(b) g(x) = 4 + 6(x 3) (x 3) 2 (= x x 2 ) M1A1 Note: Accept any alternative form that is correct. Award M1A0 for a substitution of (x + 3).

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1 Paper. Answers. (a) METHOD f (x) q x f () q 6 q 6 f() p p METHOD f(x) (x ) + 5 x + 6x q 6, p (b) g(x) + 6(x ) (x ) ( + x x ) Note: Accept any alternative form that is correct. Award A for a substitution of (x + ). [6]. (a) A a a a+ () (b) METHOD det A a + a a a a ± N METHOD det A a det A ± a ± N [5] IB Questionbank Mathematics Higher Level rd edition

2 . (a) Note: Award for intercepts of and and a concave down curve in the given domain Note: Award A if the cubic graph is extended outside the domain [, ]. (b) kx ( x+ )( x)dx () Note: The correct limits and must be seen but may be seen later. k ( x + x + x)dx k x + x + x 8 k + + () 8 k [6]. (a) AB + ( ) 88 IB Questionbank Mathematics Higher Level rd edition

3 (b) METHOD arg z, arg z Note: Allow and. Note: Allow degrees at this stage. AÔB (accept ) Note: Allow FT for final. METHOD attempt to use scalar product or cosine rule + cos AÔB A ÔB [6] 5. (a) Note: Award for each correct branch with position of asymptotes clearly indicated. If x is not indicated, only penalise once. A IB Questionbank Mathematics Higher Level rd edition

4 (b) Note: Award for behaviour at x, for intercept at x, for behaviour for large x. A [6] 6. (a) CB b c, AC b + c Note: Condone absence of vector notation in (a). (b) AC CB (b + c) (b c) b c since b c R Note: Only award the and R if working indicates that they understand that they are working with vectors. so AC is perpendicular to CB i.e. A ĈB is a right angle AG [5] 7. (a) area of AOP r sin θ (b) TP r tan θ () area of POT r(r tan θ) r tan θ (c) area of sector OAP r θ area of triangle OAP < area of sector OAP < area of triangle POT R IB Questionbank Mathematics Higher Level rd edition

5 r sinθ < r θ < r sin θ < θ < tan θ tanθ AG [5] 8. x e y y e Note: The is for switching the variables and may be awarded at any stage in the process and is awarded independently. Further marks do not rely on this mark being gained. xe y e y e y xe y + y x± x 8 e y ln x± x + 8 therefore h (x) ln x+ x + 8 since ln is undefined for the second solution Note: Accept y ln x+ x + 8. Note: The R may be gained by an appropriate comment earlier. R [6] IB Questionbank Mathematics Higher Level rd edition 5

6 9. (a) METHOD 8 P( defective in first 8) Note: Award for multiplication of probabilities with decreasing denominators. Award for multiplication of correct eight probabilities. 8 Award for multiplying by METHOD P( defective DVD players from 8) Note: Award for an expression of this form containing three combinations.!!!! 5!6! 5! 8!7! (b) P(9 th selected is th defective player defective in first 8) 7 () P(9 th selected is th 56 defective player) [7] IB Questionbank Mathematics Higher Level rd edition 6

7 . (a) let the first three terms of the geometric sequence be given by u, u r, u r u a + d, u r a + d and u r a + 6d a+ 6d a+ d a+ d a+ d a + 8ad + d a + 6ad + 9d a + d a d () AG d d 9d (b) u, ur, ur r geometric th term u r 7d arithmetic 6 th term a + 5d d+ 5d 7d Note: Accept alternative methods. [8]. (a) dy x x dx x x x, ± dy at,,,,, dx Note: Award A for all three x-values correct with errors/omissions in y-values. IB Questionbank Mathematics Higher Level rd edition 7

8 (b) at x, gradient of tangent () Note: In the following, allow FT on incorrect gradient. equation of tangent is y (x ) y x + () meets x-axis when y, (x ) x coordinates of T are, () (c) gradient of normal () equation of normal is y at x, y 8 8 ( x ) y x+ () Note: In the following, allow FT on incorrect coordinates of T and N. lengths of PN area of triangle PTN 5, PT (or equivalent e.g. ) 9 8 [5]. (a) using the factor theorem z + is a factor () z + (z + )(z z + ) IB Questionbank Mathematics Higher Level rd edition 8

9 (b) (i) METHOD z z + (z + )(z z + ) solving z z + ± ± i z therefore one cube root of is γ METHOD () AG γ i + + i γ + i + i AG METHOD i + i γ e γ e i (ii) METHOD as γ is a root of z z + then γ γ + γ γ Note: Award for the use of z z + in any way. Award R for a correct reasoned approach. R AG METHOD γ + i + i + i γ IB Questionbank Mathematics Higher Level rd edition 9

10 (iii) METHOD ( γ) 6 ( γ ) 6 () (γ) (γ ) () ( ) METHOD ( γ) 6 6γ + 5γ γ + 5γ 6γ 5 + γ 6 Note: Award for attempt at binomial expansion. use of any previous result e.g. 6γ + 5γ + 5γ + 6γ + Note: As the question uses the word hence, other methods that do not use previous results are awarded no marks. (c) METHOD γ γ γ γ + A γ γ γ γ γ γ + γ + A A + I γ + γ γ from part (b) γ γ + γ + ( γ γ + ) γ γ + + ( γ γ + ) γ γ γ hence A A + I AG METHOD + i A i Note: Award mark for each of the non-zero elements expressed in this form. verifying A A + I (d) (i) A A I IB Questionbank Mathematics Higher Level rd edition

11 A A A A I A I AG Note: Allow other valid methods. (ii) I A A A A A A A A I A Note: Allow other valid methods. AG []. (a) (i) Note: Award for correct sin x, for correct sin x. Note: Award A for two correct shapes with and/or missing. Note: Condone graph outside the domain. A (ii) sin x sin x, x sin x cos x sin x sin x ( cos x ) x, NN IB Questionbank Mathematics Higher Level rd edition

12 (iii) area (sinx sinx) dx Note: Award for an integral that contains limits, not necessarily correct, with sin x and sin x subtracted in either order. cos x + cos x cos + cos cos + cos () (b) x sin θ dx 8 sin θ cos θ dθ x sin θ 6 Note: Award for substitution and reasonable attempt at finding expression for dx in terms of dθ, first for correct limits, second for correct substitution for dx. 6 8 sin θdθ 6 cos θdθ [ sin θ] 6 θ sin () IB Questionbank Mathematics Higher Level rd edition

13 (c) (i) from the diagram above the shaded area a b f ( x)dx ab f ( y)dy R ab b f ( x)dx AG x (ii) f(x) arcsin 6 x arcsin dx f (x) sin x sin xdx Note: Award for the limit 6 seen anywhere, for all else correct. + [ cos x] 6 Note: Award no marks for methods using integration by parts. [5] IB Questionbank Mathematics Higher Level rd edition

14 Paper. Answers. area of triangle POQ 8 sin () area of sector () area between arc and chord (cm ) [5]. u u + d 7, u 9 u + 8d Note: 5d 5 gains both above marks u, d S n n ( + (n )) > n 8 [5]. (a) a e.t ()() at t, a.5 (m s ) (accept e ) (b) METHOD.t d 5( e )dt () so distance above ground 7 (m) ( s.f.) (accept 76 (m)) METHOD s 5( e.t )dt 5t + 5e.t (+ c) Taking s when t gives c 5 So when t, s 8... so distance above ground 7 (m) ( s.f.) (accept 76 (m)) [6]. (a) det A cos θ cos θ + sin θ sin θ cos (θ θ) Note: Allow use of double angle formulae if they lead to the correct answer cos θ AG IB Questionbank Mathematics Higher Level rd edition

15 (b) cos θ sin θ θ.666,.8 [6] 5. Note: Award for both vertical asymptotes correct, for recognizing that there are two turning points near the origin, for both turning points near the origin correct, (only this A mark is dependent on the M mark) for the other pair of turning points correct, for correct positioning of the oblique asymptote, for correct equation of the oblique asymptote, for correct asymptotic behaviour in all sections. [7] 6. (a) P(x <.).69 (accept.69) (b) METHOD Y ~ B(6,.85...) () P(Y ) P(Y ) ().775 (accept.778 if s.f. approximation from (a) used) IB Questionbank Mathematics Higher Level rd edition 5

16 METHOD X ~ B(6,.69...) () P(X ) ().775 (accept.778 if s.f. approximation from (a) used) (c) P(x < x <.) P( x< ) P( x<.) (accept.967) [6] 7. (a) x + x + (.6, ) ( (, )) (b) f (.59...) ( ) g (.59...) ( ) required angle arctan (accept.7 ) Note: Accept alternative methods including finding the obtuse angle first. [5] 8. let the length of one side of the triangle be x consider the triangle consisting of a side of the triangle and two radii EITHER x r + r r cos r OR x r cos IB Questionbank Mathematics Higher Level rd edition 6

17 THEN x r so perimeter r now consider the area of the triangle area r sin P A r r r r Note: Accept alternative methods [6] 9. let x distance from observer to rocket let h the height of the rocket above the ground METHOD dh when h 8 dt x h + 6 ( h + 6) dx h dh h + 6 when h 8 dx dx dh dt dh dt h h + 6 (m s ) IB Questionbank Mathematics Higher Level rd edition 7

18 METHOD h + 6 x dx h x d h dx h dh x 8 5 dh dt dx dx dh dt dh dt 5 (m s ) METHOD x 6 + h x x dh h dt dt when h 8, x dx 8 dh dt dt m s METHOD Distance between the observer and the rocket ( ) Component of the velocity in the line of sight sin θ (where θ angle of elevation) 8 sin θ component (m s ) [6] IB Questionbank Mathematics Higher Level rd edition 8

19 . x + y a dy x + y dx dy dx x y y x dy a Note: Accept from making y the subject of the equation, dx x and all correct subsequent working therefore the gradient at the point P is given by dy dx q p q equation of tangent is y q ( x p) p q (y x+ q+ q p ) p x-intercept: y, n y-intercept: x, m n + m q p + p q p+ p q q p + q q p + p+ q p + p q p + p+ q ( p + q ) a AG [8]. (a) x PQ,SR 5 y z () point S (, 6, ) IB Questionbank Mathematics Higher Level rd edition 9

20 (b) PQ PS PQ PS 7 m (c) area of parallelogram PQRS PQ PS ( ) ( ).9 (d) equation of plane is x + 7y z d substituting any of the points given gives d x + 7y z (e) equation of line is r + λ 7 Note: To get the must have r or equivalent. (f) 69λ + 9λ + λ λ (.9...) closest point is, , 7 ( (.9,.,.97)) (g) angle between planes is the same as the angle between the normals (R) + 7 cos θ 6 θ (accept θ 7. or.9 radians or.65 radians) [7]. (a) P(x ).67 IB Questionbank Mathematics Higher Level rd edition

21 (b) EITHER Using X ~ Po() OR Using ( ) 6 THEN P(X ).98 () () (c) X ~ Po(.5t) () P(x ) P(x ) () P(x ) <. e.5t <..5t < ln (.) () t > 9. months therefore months N Note: Full marks can be awarded for answers obtained directly from GDC if a systematic method is used and clearly shown. (d) (i) P( or accidents).798 E(B) $796 (accept $797 or $796.7) (ii) P() P(,, ) + P(,, ) + P(,, ) + P(, 5, 5) + P(5,, 5) + P(5, 5, ) ()() Note: Award for noting that can be written both as + and 5 +. ( ) (.7...) + (.79...) ( ).77 (accept.78) [8] IB Questionbank Mathematics Higher Level rd edition

22 . prove that n for n + LHS, RHS so true for n assume true for n k so + + now for n k + LHS: k k k k n k+ k k n+ n + ( k+ ) k R k+ + ( k+ ) k ( k+ ) k+ + (or equivalent) k k ( k+ ) + k+ (accept ( k+ ) k ) Therefore if it is true for n k it is true for n k +. It has been shown to be true for n so it is true for all n ( ). R Note: To obtain the final R mark, a reasonable attempt at induction must have been made. [8]. (a) METHOD e x sin x dx cos xe x + e x cos x dx cos xe x + e x sin x e x sin x dx 5 e x sin x dx cos xe x + e x sin x e x sin x dx 5 e x ( sin x cos x) + C AG METHOD sin xe x x x sin xe e dx cos x dx x x x sin xe e e cos x sin x dx x x 5 x e sin x cos xe e sin d x x x x e sin xdx e (sin x cosx) + C 5 AG (b) y y d x e sin xdx IB Questionbank Mathematics Higher Level rd edition

23 arcsin y 5 e x ( sin x cos x)(+ C) when x, y C 5 y sin e x ( sin x cos ) (c) (i) P is (.6, ) Note: Award for.6 seen anywhere, for complete sketch. Note: Allow FT on their answer from (b).6... (ii) V y dx.5 A Note: Allow FT on their answers from (b) and (c)(i). [5] IB Questionbank Mathematics Higher Level rd edition

24 Paper. Answers. (a) (i) P(A B) P(A) + P(B).7 (ii) P(A B) P(A) + P(B) P(A B) () P(A) + P(B) P(A)P(B) () (b) P(A B) P(A) + P(B) P(A B) P( A B) P(A B) P( B) ()..5. [7]. METHOD z ( i)(z + ) z + iz i z( i) + i + i z i + i + i z i + i i METHOD let z a + ib a+ ib i a+ ib+ a + ib ( i)((a + ) + ib) a + ib (a + ) + bi i(a + ) + b a + ib a + b + + (b a )i attempt to equate real and imaginary parts a a + b + ( a + b + ) and b b a ( a + b ) Note: Award for two correct equations. b ;a z i [] IB Questionbank Mathematics Higher Level rd edition

25 . (a) u r r (b) v 9 v d 8 d v N ( (N )) > (accept equality) N ( N) > N(5 N) > N < 7.5 N 7 Note: > N 7 or equivalent receives full marks. () () () [7]. (a) AB b a CB a + b (b) AB CB (b a) (b + a) b a since b a R Note: Only award the and R if working indicates that they understand that they are working with vectors. so AB is perpendicular to CB i.e. A Bˆ C is a right angle AG [5] IB Questionbank Mathematics Higher Level rd edition 5

26 sin θ sinθ cosθ 5. (a) + cos θ + cos θ Note: Award for use of double angle formulae. sinθ cosθ cos θ sinθ cosθ tan θ AG (b) tan 8 cot 8 sin + cos () + cos sin + + [5] 6. R is rabbit with the disease P is rabbit testing positive for the disease IB Questionbank Mathematics Higher Level rd edition 6

27 (a) P(P) P(R P) + P(R P) (.9) Note: Award for a correct tree diagram with correct probability values shown (b) P(R P) < % (or other valid argument).89. R [5] 7. METHOD area arctan x dx attempting to integrate by parts [ x arctan x] x dx + x [ x arctan x] ln(+ x ) Note: Award even if limits are absent. ln ln METHOD area tan y dy [ln cos y ] + + ln ln [6] 8. (a) (i) (g f)(x), x x+ (or equivalent) (ii) (f g)(x) x +, x (or equivalent) IB Questionbank Mathematics Higher Level rd edition 7

28 (b) EITHER f(x) (g f g)(x) (g f)(x) (f g)(x) + x+ x OR (g f g)(x) x + THEN + x + x 6x + x + 6 (or equivalent) x, y (coordinates are (, )) () [6] 9. Attempt at implicit differentiation e (x+y) dy dy + sin( xy) x + y dx dx let x, y e dy + d x d y dx let x, y e dy dy + sin( ) x + y dx dx d y so dx since both points lie on the line y x this is a common tangent Note: y x must be seen for the final R. It is not sufficient to note that the gradients are equal. R [7] IB Questionbank Mathematics Higher Level rd edition 8

29 . (a) f (x a) b () x and x a (or equivalent) (b) vertical asymptotes x, x a horizontal asymptote y Note: Equations must be seen to award these marks. maximum a, b Note: Award for correct x-coordinate and for correct y-coordinate. one branch correct shape other branches correct shape [8]. (a) AB, AC Note: Accept row vectors. IB Questionbank Mathematics Higher Level rd edition 9

30 (b) AB AC i normal n x + y + z 7 j k so r Note: If attempt to solve by a system of equations: Award for correct equations, for eliminating a variable and A for the correct answer. () 5 (c) r + λ (or equivalent) 7 (5 + λ) + ( + λ) + (7 + λ) 7 9λ 8 λ Note: λ if 8 6 is used. 6 distance + + () 6 (d) (i) area AB AC () ) (accept 576 (ii) EITHER volume area height () OR 6 volume ( AD (AB AC) ) 6 (e) AB AC IB Questionbank Mathematics Higher Level rd edition

31 AC AD EITHER i j k 6 9i j + 6k > therefore since area of ACD bigger than area ABC implies that B is closer to opposite face than D R OR correct calculation of second distance as which is smaller than Note: Only award final R in each case if the calculations are correct. R [9]. (a) (i) x ln x f (x) x x ln x x so f (x) when ln x, i.e. x e (ii) f (x) > when x < e and f (x) < when x > e R hence local maximum AG Note: Accept argument using correct second derivative. (iii) y e IB Questionbank Mathematics Higher Level rd edition

32 x ( ln x)x (b) f (x) x x x x+ x ln x x + ln x x Note: May be seen in part (a). f (x) + ln x x e since f (x) < when x < e and f (x) > when x > e R then point of inflexion e, e (c) Note: Award for the maximum and intercept, for a vertical asymptote and for shape (including turning concave up). IB Questionbank Mathematics Higher Level rd edition

33 (d) (i) Note: Award for each correct branch. (ii) all real values (iii) Note: Award ()() for sketching the graph of h, ignoring any graph of g. ()() e < x < (accept x < ) [9]. (a) (cos θ + i sin θ) cos θ + cos θ(i sin θ) + cos θ(isin θ) + (isin θ) () cos θ cos θ sin θ + i( cos θ sin θ sin θ) IB Questionbank Mathematics Higher Level rd edition

34 (b) from De Moivre s theorem (cos θ + i sin θ) cos θ + i sin θ () cos θ + i sin θ (cos θ cos θ sin θ) + i( cos θ sin θ sin θ) equating real parts cos θ cos θ cos θ sin θ cos θ cos θ ( cos θ) cos θ cos θ + cos θ cos θ cos θ AG Note: Do not award marks if part (a) is not used. (c) (cos θ + i sin θ) 5 cos 5 θ + 5 cos θ (i sin θ) + cos θ(i sin θ) + cos θ(i sin θ) + 5cos θ (i sin θ) + (i sin θ) 5 () from De Moivre s theorem cos 5θ cos 5 θ cos θ sin θ + 5 cos θ sin θ cos 5 θ cos θ ( cos θ) + 5cos θ( cos θ) cos 5 θ cos θ + cos 5 θ + 5 cos θ cos θ + 5 cos 5 θ cos 5θ 6 cos 5 θ cos θ + 5 cos θ AG Note: If compound angles used in (b) and (c), then marks can be allocated in (c) only. (d) cos 5θ + cos θ + cos θ (6 cos 5 θ cos θ + 5 cos θ) + ( cos θ cos θ) + cos θ 6 cos 5 θ 6 cos θ + cosθ cos θ (6 cos θ 6 cos θ + ) cos θ ( cos θ )( cos θ ) cos θ ; ± ; ± θ ± ; ± ; ± 6 A IB Questionbank Mathematics Higher Level rd edition

35 (e) cos 5θ 5 7 5θ... ; ; ; ; θ... ; ; ; ;... Note: These marks can be awarded for verifications later in the question. now consider 6 cos 5 θ cos θ + 5 cos θ cos θ (6 cos θ cos θ + 5) cos θ cos θ ± () () ± (6)(5) ; cos θ ± (6)(5) + (6)(5) cos ± since max value of cosine angle closest to zero cos.5+ 5 (5) cos 8 R [] IB Questionbank Mathematics Higher Level rd edition 5

36 Paper. Answers. (a) (i) median grams Note: Accept 5. (ii) th percentile 9 grams (b) 8 9 () Note: Accept answers to. []. (a) f (x) x 6x 9 ( ) () (x + )(x ) x ; x (max)(, 5); (min)(, 7) Note: The coordinates need not be explicitly stated but the values need to be seen. y 8x + 7 N (b) f (x) 6x 6 inflexion (, ) which lies on y 8x + 7 RAG [6]. METHOD sin C sin 7 5 () B ĈD 6... CD 5cos 6... Note: so allow use of sine or cosine rule. CD.6 IB Questionbank Mathematics Higher Level rd edition 6

37 METHOD let AC x cosine rule x 7 x cos x.7... x ± (.7...) x x 7.5;.8 CD is the difference in these two values.6 Note: Other methods may be seen. () () [5]. (a) f(a) a + a 7a a + a 7a + (a + ) (a 6a + 5) or sketch or GDC () a (b) substituting a into f(x) f(x) x x + EITHER graph showing unique solution which is indicated (must include max and min) OR convincing argument that only one of the solutions is real (.7,.868±.i) R R [5] 5. (a) x + x (x + )(x ) Note: Accept x + (x ). Note: Either of these may be seen in (b) and if so should be awarded. IB Questionbank Mathematics Higher Level rd edition 7

38 (b) EITHER (x + x ) 8 (x + ) 8 (x ) 8 ( 8 + 8( 7 )(x) +...)(( ) 8 + 8( ) 7 (x) +...) () coefficient of x 8 8 ( ) ( ) Note: Under FT, final can only be achieved for an integer answer. OR (x + x ) 8 ( (x x )) ( (x x )( 7 ) +...) () coefficient of x 8 ( ) Note: Under FT, final can only be achieved for an integer answer. [5] IB Questionbank Mathematics Higher Level rd edition 8

39 6..5 α arcsin ( α ) 7.5 β arcsin ( β ) 5 Note: Allow use of cosine rule. area P 7 (α sin α).8... area Q 5 (β sin β) Note: The is for an attempt at area of sector minus area of triangle. Note: The use of degrees correctly converted is acceptable. area 8.(cm ) () () () () [7] 7. (a) k ( x+ ) dx+ kdx 8 k + k k Note: Only FT on positive values of k. IB Questionbank Mathematics Higher Level rd edition 9

40 (b) (i) E(X) x ( x+ ) dx+ xdx + 9 (.) 9 (ii) median given by a such that P (X < a).5 a ( + ) d.5 x x ( x+ ) a (a + ) 6 a 6 (.8) () [7] 8. (a) 5 equation of line in graph a t a t + 5 (b) dv 5 t+ 5 dt () 5 v t + 5t + c () when t, v 5 m s 5 v t + 5t + 5 from graph or by finding time when a maximum 95 m s IB Questionbank Mathematics Higher Level rd edition

41 (c) EITHER graph drawn and intersection with v 95 m s t OR ()() 5 95 t + 5t + 5 t ;.9... ()() t (8 ) [8] 9. log x+ y log y+ x so (x + ) y ( y +) x EITHER x (x + ) x, not possible x.7, y 7.7 R IB Questionbank Mathematics Higher Level rd edition

42 OR ( x + x+ ) x attempt to solve or graph of LHS x.7, y 7.7 [6]. METHOD equation of journey of ship S r t equation of journey of speedboat S,setting off k minutes later 7 6 r + ( t k) Note: Award for perpendicular direction, for speed, for change in parameter (e.g. by using t k or T, k being the time difference between the departure of the ships). 7 6 solve t + ( t k) () Note: M mark is for equating their two expressions. t 7 6t + 6k t + t k Note: M mark is for obtaining two equations involving two different parameters. 7t 6k 7 t + k 8 k 5 latest time is :5 IB Questionbank Mathematics Higher Level rd edition

43 METHOD y O (A) time 6 5 taken t 58 S (6,5) 5 time taken t - k B (7,) x SB 5 (by perpendicular distance) SA 6 5 (by Pythagoras or coordinates) 6 5 t 5 5 t k 5 8 k leading to latest time :5 5 [7]. (a) x y z k ( + 6) + ( + ) 7 since determinant unique solution to the system planes intersect in a point Note: For any method, including row reduction, leading to the explicit 6 5k + k k solution,,, award for an attempt at a correct method, for two correct coordinates and for a third correct coordinate. R AG IB Questionbank Mathematics Higher Level rd edition

44 IB Questionbank Mathematics Higher Level rd edition (b) + + a a a a((a + )(a + ) ) ( (a + ) + 6) + ( + (a + )) () planes not meeting in a point no unique solution i.e. determinant () a(a + a ) + (a 8) + (a + ) a + a + a 7 a [5] (c) r r k r r k + + () 5 r r k + () for an infinite number of solutions to exist, + k k x + y + z y + z + λ z y x Note: Accept methods involving elimination. Note: Accept any equivalent form e.g. + + or λ λ z y x z y x. Award A if z y x or r is absent. []

45 . (a) P(X < ). P(X < 55).9 or relevant sketch () X µ given Z σ µ P(Z < z) σ () µ P(Z < z) σ () µ + (.5...) σ 55 (.8...) σ σ 6., µ. Note: Accept 6 and. Note: Working with 8 and 855 will only gain the two M marks. (b) X ~ N(..., ) late to school X > 6 P(X > 6) () number of students late () 67 (to nearest integer) Note: Accept 6 for use of and 6. (c) P( X > 6) P(X > 6 X > ) P( X > ).95 (accept anything between.9 and.9) Note: If and 6 are used.87 is obtained. This should be accepted. (d) let L be the random variable of the number of students who leave school in a minute interval since 7 L ~ Po(7) P(L 7) P(L 699) ().777 Note: Award A for P (L > 7) P(L 7) (this leads to.765). (e) (i) Y ~ B(,.7767 ) () E(Y) Note: On FT, use of.765 will lead to 5. IB Questionbank Mathematics Higher Level rd edition 5

46 (ii) P(Y > 5) P(Y 5) ().797 Note: Accept.799 from using rounded answer. Note: On FT, use of.765 will lead to.666. [7]. (a) A cosθ sinθ cosθ sinθ sinθ cosθ sinθ cosθ cos θ sin θ cosθ sinθ+ sinθ cosθ sinθ cosθ cosθ sinθ sin θ + cos θ cos θ sin θ sinθ cosθ sinθ cosθ cos θ sin θ cos θ sin θ sin θ cos θ () AG IB Questionbank Mathematics Higher Level rd edition 6

47 cosθ sinθ cos nθ sin nθ (b) let P(n) be the proposition that sinθ cosθ sin nθ cos nθ + for all n P() is true cosθ sinθ cosθ sinθ cosθ sinθ assume P(k) to be true sinθ cosθ Note: Must see the word true or equivalent, that makes clear an assumption is being made that P(k) is true. n cosθ sinθ sinθ cosθ consider P(k + ) k cos kθ sin kθ sin kθ cos kθ k+ k cosθ sinθ cosθ sinθ cosθ sinθ sinθ cosθ sinθ cosθ sinθ cosθ () cos kθ sin kθ cosθ sinθ sin kθ cos kθ sinθ cosθ cos kθ cosθ sin kθ sinθ cos kθ sinθ+ sin kθ cosθ sin kθ cosθ cos kθ sinθ sin kθ sinθ + cos kθ cosθ cos( k+ ) θ sin( k+ ) θ sin( k+ ) θ cos( k+ ) θ if P(k) is true then P(k + ) is true and since P() is true then P(n) is true + for all n R Note: The final R can only be gained if the has been gained. IB Questionbank Mathematics Higher Level rd edition 7

48 (c) EITHER A cos( θ ) sin( θ ) from formula sin( θ ) cos( θ ) cosθ sinθ sinθ cosθ A A AA cosθ sinθ cosθ sinθ cosθ sinθ cosθ sinθ cosθ sinθ Note: Accept either just A A or just AA. A is inverse of A OR A cosθ sinθ cos θ + sin θ sinθ cosθ A cosθ sinθ sinθ cosθ putting n in formula gives inverse sinθ cosθ sinθ cosθ sinθ cosθ []. (a) volume h x y () h y d d y y h h IB Questionbank Mathematics Higher Level rd edition 8

49 (b) dv dt surface area surface area x () h h V since V h dv dt dv dt V V AG d h Note: Assuming that without justification gains no marks. dt [6] (c) V 5 ( 57 cm ) dv V dt attempting to separate variables EITHER dv d V t V t+ c c 5 V 5 t hours OR 5 dv V T d t Note: Award for attempt to use definite integrals, for correct limits and for correct integrands. [ V] T 5 5 T hours [6] IB Questionbank Mathematics Higher Level rd edition 9

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