) Only one correct slope and given. 3 (b) Find the measure of the acute angle,, between l and m. Tan m m ± mm Tan ± 3 ( ) ( ) 3 (0C) Tan ± ± 45º 35º but 45º as is acute Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) Correct formula with some substitution. High partial credit: (8 marks) Two solutions given. (c) The line n cuts the x-axis at the same point as l. The line n also makes the same acute angle,, with l. Find the equation of n. the line n cuts the x-axis at the same point as l x + 3y 0 x 0 (, 0) pt to m slope y 0 (x ) y 0 x x + y 0 Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) Correct point or slope. High partial credit: (8 marks) Correct point and slope. (0C) 03 L.7/0_MS 0/48 Page 3 of 48 exams

33 4. c is a circle with centre (3, ). The lines l : x y 3 0 and l : x y 6 0 are tangents to c, as shown in the diagram. l y c l c 3 c x (a) Find the radius of c and hence, write down the equation of c. c 4 (0C) (3, ), x + y 3 0 r (3) () 3 5 () 6 3 () (x ) + (y ) ( 5 ) (x 3) + (y ) 5 Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) 5 or distance with substitution. 5 High partial credit: (8 marks) Equation of circle formula with substitution. (b) c, c 3 and c 4 are images of c under different transformations. Describe fully the transformation in each case. c axial symmetry in the line l / rotation of 70º c 3 central symmetry in the pt (0, 3) / rotation at 80º c 4 axial symmetry in the line l / rotation of 90º Scale 5C (0, 3, 4,5) Low partial credit: (3 marks) One correct substitution. High partial credit: (4 marks) Two correct substitutions. (5C) (c) Find the equations of the two circles that touch c, c, c 3 and c 4. (0C) centre (0,3) (0,3) to (3,) ( 3) () inner circle (x 0) + (y 3) ( 0 5 ) outer circle (x 0) + (y 3) ( ) Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) 0 found. High partial credit: (8 marks) and 0 5 found. 03 L.7/0_MS /48 Page 33 of 48 exams

34 5. (a) Show that sin x sinx cos x. (5B) sin(a + B) sin A cos B + cos A sin B sin (x + x) sin x cos x + cos x sin x sin x cos x Scale 5B (0, 3, 5) Partial credit: (3 marks) Sin (x) sin (x + x). Sin(A + B) sin A cos B + cos A sin B. (b) Solve the equation sin x sin x 0 in the domain 0 x, x. (0C) sinx cos x sin x 0 sinx (cos x ) 0 sinx 0 cos x cos x x 0º, 80º, 360º 60º, 300º x 0º, 60º, 80º, 300º, 360º Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) Equation in terms of x only. High partial credit: (8 marks) Sin x 0 and cos x. 03 L.7/0_MS /48 Page 34 of 48 exams

35 5(c) Verify your solution from part (b) by sketching the graphs of the functions f : x sin x and g : x sin x in the domain 0 x, x. Indicate clearly which is f and which is g. (0C) y x 0 3 y gx () fx () x Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) One correct function graphed. High partial credit: (8 marks) Two correct graphs, but answers not shown. 03 L.7/0_MS 3/48 Page 35 of 48 exams

36 6. Answer either 6A or 6B. 6A. (a) Explain the difference between an axiom and a theorem. (5B) an axiom is a statement assumed to be true // a theorem is a statement which has been proved, deducted from axioms and logical argument Scale 5B (0, 3, 5) Partial credit: (3 marks) One correct statement. (b) ABC is a triangle. Prove that, if a line l is parallel to BC and cuts [ AB ] in the ratio s : t, then it also cuts [ AC ] in the same ratio. Given Triangle ABC l \\ BC To prove AD DB Diagram AE EC A s t (5B, 5B, 0C) D E D m D l E E m D m+n B C E m+n Diagram: (5B) Proof We prove only the commensurable case. Let l cut [AB] in D in the ratio m : n with natural numbers m, n. Thus there are points. D 0 A, D, D,,D m, D m D, D m +,..., D m + n, D m +n B Equally spaced along [AB], i.e. the segments: [D 0 D ], [D D ],.[D i D i + ], [D m +n D m +n ] have equal length. Draw lines D E, D E, parallel to BC with E, E, on [AC]. Then all the segments: [AE ], [E E ], [E,E 3 ],, [E m + n C] have the same length [Theorem ] and E m E is the point where l cuts [AC]. [Axiom of Parallels] Hence E divides [AC] in the ratio m : n Scale 5B (0, 3, 5) Partial credit: (3 marks) Triangle with l drawn. Given/To prove: (5B) Scale 5B (0, 3, 5) Partial credit: (3 marks) Given correct or to prove correct. Proof: (0C) Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) One correct statement. High partial credit: (8 marks) One missing step or steps in incorrect order. 03 L.7/0_MS 4/48 Page 36 of 48 exams

37 OR 6B ACDE is a parallelogram. The points A, D and E lie on the circle which cuts [ AC ] at B. E 5 D 4 3 C B A (a) Prove that BCD is an isosceles triangle. (5C) parallelogram ACDE opposite angles of cyclic quadrilateral straight line BDC is isosceles Scale 5C (0, 8,, 5) Low partial credit: (8 marks) High partial credit: ( marks) 3. (b) Prove that BDE AED. 3 5 alternate angles 3 proved above 5 AED BDE (0C) Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) 3 5 or 3. High partial credit: (8 marks) 3 5 and L.7/0_MS 5/48 Page 37 of 48 exams

38 Section B Contexts and Applications 50 marks Answer Question 7 and Question 8 from this section. (75 marks each) 7 A confectionary manufacturer is launching a new range of chocolates. The disc-shaped chocolates, each of radius 5 cm and height 0 75 cm, are packed in two layers in a rectangular box. The configuration of chocolates in each layer is shown below. (a) Calculate the internal volume of the box. (5C) 8.5 cm.5 8 cm cm cm 3 Scale 5C (0, 8,, 5) Low partial credit: (8 marks) At least two dimensions calculated. Volume l b h given. High partial credit: ( marks) Volume of one layer only. 03 L.7/0_MS 6/48 Page 38 of 48 exams

39 A production researcher suggests that if the configuration of the chocolates in each layer of the box was altered, the packaging costs of the chocolates could be reduced. The alternative design of each layer is shown below. p (b) Using the right-angled triangle on the alternative design, calculate p and hence, find the internal volume of this box. p p 36 9 p 7 p cm cm cm 3 p 6 3 (5C) Scale 5C (0, 8,, 5) Low partial credit: (8 marks) At least two dimensions calculated. p calculated. High partial credit: ( marks) Volume of one layer only. (c) The original box costs 0 80 to produce. Calculate the ratio of the volumes of the two boxes and hence, find the potential savings of using the alternative design if the cost of producing each box is directly proportional to the volume of the box and the company projects sales of boxes annually , ,407.4 (5C) or st box ,000 0, nd box ,000, ,407.4 Scale 5C (0, 8,, 5) Low partial credit: (8 marks) 0,000 written.,59.59 written cent per new box written. High partial credit: ( marks) 0,000 and,59.59 written. 03 L.7/0_MS 7/48 Page 39 of 48 exams

40 7(d) There are three different types of chocolates in the box. Half the chocolates are milk chocolates while the ratio of dark to white chocolates is :. (i) What is the probability that a chocolate chosen at random from the box will be a white chocolate? 4 milk 6 dark 8 white (0C) Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) 4 / 6 / 8 calculated. High partial credit: (8 marks) Correct favourable outcomes. Correct total outcomes (probability 0 p ). (ii) If three chocolates are chosen at random from the box, find the probability that each of the chocolates chosen is made from the same type of chocolate. (5B) , 3, , , , 776 5, , , Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct favourable outcomes. Correct total outcomes (probability 0 p ). (iii) If three chocolates are chosen at random from the box, find the probability that each of the chocolates chosen is made from a different type of chocolate. (5B) , , , Scale 5B (0, 3, 5) Partial credit: (3 marks) Correct favourable outcomes. Correct total outcomes 6 omitted (probability 0 p ). 03 L.7/0_MS 8/48 Page 40 of 48 exams

41 7(e) The confectionary company claims that the minimum weight of chocolates in each box is 350 g. The company s quality control department has determined that the weights of chocolates in each box are normally distributed with mean equal to the specified weight and standard deviation of 8 g. 50 boxes are selected at random and it is discovered that the mean weight of chocolates in these boxes is 348 g. Use a hypothesis test at the 5% level of significance to decide whether there is sufficient evidence to validate the confectionary company s claim. Be sure to state the null hypothesis clearly, and to state the conclusion clearly. Ho the null hypothesis, that the chocolates are within the standard range Z Standard error Standard error The null hypotheses are correct and they are within the standard range (0C) Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) Hypothesis stated correctly. High partial credit: (8 marks) Standard error. 03 L.7/0_MS 9/48 Page 4 of 48 exams

42 8 Mary has a rectangular hallway measuring 6 m by 5 m. She plans on tiling the floor with regular pentagon-shaped tiles of side 6 cm using a mosaic pattern. She has decided to use a tiling contractor and he has given her a scaled drawing of the proposed work. (a) A tile on the scaled drawing is reproduced on the square grid as shown. Given the centre of enlargement, O, construct, using a compass and straight edge only, a full-size tile. (5C) cm O Scale 5C (0, 8,, 5) Low partial credit: (8 marks) At least one point extended. High partial credit: ( marks) At least three points extended. 03 L.7/0_MS 30/48 Page 4 of 48 exams

43 8(b) The tiling contractor has asked Mary to order the tiles she wants to use. She needs to find out the area that each tile covers in order to calculate the number of tiles needed to complete the work. However, the area of each tile is not shown on her drawing. She knows that the internal angles of a regular pentagon are 08 and it can be divided up into three isosceles triangles as shown. Find the area of each tile. Give your answer correct to one decimal place. 08 6cm (0C, 0C, 0B*) ab sin C (6)(6) sin 08 (6)(6) sin Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) Formula with some substitution. High partial credit: (8 marks) Formula fully substituted x (6) (6) cos 08 x ( ) x x ab sin C (9.708)(6) sin 7 (9.708)(6)( ) Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) Cosine rule with substitution. High partial credit: (8 marks) Area formula with substitution cm Scale 0B* (0, 5, 9, 0) Low partial credit: (5 marks) Three triangles added together. High partial credit: (9 marks) Not rounded off to one decimal place. 03 L.7/0_MS 3/48 Page 43 of 48 exams

44 8(c) Mary asked her friend, Sean, to check her calculations. He uses an alternative method to find the area of each tile. On the scaled drawing, Sean sketches a triangle ABC, where C is the centre of the pentagon. He calculates the area of the tile to be: A C 9 5 ( 7)()(sin54) Explain what each of the following numbers represent: B cm (0C).7: 5: 9: 54: Length of AC or BC 5 triangles 3 scale factor squared Angle CAB or CBA Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) At least one correct answer. High partial credit: (8 marks) At least three correct answers. (d) Mary s hallway measures 6 m by 5 m. The tiling contractor has asked her to purchase 0% more tiles than are required to allow for cutting and wastage. (i) Assuming the area of the triangular cut-offs required are negligible, how many boxes of tiles does Mary need to order if they are supplied in boxes of 60? tiles boxes (0C) or boxes Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) Area of hall evaluated as 7500 cm. High partial credit: (8 marks).6 tiles needed tiles needed. 03 L.7/0_MS 3/48 Page 44 of 48 exams

45 8(d)(ii) When Mary goes to order the tiles, she is informed that this particular size of tile has been discontinued. However a similar regular pentagon-shaped tile of side 5 cm supplied in boxes of 7 is available. Calculate the number of boxes of tiles she now needs to order tiles boxes (0C) Scale 0C (0, 4, 8, 0) Low partial credit: (4 marks) 5 6 or. 6 5 High partial credit: (8 marks) 36 5 or given L.7/0_MS 33/48 Page 45 of 48 exams

46 Notes: 03 L.7/0_MS 34/48 Page 46 of 48 exams

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