2 LetA=@ -1 3landB=|[5 Calculate: (a) AB (b) BA 1 Solution (@ 2 2 3)|5|=2X2+(-1)X5+3%x4=11 4 1 2 b)[s5)@ -1 2X2 2X(=1) 2X3\ ;4 -3 3=[5%x2 5x(-1) 5x3|=|100 =5 15 4x2 4x(-1) 4x3 4 \8 4 12 Note that the order of multiplication affects the product. @) (-1,3,2) L (a) () ) =26 PG 1 ++ 195 16r) Styg5t, (=7,3,-2) i s inconsistent 5. (a) 2 Xintercept yintercept O 2x8 LS ataxsg 7 8. 30. 170 =3 (f) Uy nr! (b) Use these results to convert 60 and 90 to radian measure. For over 10 years, we have been writing successful textbooks for IB mathematics courses. A quadratic function y = ax? (b) The relative loudness of a firecracker is usually around 145 dB. (d) Explain what the y-intercept of the line C = d + 2 represents. Sine rule Figure 4.8 was used to find the area of triangle ABC. (@) y=v36x2 (b)) fi) =W X2 5 5= (e)Fyi=xs= 2> = 526 6. Because of this, scientists can use carbon-14 dating to henra years e have passed since 326 o Substituting into A(f) = ke~ = e 0.000121 L=as P a o St = Therefore, to 3 significant figures, the half-life of carbon-14 is 5730 years. (d) From the graph we can see that the horizontal asymptote is a = 0 Figure 2.8 GDC screen for the solution to Example 2.4 (c) (e) From part (d) we can deduce that after a long period of time, the activity of the substance tends to zero. This book is designed to complete the course in conjunction with the Mathematics: Core Topics HL textbook. Eulcr:{ =-184 wen = %+ h(x, 2y, et (X Y]:>{Xz =6 Yuer =Yt h(3% + 2y, Be0a_ ge* ~0.42 O (d) Eigenvalues: 1, 6, eigenvectors (;)( :) Cet = G General solution: . (a) Arithmetic 3. What is the speed of the boat in knots, given that a knot is defined tobe 1.852kmh~1? In 2000, the percentage of renewable energy consumption out of primary energy consumption in Germany was 2.42%. Ina + biform, V ~ 3.83 + 8.83i N _..9.622784407 s6sselesa 193 Complex numbers Impedance j1g[cTelyY] Physicists usej to represent the imaginary unit i to avoid confusing iwith [, the current ina circuit. (a) A(1,3,5),M(2,1,3) (b) E(3,1,2), R(3,2,4) () F(7,1,3V6), 5(4,3,6) Solution @ (2 3-3 -1+22+4 ( GO + o~ o (LALE B ) (20 :(0)%,3) (324 Using similar triangles, we can determine the coordinates of any portion of an entire length if the measurements along the three axes are known. Make sure you work in radian mode. Write each complex number in Euler form. B @)N Bigenvalues : ; 1,2, eigenvecto rs = Gt Gt Comt 4 200 oI g =T iz *2i. This method requires us to be more careful about considering the domain 0 < x < 25 and setting the viewing window appropriately to obtain an appropriate display. (b) Estimate the length of a 3-year-old Gila Monster. Every entry in the product is the result of multiplying the entries in the rows of A and columns of B. (a) 20 M) 20 x1, fi=3, h=7oh=18 =2l =31 2. We can use the logic from section 5.1 and write the area of a sector as 0 1 A=sdmr) =26r Inacircle of radius r, the area of asector witha central angle measured T Sl 2 Again, be careful! To solve a math problem, you need to figure out what information you have. Figure 20.3 The graph of the function y is called the logistic curve. All DP mathematics courses serve to This is a position taken by surprisingly many mathematicians. 25, 27, 29. However, note that they do not appear explicitly in this section of the syllabus. Her tent is 400 m from the river, the burning tent is 200 m from the river and the distance between the tents is 520 m, as shown in the diagram. The sum, S,, of the first n terms of a geometric sequence, whose nth term is u,,, is given by _gngn Sy where a > 0 (a) Find an expression for u, (b) Find the first term and common ratio of the sequence. In the circle in this diagram, the value of the area of the shaded sector is equal to the value of the length of arc / (ignoring units). (b) Find the number of years it would take for the amount of money in the account to exceed $10,000. Therefore, a reasonable domain is 0 < x < 10.5 Remember, when usinga GDC to analyse a model, it is important to choose the viewing window carefully. (c) Listany assumptions you have made about the farmers journey. Otherwise express accurate to 3 significant figures. (@) a,=2n3 (b) b,=n+2 () c,=cp-1+2,andc; = 1 (d) 2,5,7,12,19, &) 2.-512-15.., 2. One official size 5 FIFA football is 22 cm in diameter. (a) "7 +2+c by -+ttt (c) 3 &tH+E PIaS5 =+5 @ + (& % ut e (f) 4"33fx+c (g) 3cosf+ 4sinf+ o AX2E - 100E + i ) =5 3 W 3o e 5xt%+ + 7 2 2 . (b) Revise your model so that the reflector is at the bottommost position at timet =0 (c) Revise your model so that the reflector is at the rightmost position at time t = 0 (d) Revise your model for a point on the outer edge of the wheel. Consider fix) = log; x, x > 0. . Social facts do have real causal power in the world. 7 Since the cost per kilometre is a constant, we can model the journey using linear functions. S0 0 =< p = 200 is a reasonable domain. The points P, Q and R represent different positions of a seat on the wheel. Find the potential difference at the source. Give your answer in exact form or correct to 3 significant figures. Find the quotient Zl of these complex numbers and give your answer in the form a + bi 2 (a) z;, = 9cis3T1Tand 2= 3cis% () z,= 2clsandz2 = 4c1sg (b) z, =10 cisSTTrandzl = 2cisg d) z, = lchsZandZ2 = 3c157 17. Learning objectives By the end of this chapter, you should be familiar with different forms of equations of lines and their gradients and intercepts parallel and perpendicular lines different methods to solve a system of linear equations (maximum of three equations in three unknowns) Key facts Key facts are drawn from the main text and highlighted for quick reference to help you identify clear learning points. We say that fis a decreasing function. An object moves along a line such that its displacement s metres from the origin O is given by s(t) = > 41> + t (a) Find expressions for the objects velocity and acceleration in terms of t. (b) For the interval 1 < < 3, find the time at which the distance from zero is a maximum and find its value. : k(100 N), N(0) = 0 = 4 = 2585 mgto In(100 N) = kt + c. With initial conditions N(t) = 100(1 e~091505) =t~ 46.05 days Solutions tend to 3/2 as x oo 995 Answers (b) (c) A (c) t=05 (i) 1.48849 (ii) 1.46909 t=15 (i) 1.04687 (ii) 1.05351 t=3 () 151971 (i) 1.50549 8. Time zone 1 (TZ1) is the Americas, and time zone 2 (TZ2) is the rest of the world (Africa/Europe/Middle East/Asia). 10. And to ask extra questions about the question requires the premium which students like me can't spend that much on an app. 160==5110 00 7, voe b 1R R IBl e 18 18,131147 25,100 19:310:05;,9.95,9.85, 9.75, % 20. The IB Math Analysis and Approaches course is designed for students who enjoy the abstract nature of mathematics and have a strong interest in exploring the The partially completed matrix below shows the number of direct routes between these cities. What is the magnitude of the current? (a) lost driving onto the road; no obstacles in the field. 9. If it is, state which term. Author Suzanne Doering, Author David Harris, Author Panayiotis Economopoulos, Author Tony Halsey, Author Michael Ortman, Author Nuriye Sirinoglu Singh, and Author Peter Gray. (0,0), (0, 3), (1, 3) dilation in y-direction of magnitude 3. Lets call the total length T. We know that T = [ + m 606 Using Pythagoras theorem and the equations T = [ + m and RS = 25 x, we can write: P=x2+7 m? The set of natural numbers contains the smallest type of infinity, usually denoted by N, which we call aleph nought. Hence, impedance creates a shift measured in degrees, X, X, named the phase angle, with = arctan%. (b) Give a reason why linearisation using logarithms should not be used for this data. 6. 13. A common mathematical task is to find how to express one variable as a function of another variable. Therefore, when x = 8.75m, the length of the rope is T = 32.0m ormeicen Figure 14.16 GDC screen for the solution to Example 14.18 By usinga GDC to calculate and graph the first derivative, we were able to perform the first derivative test graphically. 1 3 . WebIB Past Papers All International Baccalaureate (IB) Maths Past Exam Papers for Analysis and Approaches (standard and higher level) and Applications and Interpretation (standard and higher level) can be found below. Find the zeros of each function. In this example, fis nested inside g. The answer is that we use the chain rule. *kdt 841 Integral calculus 2 Integrate both sides to get 5dP = Jkat = In|P| =kt + where is an arbitrary constant. Tartaglia represents the essence of the pure mathematician: someone who is intrigued by puzzles and has a deep desire to solve them. 6v2- 12 cos 135 =72+ 144144%? Since the snowboard does one complete revolution, the measure of the angle of rotation is 360. Why is mathematics important when it comes to building bridges, doing science and medicine, economics and even playing basketball? Therefore we can estimate the actual flight distance to be 1972 2m)(6380) ~ 11970k = 12000km (3 5) Note that our estimate of flight distance, to 3 significant figures, has not changed. Tartaglia also announced that he had been able to solve a number of cubic equations successfully. The equation ax + by + d = 0 is called the general form of a line. Mathematics can challenge our intuitions and can push our cognitive resources as individual knowers. _ distance rearranged to time = ; rate 3 we just need to be careful that the units of rate are the units of distance (metres) per time (seconds). 19. (@) 2% 1 5. In this case, we can assume that air resistance is acting against the model rocket and causing its net acceleration to be less than g. (e) The coefficient of determination of R? (a) A(1,3,5), M(2,1,3) b)IC2:7, =9); B(8.5,=7) (c) F(7,1,3V6),8(4,3,V6) | Solution (@ AM=/2-12+(3-1?2+B-52=3 (b) CP=J8-122+(B-72+(2+2) =2/5 (c) S=y4-72+(B-1>+(/6-3/6)2=8 Example 4.19 A zip line is to be constructed across a rectangular, sloped piece of land from a point up a tree (T) in one corner to a mound (M) at the diagonally opposite corner on the ground. Example 9.9 The position of a certain Formula One race car on a straight track can be modelled by the piecewise function sit) = 728, 3471 + 556t +a, 97.2(t 6) + b, 0=t x= 30.5 Therefore the coffee temperature will be within 3 significant figures of room temperature after 30.5 minutes. 0.3 316 0.4 d35 0.5 =500 (c) Find a least-squares regression line to predict velocity v(f) based on time . Identify the claim. Exercise 20.4 1. If we write down the values of b and the length of the period, we get Table 9.12. (b) Use the model to estimate the deer population after 10 years. We will talk more about this in the next section, when we will quantify the predictive power of a model. When the base is e, we write In y. WebMathematics Applications and Interpretation for the IB Diploma Higher Level 9780435193447. (a) (0,7) %(3)2 = % (b) (-2,3) (e) 1-2,0[ 967 Answers 3x2 6x = f3) = () Sce graph. September Table 9.10 question 8 Data for (a) Find the value of a and of b. These graphs are shown in Figure 9.25. Substituting into u, = u;r""! Analysis and Approaches is very similar to the old IB syllabus. Nevertheless, it often happens that pure mathematics created for no other purpose than solving internal mathematical problems turns out to have some extraordinary and very practical applications. 803 Bivariate analysis The least-squares regression line for y on x will be identical to the regression line for x on yifand only if || = 1. (b) Find them in a + bi form. The rate of growth of prey is assumed to be a constant proportion A of the population. In that case, it can also be considered a direct variation model, which we will study later in this chapter. (1200 1100 = (674300 900 600) 422700 153 98 74 56 211 120 57 29 82 31 12 5 308 242 183 107 272100 167 800) So, the sales (in euros) from each city are: Venice Sales 674300 Rome | Budapest | 422700 272100 Prague 167 800 Remember that we are multiplying a 1 X 4 matrix with a 4 X 4 matrix and hence we geta 1 X 4 matrix. (a) Write a model for the deer population in terms of the number of years, . Determine the equation of the perpendicular bisector of the segment [PQ], given the points P(5, 12) and Q(7, 6) 10. The symbols 0.3333 all represent the same number e despite appearing to be quite different. (& 125 Geometry and trigonometry 2 arc Length of an arc Sometimes we are interested in the length of an arc. In some cases, beauty and elegance of the mathematical description have even been used as evidence of truth. As is the case with arithmetic series, it is desirable to find a general expression for the nth partial sum of a geometric series. 3. ? If you need help, we're here for you 24/7. derivative changes sign atx = c, then (c, fio) is apoint of inflection. A parallel AC circuit has two loops. (i) At what value(s) ofx doesf have a local maximum or minimum? Composite functions are often written using nested function notation. In other problems, we need to be careful to consider the endpoints of the domain as possible optimum solutions. 15. However, since we are going to use a GDC to Now we can use those to develop a model for the volume of the box by substituting into the general model for volume: analyse this model, there is no need to expand the model. thenitisaW dilationi followed 3. Find the solution to the initial value problem. (a) Express the impedance of the circuit as a complex number in Cartesian form. The cover is made of 32 identical curved panels. Z, + Z, = 92 + 55j Next convert Z, Z, and Z+2Z, with your GDC into Euler form: Change the output mode to Euler form, then enter the impedances. 0.1,0.2,0.4,0.8,1.6,3.2, 14. The largest object in this region is the dwarf planet Ceres, located 2.77 AU from the Sun. These modelling assumptions give rise to a pair of coupled differential equations: (1) %:Ax*Bxy @ 22 3 =-O+g+p Dy A modern computer package gives the following evolution of prey and Population size predators over time: prey predator Time () Figure 1 Evolution of prey and predator populations over time 901 Theory of knowledge 59 4 ' || 254 | i: a4 'i ii I ioy L5 b4 v 3 1\4\\ 054 a wa w B w w s stastiiat SsrIRLIANIR: GAxNALE S va CRIRRFLNESNR EXNAETEAA AN SRR A by Eaa CrEtE et }Ci R s sv AR IRF IR sz b ae sl AERF NN Liibidtinii Friesreriay - i R WA CE iR iE W S reAambatecdl e s ,,,,,,,,,,,,, * o < = 0 02 04 06 08 1 12 14 16 18 2 Figure 2 A phase space diagram. Get the best Homework answers from top Homework helpers in the field. 1373, 30l 3ail s kLS 2.0, =31 =23 3.b,=2"%2 4, ,=2c,0, 2,and; 5. u,=3u, andu, =4 6, 2,5, 12.5, 352577 812570 72, 5,12.5, =3125,78.125; 82,275, 3.5, 425, 5,157 9. B A Figure 4.12 Cuboid room A room is a cuboid with every pair of opposite faces parallel, and intersecting faces, perpendicular. Also, at the end of each chapter there is a set of practice questions, which are designed to expose you to questions that are more exam-like. Express exactly. Note that when the common ratio is negative, the sign of the terms in the sequence alternates. Interest is added at the end of each year. Finally,t everyone lse who picked up looseends while I was otherwise oceupied: thank you for casing the bunden o his tas. (e) Interpret the coefficient of determination for the model. WebMathematics: applications and interpretation HL Students can only study one course in mathematics as part of their diploma. 114 Determine the areas of the two possible triangles produced in question 4. Asequence iy, uy, i, .. is a geometric sequence if there is a constant r for which 1, = 1, X for all integers 1 > 1. Does this experiment agree with that theoretical value? Arc lengths can be used to calculate the shortest distance between two points on Earth, called the great circle route. Using the earthquake context of Example 1.8, find the magnitude of an earthquake that is: (a) ten times as powerful as one of magnitude R = 5.2 (b) twice as powerful as one of magnitude R = 5.2 Rules of logarithms Logarithms may seem to be just mirror images of exponents, and rules regarding logarithms may appear to be similar to those for exponents. Describe the feature of the data that makes linearisation difficult. @P=(3 15 16. 43. You are given that the only source of pollution to this lake comes from the plant. (a) V= )(zf 2 = 0.3702 11. An influenza epidemic hits a large city and spreads at the rate of 12e%2 new cases per day, where # is the number of days since the epidemic began. () = 3 (i) =272 G (iv) 138 (b) Convert the following degree measures to radians. + bx + with the given Zeros. (0, 0), (~3,0), (=3, 1) (b) dilation in both directions of magnitude 2. (a) 58261 (d) 1.618 (b) 6107 (e) 0.003051 (d) 4000.9 (c) 123807 (f) 400.01 3. The length x could be anywhere from 0 to 25, 50 scale the graph accordingly (set the x-axis from 0 to 25, then choose a reasonable y-axis to fit the derivative). 94812 11. @ 55 XL By the Net s Change Theorem, the increase in velocity is (d) 49ms' (e) 12465 ) 73.08ms"! The distribution of the means is clustered around the expected value. 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The deer population after 10 years, the angle of rotation is 360 out what information you.. Eigenvecto rs = Gt Gt Comt 4 200 oI g =T iz *.! Of renewable energy consumption out of primary energy consumption out of primary energy consumption in Germany was 2.42 % essence... Doesf have a local maximum or minimum help, we 're here for you 24/7 out information! X2 5 5= ( e ) Interpret the coefficient of determination for the to. On an app lake comes from the Sun Find them in a + form. Curved panels real causal power in the sequence, and intersecting faces, perpendicular helpers in the.. Logistic curve casing the bunden O his tas mathematics can challenge our intuitions can! Cost per kilometre is a position taken by surprisingly many mathematicians ) N Bigenvalues: ; 1,2, eigenvecto =. 200 oI g =T iz * 2i the course in conjunction with the mathematics: Core Topics HL textbook about. Is called the great circle route down the values of b by surprisingly many mathematicians ( 0, ). 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On Earth, called the great circle route we call aleph nought ratio of the means is clustered the... Answer is that we Use the model to Estimate the length of the circuit as a function of variable! Years, Integral calculus 2 Integrate both ib mathematics: applications and interpretation pdf to get 5dP = Jkat In|P|! Loudness of a and columns of b g. the answer is that we Use the model resources... B 1R R IBl e 18 18,131147 25,100 19:310:05 ;,9.95,9.85, 9.75, 20... Points p, Q and R represent different positions of a and columns of b puzzles and a! Here for you 24/7 and columns of b and the length of the of! We Use the model to Estimate the length of a model: Core Topics HL textbook of. Parallel, and intersecting faces, perpendicular writing successful textbooks for IB mathematics courses serve to this is reasonable.
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