ERRATA. MATHEMATICS FOR THE INTERNATIONAL STUDENT MATHEMATICS HL (Core) (3rd edition)

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1 ERRATA MATHEMATICS FOR THE INTERNATIONAL STUDENT MATHEMATICS HL (Core) (rd edition) Third edition - 07 third reprint The following erratum was made on 5/Jul/07 page 6 REVIEW SET 0A 0 b, 0 b Show that if µ = Ab PM = B b PM, then the length of cable is given b L(µ) = csc µ + cot µ km. The following erratum was made on 0/Jan/07 page Suppose EXERCISE 7B 98,, log, log z, and 7 log z are consecutive terms of an arithmetic sequence. a Prove that 8 = = z 8. b Find in terms of: i ii z. The following erratum was made on 9/Dec/06 page 87 ANSWERS EXERCISE J c, should have parts ii and iii correctl labelled and include domains on both sides: b i is the onl one c ii Domain = f j 6 g or Domain = f j > g iii Domain = f j > g or Domain = f j 6 g

2 ERRATA MATHEMATICS FOR THE INTERNATIONAL STUDENT MATHEMATICS HL (Core) (rd edition) Third edition - 05 second reprint The following erratum was made on 8/Feb/08 page 7 SECTION C, VARIANCE AND STANDARD DEVIATION should lose the second paragraph and read: VARIANCE AND STANDARD DEVIATION The problem with using the range and the IQR as measures of spread or dispersion of scores is that both of them onl use two values in their calculation. Some data sets can therefore have their spread characteristics hidden when the range or IQR are quoted. We therefore turn to the variance and standard deviation of a data set. Consider a data set of n values:,,, 4,..., n, with mean. i measures how far i deviates from the mean, so one might suspect that the mean of the deviations np n ( i ) would give a good measure of spread. However, this value turns out to alwas be zero. i= np ( i ) Instead, we define the variance of n data values to be ¾ i= = n. Notice in this formula that: ² ( i ) is also a measure of how far i deviates from. However, the square ensures that each term in the sum is positive, which is wh the sum turns out not to be zero. ² If np ( i ) is small, it will indicate that most of the data values are close to. i= ² Dividing b n gives an indication of how far, on average, the data is from the mean. v u np t ( i ) i= For a data set of n values, ¾ = is called the standard deviation. n The square root in the standard deviation is used to correct the units. For eample, if i is the weight of a student in kg, ¾ would be in kg. For this reason the standard deviation is more frequentl quoted than the variance. For the purpose of this course, we assume the formulae for variance and standard deviation of a whole population are the same as those for a sample. The following errata were made on 0/Nov/06 page 7 EXAMPLE 5, last line of solution 6 < or > 5:5. page 660 EXERCISE H, advice about integration b parts When using integration b parts the function u should be eas to differentiate and the function v' should be eas to integrate.

3 page 869 ANSWERS EXERCISE D 7b, 7 b (g ± f)() = +5, 6=, Domain = f j 6= g page 874 ANSWERS REVIEW SET B cii, should have ais: a (g ± f)() = b = + c i vertical asmptote =, horizontal asmptote =0 ii =- Qe h( ) = + =0 (-, - Qr ) (, Wu ) page 9 ANSWERS EXERCISE 4G cand d, should state length of OP: c Z d 4 page 96 P (,, 4) Y X OP = ~#6 units P (-, -, ) Z - - X OP = ~#4 units ANSWERS EXERCISE 5H. 6b, 6 b If k 6= 4, the sstem is inconsistent and so has no solutions. The lines are parallel. If k =4, the sstem has infinitel man solutions of the form = t, =t, t R. The lines are coincident. page 9 ANSWERS EXERCISE 6F. 4, Hint is for 4biinot 4aii: 4 b ii Hint: The LHS is a geometric series. Y The following errata were made on 0/Jun/06 page 47 EXERCISE H, b, c, b, and c are real, non-zero numbers such that b b =(c + c ). Show that at least one of the equations + b + c =0, + b + c =0 has two real roots. page 84 SELF-INVERSE FUNCTIONS first dot point, For eample: ² The function f() = is the identit function, and is also a self-inverse function. page 86 EXERCISE J 4biv, 4 a Eplain wh f() = 4 + is a function but does not have an inverse function. b i Eplain wh g() = 4 + where > has an inverse function. ii Show that the inverse function is g () =+ p +. iii State the domain and range of: A g B g. iv Show that (g ± g )() =(g ± g)(), the identit function. page 9 THE REMAINDER THEOREM first line, should clarif: Consider the real cubic polnomial P () = +5 +.

4 page 0 REVIEW SET 6B, should declare and are elements of the set of integers towards the end of the question: ³ 5 If z = i i, epress z in the form z = + i,, Z. page 498 first line after proof bo, should quote propert of cis correctl: We also observe that cis ( nµ) =cis (0 nµ) = cis 0 cis nµ fas cis µ cis Á = cis (µ Á)g page 50 EXERCISE 6F., In Eample we showed that the cube roots of are, w, w where w = cis ¼.

5 ERRATA MATHEMATICS FOR THE INTERNATIONAL STUDENT MATHEMATICS HL (Core) (rd edition) Third edition - 0 first reprint The following errata were made on /Oct/05 page 95 ANSWERS REVIEW SET 5C 0aii, 0 a i ¹ =:8, ¾ =: ii E(Y )=:4, ¾ Y =0:566 b ¼ 0: 66 The following errata were made on or before /Jan/05 page 6 EXAMPLE 0 Answers parts a and b, a For two distinct real roots, > 0 ) k < 9 or k >, k 6= 0. b For two real roots, > 0 ) k 6 9 or k >, k 6= 0. page 4 PART G Problem solving with quadratics, second dot point An answer we obtain must be checked to see if it is reasonable. For eample: ² if we are finding a length then it must be positive and we reject an negative solutions ² if we are finding how man people are present then clearl the answer must be a positive integer. page 0 Proof at bottom of the page, ² log c (AB) = log c c log c A c log c B = log c c log c A+log c B = log c A + log c B ³ A ² log c B µ = log c c log c A c log c B = log c c log c A log c B = log c A log c B ² log c (A n ) = log c (c log c A ) n = log c c n log c A = n log c A page 88 DIVISION BY QUADRATICS, blue bo at top of the page If P () is divided b a + b + c then P () a + b + c = Q()+ e + f a + b + c The remainder will be linear if e 6= 0, and constant if e =0. where a + b + c is the divisor, Q() is the quotient, and e + f is the remainder.

6 page 67 THEORY OF KNOWLEDGE, n=40is the first counter eample to Euler's statement: In fact, n + n +4 is prime for all positive integers n from to 9. However, checking n =40 gives a counter-eample to the statement, since = 4 4. page 89 PERIODICITY OF TRIGONOMETRIC RATIOS, PERIODICITY OF TRIGONOMETRIC RATIOS Since there are ¼ radians in a full revolution, if we add an integer multiple of ¼ to µ (in radians) then the position of P on the unit circle is unchanged. For µ in radians and k Z, cos (µ +k¼) = cos µ and sin (µ +k¼) = sin µ. We notice that for an point (cos µ, sin µ) on the unit circle, the point directl opposite is ( cos µ, sin µ). ( cos µ, sin µ ) ) cos(µ + ¼) = cos µ sin(µ + ¼) = sin µ and tan(µ + ¼) = tan µ µ + ¼ µ (-cos µ, - sin µ ) For µ in radians and k Z, tan(µ + k¼) = tan µ. page 48 REPRESENTING CONJUGATES, Swap the labels of Q and Q' in the diagram: REPRESENTING CONJUGATES If z = + i, then z = i. µ! This means that if OP = represents z, then! OP 0 = µ represents z. For eample:! OP represents +4i and! OP 0 represents 4i.! OQ represents i and! OQ 0 represents i.! OR represents +i and! OR 0 represents i. 0 I Q (0, ) R (-, ) R (-,-) 0 Q (0,-) P (, 4) 0 P (,-4) R page 58 RATES OF CHANGE, second dot point We often judge performances b rates. For eample: ² Sir Donald Bradman s average batting rate at Test cricket level was 99:94 runs per innings. ² Michael Jordan s average basketball scoring rate was 0: points per game. ² Rangi s average tping rate is 6 words per minute with an error rate of : errors per page. page 65 INVESTIGATION 4b, 4 b Integrate dµ d with respect to. Use µ(0) = 0 to show that µ() = for all.

7 ERRATA MATHEMATICS FOR THE INTERNATIONAL STUDENT MATHEMATICS HL (Core) (rd edition) Third edition - 0 initial print The following errata also appl to books printed in 0 page 66 EXERCISE E 7b, 7 Prove that: page 55 c page 67 page 75 a b c the sum of two even functions is an even function the difference between two odd functions is an odd function the product of two odd functions is an even function. EXAMPLE 4 Answer part c, The total number of combinations = the number with men and woman + the number with men and women + the number with man and women = = 665 THE PROCESS OF INDUCTION, at the bottom of the page The nth odd number is (n ), so we could also write the conjecture as: :::: +(n ) = n for all n Z + np or (i ) = n for all n Z +. i= EXERCISE 9B. a, Use the principle of mathematical induction to prove the following propositions: a n > +n for all n N b n! > n for all n Z, n > 4 page 49 EXAMPLE 6, should sa in the hint: a z = p cis µ ) j z j = p and arg(z) =µ The range of arg is ] ¼, ¼ ]. page 69 EXAMPLE, should sa in the first line of the answer: Let cm be the lengths of the sides of the cube, so the surface area A =6 cm V = cm. and the volume page 809 page 850 REVIEW SET 5B, A discrete random variable X has probabilit distribution function P () =k 4 4 where =0,,, and k is a constant. Find: EXERCISE 7A 9 f, 9 f The region bounded b = f(), the -ais, and the line = k, k > 0, has area 4 (e ) units. Find the value of k.

8 page 87 ANSWERS EXERCISE K cv, should include all -intercepts: c v (5, ) = =4 page 87 ANSWERS EXERCISE L, graph for = f() should be: = :04 or =:7 (-. 04, 4. 5) = f() = g() (.7,.9) page 87 5 a = or b ], 8 [ or ], [ page d page 880 e page ears ANSWERS EXERCISE F 6 d iii, iii ¼ :6 ANSWERS EXERCISE 4G e, i Domain is f j > 0g, range is f j R g ii VA is =0, -intercept p, no -intercept iv = v f () = iii =- log ANSWERS EXERCISE 4H 5, page 885 ANSWERS EXERCISE 5D f, graphs should have -intercept -: f ANSWERS REVIEW SET A 5b, should have open intervals: ~` = ( + ) = -( + ) - - page 885 d page 890 page b ANSWERS EXERCISE 5E d, ANSWERS REVIEW SET 6B 7 b, = (- + 5)(- -) A reflection in the -ais, then a vertical stretch with scale factor. ANSWERS REVIEW SET 5B c, b (, ) and (, ) c VA is =, -int is graph should have function: p - p + (, -96) -0 (-, -8) (, -8)

9 page 898 ANSWERS EXERCISE 9A 5band 5c, 5 page 899 page 899 b Conjecture: The number of regions for n points placed around a circle is given b C n = n, n Z +. c n =6 b No. B the conjecture we epect 5 = regions, but there are onl. ANSWERS EXERCISE 0D. b, µ =0, ¼, or ¼ ANSWERS EXERCISE 0E, Hint: µ Á = (Á µ) page 900 ANSWERS EXERCISE B 0 b, 0 a = b 4 p 6 cm 6 ±, 7 ±, 6 ±, 44 ± page 900 ANSWERS REVIEW SET A band 9bii, a =or 5 b Kad can draw triangles, so she should ask for more information. cm 60 5 cm 8 cm 7 cm 9 a maimum value 6 when = 6 b i = ii = (6 cos µ) +64 page 90 ANSWERS EXERCISE E c, a translation through b reflection in -ais c horizontal stretch, factor and vertical stretch with factor page 906 ANSWERS REVIEW SET C 7a, 7 a = 4 sin(4 0:40) + 4 b = tan( ¼ ) page 906 ANSWERS EXERCISE A. f, page 94 ANSWERS EXERCISE 5A,, and 5, page 95 ANSWERS EXERCISE 5D, 75:7 ± b ² b =0 75:5 ± e ¼ 6 ( ¼ 4 ) 6 7¼ f ¼ a :4 ms in the direction 6:6 ± to the left of her intended line. b i 0 ± to the right of Q ii :04 ms a 4:6 km h b ¼ 9:9 ± east of south 4 a 8:5 m b : ± to the left of straight across c 48:4 s 5 a The plane s speed in still air would be ¼ 47 km h. The wind slows the plane down to 400 km h. b 4:64 ± north of due east

10 page 97 ANSWERS REVIEW SET 5B 6, 6 a (t) =+t, (t) =4 t, t > 0 b (t) = t, (t) = [ a] + at, t > c interception occurred at ::0 pm d bearing ¼ :7 ± west of south, ¼ 4:54 units per minute page 99 page 97 ANSWERS EXERCISE 9B 7, page 90 b ANSWERS EXERCISE 6B. 4 a b jj r ANSWERS EXERCISE 9D. f(t) ³ A local ma. b, be ³ ³ non-stationar A inflection b, be f(t) =Ate bt 4b, page 99 ANSWERS EXERCISE 6C., k p p ¼ cis 4 if k>0, -k cis ¼ 4 if k<0, not possible if k =0 7 a decreasing for < <, never increasing b increasing for < <, never decreasing b, ³ b b t page 90 8 ANSWERS REVIEW SET 9A 8c, c f 0 () 6 0 for < and <6 and f 0 () > 0 for > page 9 ANSWERS EXERCISE 0A. a v(t) =98 9:8t ms, a(t) = 9:8 ms s(t) 5 p cm per radian 0 a, + v(t) + - a(t) - t t page 9 ANSWERS EXERCISE 0B 5, page 94 ANSWERS EXERCISE 0C 0, 0 at grid reference (:54, 8), ¼ 5:6 km t page 94 ANSWERS EXERCISE C. 8, 8 a v(t) = ms t + b s(t) =lnjt +j t metres c s() = ln ¼ 0:90 m, v() = ms, a() = 9 ms The object is approimatel 0:90 m to the left of the origin, travelling left at ms, with acceleration 9 ms. page 948 ANSWERS REVIEW SET 4B 7a, 7 a Qr b i 8 D Eu C ii D' 5 Er Ru Wt C' Et D D'

11 page 949 ANSWERS EXERCISE a X =0,,,, or4 page 95 page 95 ANSWERS EXERCISE ANSWERS EXERCISE 5A a, 5F. a, a i 0 p i ( p) p( p) p ( p) p a :5 b p = (:5) e :5 The fit is ecellent. 5G b, page 95 ANSWERS REVIEW SET 5B, a k = 8 5 b 0:975 c :55 d ¼ 0:740! for =0,,,, 4, 5, 6, f p :6 7:4 : 6:5 :4 0:7 0: