UNIT 3 Analysis Task (SAC 1) 2015

NAME: Student Number:

SPECIALIST MATHEMATICS unit 3

School Assessed Coursework 1(SAC 1)

ITEM RESPONSE ANALYSIS TASK

Reading time: 5 minutes

Writing time: 70 minutes

QUESTION & ANSWER BOOK

Structure of Book

Number of Questions / Number of questions to be answered / Number of marks
8 / 8 / 54

Students are permitted to bring into the SAC room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set-squares, aids for curve sketching, one bound reference, one approved CAS calculator (memory DOES NOT need to be cleared) and, if desired, one scientific calculator. For approved computer-based CAS, their full functionality maybe used.

Students are NOT permitted to bring into the SAC room: blank sheets of paper and/or white out liquid/tape.

Materials supplied

Question and answer book.

Instructions

Write your name in the space provided above on this page.

All written responses must be in English.

Students are NOT permitted to bring mobile phones and/or any other unauthorized electronic devices into the test room.

At the end of the year Examination 2 includes 22 multiple choice questions to which you must select the correct response only. No working or justification is required.

However, in this analysis task you are required to analyse the incorrect responses (distractors), justify why they are incorrect and what concepts are being evaluated, to assist in determining the correct response. Each item for analysis includes a set of related questions.

Item 1 (6 marks)

The diameter of a circle has end points at (-1, 5) and (5, -3). The equation of the cirle is

A. x-22+y-12=5

B. x-22+y-12=25

C. x-22+y-12=100

D. x-32+y-42=5

E. x-32+y-42=25

(a) (i) Find the length of the diameter. (1 mark)

(ii) Hence find the radius of the circle. (1 mark)

(iii) Which responses must be eliminated and why? (1 mark)

(b) (i) Find the midpoint of (-1, 5) and (5, -3). (1 mark)

(ii) What significance does this midpoint have in relation to the equation of the circle? (1 mark)

(iii) Which response is correct and why? (1 mark)

Item 2 (5 marks)

The rule for the relation determined by the parametric equations x=3tanθ-1 and y=2secθ+1 is

A. x+123-y-122=1

B. y-122-x+123=1

C. x+129+y-124=1

D. x+129-y-124=1

E. y-124-x+129=1

(a) (i) Find an expression for tan2θ in terms of x. (1 mark)

(ii) Find an expression for sec2θ in terms of y. (1 mark)

(iii) Explain why responses A and B are incorrect. (1 mark)

(b) Use a trigonometric identity to explain why response C is incorrect. (1 mark)

Item 3 (9 marks)

The ellipse given by 4x2-16x+y2+6y+23=0 has centre, length of horizontal semi-axis and length of vertical semi-axis respectively of

A. 2,-3,12,2

B. 2,-3,2,12

C. -2,3,12,2

D. -2,3,12,2

E. 2,-3,12,2

(a) Express response A

(i) as a Cartesian equation in the form x-h2a2+y-k2b2=1 (1 mark)

(ii) in the simplest expanded form of mx2+nx+py2+qy+s=0 where m, n, p, q, s ∈Z.

(1 mark)

(iii) Explain why another response must also be incorrect. (2 marks)

(b) Show that response B is also incorrect by expressing the ellipse

(i) as a Cartesian equation in the form x-h2a2+y-k2b2=1 (1 mark)

(ii) in the simplest form of mx2+nx+py2+qy+s=0 where m, n, p, q, s ∈Z. (1 mark)

(c) (i) Given the equation of the ellipse is 4x2-16x+y2+6y+23=0, explain the significance of the sign of the coefficient of x and the sign of the coefficient of y in terms of the coordinates of the centre of the ellipse. (1 mark)

(ii) Use this to explain which response is correct? (2 marks)

Item 4 (7 marks)

The principal argument of -3+3i2-23i is

A. -13π12

B. -11π12

C. 5π12

D. 11π12

E. 13π12

(a) Explain why A and E are incorrect. (1 mark)

(b) (i) Let the principal argument of -3+3i be . Find (1 mark)

(ii) Let the principal argument of 2-23i be . Find . (1 mark)

(iii) Explore how the values of and can be used to arrive at responses B, C and D?

Evaluation / Response

(3 marks)

(iv) Which is the correct response and why? (1 mark)

Item 5 (6 marks)

If z=x+yi, where x and y are non-zero real numbers, which of the following represents a real number?

(a) Express response A in the form a + bi. (1 mark)

(b) Based on your answer to part (a), which other response must be incorrect? (1 mark)

(d) Express response D in terms of x and y. (1 mark)

(e) Based on your answers to parts (a) through (d), which one is the correct response? Justify that this answer is indeed the correct response. (2 marks)

Item 6 (7 marks)

The set of points in the complex plane defined by z=z-2i corresponds to

A. the point given by z=i.

B. the line Imz=i.

C. the line Imz=1.

D. the line Rez=1.

E. the circle with centre 2i and radius 1.

(a) Using the substitution =x+yi , where , convert z=z-2i to a Cartesian equation. (2 marks)

(b) Use the wording of the question to explain why response A is not correct. (1 mark)

(d) Determine the Cartesian equations for each of the responses B through to E. Hence determine the appropriateness of each multiple choice response. (3 marks)

Response / Cartesian equation / Response Correct? (Y/N)
C
D
E

Item 7 (5 marks)

The graph of is shown below for 0≤x≤π. The values for a and b could be:

A.
B.
C.
D.
E. /

(a) Find the period of the graph. Hence determine an appropriate value for a. (2 marks)

(b) (i) Plot and label the points and on the above diagram. (1 mark)

(ii) Explain how b can be evaluated by using either of these points. (1 mark)

(iii) Find the value of b and hence identify the correct response. (2 marks)

Item 8 (8 marks)

Let . Given that , all the solutions of the equation are:

a) i) State the conjugate root theorem. (1 mark)

ii) Given that is one root, write down the other root. (1 mark)

iii) List the responses which can be eliminated now. (1 mark)

b) i) Expand (1 mark)

ii) Show how we can find the linear factor by ‘equating the coefficients’ method.

(2 marks)

iii) State the linear factor. (1 mark)

iv) Hence state the real root and identify the correct response. (1 mark)

c) A student has selected response C. What mistake may have been made? (1 mark)

END OF SAC 1