2(x)=10000 (43.1,10000) 1(x)=1500- (1.05)% 3 (43.1,10000) 2(x)=10000 1(x)=1500- (1.05)* 43.1, so the number of years is 44. A sa.n'lple of live tissue of the same After another 3 days, Earth continuously take in carbon. Observed values: Expected values: 12 4 8 6 12 6.8 | 52 6 18 13 5 9 18 | 102 | 7.8 9 2] [12455 18135 9 In order to calculate the y? (a) 9:g,r:20mm (c) 6=2,r=3km (b) 9:g,r:70m (d) 6=57r=3cm 38. Given that & can be expressed in the form h(t) = a sin(blt d)) + c, find the value of d. / Bucket " Figure 9.34 Water level Diagram for question 5 . Figure 5.4 Anarcand its We use the same logic we used for the area of a sector: we need the size of the central angle for the arc, and then we calculate the portion of the circumference using a ratio. [S7ET oI ECXS The dimensions of a piece of A4 paper, to the nearest centimetre, are 21 X 30 cm. The death rate of predators, C, is assumed to be a constant proportion of the population, and there is a rate of generation of new predators, D, dependent on the product of prey and predators. For the first 8 km, the cost is $0.50 per km and the starting cost is $3. This is called the Platonic view because the philosopher Plato (427-347 Bc) took the view that mathematical objects belonged to the real world, underlying the world of appearances in which we lived. -5 - 2 3 and B=| 5 =9 206 2 IS 8 0] 4 503 0 0 | Solution Aisa2 X 3 matrix, Bis a 3 X 4 matrix, so the product will bea 2 X 4 matrix. 70km| caysoll 1 sandysoll o B 100km Find a function to model the cost of the general route from A to B and use it to find the minimum cost of building the pipeline. 670 [ 665 | 609 [ 679 [ 625 | 580 | 607 | 605 f&i;;:;::t; 56 | 46 | 54 | 70 | 51 | 40 | 51 | 48 (a) Generate a scatter diagram and describe the association between speed and fuel consumption. Find the area of the sector and the arc length determined by each given radius and central angle 6. Determine the equation of the perpendicular bisector of the segment [PQ], given the points P(5, 12) and Q(7, 6) 10. (b) 0.439cmss! Calculate the intensity of the sound. In this chapter we first look at circles, where we learn to calculate the area of a sector and the length of an arc. \(oll(=Tale]ig} For any angle 6, its radian measure is given by = % Simple re-arrangement of this formula leads to a simple formula for arc length. (a) product of matrices = (1 = ) = cfsgg "Zg) cos sin%0 (b) product of matrices = (_? Thats certainly what makes a lot of people hate mathematics () There are various kinds of Platonists. (a) (a) (a) (a) (a) (a) S=075d 9.8 1.62 4.00 X 10" 7.664 0.613 () x=16 (b) x=10 (b) x=4 (b) x (b) x=10 (b) (d) (b) (d) sometimes never never never =100 () x=3 (b) (b) (b) (b) (b) (b) 5.25m 39.2 40.5m 7560ms ! R 997 Answers (f) Eigenvalues: , 10, eigenvectors 5\ (1 % 5C,eH + Ce General solution: { 2C,e% + 4C,e Y i) Saddle point. (a) A bank offers loans of $P at the beginning of a particular month at a monthly interest rate of I . The wheel rotates at a constant rate in an anticlockwise direction. embeddedin the unit circle tennhoes will always be equal to the y coordinate when we draw a right-angled triangle in this manner. Now, instead of merely keeping a tally, after rolling the dice 6 times, find the mean of the outcomes. (b) Indicate the degrees of freedom, v, in this analysis. Mathematical objects such as perfect circles and numbers existed in this real world; circles on Earth were mere inferior shadows. As one approaches the tree, the angle of elevation changes to 40. The cone must be no longer than three times the height of the hemisphere. Such an investment is called an annuity. A story that makes this statement is describing a world that is selfcontradictory that is, an absurd and unintelligible world. Afternoon exams must start after 12 PM and finish by 6 PM (the usual start time is 12 PM or 1 PM). One applies the rules (these are rules of algebra typically) to a line in the proof to get the next line. 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. V=27 fz xfx)dx; we use Trap. Find the radius of the circle. 139 Geometry and trigonometry 2 Definition of the trigonometric functions Let be an angle in standard position and (x,) the point where the terminal side of angle Gintersects the unit circle. The temperature of the room is 20 C. Experiment with different values of a to see how the graph changes. 5.22-2/3~185 circle with centre (0, 0) and radius 3 y-axis linex=4 circle with centre (3, 0) and radius 2 aline segment between (1, 0) and (3, 0) L @) z=3ZcsT ) z= 3\7m341 z= 3&:3% @ z= 3\7&5541 () z= wmg ) @ z= wm% z= lOcisl; e wc;s% 2. (a) Given that fixed costs are 339, find the cost function, C(x). (iii) Explain the meaning of the percentage difference in context. That is, there is a first number, a second, and so forth. x>~ 443 and itsp value ~ 0.218. V3 - V8T oy 32 19. However, from point C, the distance AC = 20 m and BC=32mand C= 60 Determine the distance AB. To model the situation above: We know that the acceleration due to gravity is 9.8 m s~2, and we also know that the distance travelled s is given by the equation: s= %utz, where a = acceleration and = time So we substitute the known values into the equation and get: 30 =Les)r 2 Rearranging the equation gives us: S0 _ 9.8 2,50t 60 (o8 = 2.47 seconds (3 s.f.) adjacent The same logic applies for the cosine, so that cos (45) = = hypotenuse which implies that the value of the cosine ratio is always equal to the . 0.1,0.2,0.4,0.8,1.6,3.2, 14. 94812 11. ()1>7+(1)679 0 5 10 300 w? In contrast to the science of the day, the mathematical truths of Pythagoras are just as true today as they were then - indeed his famous theorem is still taught today as can be seen in this book. (b) Find the size of the population in China in 2020. A geometric sequence has a second term of 2, a seventh term of 243 1nd 19 683 an nth term of 3107 512 Find the value of n. 38. Finding the right substitution is abit of an art, and you need to acquire it. Has it changed? 36. It is a result of our mathematics game - our deduction from axioms. Mathematicians discuss 11-dimensional hypercubes, infinite sets of numbers, infinite numbers, surfaces that turn you from being right-handed to left-handed as you traverse them, spaces where the angles of a triangle add up to more than 180 degrees, spaces where parallel lines diverge, systems where the order of the operation matters (where A * B is not the same as B * A), vectors in infinite-dimensional space, series that go on forever, and geometric figures that are self-similar called fractals (where you can take a small piece of the original figure then enlarge it and it looks identical - truly identical - to the original). (b) Give a reason why logarithms should not be used to linearise these data. d We sce from Equation (1) that if y < L, then d}; > 0 and the population increases, whereas if y > L, then d_)t/ < 0 and the population decreases. WebMathematics HL - Applications and Interpretations - OXFORD 2019.PDF - Free ebook download as PDF File (.pdf) or read book online for free. 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. On any wall, floor, or ceiling, a diagonal drawn is the hypotenuse of a rightangled triangle. Analysis and Approaches Paper 1 (HL and SL), Analysis and Approaches Paper 2 (HL and SL), Applications and Interpretation Paper 1 (HL and SL), Applications and Interpretation Paper 2 (HL and SL), Applications and Interpretation Paper 3 (HL). (e) Eigenvalues: . 10. Hours since midnight Temperature (C) Hours since midnight Temperature (C) 0 2 233 | 237 | 14 243 | 16 238 | 4 242 | 18 233 | 6 247 | 20 23.0 | 8 249 | 22 231 | 10 250 | 12 247 24 233 (a) Generate a scatter diagram for the data and describe it. (a) Given that fixed costs are 2500, find the cost function, C(x). (b) Atwhat speed is the bicycle travelling along the ground, in kmh~'? 197kmh! (a) Find an equation to model the total cost C in terms of the distance travelled d. (b) Find the total cost for a journey of distance 12 km. (a) V8 o) /0 + /39 (c) V=16 6. Arc length and area of a sector A sector can be thought of as a pizza slice. All courses, with the exception of Math, Environmental Systems and Societies and ab initio language, are taught at the HL level. 7. ty! At x = 0 sine begins on the principal axis and then increases as x increases. = /(100 d)? We see that a person would be at least 100 feet above the ground for 2.31 = t < 7.69 minutes. The diameter of the wheel is 8 m. The centre of the wheel, 4, is 2m above the water level. While generalising is an important part of mathematics, we have some very good reasons to do it: as you will see, trigonometric functions work very well for describing periodic behaviour - that is, any phenomenon that repeats itself with a fixed interval. @ -2 = 2u, 30. The building measures 5 m? 55,667 1(3)= V94 V(2516 As before, we find that the rope should be anchored 8.75 m from the post, and that the minimum length of the rope is 32m. . Tartaglia did not take Fiors money though; the honour of winning was enough. 108 (a) Find, to the nearest metre: (i) the height of the building (ii) the height of the top of the antenna above ground (iii) the height of just the antenna. WebMathematics: applications and interpretation HL Students can only study one course in mathematics as part of their diploma. Kevin Frederick Iyvould like t thank my familyfor their fove and invaluable support. The inflation rate during the time of the investment is 2%. Here is an example to illustrate how we might have to correct our intuition by careful mathematical reasoning. _ Solution Since the fifth term is given, using the explicit general form: as=a;+((51d =11=5+4d =d= _32 This leads to the general term a,=5+ 3 E(n =al Therefore, a,y = 5 + %(19) = % Sometimes a real-life situation can be modelled by an arithmetic sequence, even though it does not follow the sequence exactly. The general (nth) term ofa geometric sequence, , with common ratio r and first term u; may be expressed explicitly as Uy =yt This result is useful in finding any term of the sequence without knowing all the previous terms. I function is either increasing or decreasing, it is said to be monotonic. Dont extrapolate. 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. In the Kelvin temperature scale, the freezing point of water is 273 K and the boiling point of water is 373 K. (d) Find a linear function that converts Kelvin to Fahrenheit. All points on the floor are mapped by the x- and y-axes. In mathematics, concavity is whether a graph curves up (concave up) or curves down (concave down), as shown in Figure 14.9. 305 Modelling real-life phenomena (e) The ball hits the ground when the height is zero. Is the flying speed of animals related to their overall body length? If you want to save time, do your research and plan ahead. 2 X2+ What about the domain? (a) Eigenvalues: v s vy sfov v v p Hovl (h) Eigenvalues: 1, 4, eigemmurs( J %),(2)1 General solution: x=Ce= ' = 2Ce2 y=VICe + Gt Eigenvalues are complex and the real part is positive, the trajectories move away from the origin in a spiral manner, the origin is an unstable focus, also known as asa node. Typically in chess, a move involves the movement of only one piece. The mathematics it produces can be just as interesting from an insiders viewpoint as the problems of pure mathematics (and often the two are inseparable), but a piece of applied mathematics is judged by whether it can be usefully applied in the world. = 0.297(80)(1.225)V?3 = 29.1V? For example, in the second sequence, each term is 3 times the previous term. Each waveform: * is periodic, repeating after a cycle is completed, over a span of n degrees or radians. The potential differences across a resistor (V; = 6 V), an inductor (V; = 11.5V), and capacitor (V. = 3.5V) are individually measured in a series circuit. A moneychanger exchanges 250 THB (Thai Baht) into 400 PHP (Philippine Pesos). (a) M= 035 cus(jt) +4 ) d"= 0.1307sin () Increamgfmo 0 hence production costs increase for all x > 0. This textbook has been designed so that the chapters proceed in a manner that supports effective learning of the course content. Write out what is meant by: @ 3 3 Yoo, Solution 5 {a) > F =154 25434 =1 45 4 50 n (c) ijpj =xaptxipt =1 7 (b)Y 3 =2 k34 3+ 7=3 t xp, Example 3.16 5 Evaluate ) 2" n=0 Solution 5 D27 =204 2142242324+ 25=63 n=0 Example 3.17 ; 12 GP o Write the sum 5T 78 15 S [ 100 in : sigma notation. (b) To generate the two pieces, we partition the data into two parts, before and after the breakpoint we have chosen visually. (a) Write down the temperature of the potato at the moment it is placed in the oven. Their marginal cost per day is given by the following model: MC(x) = 0.0006x2 0.02x 10 where x is the number of wallets produced. We are told that the test is 99% accurate so, of the 100 cases of the disease the test would show positive in 99 cases and negative in one. Maryam was the first woman to receive the Fields medal (the equivalent of the Nobel prize in mathematics). When using linear regression models, or any other model, we need to be careful to avoid a few common pitfalls. It has a length [cm, width wem and height 20 cm. Estimate the number of pages in this textbook. This is because the maximum value that T can achieve is S. For example, if a can of soda is taken out of a fridge and left in a room with constant temperature of 21C, the temperature of the soda will increase to reach the room temperature. anay=dn D& ap=a,+n1d For the sequence 2, 11, 20, 29, 38, 47, , find: (a) aformula for the nth term (b) the 50th term. Here the evolution of the system over time appears as a closed loop around the stationary point (%, %), which is an attractor of the dynamical system. For thetrigonometric models y = a sinfbtx ) + dand y = a coslbx ) + d + amplitude = o + The amplitude i equal to halfthe wave height. The aims of these courses are to enable students to: Students are also encouraged to appreciate the international dimensions of mathematics and the multiplicity of its cultural and historical perspectives. 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