mtm315109 part 1 sample 2010 - tqa.tas.gov.au · 6 use integral calculus in the study of functions....

Pages: 16 Questions: 20 ©Copyright for part(s) of this examination may be held by individuals and/or organisations other than the Tasmanian Qualifications Authority.

Tasmanian Certificate of Education

MATHEMATICS – METHODS

Senior Secondary

Subject Code: MTM31509

FINAL Sample External Assessment

2010

Part 1 Calculators are NOT allowed to be used

Time: 80 minutes

On the basis of your performance in this examination, the examiners will provide a result on the following criteria taken from the syllabus statement:

Criteria Description For Marker Use Only

3 Demonstrate an understanding of polynomial, hyperbolic, exponential and logarithmic functions.

4 Demonstrate an understanding of circular functions.

5 Use differential calculus in the study of functions.

6 Use integral calculus in the study of functions.

7 Demonstrate an understanding of binomial, hypergeometric and normal probability distributions.

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FINAL Sample Paper 2010 – Mathematics – Methods (Part 1 – No Calculators)

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CANDIDATE INSTRUCTIONS You MUST address all the externally assessed criteria in this examination paper. Calculator(s) MUST be placed on the floor to the right hand side of your desk before reading time commences. Calculator(s) CANNOT be used during reading time nor during the 80 minutes allocated to complete Part 1 of the examination paper. You may commence Part 2 during this time but are not permitted to use a calculator. Part 1 will be collected after 80 minutes (the time allocated to complete this part). There will be a 10 minute break while this occurs and you are NOT permitted to work on the exam during this time. The exam supervisors will instruct you when you are permitted to use your calculator(s). You will have 100 minutes to complete Part 2 and calculators CAN be used during this time. You may use your calculator without restriction (to its full capacity) during Part 2 of the examination. The 2010 Mathematics Methods Information Sheet provided can be used throughout the examination. No other printed material is allowed into the examination. For questions worth 1 or 2 marks, working does not need to be shown. For questions worth 3 or more marks, you are required to show relevant working. Answers and any working MUST be written in the spaces provided on the examination paper. You are expected to provide a calculator(s) as approved by the Tasmanian Qualifications Authority. All written responses must be in English.


Section A

Answer ALL questions in this section. This section assesses Criterion 3. Question 1 For the function shown on the right: (a) Write down the domain. (1 mark) ............................................................... (b) Write down the range. (1 mark) ............................................................... (2 Marks) Question 2 Given that and , evaluate . (2 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

Section A continues opposite.

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Section A (continued) Question 3 The graph of a function g(x) is shown below. Sketch a graph of the function on the same axes. (3 marks)

Question 4 Solve for x the equation . (5 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Section B

Answer ALL questions in this section. This section assesses Criterion 4. Question 5 Put appropriate scales on both the x and y axes given the graph below is of the trigonometric

function . (2 marks)

Question 6

The function g is defined by the equation for all real x.

(a) Determine the minimum value of g. (1 mark) ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. (b) Determine the period of g. (1 mark) ............................................................................................................................................. ............................................................................................................................................. .............................................................................................................................................

Section B continues opposite.

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Section B (continued) Question 7

If , use symmetry properties to find the exact value of

. (3 marks) ....................................................................................................................................................... .......................................................................................................................................................

....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

....................................................................................................................................................... Question 8

Solve for x the equation , giving all exact solutions over the domain .

(5 marks) ....................................................................................................................................................... .......................................................................................................................................................

....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Section C

Answer ALL questions in this section. This section assesses Criterion 5. Question 9 Find the derivative of with respect to x. (2 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 10

Find the equation of the tangent to the curve at the point . (2 marks)

....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Section C (continued) Question 11 Determine, from first principles, the derivative of the function . (3 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 12

Determine the derivative of , where a and b are constants and , and hence

explain why the function has no stationary points. (5 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Section D

Answer ALL questions in this section. This section assesses Criterion 6. Question 13 Determine the indefinite integral for . (2 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 14

If , what is ? (2 marks)

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Section D (continued) Question 15

If a > 0, determine the two smallest values for a if . (3 marks)

....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 16

Differentiate and hence, find the value of dx (5 marks)

....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Section E

Answer ALL questions in this section. This section assesses Criterion 7. Question 17 The diagram below shows two normal probability distribution functions comparing the results achieved on a Methods test for year 11 and year 12 students.

Compare the means and variances for the year groups. (2 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 18 A spinner is constructed as shown. When the arrow is spun about the centre, it is equally likely to stop at any position. If the arrow is spun many times, what is the expected value of the number indicated by the arrow? (2 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Section E (continued) Question 19 The discrete random variable, X, has a probability distribution as given in the table below:

x 0 2 4 6 8 Pr(X=x) a 0.1 0.3 0.2 b

Write down two equations involving a and b, that could be used to find the values of a and b. (Solutions of the equations not required) (3 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 20 A binomial distribution graph for is provided below. (5 marks) (a) Show the effect of altering p from 0.35 to 0.65 by completing a binomial distribution

graph on the right-hand grid below.

(b) For the binomial distribution graph of

€

X ~ Bi (6, 0.35) shown above (left), describe what would change and what would stay relatively the same if n equals 10 so that

. ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Pages: Questions: 20 ©Copyright for part(s) of this examination may be held by individuals and/or organisations other than the Tasmanian Qualifications Authority.

Tasmanian Certificate of Education

MATHEMATICS – METHODS

Senior Secondary

Subject Code: MTM31509

FINAL Sample External Assessment

2010

Part 2 Calculators are allowed to be used

Time: 100 minutes

On the basis of your performance in this examination, the examiners will provide a result on the following criteria taken from the syllabus statement:

Criteria Description For Marker Use Only

3 Demonstrate an understanding of polynomial, hyperbolic, exponential and logarithmic functions.

4 Demonstrate an understanding of circular functions.

5 Use differential calculus in the study of functions.

6 Use integral calculus in the study of functions.

7 Demonstrate an understanding of binomial, hypergeometric and normal probability distributions.

TA

SM

AN

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ION

S A

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HO

RIT

Y

FINAL Sample Paper 2010 – Mathematics – Methods (Part 2 – Calculators)

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CANDIDATE INSTRUCTIONS You MUST address all the externally assessed criteria in this examination paper. Calculator(s) MUST be placed on the floor to the right hand side of your desk before reading time commences. Calculator(s) CANNOT be used during reading time nor during the 80 minutes allocated to complete Part 1 of the examination paper. You may commence Part 2 during this time but are not permitted to use a calculator. Part 1 will be collected after 80 minutes (the time allocated to complete this part). There will be a 10 minute break while this occurs and you are NOT permitted to work on the exam during this time. The exam supervisors will instruct you when you are permitted to use your calculator(s). You will have 100 minutes to complete Part 2 and calculators CAN be used during this time. You may use your calculator without restriction (to its full capacity) during Part 2 of the examination. The 2010 Mathematics Methods Information Sheet provided can be used throughout the examination. No other printed material is allowed into the examination. For questions worth 1 or 2 marks, working does not need to be shown. For questions worth 3 or more marks, you are required to show relevant working. Answers and any working MUST be written in the spaces provided on the examination paper. You are expected to provide a calculator(s) as approved by the Tasmanian Qualifications Authority. All written responses must be in English.


Page 4

Section A

Answer ALL questions in this section. This section assesses Criterion 3. Question 21

Sketch the graph of on the grid provided below, clearly labelling any

asymptotes, the y intercept and the zeros. (2 marks)


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Page 5

Section A (continued) Question 22 The graphs of an absolute value function and a square root function are shown below: (4 marks) (a) Determine the equation of the function . ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. (b) Determine the equation of the function .

............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. .............................................................................................................................................

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Page 6

Section A (continued) Question 23 A graph of the function is shown. Determine a possible equation for the inverse function, . (4 marks) ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. .............................................................................................................................................


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Page 7

Section A (continued) Question 24 A metal ball is heated to 98˚C and then allowed to cool towards room temperature (18˚C). Its temperature T˚C after cooling for t minutes is given by the equation , where c and k are constants. When the ball has cooled for 5 minutes, its temperature is 48˚C. (6 marks)

(a) Determine the values of c and k. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. (b) Predict the temperature of the ball after it has cooled for 22 minutes. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. .............................................................................................................................................

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Page 8

Section B


Find the number of solutions to the equation for . (2 marks) ....................................................................................................................................................... ....................................................................................................................................................... Question 26

Determine the exact values of all solutions of the equation

over the interval . (4 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 27

If and find, in terms of a, the value of . (4 marks)

....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Page 9

Section B (continued) Question 28 The depth of water at the entrance to a harbour hours after low tide is D metres and can be modelled by the equation for the constants a, b, c and d. At , it is low tide and the depth is 2 metres. At the next high tide, 6 hours later, the depth is 8 metres. (6 marks) (a) Find exact values for a, b, c and d, given that all constants are positive. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. (b) Hence, write and solve an equation to find how soon after high tide vessels requiring a

depth of at least of water will not be able to enter the harbour.

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Page 10

Section C


Find all stationary points of . (2 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 30

Sketch a possible graph of given the following information. (4 marks)

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Page 11

Section C (continued) Question 31

A graph of the function , for constant a, has a local maximum at P and a local minimum at Q (see the diagram). Determine the coordinates of P and Q in terms of a. (4 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Page 12

r

20 cm

h

30 cm

Section C (continued) Question 32 A cylinder of radius r and height h is inscribed in a right circular cone of radius 20 cm and height 30 cm, as shown in the diagram.

Given that , use calculus to determine the value of r for which the cylinder has maximum volume. Justification of the maximum must be included. [Volume of cylinder = ] (6 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Page 13

Section D


Evaluate dx correct to 3 decimal places. (2 marks)

....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 34

Velocity, v, is the instantaneous rate of change of displacement, s, that is, .

The velocity (in metres per second) of a vehicle t seconds after the brakes are applied is given by the equation . How far does the vehicle travel in the 2.2 seconds after the brakes are applied? (4 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Page 14

Section D (continued) Question 35 In the graph shown, B and O are points on the curve: If the point A has coordinates (x, 0), and the point B has co-ordinates

€

x, x( )

Show that the area of the shaded region is the area of the rectangle OABC. (4 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Page 15

Section D (continued) Question 36

In the domain , the curves and intersect when (see diagram). (6 marks) (a) Write down an expression using integrals

for the area bound between the curves on the interval .

................................................................. ................................................................. ................................................................. ................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. (b) Use a calculator to determine the values of a and b correct to 3 decimal places and hence

evaluate the area enclosed between the curves on the interval correct to 3 decimal places.

............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. .............................................................................................................................................

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Section E

Answer ALL questions in this section. This section assesses Criterion 7. Question 37 It is known that the probability of a particular drug causing side-effects in a person is 0.25. What is the probability that at least two of a random sample of 12 people taking the drug will experience side-effects? [Express answer correct to 2 decimal places]. (2 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... Question 38 A barrel contains six green marbles and four yellow marbles. A single marble is drawn from the barrel, its colour noted, and it is then returned to the barrel. The contents of the barrel are then thoroughly mixed before another marble is drawn out. This process is repeated until a total of twelve draws is made. What is the probability that a green marble is drawn at least nine times? (4 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Section E (continued) Question 39 A trick die has five of its six faces numbered 2, 3, 5, 8 and 12, and all six faces are equally likely to be uppermost when the die is rolled. What possible number(s) could be on the sixth face if the variance of the outcomes of rolling the die is 16? (4 marks) ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... ....................................................................................................................................................... .......................................................................................................................................................

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Section E (continued) Question 40 Anna’s engineering firm has imposed acceptable limits on the lengths of steel rods made at 1.930 cm and 2.080 cm. It is observed that, on average, 4% of rods are rejected as undersized and 5% of rods are rejected as oversized. (6 marks) Assuming that the lengths are normally distributed: (a) Complete the normal distribution sketch below, marking in areas, lengths and

probabilities for the undersized and oversized steel rods. (b) Find the mean and standard deviation of the distribution (correct to 4 decimal places).

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mtm315109 part 1 sample 2010 - tqa.tas.gov.au · 6 use integral calculus in the study of functions....

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