ib hl math nov 95_p2

8/7/2019 IB HL Math Nov 95_P2

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INTERNATIONAL BACCALAUREATEMATHEMATICS

H igher Leve lT ue sd ay 1 4 Nov ember 1 99 5 (mo rn in g)

Paper 2 2 hou rs 3 .0m i nu te s

T his e xam in atio n p ap er C on sists o f 2 s ec tio ns , Sectio n A a nd Sec tio n B .Sec tio n A c on sis ts o f 4 q ue stio ns .Sec tio n B co ns ists o f 4 q ue stio ns .T he m axim um mark for S ection A is 80.T he m axim um mark for each question in S ection B is 40.T he m axim um mark for th is paper is 120.T his e xam in atio n p ap er c on sis ts o f 1 3 p ag es .

INSTRUCTIONS TO CANDIDATESDO NOT open this examination paper until instructed to do so.Answer all FOUR questions from Section A and ONE question fromSection B.Unless otherwise stated in the question, all numerical answers mustbe given exactly or to three Significant figures as appropriate.

EXAMINAT ION MATERIALS.Required/Essential:

IB S tat is tic a l TablesM illime tre s quar e g raph paperElectronic calculatorRu le r and compas sesAllowed/Optional:

A s imp le tra ns la tin g d ic tio na ry fo r c an did ate s n ot w ork in g in th eir own la ng ua ge

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FORMULAE

Trigonometrical Identities; . sin(a + {3)= sinacos {3+cosasin{3cos(a + 1 3 ) = cosa cos 1 3 - sina sin f J

tana+tanRtan(a + {3)= ---.....:p~1-tanatan{3. . R 2' a+f3 a-f3sma+smp= sm--cos--2 2. . {3 2 a+f3. a-f3sma- sm = cos --- SIn--

2 2cosa + cos 1 3 = 2cos a + f3cos a - 1 3. 2 2

J 3 2 a+f3 . f3-acosa-cos = sm--sm--2 2cos 28 = 2cos2 8 -1= 1-2sin2 8 = cos2 8 - sin2 (j

8 . 2t I-t2If tan- =t thensm8 =-- and cos8=--2 1+ t2 1+ t2

Integration by parts:

Standard integrals:

Statistics: If (xt ' x2 ' ,xn) occur with frequencies ift ,J;, ... ,I,,) then the mean m andstandard deviation s are given by!./;x.m=-'-'!./; s=

!./;(X;-mi!./; i=1,2, ... ,n

Binomial distribution: r, = ( : ) pX(l_ pt-x, x = 0, I, 2, ... .nSA22SS282


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A correct answer with DO indication of the method used will normally receive DO marks. You aretherefore advised to show your working.

SECI10NA.Answer all FOUR questions from this section.

1. [Maximum mark: 18](i) For a function y, d"y represents the nth derivative of y with respect to x .dx"

d4 .(a) If y = e X sin x, show that - - - f = -4ydx..(b) Hence, find the' expression for d8~ in terms of y, and conjecture adx

d4 1 1formula for .~, n e N*.dx(c) Use mathematical induction to prove your conjecture.

(ii) A rectangular fish tank, of width w, length I and height h as shown, is fullof water and is tipped along its length by an angle 8 to the horizontal.

h

wh w 21tan8(a) If tan 8 < -, show that the volume of water spilled is ---w 2

(b) Find an expression for the volume spilled when tan 8 > ! ! _ .W

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2. [Maximum mark: 20](i) Consider the two lines LI and L2 .

L1: x-I y-3 z -1--=--=--2 3 2L2 : 3-x 2y-3 z+l--=--=--4 3 2

-+ -+ -+(a) Write the vector equation of each line in the form r = ro + A . U (b) Show that the two lines do not intersect, and state whether or not theyare parallel.(c) Find, in the form ax + by + cz = d , the equation of a plane that isperpendicular to L2 and intersects LI .(d) Find a vector that is perpendicular to both lines.(e) Hence or otherwise, find the ~tance, between L) and ~ .

(ii) An aeroplane is flying in a straight line at an altitude of SOOOm. It passesover a point on the ground that is 6000m due south of a radio beacon atground level, and then over a point that is 8000m due east of the beacon.By taking a coordinate system whose origin is at the beacon, derive a vectorin the direction of the Bight path.Hence, or otherwise, calculate how close the aeroplane gets to the beacon.

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3. -[Maximum mark: 18J

. (i) A continuous probability distribution is defined as follows:

{0,

p(x)= _1_2,l+x0,

: x < 0,O :e ; ; x : e ; ; k ,

x z - k .

Find k , the mean u , and the standard deviation a of p(x) . [8 marks](ii) Two versions of an electric heater are made, one with four heatingcomponents and one with two heating components. There is a probability

q that any component will fail when a heater is switched on, independent ofthe other components. Heaters will not operate ifmore than half of theircomponents fail.(a) Deduce that the probability that the two component heater will'operate is I - q 2 and derive an expression for the probability that thefour component heater will operate .

. (b) What values of q makes the.heaters equally reliable?(c) What values ofq will make the two component heater more reliablethan the four component heater? [10 marks]

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4. .[Maximum mark: 24J(i) (a) Sketch the region enclosed between the curves of y = ~ ,y = 6 - xand y = c, where I . s ; ; c . s ; ; 2 .

(b) Find, in terms of c , the area of this region.(c) Verify that your answer to (b) is correct by considering the case when

c = 2. [12marks](ii) A manufacturer of cans for soft drinks is asked to supply cans which hold

500 em3 (one half of a litre) of the drink. The cans, when full, are closedcylinders made from very thin sheet metal. For economic reasons thedimensions of the can are such that the amount of sheet metal used tomake each can is a "minimum.(a) Denoting the .radius and the height of a can by r em and h e m

respectively, write down an expression for A , the area of sheet metal.required to .make one can.

(b) Express A in terms of r only and deduce that your expression has nomaximum value.

10(c) Show that the radius of the required can is V;; em, and find theheight of the can. [12 marks]

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SECTIONBAnswer ONE question from this section.

. Abstract AJgebra.5. [Maximum mark: 40J

(a) Let M be the set of 2 x 2 matrices {I, A, B, C, D, E} where I is the identitymatrix and the other matrices are:A = ( 0 1 ) ; B = ( 0 1 ) ; C = ( - 1 - 1 ) ; D = ( - 1 - 1 ) ; E = ( 1 0 ) .1 0 - 1 - 1 0 1 1 0 - 1 - 1

Given that matrixmultiplication is associative show, by presenting theoperation table, that M is a group under matrix multiplication. (Use theletters I, A, B, C, D andE for the elements of the table.)Show by example that the group is not abelian.

(b) Let S3 be the set of all permutations of the elements of the set {I, 2, 3} .T~o of the elements of S3 are thus:

( 1 2 3 )P2=231'Write down the remaining elements, denoting the elements byPi' i= 3,4 ...Given that S3 is a group under the operation of composition, write downthe group table. (Again, use letters Pi for the elements.)

(c) Explain what is meant by saying that two groups are isomorphic.Show, by finding an isomorphism, that the groups in (a) and (b) areisomorphic.

[Question 5 continues on the next page J

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[Question 5 continued](d) . Let S" be the set of all permutations of the set {I, 2, 3, ... , n} where

n;;'3.By considering the two elements of S"

_ ( 1 2 3 . . . ) d _ ( 1 2 3 . . . )SI- an ~-132... 321 ...in which each number after 3 is unchanged, deduce that S" under theoperation of composition is not abelian. .Hence verify that the group in (a) is not abelian.

. (e) Using either of the group tables in (a) and (b), find an example to suggestan expression for the inverse of (x * y) in terms. of the inverse of x and y ,where x and y. are group elements and denotes the relevant operation.Prove that your result is true for any group.

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Grap. and. Trees6. [Maximum mark: 40J

(i) Nine intersections in the road system of an American city are connected by. unpaved roads. The distances between the intersections, in kilometres, aregiven in the diagram below.

3D~----~----'_------~--~~F

4

H IGWhat is the minimum length of road that should be paved so that there is apaved route between any two intersections? [10 marks]

(ii) (a) Write down the adjacency matrix of the graph below.

Hence, or otherwise, calculate the number of ways of getting from VIto V4 in the graph above using either 2 or 4 edges in the path.(b) State clearly a property of the adjacency matrix of a graph with nvertices, that you possibly used to obtain your answer to (a).

Explain how this property allows the number of edges in the shortestpath between two vertices to be calculated. [10 marks]

[Question 6 continues on the next page J

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[Question 6 continued]. (iii) Draw a graph with incidence matrix B, given by

1 1 1 1 0 01 0 0 0 1 1

B= 0 0 0 (j ,0 0 0 0 0 10 0 0 1 0 0

explaining clearly the relationship between the matrix and the graph. [10 marks]

(iv) The column totals of the adjacency matrix of a connected graph with noloops are:

5 4 1 2 6 4 4 8 6 2Does the graph have either an Eulerian path or a Hamiltonian circuit?Explain your answers.Deduce the number of edges in the graph, and the number of regions in anyplanar representation of the graph. [10 marks]

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Statistics7. [Maximum mark: 40]

. (a) The number XI ofa particular model of bicycle sold per week by a shophas a Poisson distribution with mean 5. Determine the probability of thefollowing events:(i) none of the bicycles of that type are sold ina given week;(ii) more than ten such bicycles are sold in a given week.If each bicycle is sold for $200, determine the mean and variance of thetotal weekly amount in sales on that model. (Results on means andvariances for Poisson distributions may be stated without proof.)For a second model of bicycle the number X 2 sold per week is Poisson withmean 3 and is independent of XI' The cost of this second type of bicycle is$250. Determine the mean and the standard deviation for the totalcombined weekly sales of the two models. (Again, any results used toderive your answer must be clearly stated but need not be proved.) [13marks]

(b) A record was kept of the time taken for a particular mechanic, A, toservice each bicycle before sale. In a sample of nA = 25 bicycles the mean'and standard deviation of the service times were 57.5 and 3.7 minutesrespectively.On the assumption that service times are normally distributed, explain howto obtain a 95% confidence interval for ..Il,., the mean service time.Distributive results used in your answer should be clearly stated:Calculate the end points of the 95% confidence interval from the abovedata. Would your results lead to the assumption of a mean service time of60 minutes when testing against a two-tailed alternative hypothesis at the5% significance level? [12marks]

(c) For mechanic B a sample of nB = 20 service times had a mean and standarddeviation of 61.7 and 3.1 minutes respectively.Obtain a 95%.confidence interval for J .L A - J . L B ' the difference in mean servicetimes for the two mechanics. Any assumptions made and distributive resultsused in your answer should be clearly stated.State, with reasons, whether or not you would accept that 1 l . o 4 is equal toIlB [15 marks]

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-12- N9S I SI0/H (2 )Analysis and Appro ximat io n8. [Maximum mark: 40]

. (i) .(a) State clearly the conditions for the integral test to be applied to the-series Lt Z n 11.. 1

Apply the integral test to the series f+.11.. 1 n +1

- 1(b) State clearly the values ofp > 0 for which the series l:r? converges.11.1

Setting p = ~ , what does the comparison tell you about the convergence- 1or divergence of the series L ?

11=12 +.,InUse an .alternative value of p, if necessary, to establish whether the seriesconverges or diverges. [16 marks]

(ii) (a) Write down the first two terms of the Taylor series for f(x) about. apoint X I I ' and denote the expression by p(x). Let xlI+I be a realnumber such that p(xII+I ) = 0 .Find an expression for xlI+I in terms of X II and so, by lettingn = 0, 1,2, ... , derive the Newton-Raphson iterative method forsolvingf(x) = 0 .

(b) Show that when the result in (a) is applied to the equation x 2 - c = 0 itbecomes

XII+1 = . ! . . { X I I + ~ } ,2 X II n = 0, I, 2, ...Show further that

forn=0,1,2, ...[Question 8 continues on the next page]

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[Question 8 continued](c) Deduce that

Hence show that for any X o >0, the sequence of iterates convergesto . . f c . . . . [1411Ulrks]

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ib hl math nov 95_p2

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