Kilbaha 2021 Exam 1(with detailed solutions and explanation) PDF

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Solutions for iTute exam 1 2021. Full detailed explanation from itute....

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2021 VCE Specialist Mathematics Trial Examination 1

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Victorian Certificate of Education 2021 STUDENT NUMBER Letter Figures Words

SPECIALIST MATHEMATICS Trial Written Examination 1 Reading time: 15 minutes Total writing time: 1 hour

QUESTION AND ANSWER BOOK Structure of book Number of questions 11 • •

Number of questions to be answered 11

Number of marks 40

Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers. Students are NOT permitted to bring into the examination room: any technology (calculators or software), notes of any kind, blank sheets of paper and/or white out liquid/tape.

Materials supplied •

Question and answer book of 20 pages with a detachable sheet of miscellaneous formulas at the end of this booklet.

Instructions •

Detach the formula sheet from the end of this book during reading time.

• •

Write your student number 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 unauthorised electronic devices into the examination room.

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1

Page 4

Instructions Answer all questions in the spaces provided. Unless otherwise specified an exact answer is required to a question. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Take the acceleration due to gravity to have magnitude g m/s2, where g = 9.8 . Question 1

(3 marks)

The cubic equation P ( z ) = z3 − 12 z 2 + bz + c , where b and c are real constants, has amongst its roots 2u and u + 2 2 i , where u is a real constant. Find the values of u, b and c. _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1 Question 2

Page 5

(3 marks)

9 −1 2 ms . 81 − 4t Find the distance travelled in metres by the particle over the first three seconds. A particle moves, so that at time t seconds, its velocity is given by

________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1 (5 marks)

Question 3 Let f ( x) = a.

Page 6

1  3x − 2  sin −  .   4 

4

If f ( x) =

a b

 a + ax −bx2

find the values of the real constants a and b.

2 marks _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1 b.

Page 7

State the maximal domain of f ( x) , and sketch the graph of the function on the axes below. Label the coordinates of the endpoints, the point of inflexion and the axial intercepts. 3 marks

________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1 Question 4

Page 8

(2 marks)

The amount of sauce in bottles packed by a machine varies normally with a mean of 500 mL, and a standard deviation of 3 mL. Let p be the probability that in three randomly selected sauce bottles, the total amount of sauce is between 1497 and 1503 mL. 1− p If Z is the standard normal, find the value of b, if Pr ( Z  b ) = . 2 _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1 Question 5

(4 marks)

f ( x) f −x   ( ) dx. dx =  Show that  −a f ( x) + f ( − x) −a f ( x ) + f ( − x ) a

a.

Page 9

a

2 marks _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ b

b.

ex  Hence or otherwise evaluate  x − x dx giving your answer in simplest form. −b e + e

2 marks _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________ _______________________________________________________________________________

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1 Question 6

Page 10

(3 marks)

The diagram shows the shaded area bounded by the y-axis, the lines y = 1 and y = 3 and part of the graph of y =

1 − 3 x2 . x

The shaded area is rotated around the y-axis to form a solid of revolution. Find the volume, giving your answer in form

2 a where a and b are positive integers. b

________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1

Page 11

Question 7 (5 marks)

dy + 4 x 2 − x 2y 2 = 0 , y ( 2 ) = 1 dx Express y in terms of x for x  0.

Given the differential equation a.

3 marks ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ b.

Find the value of

d 2y x = 2 and y = 1. 2 when dx

2 marks ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1 Question 8

Page 12

(4 marks)

AOB is a triangle, where O is the origin. The coordinates of the points A and B are ( −2,2, −1) ,

( 2, m,2 ) respectively, where m is a real number. The angle between the vectors

OA and OB

 8 is cos−1  −  .  9 a. Find the value(s) of m.

2 marks ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ b.

When m is an integer, find the area of the triangle AOB.

1 mark ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ When m is an integer, find the magnitude of the vector resolute of OA perpendicular to OB . 1 mark ________________________________________________________________________________

c.

________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________

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2021 Kilbaha VCE Specialist Mathematics Trial Examination 1 Question 9 a.

Page 13

(4 marks)

Solve cos ( 2 x ) = cos ( x ) for x 0, 2  .

2 marks ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ b.

  3  Solve sec ( 2x )  sec ( x ) for x 0 ,   \  , ,  . 4 2 4

2 marks ________________________________________________________________________________ ________________________________________________________...