Mathematical Methods

2005

Trial Examination 1

2

Part I Multiple-choice questions (27 marks) Question 1 The function represented by the graph on the right is y

A. ( ) bxabxfRRf +=→ ,: a 0 x

B. [ ] ( ) bxabxfRaf −−=→− ,0,:

C. [ ] ( ) bxabxfRaf +−=→ ,0,:

D. [ ] ( ) bxabxfRbaf +−=→ ,,: b

E. [ ] ( ) bxbaxfRbf +=→ ,,0:

Question 2 The quartic represented by the graph on the right has y A. 2 turning points and 1 inflection point B. 1 turning point and 2 inflection points C. 1 turning point and 1 inflection point D. 2 turning points and 2 inflection points E. 2 stationary points and 1 inflection point 0 x Question 3 Which one of the following correctly describes the asymptotic behaviour of

32

1+

+−

=x

y , 2−<x ?

A. As −∞→x , 0→y B. As 2−→x , −∞→y C. As 2→x , +∞→y D. As +∞→x , 3→y E. As −∞→x , 3→y Question 4 The domain and range of ( ) ( ) BbxAxf e ++= log are respectively A. [ )∞,b , ( ]B,∞− B. ( )∞− ,b , R C. [ )∞− ,b , R D. ( )∞,b , ( )B,∞− E. R, ( )∞,b

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Question 5 The graph of ( )xfy = is shown on the right. y • 2 –1 0 x Which one of the following is the graph of ( )( )21 −−−= xfy ? y A. y B. y C. 0 x 0• x • 0 x –4 • –2 D. y E. y –2 0 x 0 • x • –4 y Question 6 The graph of the inverse function 1−f is shown on the right. 0 x Which one of the following is the graph of the original function f ? A. y B. y C. y 0 x 0 x 0 x D. y E. y 0 x 0 x

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Question 7 The graph of ( )( ) BbxaAy ++= cos , for a

x π20 ≤≤ and +∈ RBAba ,,, , cuts the x-axis more

than once if

A. BA < B. ba < C. BA > D. ba > E. 1≥BA

Question 8 The number of solutions of the equation axax cossin −= , where π≤≤ x0 and a is a positive interger, is A. a B. 2a C. 0 D. 1 E. 2 Question 9 The derivative of ( )2log axe is

A. x2 , +∈ Rx B.

ax2 , +∈ Rx C.

ax2 , { }0\Rx∈ D.

x2 , { }0\Rx∈ E. undefined

Question 10 The graph of ( )xf is shown on the right. y 0 x The graph of ( )xf ′ is most likely to be A. y B. y C. y 0 x 0 x 0 x D. y E. y 0 x 0 x

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Question 11 Given ( ) ( )xfexg x−= , the point ( )( )aga, is a stationary point if A. ( ) ( )afaf −=′ B. ( ) ( )afaf =′ C. 0=−ae D. 1=−ae E. ( ) 0=′ af

Question 12 The best approximation of the gradient of the tangent to the curve ( )xxxxy

sin2sin−

+=

π at π=x is

A. – 0.127 B. – 0.125 C. – 0.120 D. 1.25 E. 1.27 Question 13 For the function ( ) 21cos2 xexxf −−= , the rate of change with respect to x is positive in the interval A. [ ]1,1− B. ( )1,1− C. [ ]1,0 D. [ )1,0 E. ( )1,0

Question 14 Using ( ) ( ) ( )afhafhaf ′+≈+ , the estimated value of 5

sin π in exact form is

A. 60

330 π+ B. 20

310 + C. 1000683 D.

625427 E.

10059

y Question 15 The graph of ( )xfy = is shown on the right. 0 x Which one of the following is a possible graph of ( )dxxfy ∫= ?

A. y B. y C. y 0 x 0 x 0 x D. y E. y 0 x 0 x

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Question 16 y 0 x

The curve shown above is the graph of ( )2416 −−= xy . Using right rectangles of 2-unit width for approximation the area between the curve and the x-axis in exact form is A. ( )328 + B. ( )318 + C. ( )3342 + D. 25 E. π8

Question 17 Given ( ) ( )xFxf ′= , ( )dxxfa

∫0

is equal to

A. ( ) ( )aFF ′−′ 0 B. ( ) ( )0FaF ′−′ C. ( ) ( )aFF −0 D. ( ) ( )0FaF − E. ( )aF Question 18 An antiderivative of ( ) ( )21 11 +−+ − axax , where 01 >+ax , is

A. ( )3

1 3+ax

B. ( )a

ax3

1 3+

C. ( ) ( )a

axaxe 311log

3+−+

D. ( ) ( ) a

aax

aaxe 2

311log 3

++

−+

E. ( ) ( )

aax

aaxe

311log 3+

−+−

Question 19 Given ( ) cdxxf =∫1

0

where +∈Rc and ( ) ( ) 112 −+= xfxg , the value of ( )dxxg∫−

0

1

is equal to

A. 2−c B. 2+c C. 12 −c D. 12 +c E. 1−c Question 20 The solution(s) of ( ) ( ) 07log7log =−++ xx ee is/are A. – 7, 7 B. 0 C. 25− , 25 D. 25− E. 25

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Question 21 If ( ) 221 1 += −xexf and 1≥x , then the inverse of ( )xf is

A. ( ) ( ) 12log1 +−=− xxf e , 2>x

B. ( ) 22

1log1 +−

=− xxf e , 1>x

C. ( ) ( ) 122log1 +−=− xxf e , 5.2≥x D. ( ) ( ) 122log1 +−=− xxf e , 2>x

E. ( ) 22

2log1 +−

=− xxf e , 2>x

Question 22 The seventh row in Pascal’s triangle is “1 6 15 20 15 6 1”. The entry 15 is a factor of the coefficient of the A. 3x term in the expansion of ( )7bax + B. 3x term in the expansion of ( )6bax + C. 3x term in the expansion of ( )5bax + D. 2x term in the expansion of ( )7bax − E. 2x term in the expansion of ( )6bax − Question 23 Which one of the following is a discrete random variable? A. Getting 2 tails in tossing a coin 5 times B. Volume of soft drink in an unopened 375ml can C. The sum is 5 when 2 dice are rolled at the same time D. The number of times that the sum is 5 in rolling 2 dice together 5 times E. The chance that your answer is correct Question 24 The following table shows a probability distribution.

x 0 1 2 3 4

( )xX =Pr 71

61

51 p

31

The values of p and ( )XE are respectively

A. 7011 ,

3583 B.

41 , 2 C.

7011 , 2 D.

41 ,

2053 E. 0.16, 2

Question 25 In rolling a fair die once the mean and variance of the number of times that 5 occurs are respectively

A. 21 ,

365 B.

65 ,

65 C.

65 ,

365 D.

61 ,

65 E.

61 ,

365

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Question 26 A bag contains 3 red, 4 blue and 5 green marbles. 6 marbles are randomly taken out at the same time. The probability that 3 of them are red is

A. 1 B. 21 C.

41 D.

91 E.

111

Question 27 pf pf 0.95 0 3.5 7.5 11.5 X – 1.3 0 Z The above graphs show the normal distribution of random variable X and the standard normal distribution. The value of X that corresponds to 3.1−=Z is A. 2.3 B. 4.4 C. 4.9 D. 10.1 E. – 4.4 Part II Short-answer questions (23 marks) Question 1 The cubic function ( ) xxxxf 662 23 +−= can be changed to the form ( ) ( ) cbxaxf +−= 3 . a. Find the values of a, b and c.

b. Hence sketch the graph of ( ) ( )( )2bx

cxfxg−−

= .

2 + 2 = 4 marks Question 2 Consider the quadratic function ( ){ }1:, 2 += xyyx . a. State why the inverse function of ( ){ }1:, 2 += xyyx does not exist? b. Write down the inverse of ( ){ }1:, 2 += xyyx . c. Write down a function that is a subset of the inverse of ( ){ }1:, 2 += xyyx . d. Sketch the graph of the function in part c. 1 + 1 + 1 + 1 = 4 marks Question 3 a. Without using calculator solve 02112 =−− ++ eee xx for x. Write answer in exact form. b. Use calculator to solve xee x 22 = for x, correct to 3 decimal places. 2 + 1 = 3 marks

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Question 4

Imagine the graph of the function with equation ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛= 1

43sin

23 xy π ,

3380 ≤≤ x .

a. Calculate the exact distance between the maximum and minimum points of the function. b. Find the exact x-coordinate of a point of the function where the gradient of the tangent is equal to the

gradient of the line segment joining the maximum and minimum points. 2 + 2 = 4 marks Question 5 The following diagram shows the graph of the function with equation xy elog= . y 0 1 e x a. On the diagram above sketch accurately the graph of the inverse of the function with equation xy elog= .

Write down the equation of the inverse.

b. Hence use calculus to find the exact value of dxxe

e∫1

log .

2 + 2 = 4 marks Question 6 a. If you picked all your answers randomly in Part I: Multiple-choice questions, what would be the

probability (3 decimal places) that more than a third of the total number of questions were correct? b. Give a reason why the normal distribution approximation would or would not be a good method in finding

the probability in part a. Justify your answer by finding the probability (3 decimal places) using normal approximation.

2 + 2 = 4 marks

End of trial examination 1