IB Math Standard Level Year 1: Final Exam Review Alei - Desert Academy

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1 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0- Standard Level Year Final Exam Review Name: Date: Class: You may not use a calculator on problems #- of this review.. Consider the arithmetic sequence, 5, 8,,... (a) Find u 0. Find the value of n so that u n = 5.. Consider the infinite geometric sequence, (0.9), (0.9), (0.9),. (a) Write down the 0 th term of the sequence. Do not simplify your answer. () Find the sum of the infinite sequence.. Let f (x) = ln (x + 5) + ln, for x > 5. (a) Find f (x). Let g (x) = e x. Find (g f) (x), giving your answer in the form ax + b, where a, b,. (Total 5 marks) (Total 7 marks) 4. Let f (x) = (x + ). (a) Show that f (x) = x + 6x 9. For the graph of f (i) write down the coordinates of the vertex; (ii) write down the equation of the axis of symmetry; (iii) write down the y-intercept; (iv) find both x-intercepts. (c) Hence sketch the graph of f. (d) Let g (x) = x. The graph of f may be obtained from the graph of g by the two transformations: a stretch of scale factor t in the y-direction p followed by a translation of. q p Find and the value of t. q 5. Consider f(x) = x 5. (a) Find (i) f(); (ii) f(86); (iii) f(5). (c) Find the values of x for which f is undefined. Let g(x) = x. Find (g f)(x). (8) (Total 5 marks) (Total 7 marks) Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page of 0

2 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0-6. The quadratic function f is defined by f(x) = x x +. (a) Write f in the form f(x) = (x h) k. The graph of f is translated units in the positive x-direction and 5 units in the positive y-direction. Find the function g for the translated graph, giving your answer in the form g(x) = (x p) + q Let M =, and O =. Given that M 6M + ki = O, find k Let f (x) = x x + 5. (a) Express f(x) in the form f(x) = (x h) k. Write down the vertex of the graph of f. (c) Write down the equation of the axis of symmetry of the graph of f. () (d) Find the y-intercept of the graph of f. (e) The x-intercepts of f can be written as Find the value of p, of q, and of r. 9. Let f(x) = x, x 0. (a) Sketch the graph of f. p± r q, where p, q, r. The graph of f is transformed to the graph of g by a translation of. Find an expression for g(x). (c) (i) Find the intercepts of g. (ii) Write down the equations of the asymptotes of g. (iii) Sketch the graph of g. 0. The following diagram shows part of the graph of f, where f (x) = x x. (7) (Total 5 marks) (0) (Total 4 marks) (a) Find both x-intercepts. Find the x-coordinate of the vertex. Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page of 0

3 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0-. In an arithmetic sequence u = 7 and u 4 =. (a) Find (i) the common difference; (ii) the first term. Find S 0.. Let u n = n. (a) Write down the value of u, u, and u. 0 Find ( n ). n=. Solve the following equations. (a) log x 49 = log 8 = x (c) log 5 x = (d) log x + log (x 7) = (Total 7 marks) (5) (Total marks) 4. Part of the graph of a function f is shown in the diagram below. y x 4 (a) On the same diagram sketch the graph of y = f (x). Let g (x) = f (x + ). (i) Find g ( ). (ii) Describe fully the transformation that maps the graph of f to the graph of g. Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page of 0

4 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0-5. Let f be the function given by f(x) = e 0.5x, 0 x.5. The diagram shows the graph of f. (a) On the same diagram, sketch the graph of f. Write down the range of f. (c) Find f (x). () (Total 7 marks) 6. Let f(t) = a cos b (t c) + d, t 0. Part of the graph of y = f(t) is given below. When t =, there is a maximum value of 9, at M. When t = 9, there is a minimum value of 5. (a) (i) Find the value of a. (ii) Show that b = 6 π. (iii) Find the value of d. (iv) Write down a value for c. The transformation P is given by a horizontal stretch of a scale factor of, followed by a (7) translation of. 0 Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 4 of 0

5 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0- Let M be the image of M under P. Find the coordinates of M. The graph of g is the image of the graph of f under P. (c) Find g(t) in the form g(t) = 7 cos B(t C) + D. (d) Give a full geometric description of the transformation that maps the graph of g to the graph of f Let A = and B =. Find (a) A + B; A; (c) AB. 8. Let M =. (a) Write down the determinant of M. Write down M. x 4 (c) Hence solve M =. y 8 (Total 7 marks) () 9. Let A = and B =. p q (a) Find AB in terms of p and q. Matrix B is the inverse of matrix A. Find the value of p and of q. 0. Let X be normally distributed with mean 00 cm and standard deviation 5 cm. (a) On the diagram below, shade the region representing P(X > 05). (5) (Total 7 marks) Given that P(X < d) = P(X > 05), find the value of d. (c) Given that P(X > 05) = 0.6 (correct to two significant figures), find P(d < X < 05). Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 5 of 0

6 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0- You may use a calculator on problems #-8 of this review (though it may not be needed). *. Two boxes contain numbered cards as shown below. Two cards are drawn at random, one from each box. (a) Copy and complete the table below to show all nine equally likely outcomes., 9, 0, 0 Let S be the sum of the numbers on the two cards. Write down all the possible values of S. (c) Find the probability of each value of S. (d) Find the expected value of S. (e) Anna plays a game where she wins $50 if S is even and loses $0 if S is odd. Anna plays the game 6 times. Find the amount she expects to have at the end of the 6 games. *. Consider the infinite geometric sequence 000, 800, 080, 648,. (a) Find the common ratio. (c) Find the 0 th term. Find the exact sum of the infinite sequence. *. Find the term in x in the expansion of x. 4*. (a) Expand (x ) 4 and simplify your result. Find the term in x in (x + 4)(x ) 4. 5*. The function f is defined by f(x) =, for < x <. 9 x (a) On the grid below, sketch the graph of f. 8 (Total marks) (Total 5 marks) Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 6 of 0

7 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0- Write down the equation of each vertical asymptote. (c) Write down the range of the function f. 6*. The functions f and g are defined by f : x α x, g : x α x +. (a) Find an expression for (f g)(x). Find f (8) + g (8). 7*. A theatre has 0 rows of seats. There are 5 seats in the first row, 7 seats in the second row, and each successive row of seats has two more seats in it than the previous row. (a) Calculate the number of seats in the 0th row. Calculate the total number of seats. 8*. A sum of $ 5000 is invested at a compound interest rate of 6. % per annum. (a) Write down an expression for the value of the investment after n full years. (c) What will be the value of the investment at the end of five years? The value of the investment will exceed $ after n full years. (i) Write down an inequality to represent this information. (ii) Calculate the minimum value of n. 9*. Let f (x) = x e x 4, for x 5. (a) Find the x-intercepts of the graph of f. On the grid below, sketch the graph of f. y () () x Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 7 of 0

8 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0-0*. A city is concerned about pollution, and decides to look at the number of people using taxis. At the end of the year 000, there were 80 taxis in the city. After n years the number of taxis, T, in the city is given by T = 80. n. (a) (i) Find the number of taxis in the city at the end of 005. (ii) Find the year in which the number of taxis is double the number of taxis there were at the end of 000. (c) At the end of 000 there were people in the city who used taxis. After n years the number of people, P, in the city who used taxis is given by P =. 0.n 0+ 90e (i) Find the value of P at the end of 005, giving your answer to the nearest whole number. (ii) After seven complete years, will the value of P be double its value at the end of 000? Justify your answer. Let R be the ratio of the number of people using taxis in the city to the number of taxis. The city will reduce the number of taxis if R < 70. (i) Find the value of R at the end of 000. (ii) After how many complete years will the city first reduce the number of taxis? *. Let f(x) = x + 4x 6. (a) Express f(x) in the form f(x) = (x h) + k. Write down the equation of the axis of symmetry of the graph of f. (c) Express f(x) in the form f(x) = (x p)(x q). *. Let f(x) = x cos (x sin x), 0 x. (a) Sketch the graph of f on the following set of axes. (6) (6) (5) (Total 7 marks) () The graph of f intersects the x-axis when x = a, a 0. Write down the value of a. () (Total 4 marks) Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 8 of 0

9 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0- *. Let A =. 0 (a) Write down A. The matrix B satisfies the equation I B = A, where I is the identity matrix. (i) Show that B = (A I). (ii) Find B. (iii) Write down det B. (iv) Hence, explain why B exists. x Let BX = C, where X = y and C =. z (c) (i) Find X. (ii) Write down a system of equations whose solution is represented by X. 4*. In any given season, a soccer team plays 65 % of their games at home. When the team plays at home, they win 8 % of their games. When they play away from home, they win 6 % of their games. The team plays one game. (a) Find the probability that the team wins the game. (6) (5) (Total marks) If the team does not win the game, find the probability that the game was played at home. (Total 8 marks) 5*. The following diagram is a box and whisker plot for a set of data. The interquartile range is 0 and the range is 40. (a) Write down the median value. Find the value of (i) a; (ii) b. (Total 5 marks) 6*. A box contains a large number of biscuits. The weights of biscuits are normally distributed with mean 7 g and standard deviation 0.5 g. (a) One biscuit is chosen at random from the box. Find the probability that this biscuit (i) weighs less than 8 g; (ii) weighs between 6 g and 8 g. Five percent of the biscuits in the box weigh less than d grams. (i) Copy and complete the following normal distribution diagram, to represent this information, by indicating d, and shading the appropriate region. () (ii) Find the value of d. (5) Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 9 of 0

10 IB Math Standard Level Year : Final Exam Review Alei - Desert Academy 0- (c) The weights of biscuits in another box are normally distributed with mean µ and standard deviation 0.5 g. It is known that 0% of the biscuits in this second box weigh less than 5 g. Find the value of µ. (Total marks) 7*. The following table shows the probability distribution of a discrete random variable X. x 0 P (X = x) 0. 0k 0.4 k (a) Find the value of k. Find the expected value of X. (Total 7 marks) 8*. The following is a cumulative frequency diagram for the time t, in minutes, taken by 80 students to complete a task. (a) (c) Write down the median. Find the interquartile range. Complete the frequency table below. Time (minutes) Number of students 0 t < t < 0 0 t < t < t < t < 60 6 () Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 0 of 0

11 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- Standard Level Year Final Exam Review (MarkScheme). (a) d = () evidence of substitution into u n = a + (n ) d e.g. u 0 = + 00 u 0 = 0 N correct approach e.g. 5 = + (n ) correct simplification () e.g. 50 = (n ), 50 = n, 5 = + n n = 5 N. (a) u 0 = (0.9) 9 N recognizing r = 0.9 () correct substitution e.g. S = S = S = 0 N [5]. (a) METHOD ln (x + 5) + ln = ln ((x + 5)) (= ln (x + 0)) () interchanging x and y (seen anywhere) e.g. x = ln (y + 0) evidence of correct manipulation () e.g. e x = y + 0 x e f ( x) = 0 N METHOD y = ln (x + 5) + ln y ln = ln (x + 5) () evidence of correct manipulation () e.g. e y ln = x + 5 interchanging x and y (seen anywhere) e.g. e x ln = y + 5 f (x) = e x ln 5 N METHOD evidence of composition in correct order e.g. (g f) (x) = g (ln (x + 5) + ln ) = e ln ((x + 5)) = (x + 5) (g f) (x) = x + 0 N METHOD evidence of composition in correct order ln(x + 5) + ln e.g. (g f) (x) = e = e ln (x + 5) e ln = (x + 5) (g f) (x) = x + 0 N [7] 4. (a) f (x) = (x + x + ) = x + 6x + = x + 6x 9 AG N0 (i) vertex is (, ) N (ii) x = (must be an equation) N (iii) (0, 9) N () Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page of 4

12 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- (c) (iv) evidence of solving f (x) = 0 e.g. factorizing, formula, correct working 6 ± e.g. (x + )(x ) = 0, x = 6 (, 0), (, 0) NN y x 9 N Notes: Award for a parabola opening upward, for vertex and intercepts in approximately correct positions. (d) p =, t = q (accept p =, q =, t = ) N [5] 5. (a) (i) 6 N (ii) 9 N (iii) 0 N x < 5 A N (c) (g f)(x) = ( x 5 ) = x 5 N [7] 6. (a) For a reasonable attempt to complete the square, (or expanding) e.g. x x + = (x 4x + 4) + f(x) = (x ) (accept h =, k = ) N METHOD Vertex shifted to ( +, + 5) = (5, 4) M so the new function is (x 5) + 4 (accept p = 5, q = 4) N METHOD g(x) = ((x ) h) + k + 5 = ((x ) ) + 5 M = (x 5) + 4 (accept p = 5, q = 4) N k = () M 7 6 = 8 9 A 6 6M = k = k Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page of 4

13 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- k = 5 N 8. (a) Evidence of completing the square f(x) = (x 6x + 9) () = (x ) (accept h =, k = ) N Vertex is (, ) N (c) x = (must be an equation) N (d) evidence of using fact that x = 0 at y-intercept y-intercept is (0, 5) (accept 5) N (e) METHOD evidence of using y = 0 at x-intercept e.g. (x ) = 0 evidence of solving this equation e.g. (x ) = 9. (a) (x ) = ± x = ± 6 = ± 6± 6 x = p = 6, q = 6, r = N4 METHOD evidence of using y = 0 at x-intercept e.g. x x + 5 = 0 evidence of using the quadratic formula ± x = ± x = ± 6 = 4 p =, q = 04, r = 4 (or p = 6, q = 6, r = ) N4 [5] N Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page of 4

14 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- g(x) = x Note: Award for the left branch, and for the right branch. + N (c) (i) Evidence of using x = 0 g (0) = + y = 5 (=.5) evidence of solving y = 0 ( + (x ) = 0) M + x 6 = 0 () x = 5 5 x = Intercepts are x =, y = (accept, 0 0, ) N (ii) x = N y = N (iii) Note: Award for the shape (both branches), for the correct behaviour close to the asymptotes, and 5 5 for the intercepts at approximately, 0 0,. 0. (a) evidence of attempting to solve f (x) = 0 evidence of correct working ± 9 e.g. ( x + )( x ), intercepts are (, 0) and (, 0) (accept x =, x = ) NN evidence of appropriate method N [4] Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 4 of 4

15 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- x+ x b e.g. xv=, xv=, reference to symmetry a x v = 0.5 N. (a) (i) attempt to set up equations 7 = u + 0d and = u + d 4 = 7d d = N (ii) = u 6 u = N u 0 = + 9 = 5 () 0 S 0 = ( + ( 5)) M = 60 N [7]. (a) u =, u =, u = N Evidence of using appropriate formula M correct values S 0 = 0 ( + 9 ) (= 0( 8)) S 0 = 60 N. (a) x = 49 x = ±7 () x = 7 N x = 8 x = N 4. (a) (c) x = 5 x = 5 x = 5 (d) log (x(x 7)) = log (x 7x) = = 8 (8 = x 7x) () x 7x 8 = 0 (x 8)(x + ) = 0 (x = 8, x = ) () x = 8 N [] () N Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 5 of 4

16 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- y x 4 5. (a) M N Note: Award M for evidence of reflection in x-axis, for correct vertex and all intercepts approximately correct. (i) g ( ) = f (0) () f (0) =.5 N (ii) translation (accept shift, slide, etc.) of 0 N N Note: Award for approximately correct (reflected) shape, for right end point in circle, for through (, 0). 0 y.5 N (c) interchanging x and y (seen anywhere) M e.g. x = e 0.5y evidence of changing to log form e.g. ln x = 0.5y, ln x = ln e 0.5y (any base), ln x = 0.5 y ln e (any base) f (x) = ln x N [7] 6. (a) (i) attempt to substitute 9 5 e.g. a = Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 6 of 4

17 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- a = 7 (accept a = 7) N (ii) period = () π b = b = 6 π (iii) attempt to substitute 9+5 e.g. d = d = N (iv) c = (accept c = 9 from a = 7) N Note: Other correct values for c can be found, c = ± k, k. stretch takes to.5 () translation maps (.5, 9) to (4.5, 9) (so M is (4.5, 9)) N (c) g(t) = 7 cos π (t 4.5) + A N4 Note: Award for π, A for 4.5, for. Other correct values for c can be found c = 4.5 ± 6k, k. (d) translation 0 () horizontal stretch of a scale factor of () completely correct description, in correct order N e.g. translation then horizontal stretch of a scale factor of 0 7. (a) evidence of addition e.g. at least two correct elements 4 A + B = 0 N evidence of multiplication e.g. at least two correct elements 6 A = 9 N (c) evidence of matrix multiplication (in correct order) ( ) + ( ) ( 0) + ( ) e.g. AB = ( ) ( )( ) ( ) ( )( ) AB = A N [7] 8. (a) det M = 4 N M = = N Note: Award for and for the correct 4 matrix. AG N0 Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 7 of 4

18 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- (c) X = M 4 4 X = M X = ( x =, y= ) N0 Note: Award no marks for an algebraic solution of the system x + y = 4, x y = (a) evidence of correct method e.g. at least correct element (must be in a matrix) q 0 AB = p 6+ pq + N METHOD evidence of using AB = I correct equations e.g. q = and + p =, 6 + pq = 0 0. (a) p = 4, q = NN METHOD finding A p = p+ 6 evidence of using A = B (M) p e.g. = and = q, = and = q p+ 6 p+ 6 p+ 6 p+ 6 p = 4, q = NN [7] N Note: Award for vertical line to right of mean, for shading to right of their vertical line. evidence of recognizing symmetry e.g. 05 is one standard deviation above the mean so d is one standard deviation below the mean, shading the corresponding part, = 00 d d = 95 N (c) evidence of using complement e.g. 0., p P(d < X < 05) = 0.68 N. (a), 9 4, 9 5, 9, 0 4, 0 5, 0, 0 4, 0 5, 0 A N,, 4, 5 (accept,,,, 4, 4, 4, 5, 5) A N Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 8 of 4

19 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- (c) P() = 9, P() = 9, P = 9, P(5) = 9 (d) correct substitution into formula for E(X) e.g. E(S) = E(S) = 9 A N (e) METHOD correct expression for expected gain E(A) for game () 4 5 e.g E(A) = 9 amount at end = expected gain for game 6 = 00 (dollars) N METHOD attempt to find expected number of wins and losses 4 5 e.g. 6, attempt to find expected gain E(G) e.g E(G) = 00 (dollars) N []. (a) evidence of dividing two terms e.g. 800, r = 0.6 N evidence of substituting into the formula for the 0 th term e.g. u 0 = 000( 0.6) 9 u 0 = 0. (accept the exact value 0.088) N (c) evidence of substituting into the formula for the infinite sum 000 e. g. S=.6 S = 875 N. evidence of using binomial expansion e.g. selecting correct term, a b + a b+ a b +... evidence of calculating the factors, in any order 5 8 e.g. 56,,, ( ) 5 5 x 40x (accept = 400x to s.f.) N [5] 4. (a) evidence of expanding M e.g. (x ) 4 = x 4 + 4x ( ) + 6x ( ) + 4x( ) + ( ) 4 (x ) 4 = x 4 8x + 4x x + 6 A N finding coefficients, 4 (= 7),4 ( 8)(= ) ()() term is 40x N 5. (a) A N Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 9 of 4

20 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- N Note: Award for the general shape and for the y-intercept at. x =, x = NN (c) y A N 6. (a) (f g): x α (x + ) (= x + 6) A N METHOD Evidence of finding inverse functions M e.g. f x (x) = g (x) = x f 8 (8) = (= 6) () g (8) = 8 (= 6) () f (8) + g (8) = = N METHOD Evidence of solving equations M e.g. x = 8, x + = 8 x = 6, x = 6 ()() f (8) + g (8) = = N 7. (a) Recognizing an AP u = 5 d = n = 0 () substituting into u 0 = 5 + (0 ) M = 5 (that is, 5 seats in the 0th row) N 0 0 Substituting into S 0 = ((5) + (0 )) (or into (5 + 5)) M = 680 (that is, 680 seats in total) N 8. (a) 5000(.06) n N Value = $ 5000(.06) 5 (= $ ) = $ 6790 to s.f. (accept $ 6786, or $ ) N (c) (i) 5000(.06) n > or (.06) n > N (ii) Attempting to solve the inequality nlog(.06) > log n >.45 () years N Note: Candidates are likely to use TABLE or LIST on a GDC to find n. A good way of communicating this is suggested below. Let y =.06 x Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 0 of 4

21 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- When x =, y =.958, when x =, y =.086 () x = i.e. years N 9. (a) intercepts when f (x) = 0 (.54, 0) (4., 0) (accept x =.54 x = 4.) N y x N Note: Award for passing through approximately (0, 4), for correct shape, for a range of approximately 9 to.. (c) gradient is N [7] 0. (a) (i) n = 5 () T = T = 49 N (ii) evidence of doubling () e.g. 560 setting up equation e.g. 80. n = 560,. n = n = () in the year 007 N (i) P = 0.( 5) e () P = () P = 9 66 N (ii) P = 0.( 7) e P = not doubled N0 valid reason for their answer R e.g. P < 500 (c) (i) correct value A N 5600 e.g., 9.4, 640:7 80 (ii) setting up an inequality (accept an equation, or reversed inequality) M Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page of 4

22 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0- e. g. P T < 70, n ( 0+ 90e ) 80. n < 70 finding the value 9... () after 0 years N [7]. (a) evidence of obtaining the vertex b e.g. a graph, x =, completing the square a f(x) = (x + ) 8 A N x = (equation must be seen) N (c) f(x) = (x )(x + ) N. (a) A N Notes: Award for correct domain, 0 x. Award A for approximately correct shape, with local maximum in circle and right endpoint in circle. a =. N f ( x) dx fully correct integral expression A.. e.g. V = π [ x cos( x sin x)] dx, V = π [ f ( x)] dx 0 0 V = 5.90 N [8]. (a) A = A N (c) evidence of using V = π [ ] (i) I B = A B = A I B = (A I) AG 4 (ii) B = A N (iii) det B = N (iv) det B 0 R N (c) (i) evidence of using a valid approach M e.g. X = B C Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page of 4

23 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0-0. X = =. 4 N (ii) 4x y + z =, x + y.5z =, x + y + 0.5z = N [] 4. (a) appropriate approach e.g. tree diagram or a table P(win) = P(H W) + P(A W)) = (0.65)(0.8) + (0.5)(0.6) = (or 0.6) N evidence of using complement e.g. p, choosing a formula for conditional probability P( W H ) e.g. P(H W ) = P( W ) correct substitution (0.65)(0.7) e.g. = P(home) = 0.99 N [8] 5. (a) 8 N (i) 0 A N (ii) 44 A N [5] 6. X ~ N (7, 0.5 ) (a) (i) z = P(X < 8) = P(Z < ) = N (ii) evidence of appropriate approach e.g. symmetry, z = P(6 < X < 8) = (tables 0.955) N Note: Award M(AP) if candidates refer to standard deviations from the mean, leading to (i) d N Note: Award for d to the left of the mean, for area to the left of d shaded. (ii) z =.645 () d 7 = d = 6.8 N (c) Y ~ N(µ, 0.5 ) P(Y < 5) = 0. z = µ = µ = 5.4 N Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page of 4

24 IB Math SL Final Exam Review: MarkScheme Alei - Desert Academy 0-7. (a) evidence of using p i = correct substitution e.g. 0k + k =, 0k + k 0.4 = 0 k = 0. A N evidence of using E(X) = p i x i correct substitution () e.g E(X) =.5 N [7] 8. (a) median m = N lower quartile Q =, upper quartile Q = 40 ()() interquartile range = 8 N (c) Time (minutes) Number of students 0 t < t < 0 0 t < t < t < t < 60 6 N [] Z:\Dropbox\Desert\SL\SLFinalExamReview-.doc on 5/7/0 at 5:8:05 PM Page 4 of 4

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