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MCD 4490 Advanced Mathematics Practice Test 1 Question and Answer Booklet Reading and Writing Time: 2 hours and 10 minutes Structure of Test Questions Marks

1 10

2 10

3 10

4 10

5 10

Total 50

Actual Marks

INSTRUCTIONS TO CANDIDATES During a test, you must not have in your possessions, a book, notes, paper, electronic device/s, calculator, pencil case, mobile phone, smart watch/device of other material/ item which has not been authorised for the test or specifically permitted as noted below. Any material or item on your desk, chair or person will be deemed to be in your possession.

(1) (2) (3) (4) (5)

Give clear explanations and show all working to support each of your answers. Do not use pencil Write clearly. Marks cannot be awarded if your handwriting cannot be read. Answer all questions in the space provided. A formulae sheet is included.

AUTHORISED MATERIALS CALCULATORS

NO

OPEN BOOK

NO

SPCIFICALLY PERMITTED ITEMS

NO

STUDENT ID _____________________DESK NUMBER _____________________ STUDENT NAME _________________TUTOR _____________________________

THIS ENTIRE TEST PAPER MUST BE HANDED IN AT THE END OF THE TEST. DOD NOT OPE THIS BOOKLET UNTIL INSTRUCTED TO DO SO.

Question 1 (a) Determine the domain and range of f  x  

x2  x  6 x2  2 x  3

in the interval notation. [2 marks]

(b) Given that f  x  

(c) i.

1 x 2 1

and g  x   x  1 . Find f  g  x   and its domain.

Find the smallest possible value of k such that f :  k ,  

[3 marks] 1  is x 1

a one to one function. [1 mark] ii. Hence find the expression of the inverse function of f  x  and state its domain iii.

and range. Sketch both f and f 1 on the same set of axes

[2 marks] [2 marks]

Question 2 (a) i.

Find the inverse of f  x  3ln  2 x  1  7 and state its domain and range.

[3 marks] ii. Clearly showing axial intercepts, sketch both f and f on the same set of axes. [2 marks] 1 (b) The value of a house that originally cost $500, 000 increases every year by . 3 i. Find a formula for V t  , the value of the house after t year. [2 marks] 1

ii. iii.

Find the value of the house after 3 years. How long does it take for the price to reach $1 000, 000?

[1 mark] [2 marks]

Question 3 (a) Solve sin  4 x  sin  2 x   2 cos  2 x   1,

0 x

(b) Find the domain and range of f  x   3sin 1  2 x  5  

[4 marks]



[2 marks] 6 (c) Express f  x   27 sin 3 x   3cos 3 x   2 of the form A cos 3 x     B, A  0 and hence find the amplitude, phase and range of f

[4 marks]

Question 4 3  3 cis    6  by using Exponential form and express the final answer (a) Simplify  i e3 in Cartesian form [2 marks] 2 (b) Factorise z  2 z  3 by using the method of completing the square and then find all the [2 marks] solutions to z 2  2 z  3  0

1 



3i

[3 marks]

(d) Sketch A  z  C : z  2i  1  2 z 

[3 marks]

(c) Calculate

3

Question 5 (a) Consider the position vectors a  3i  2 j  k and b  i  2 j  3k i. ii. iii.

Find u  a  2 b and v  a  b [2 marks] [2 marks] Find the acute angle between u and v Find the vector components of u parallel and perpendicular to v [2 marks]

(b) Check the continuity of f  x   x  2  3x  1 (c) Find lim x 

3x 2  2x  x x 1

[3 marks] [1 mark]

End of practice test 1

MCD4490

FORMULAE SHEET

This is not a complete list, nor does it state theorems in full; you are expected to know the context of, and under what conditions you may apply these formulae.

MCD4490

FORMULAE SHEET...