Exemplar Booklet - Manitoba Department of Education and Training

Grade 12 

Pre-Calculus Mathematics 

Achievement Test 

Exemplar Booklet 

June 2013

Manitoba Education Cataloguing in Publication Data 

Grade 12 pre-calculus mathematics achievement test. 

Exemplar booklet (June 2013) [electronic resource] 

ISBN: 978-0-7711-5428-7 

1. Mathematics—Examinations, questions, etc. 

2. Mathematics—Examinations. 

3. Educational tests and measurements—Manitoba. 

4. Mathematics—Study and teaching (Secondary)—Manitoba. 

5. Calculus—Study and teaching (Secondary)—Manitoba. 

6. Mathematical ability—Testing. 

I. Manitoba. Manitoba Education. 

515.076 

Manitoba Education 

School Programs Division 

Winnipeg, Manitoba, Canada 

Permission is hereby given to reproduce this document for non-profit educational 

purposes provided the source is cited. 

After the administration of this test, print copies of this resource will be 

available for purchase from the Manitoba Text Book Bureau. Order online at 

. 

This resource will also be available on the Manitoba Education website at 

. 

Websites are subject to change without notice. 

Disponible en français. 

Available in alternate formats upon request.

Table of Contents 

Instructions .................................. 1 

Communication Errors ........................ 1 

Booklet 1 Questions .................... 3 

Question 1........................................ 5 

Question 2........................................ 6 

Question 3........................................ 8 

Question 4...................................... 11 

Question 5...................................... 13 

Question 6...................................... 14 

Question 7...................................... 15 

Question 8...................................... 17 

Question 9...................................... 20 

Question 10 .................................... 21 

Question 11 .................................... 25 

Question 12 .................................... 27 

Question 13 .................................... 28 

Question 14 .................................... 30 

Question 15 .................................... 32 

Question 16 .................................... 37 

Booklet 2 Questions .................. 41 

Question 25 .................................... 42 

Question 26 .................................... 44 

Question 27 .................................... 45 

Question 28 .................................... 47 

Question 29 .................................... 49 

Question 30 .................................... 53 

Question 31 .................................... 54 

Question 32 .................................... 56 

Question 33 .................................... 58 

Question 34 .................................... 60 

Question 35 .................................... 62 

Question 36 .................................... 64 

Question 37 .................................... 66 

Question 38 .................................... 69 

Question 39 .................................... 71 

Question 40 .................................... 74 

Question 41 .................................... 78 

Question 42 .................................... 80 

Question 43 .................................... 82 

Question 44 .................................... 85 

Question 45 .................................... 87 

Appendices ................................. 91 

Appendix A: Marking Guidelines ........ 93 

i

Instructions 

This booklet contains exemplars of student responses to train teachers to mark the Grade 12 Pre-Calculus 

Mathematics Achievement Test. For each short and long-answer question, exemplars of student responses 

are provided, when necessary. These exemplars were selected for their representation of a variety of 

strategies most commonly used by students during pilot testing. 

The recommended procedure for scoring student exemplars is as follows: 

1. Read the Grade 12 Pre-Calculus Mathematics Achievement Test: Marking Guide (June 2013). 

2. Determine the mark for the exemplar by comparing its features with those in the Marking Guide. 

3. Compare this mark with the one provided and study the rationales for the allotted score. 

Communication Errors 

The marks allocated to questions are primarily based on the concepts and procedures associated with the 

learning outcomes in the curriculum. For each question, shade in the circle on the Answer/Scoring Sheet 

that represents the marks given based on the concepts and procedures. A total of these marks will 

provide the preliminary mark. 

Errors that are not related to concepts or procedures are called “Communication Errors” (see Appendix A) 

and will be tracked on the Answer/Scoring Sheet in a separate section. There is a ½ mark deduction for 

each type of communication error committed, regardless of the number of errors per type 

(i.e. committing a second error for any type will not further affect a student’s mark), with a maximum 

deduction of 5 marks from the total test mark. 

The total mark deduction for communication errors for any student response is not to exceed the marks 

given for that response. When multiple communication errors are made in a given response, any 

deductions are to be indicated in the order in which the errors occur in the response, without exceeding 

the given marks. 

The student’s final mark is determined by subtracting the communication errors from the preliminary 

mark. 

Example: A student has a preliminary mark of 72. The student committed two E1 errors (½ mark 

deduction), four E7 errors (½ mark deduction), and one E8 error (½ mark deduction). Although 

seven communication errors were committed in total, there is a deduction of only 1½ marks. 

36 9 45 90 

Pre-Calculus Mathematics: Exemplar Booklet (June 2013) 1

2 Pre-Calculus Mathematics: Exemplar Booklet (June 2013)

Booklet 1 Questions 



0BQuestion 1 

A central angle of a circle subtends an arc length of 5π cm. 

Given the circle has a radius of 9 cm, find the measure of the central angle in degrees. 

Exemplar 1 

1 out of 2 

+ ½ mark for substitution into correct formula 

+ ½ mark for solving for θ 



award full marks 

 

E7 (notation error in line 5 for equating 5 π and 100 ) 


1BQuestion 2 

2 

Solve the equation csc + 3csc − 4 = 0 

0, 2 π . 

Express your answers as exact values or correct to 3 decimal places. 

θ θ over the interval [ ] 



Method 2 

+ 1 mark for solutions 

E6 (rounding error in line 3) 


2½ out of 4 

Method 1 

+ 1 mark for solving for cscθ 

+ 1 mark for reciprocal of cscθ 

+ 1 mark for consistent solution 

− ½ mark for arithmetic error in line 2 




Method 1 

+ 1 mark for solving for cscθ 

+ 1 mark for reciprocal of cscθ 

+ 1 mark for consistent solution 

E3 (variable omitted in line 1) 


2BQuestion 3 

Jess invests $12 000 at a rate of 4.75% compounded monthly. 

How long will it take for Jess to triple her investment? 

Express your answer in years, correct to 3 decimal places. 



Method 1 

+ ½ mark for substitution 

+ ½ mark for applying logarithms 

+ 1 mark for power rule 

+ ½ mark for isolating t 

− 1 mark for concept error ( ln1 ≠ ln e ) 

E6 (rounding errors in lines 3 and 4) 

E7 (notation errors in lines 3, 4, and 5) 




Method 1 

+ 2 marks for the formula 

rt 

A = Pe used correctly (see note in Marking Guide) 



1 


Method 1 


3BQuestion 4 

The 4th term in the binomial expansion of 

Determine the value of q algebraically. 

10 

⎛ 2 3⎞ 

⎜qx 

− 

⎝ 

⎟ 

x⎠ 

is 

11 

414 720 x . 








+ 1 mark for consistent factors 

+ ½ mark for comparing coefficients 


E2 (changing an equation to an expression in line 2) 


4BQuestion 5 

Bella has 2 pairs of shoes, 3 pairs of pants, and 10 shirts. 

Carey has 4 pairs of shoes, 4 pairs of pants, and 4 shirts. 

An outfit is made up of one pair of shoes, one pair of pants, and one shirt. 

Who can make more outfits? Justify your answer. 




½ out of 1 




5BQuestion 6 

In the binomial expansion of ( x y) 10 , 

Justify your answer. 

− how many terms will be positive? 



+ 1 mark for justification 


There are 11 terms. Since the first and last terms 

are positive and every other (alternate) term is 

positive, that would give a total of 6. 



1 st , 3 rd , 5 th , 7 th , 9 th and 11 th term will be 

positive. 


+ 1 mark for six terms 


6BQuestion 7 

 

Solve the following equation algebraically where 180 ≤ θ ≤ 360 . 

2 

2sin θ + 5cosθ 

+ 1 = 0 



+ 1 mark for identity 

+ 1 mark for solving for cosθ 



E7 (transcription error in line 1) 



+ 1 mark for identity 


+ 1 mark for indicating no solution 





+ 1 mark for indicating no solution 

+ 1 mark for solving for θ 


E1 (answer stated in radians instead of degrees in line 6) 


E8 (answer given outside the domain in line 6) 


7BQuestion 8 

Solve the following equation algebraically: 

log ( x − 4) + log ( x − 2) 

= 1 

3 3 



Method 1 

+ 1 mark for product rule 

+ 1 mark for exponential form 

+ ½ mark for solving for x within a quadratic equation 




Method 1 





Method 2 


+ ½ mark for logarithmic form 

+ ½ mark for equating arguments 


E7 (notation error in line 5 of missing “ x = ”) 



Method 2 


+ ½ mark for equating arguments 

+ ½ mark for solving for x within a quadratic equation 





Method 2 

+ ½ mark for rejecting extraneous root 


8BQuestion 9 

{ } 

Given that f ( x ) = ( 1, 3 ), ( 2, 5 ), ( 3, 4 ), ( 4, 2 ) , find f ( f ( )) 

3 . 










9BQuestion 10 

Given the graphs of f ( x ) and g( x ) below, 

f ( x) 

y 

g( x) 

y 

1 

1 

x 

1 

1 

x 

sketch the graph of y = f ( x) − g( x). 



+ 1 mark for subtraction of f ( x) − g( x) 





− ½ mark for arithmetic error [see the point ( 1, 0 ) ] 




− ½ mark for arithmetic error ( −2−3≠− 

6) 

E7 [transcription error g ( − 2) 

= 2 ] 

( ) 










10BQuestion 11 

Given the graph of y = f ( x), 

describe the transformations to obtain the graph of the function 

( ) 

y = f 2x 

− 6. 


Divide the x-values by 2 

Shift to the right 6 units 


Method 2 


− 1 mark for concept error (order of transformations) 


Multiply the x-values by 2 

Horizontal shift of 3 units to the right 


Method 1 

+ 1 mark for ending with a horizontal shift of 3 units to the right 



Method 1 

+ 1 mark for ending with a horizontal shift of 3 units to the right 



Horizontally stretch by 2 

Translation of 6 units left 



Factor out the 2 from the bracket. 

Translate 3 units to the right. 

Horizontal compression by 2. 


Method 1 


− 1 mark for concept error (order of transformations) 



{ } 

Given f ( x ) = ( − 3,4 ),( 2,7 ),( 8,6 ) , state the domain of the resulting function after f ( ) 

reflected through the line y = x. 

x is 



+ ½ mark for stating inverse correctly (see note in Marking Guide) 



+ ½ mark for stating inverse correctly (see note in Marking Guide) 





Determine the value of y in the following equation: 

log 27 log 3 2log 

x − x = x y 



+ 1 mark for quotient rule 


+ ½ mark for positive value of y 

















Angle θ, measuring 5 π 

, is drawn in standard position as shown below. 

4 

−4 π,2π that are coterminal with θ. 

Determine the measures of all angles in the interval [ ] 

y 

θ 

x 




− ½ mark for arithmetic error in line 1 (calculation of domain) 






E7 (notation error in line 2) 

E8 (answer given outside the domain) 

Note: When multiple communication errors are made, any deductions are to be indicated in the 

order in which they occur in the response, without exceeding the given marks (see page 1). 



Prove the identity below for all permissible values of x: 

2 

sin x 

2 

= cos x − cos x 

sec x + 1 



Method 1 

+ 1 mark for correct substitution of identities 

+ 1 mark for algebraic strategies 




Method 1 


E3 (variable omitted) 

E3 (variable introduced without being defined) 

E7 (notation error in line 2 of right-hand side) 




Method 1 





Method 1 


+ 1 mark for algebraic strategies 




Method 1 

E2 (equating the two sides when proving an identity) 

E3 (variable omitted in line 4) 

E3 (variable introduced without being defined in line 1) 

E4 (missing brackets but still implied in line 2) 

Note: No communication error deductions because no marks were given. 



Solve algebraically: 

4 5 

n C 2 

= n + 



+ ½ mark for factorial notation 

+ ½ mark for simplification of factorial in line 2 

+ ½ mark for simplification in line 5 





+ ½ mark for factorial expansion 


+ ½ mark for simplification in line 4 

+ ½ mark for both values of n 





+ ½ mark for factorial expansion 


+ ½ mark for both values of n 




E2 (changing an equation to an expression) 

E4 (missing brackets but still implied in line 2) 

E7 (notation error in line 2: missing !) 

Note: No communication error deductions because no marks were given. 


Booklet 2 Questions 



Given the graph of y = 2cos π x + 1 below, determine another equation that will produce the 

same graph. 

y 

1 

1 

x 




E7 [notation error (missing bracket)] 








Given f ( x ) = 3 and g( x) = x + 2, determine the domain and range of h( x) 

= 

( ) 

( ) . 

f x 

g x 


R 


+ 1 mark for domain 


Domain : x ≠−2 

Range : y ≠ 0 



Domain 

Range 



E7 (transcription error) 



⎛19π⎞ 

Explain how to find the exact value of sec ⎜ . 

⎝ 

⎟ 

6 ⎠ 



+ 1 mark for 

⎛19π⎞ 

cos ⎜ 

⎝ 

⎟ 

6 ⎠ 







− ½ mark for terminology error 



+ 1 mark for 

⎛19π⎞ 

cos ⎜ 

⎝ 

⎟ 

6 ⎠ 



− 

Given f ( x) = 4 − x, 

verify that f 1 ( x) = f ( x) 

. 




−1 

E1 [final answer not stated: f ( x) f ( x) 

= ] 




−1 


= ] 





−1 


= ] 





Sketch the graph of: 

( ) = ( 2 − )( + 3)( + 1) 2 

f x x x x 

Label the x-intercepts and y-intercept. 



+ 1 mark for x-intercepts 

+ ½ mark for y-intercept 

− ½ mark for incorrect shape of graph 




+ ½ mark for end behaviour 






+ 1 mark for multiplicity of 2 at x =− 1 only 

− ½ mark for incorrect shape of graph (graph touched the x-axis more than 3 times) 








Which expression has a larger value? 

log 36 or log 80 

2 3 





E7 [notation error ( used = instead of ≈)] 




− ½ mark for arithmetic error 





The graph below represents the equation 

y 

3 2 

y = ax + 6x + 5x 

− 10. 

x 

What must be true about the value of a? Explain your reasoning. 







Must be a negative value 

so that the two ends look 

like the graph’s end behaviour. 



23B0B 


The terminal arm of an angle θ, in standard position, intersects the unit circle in Quadrant IV 

⎛ 5 ⎞ 

at a point P ⎜ , y . 

4 

⎟ Determine the value of sinθ. 

⎝ ⎠ 



Method 2 


+ ½ mark for using the value of y to find the value of sinθ 




Method 2 





( ) 


Method 2 


+ ½ mark for solving for y 

+ ½ mark for using the value of y to find the value of sinθ 

E3 (variable omitted) 



Given the sinusoidal function f ( x ) below, sketch the graph of g( x) = f ( x) − 1. 

f ( x) 

y 

1 

 

90 

 

x 


− 



E9 (scale value on x-axis not indicated) 




+ 1 mark for absolute value 

E9 (scale value on x-axis not indicated) 



+ 1 mark for vertical shift 



The graph of a rational function, f ( x ), has a point of discontinuity when x = 2 and an 

asymptote when x = 4. Write a possible equation for f ( x ). 






− ½ mark for procedural error of not indicating that x ≠ 2 after the simplification 





E2 (changing an equation to an expression) 



+ 1 mark for 

x − 2 

x − 2 



Given that ( x − 1) 

is one of the factors, express 

3 

x 

− 57x 

+ 56 as a product of factors. 






− ½ mark for arithmetic error (factoring) 

E2 (changing an expression to an equation) 




+ ½ mark for x = 1 



+ ½ mark for 1 x = 

+ ½ mark for consistent factors 



Give an example using values for A and B, in degrees or radians, to verify that 

cos A+ B = cos A+ cos B is not an identity. 

( ) 



Method 2 



Method 1 

+ 1 mark for simplification of cos A+ 

cos B 

− ½ mark for arithmetic errors in lines 3 and 4 




Method 2 



E2 (equating the two sides when proving an identity in line 6) 



Method 2 



E2 (equating the two sides when proving an identity) 


28B 

BQuestion 37 

Sketch the graph of y = x + 1 − 2 and verify that the value of the x-intercept is the same as 

the solution to the equation x + 1 − 2 = 0. 



+ ½ mark for horizontal shift 

+ ½ mark for vertical shift 

+ 1 mark for verification 









+ ½ mark for vertical shift 





+ 1 mark for general shape 


− ½ mark for not including a minimum of 2 points 





Mohamed is asked to sketch the graph of y = tan x. 

His graph is shown below. 

y 

1 

π 

2 

x 

Explain why his graph is incorrect. 


Graph of 

y = tanx 




If Mohamed’s graph 



− ½ mark for terminology error (incorrect transformation) 







On the interval 0 ≤ θ< 2π, 

identify the non-permissible values of θ for the trigonometric 

identity: 

tanθ 

= 

1 

cotθ 



+ 1 mark for restrictions of cosθ = 0 (½ mark for cosθ = 0, ½ mark for solving for θ ) 




E8 (answer included outside the given domain) 




+ 1 mark for restrictions of cosθ = 0 (½ mark for cosθ = 0, ½ mark for solving for θ ) 




+ 1 mark for identifying non-permissible values 

E7 (notation error in line 4) 



a) Sketch the graph of y = ( x) 

ln . 

b) Sketch the graph of y =− ( x − ) 

ln 2 . 


a) 


b) 



E9 (scale values on axes not indicated) 



a) 


+ ½ mark for increasing logarithmic function 

+ ½ mark for x-intercept at ( 1, 0 ) 

+ ½ mark for vertical asymptotic behaviour 

b) 


+ 1 mark for horizontal shift 

− ½ mark for incorrect shape of graph (asymptotic behaviour at y = 0 ) 



a) 


b) 



− ½ mark for incorrect shape of graph (missing asymptote) 



a) 


+ ½ mark for x-intercept at ( 1, 0 ) 

E9 (scale values on axes not indicated) 

b) 



32B32B32BQuestion 41 

( ). 

Given f ( x) = x − 2 and g( x) = 3, x write the equation for hx ( ) = f gx ( ) 

What are the restrictions on the domain of h( x )? 

Explain your reasoning. 



( ) 

+ 1 mark for h( x) = f g( x) 

+ ½ mark for explanation 



( ) 


+ ½ mark for identifying restriction 




( ) 




( ) 




+ ½ mark for identifying restriction 



⎡π 

⎤ 

⎢ 

⎣2 

⎥ 

⎦ 

Sketch the graph of y = 10cos ( x − 2) 

over the interval [ 0, 6 ]. 



+ 1 mark for amplitude 

+ 1 mark for horizontal shift 

− ½ mark for not including interval [ 0, 6 ] (see note in Marking Guide) 













+ 1 mark for amplitude 

− ½ mark for not including interval [ 0, 6 ] (see note in Marking Guide) 



Sketch the graph of the function f ( x) 

= 

2 

x 

2 

x − 

. 

x 



+ 1 mark for vertical asymptote at 1 x = 

+ ½ mark for graph right of the vertical asymptote 




+ 1 mark for vertical asymptote at x = 1 

+ 1 mark for horizontal asymptote at y = 1 

+ ½ mark for graph left of vertical asymptote 

E10 (asymptotes drawn as solid lines) 



+ 1 mark for horizontal asymptote at 1 y = 


+ ½ mark for graph right of vertical asymptote 

− ½ mark for not including a minimum of 2 points 




+ 1 mark for vertical asymptote at 1 x = 

+ 1 mark for point of discontinuity consistent with graph 


+ ½ mark for graph right of vertical asymptote 

E10 (graph curls away from asymptote) 



Is ( x − 3) 

a factor of 


4 3 2 

x − x − 3x + x − 1? 



Method 2 





Method 1 

+ 1 mark for remainder theorem 





Method 2 

+ ½ mark for 3 x = 




Given f ( x) = x − 1 and g ( x ) = x 

2 , write the equation of y f ( g ( x )) 


= and sketch the graph. 




+ 1 mark for consistent graph 




+ 1 mark for composition 


2 

f ( g ( x) 

) = x − 

1 









Appendices 



Appendix A 

Appendix A: Marking Guide 

MARKING GUIDELINES 

Errors that are conceptually related to the learning outcomes associated with the question 

will result in a 1 mark deduction. 

Each time a student makes one of the following errors, a ½ mark deduction will apply. 

• arithmetic error 

• procedural error 

• terminology error 

• lack of clarity in explanation 

• incorrect shape of graph (only when marks are not allocated for shape) 

Communication Errors 

The following errors, which are not conceptually related to the learning outcomes associated 

with the question, may result in a ½ mark deduction and will be tracked on the 

Answer/Scoring Sheet. 

E1 

E2 

E3 

• answer given as a complex fraction 

• final answer not stated 

• answer stated in degrees instead of radians or vice versa 

• changing an equation to an expression or vice versa 

• equating the two sides when proving an identity 

• variable omitted in an equation or identity 

• variables introduced without being defined 

E4 • 

2 

2 

“ sin x ” written instead of “ sin x ” 

• missing brackets but still implied 

E5 

E6 

E7 

E8 

E9 

E10 

• missing units of measure 

• incorrect units of measure 

• rounding error 

• rounding too early 

• transcription error 

• notation error 

• answer given outside the domain 

• bracket error made when stating domain or range 

• domain or range written in incorrect order 

• incorrect or missing endpoints or arrowheads 

• scale values on axes not indicated 

• coordinate points labelled incorrectly 

• asymptotes drawn as solid lines 

• graph crosses or curls away from asymptotes

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