Abbotsleigh 2016 2U Trials & Solutions PDF

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Summary

HSC Maths 2u practice past papers...

Description

Student’s Name: Student Number: Teacher’s Name:

ABBOTSLEIGH

2016 HIGHER SCHOOL CERTIFICATE

Assessment 4

Mathematics General Instructions

Total marks - 100

 Reading time – 5 minutes

 Attempt Sections 1 and 2.

 Working time – 3 hours  Write using black pen.

Section I

Pages 3 - 6

 Board-approved calculators may be used.  A reference sheet is provided.  In Questions 11–16, show relevant mathematical reasoning and/ or calculations.  Make sure your HSC candidate Number is on the front cover of each booklet.

10 marks  Attempt Questions 1–10.  Allow about 15 minutes for this section.

Section II

Pages 7 - 15

 Start a new booklet for Each Question.  Answer the Multiple Choice questions on the answer sheet provided.  If you do not attempt a whole question, you must still hand in the Writing Booklet, with the words 'NOT ATTEMPTED' written clearly on the front cover.

90 marks  Attempt Questions 11- 16.  Allow about 2 hrs and 45 minutes for this section.

Outcomes to be assessed: Mathematics Preliminary Outcomes: P2 P3 P4 P5 P6 P7 P8

Provides reasoning to support conclusions which are appropriate to the context Performs routine arithmetic and algebraic manipulation involving surds, simple rational expressions and trigonometric identities Chooses and applies appropriate arithmetic, algebraic, graphical, trigonometric and geometric techniques Understands the concept of a function and the relationship between a function and its graph Relates the derivative of a function to the slope of its graph Determines the derivative of a function through routine application of the rules of differentiation Understands and uses the language and notation of calculus

HSC Outcomes: H1 H2 H3 H4 H5 H6 H7 H8 H9

Seeks to apply mathematical techniques to problems in a wide range of practical contexts Constructs arguments to prove and justify results Manipulates algebraic expressions involving logarithmic and exponential functions Expresses practical problems in mathematical terms based on simple given models Applies appropriate techniques from the study of calculus, geometry, trigonometry and series to solve problems Uses the derivative to determine the features of the graph of a function Uses the features of a graph to deduce information about the derivative Uses techniques of integration to calculate areas and volumes Communicates using mathematical language, notation, diagrams and graphs

Mathematics Task 4 2016

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SECTION I 10 marks Attempt Questions 1 – 10 Use the multiple-choice answer sheet Select the alternative A, B, C or D that best answers the question. Fill in the response oval completely. Sample

2 + 4 =

(A)

(A)

2

(B)

6

(B)

(C)

8

(C)

(D)

9

(D)

If you think you have made a mistake, put a cross through the incorrect answer and fill in the new answer. (B) (C) (D) (A) If you change your mind and have crossed out what you consider to be the correct answer, then indicate this by writing the word correct and drawing an arrow as follows. correct

(A)

(B)

(C)

(D)

_______________________________________________________________________________________

1.

Evaluate 4e 3  ln  2016  correct to 3 significant figures.

(A) 87.9 (B) 87.951 (C) 88.0 (D) 88.951 2.

The equation of the line passing through the point (0, 2) and perpendicular to the line 2 x  3 y  10 is: (A)

3x  2 y  4  0

(B)

3x  2 y  6  0

(C)

2x  3y  6  0

(D)

2x  3y  6  0

Mathematics Task 4 2016

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3.

Flora notices that her household expenses are increasing by $10.50 each month. If in July 2016 her expenses were $455, then her anticipated expenses for the month of August 2017 will be? (A) $465.50 (B) $570.50 (C) $581 (D) $591.50

4.

Which set of inequations represent the shaded region shown below?

y  x2  2x 1

x y4

(A)

x  y  4 and y  x 2  2x  1

(B)

x  y  4 and y  x2  2x  1

(C)

x  y  4 and y  x 2  2x  1

(D)

x  y  4 and y  x2  2 x  1

Mathematics Task 4 2016

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5.

Consider the function f (x ) 

x2 . x3

Which expression represents the largest possible domain for f ( x) ?

6.

(A)

x3

(B)

x3

(C)

x2

(D)

x2

A composite shape is made up of a parallelogram and an isosceles triangle, as shown in the diagram.

Which of the following is correct? (A) a  2b  180 (B)

2a  b  180

(C)

a  b  180

(D) b  2a

7.

If f ( x  1)  x 2  2 x  3 , then f ( x) is equal to? (A)

x2  2

(B)

x2  2

(C)

x2  2x  2

(D)

x2  2x  4

Mathematics Task 4 2016

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8.

The graph of y  kx  4 intersects the graph of y  x 2  2 x at two distinct points. Which of the following statements is true? (A) k 2  4k  16  0 (B)

k 2  4k  16  0

(C)

k 2  4k  12  0

(D) k 2  4k  12  0

9.

Consider the tangent to the graph y  x2 at the point  2, 4  . Which of the following lines is parallel to the tangent?

10.

(A)

x  4y 1 0

(B)

4x  y 1  0

(C)

y  4 x  6

(D)

y  2x  4

The limiting sum of 1 

(A)

3 5

(B)



(C)

3 2

2 4 8    ... is? 3 9 27

2 3

(D) 3

End of Section 1

Mathematics Task 4 2016

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SECTION II

90 marks Attempt Questions 11–16 Allow about 2 hours and 45 minutes for this section. Answer each question in the appropriate writing booklet. Extra writing booklets are available. In Questions 11–16, your responses should include relevant mathematical reasoning and/or calculations.

(15 marks) Use a SEPARATE writing booklet.

Question 11

(a)

Factorise fully 4 x3  108 .

(b)

Show that the derivative of

(c)

Find the derivative of y  2  tan 3x .

(d)

If y 

(e)

2

 x 3  e2x

is e 2x  2x  5 .

48 in the form k 3 where k is an integer.

(ii) Hence or otherwise, simplify

2

1

sin x 1 dy , show that .  1  cos x dx 1  cos x

(i) Express

Marks

48  2 27 , giving your answer in 6

3

1

2

simplified surd form.



(f)

4 Evaluate  cos 2x dx . 0

2

(g)

x2  dx . Find  3  x 6

2 End of Question 11

Mathematics Task 4 2016

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(15 marks) Use a SEPARATE writing booklet.

Question 12 (a)

Marks

The line l has the equation 7 x  3 y  13  0 and the points C and D are  1, 3  and

3   ,  1  respectively. 2 

NOT TO SCALE C

D

7x  3 y  13  0

(i) Find the gradient of line l.

1

(ii) Find the equation of the line which passes through C and is parallel to l.

2

(iii) The point A lies on l and D is the midpoint of AC. Find the coordinates of A.

2

(iv) Without finding the point of intersection, find the equation of the line which

3

passes through the point of intersection of 7 x  3 y  13  0 and

3x  2 y  12  0 and also passes through D.

(b)

The equation 4 x2  6 x  3  0 has roots  and  . (i) Write down the values of    and  .

3 (ii) Show that  2   2  . 4

2

(iii) Show that 3   3    

21 . 4

(iv) Explain why 4 x 2  12 x  21  0 has the roots 3    and 3    .

End of Question 12

Mathematics Task 4 2016

2

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2

1

(15 marks) Use a SEPARATE writing booklet.

Question 13

(a)

Marks

The curve with the equation y  x3  2 x 2  3 is shown in the diagram.

NOT TO SCALE

The curve cuts the x-axis at the point A ( 1, 0) and passes through the point

B (1, 2) . 1

 3 2 (i) Evaluate  x  2 x  3 dx . 1

(ii) Hence, find the area of the shaded region bounded by the curve

2

1

y  x 3  2 x 2  3 and the line AB.

(b)

The displacement x metres from the origin at time t seconds, of a particle travelling in a straight line is given by the formula x  t 3  21t 2 . (i) Find the acceleration of the particle at time t seconds.

(ii) Find the time(s) at which the particle is stationary.

(c)

2

2

The volume, V m3, of water in a tank after time t seconds is given by the equation

V  3ln(2t 1)  5t  2 (i) Find

dV . dt

2

(ii) Explain why the volume of water is decreasing at t  1 . Question 13 continues on the next page. Mathematics Task 4 2016

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2

Question 13 continued. (d)

There are plans to construct a series of straight paths on the flat top of a mountain. A straight path will connect the cable car station at C to a communication tower at T, as shown in the diagram below. The bearing of the communication tower to the cable car station is 060 . The length of the straight path between the communication tower and the cable car station is 950 m.

Paths will also connect the cable car station and the communication tower to the camp site at E. The length of the straight path between the cable car station and the camp site is 1400 m. The angle TCE is 40 . (i) Calculate the length of the path between the communication tower and the

2

camp site, correct to the nearest metre.

(ii) Find the bearing of the camp site from the communication tower, correct to the nearest degree.

End of Question 13

Mathematics Task 4 2016

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2

(15 marks) Use a SEPARATE writing booklet.

Question 14 (a)

Marks

The area bounded by the curve

y

3

1  x and the lines x  1 and x

x  3 is rotated about the x-axis. Find the volume of the solid of revolution formed. Leave your answer in exact form.

(b)

A particle moves in a straight line. At time t seconds its displacement x metres from a fixed point O on the line is given by x  3  cos 2t , 0  t  2 .

(i) Sketch the graph of x as a function of t, showing all the important features.

2

(ii) Explain how you can use your graph to determine the times that the particle

1

is at rest. (iii) Find the time when the particle first reaches its maximum speed.

(c)

2

A company is designing a new NOT TO SCALE

logo in the shape of a circle with a small segment taken out as shown to the right. The radius of the circle is 4 cm and the length of AB is also 4 cm. (i) Explain why AOB 

 3

1

.

(ii) Find the area of the logo correct to 3 significant figures.

Question 14 continues on the next page.

Mathematics Task 4 2016

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2

Question 14 continued.

(d)

The cubic function y  ax3  bx 2  c where a, b and c are real constants with a  0 is shown in the diagram. The derivative of this function is f '( x)  3ax2  2bx .

NOT TO SCALE

Two tangents are drawn to this function such that their equations are: 

y  3x  3 at the point 1, 0  and



y  9 x  25 at the point 3, 2  .

(i) Show that 9a  2b  3 and 3a  2b  3 . (ii) Hence find the values of a, b and c.

End of Question 14

Mathematics Task 4 2016

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2

2

(15 marks) Use a SEPARATE writing booklet.

Question 15 (a)

Marks

The rate of increase of a population P t  of people in a certain country is

dP  kP and P  P0e kt , where k is a constant, P0 is dt the original population and t is the time in years. The population of the country determined by the equations

doubles every 20 years. (i) Show that k 

1 ln 2 . 20

2

(ii) Data is first collected about the population of this country in the year 2000.

2

In which year will the country reach a population three times that it had at the beginning of 2000? (iii) Given that at the beginning of the year 2000 the population was 15.1 million,

2

what will be the population of the country at the beginning of the year 2050? Give your answer correct to 3 significant figures.

(b)

On the 1st of January 2016 the population of a particular country town was 30 000. At the end of each year 2500 people leave the town to live in the city. During the period between January and the people leaving in December each year, the population increases by 5%. (i) Show that the number of people in the country town just after the first

1

group of 2500 left in December 2016 is 29 000. (ii) Show that the expression for the number of people in the country town

1

just after the second group of 2500 left in December 2017 is given by P2  30000  1.05 2  2500(1.05  1) .

(iii) Show that Pn , the population after the nth group left is given by

2

Pn  30000  1.05n  50000 (1.05 n  1) .

(iv) Hence, determine in which year the population of the town will be zero.

(c)

A parabola has its focus at 5, 1  and vertex at 3, 1  . Show that the equation of the parabola is y  2 y  8x  25  0 . 2

End of Question 15 Mathematics Task 4 2016

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3

2

Question 16 (a)

(15 marks) Use a SEPARATE writing booklet.

Marks

An irrigation channel has a cross-section in the shape of a trapezium as shown in the diagram. The bottom and sides of the trapezium are 4 metres long. Suppose that the sides of the channel make an angle of  with the horizontal where  

 . 2

(i) Show that the cross-sectional area is given by A 16 sin   cos  sin  .

(ii) Show that

dA  16  2cos 2   cos   1 . d

(iii) Hence, show that the maximum cross-sectional area occurs when  

2

 . 3

(iv) Hence, find the maximum area of the irrigation channel, correct to the nearest square metre.

Question 16 continues on the next page.

Mathematics Task 4 2016

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2

3

1

Question 16 continued.

(b)

When a radial arm saw (as shown on the right) is used, its cutting edge (as indicated by the dot) moves forwards and then backwards along a straight line. During a particular cutting procedure, the velocity of the cutting edge of the saw, in metres per second, can be modelled by the function

v (t )  0.05t 3  0.38t 2  0.624t ,

where t represents the time in seconds from the start of the cutting procedure and

0  t  5.2 .

(i) For what values of t is the cutting edge of the saw at rest?

2

5.2

 (ii) Calculate  0

v(t ) dt , correct to 3 decimal places.

2

(iii) Interpret your answer to part (ii) in the context of the motion of the cutting

1

edge of the saw.

(iv) Write an expression to find the total distance travelled by the cutting edge of the saw during the cutting procedure. (There is no need to evaluate this).

End of Paper

Mathematics Task 4 2016

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2

BLANK PAGE

Mathematics Task 4 2016

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Abbotsleigh 2 unit Mathematics Task 4 2016 Solutions: Question 1

2

Working 88.0

Solution C

2x  10 3 2 m 3 y

 m  line  

3 2

A

3 Equ of line is y   x  2 2  2 y  3x  4  0 3

a  455, d  10.50, n  14 T14  455  13  10.50  $591.50

4

Test pointt (0,0) x y 4 y  x 2  2 x 1

5

x3 0 x 3

D

D

A

6

B

180  b 2 360  2a  180  b 2a  b  180

180  a 

7

Trial and error for B f ( x -1)  ( x  1) 2  2  x2  2 x  1  2  x 2  2x  3 OR sub in (x  1)

B

(x  1)2  2( x  1)  3  x2  2x  1  2x  2  3 x 2 2

8

Solve y  kx - 4   y  x2  2 x kx  4  x 2  2 x x2  x(2  k )  4  0 For two distinct roots,   0

C

(2  k )2  4(1)(4)  0 4  4 k  k 2  16  0 k2  4 k  12  0 9

y  x2 y'  2x At x  2, y '  4 m of tangent = 4  parallel line has same gradient ie, 4 x  y  1  0

B

10

a  1, r  

2 3

a 1 r 1   2 1     3 3  5

S 

A

Question 11(a)

Working

Marks

4 x  108 3

2

 4  x3  27   4  x  3 x2  3 x  9 11(b)

RTShow: derivative of ( x - 3) e2 x is e2 x (2 x  5) Proof: d 2  ( x  3)2 e2x  e2x (1) ( x - 3) e x   dx  e2x  2( x  3)  1

2

 e 2 x  2 x  6  1  e2x (2x  5) as required 11(c)

y  tan 3 x  2 1

dy  3sec2 3x dx 11(d)

RTShow: If y 

dy sin x dy 1  then dx 1  cos x 1  cos x dx

Proof: dy (1  cos x )(cos x )  sin x (  sin x )  dx 1  cos x 2   

11(e)(i)

3

cos x  cos 2 x sin 2 x

1  cos x 

2

1 cos x

1 cos x 

2

(since cos 2 x  sin 2 x  1)

1 1  cos x as required

48  16  3 4 3

1

11(e)(ii)

48  2 27 6 4 3  23 3  6

2

10 3 6 10  2     2  2  

10 2 2 5 2 

11(f)



4  cos 2x dx 0 

1 ...