PaperPractice [89 marks] INSTRUCTIONS TO CANDIDATE Write your session number in the boxes above. Do not open this examination paper until instructed to do so. You are not permitted access to any calculator for this paper. Section A: answer all questions in the boxes provided. Section B: answer all questions in the answer booklet provided. Fill in your session number on the front of the answer booklet, and attach it to this examination paper and your cover sheet using the tag provided. Unless otherwise stated in the question, all numerical answers should be given exactly or correct to three significant figures. A clean copy of the Mathematics SL formula booklet is required for this paper The maximum mark for this examination paper is [90 marks].. Let cosxdx =, where < a <. Find the value of a. a [7 marks] correct intration (ignore absence of limits and +C ) (A) (A) setting their result from an intrated function equal to correct value sin(x), cosx = [ sin(x)] substituting limits into their intrated function and subtracting (in any order) a = 5 A [7 marks] a sin(a) sin(), sin() sin(a) sin() = 0 sin a =, sin(a) = recognizing = (A) a =, a = sin (A) +, a =, a = + N 5 a M (M) Let f(x) = p x + p x + qx. a. Find f (x). [ marks] f (x) = p x + px + q A N Note: Award A if only error. [ marks] b. Given that f (x) 0, show that p pq. [5 marks]
b. evidence of discriminant (must be seen explicitly, not in quadratic formula) (M) correct substitution into discriminant (may be seen in inequality) f (x) 0 then f has two equal roots or no roots (R) recognizing discriminant less or equal than zero R correct working that clearly leads to the required answer p b [5 marks] ac (p) p q, p pq Δ 0, p pq 0 p pq 0, p pq pq AG N0 A A Find the value of each of the following, giving your answer as an inter. a. log 6 6 [ marks] correct approach A [ marks] 6 x = 6, 6 N (A) b. + 9 log 6 log 6 [ marks] correct simplification (A) log 6 6, log( 9) A N [ marks] c. log 6 log 6 correct simplification correct working (A) (A) log 6, log( ) log 6, 6 =, = 6 6 6x 6 A N
The following diagram shows the graph of y = f(x), for x 5. a. Write down the value of f( ). [ mark] f( ) = [ mark] A N b. Write down the value of f (). [ mark] f () = 0 (accept y = 0) A N [ mark] c. Find the domain of f. [ marks] domain of f is range of \(f\) (R) Rf = Df correct answer A N x, x [, ] (accept < x <, y ) [ marks] d. On the grid above, sketch the graph of f.
AA N Note: Graph must be approximately correct reflection in y = x. Only if the shape is approximately correct, award the following: A for x-intercept at, and A for endpoints within circles. [ marks] The line L is parallel to the vector ( ). 5a. Find the gradient of the line L. [ marks] attempt to find gradient (M) reference to change in x is and/or y is, gradient A N [ marks] = The line L passes through the point (9, ). 5b. Find the equation of the line L in the form y = ax + b. attempt to substitute coordinates and/or gradient into Cartesian equation for a line (M) y = m(x 9), y = x + b, 9 = a() + c correct substitution (A) = (9) + c, y = (x 9) y = x (accept a =, b = ) A N
5c. Write down a vector equation for the line L. [ marks] any correct equation in the form r = a + tb (any parameter for t), where a indicates position ( 9 0 ) or ( ), and b is a scalar multiple of ( ) 9 x t + 9 r = ( ) + t ( ),( ) = ( ), r = 0i j + s(i + j) A N y t + Note: Award A for a + tb, A for L = a + tb, A0 for r = b + ta. [ marks] 6. The graph of a function h passes through the point (,5). Given that h (x) = cosx, find h(x). evidence of anti-differentiation correct intration (A) (M) attempt to substitute (,5) into their equation (M) correct working (A) ( ) + c = 5, c = h(x) = sin x + A [6 marks] h (x), cosxdx sin x h(x) = sin x + c, sin( ) + c = 5, sin( ) = 5 N5 6 [6 marks] The following diagram shows part of the graph of y = f(x). The graph has a local maximum at A, where x =, and a local minimum at B, where x = 6. 7a. On the following axes, sketch the graph of y = f (x). [ marks]
AAAA N Note: Award A for x-intercept in circle at, A for x-intercept in circle at 6. Award A for approximately correct shape. Only if this A is awarded, award A for a native y-intercept. [ marks] 7b. Write down the following in order from least to greatest: f(0), f (6), f ( ). [ marks] f ( ), f (6), f(0) A [ marks] N The sums of the terms of a sequence follow the pattern S = + k, S = 5 + k, S = + 7k, S = + 5k,, where k Z. 8a. Given that u = + k, find u, u and u. [ marks] valid method [ marks] (M) u = S S, + k + u = 5 + k u = + k, u = 7 + k, u = 0 + 8k AAA N 8b. Find a general expression for u n. [ marks]
correct AP or GP (A) finding common difference is, common ratio is valid approach using arithmetic and geometric formulas + (n ) and r n k u n = n + n k AA N (M) Note: Award A for n, A for n k. [ marks] Consider the vectors a = ( ) and b = ( ). (a) 9a. Find (i) a + b ; (ii) a + b. [6 marks] Let a + b + c = 0, where 0 is the zero vector. (b) Find c. (a) (i) a = ( ) 6 (A) correct expression for a + b A N 5 ( ), (5, ), 5i j (ii) correct substitution into length formula (A) 5 +, 5 + a + b = 9 A N [ marks] (b) valid approach (M) c = (a + b), 5 + x = 0, + y = 0 5 c = ( ) [ marks] A N Find 9b. (i) a + b ; (ii) a + b. [ marks]
(i) a = ( ) 6 (A) correct expression for a + b A N 5 ( ), (5, ), 5i j (ii) correct substitution into length formula (A) 5 +, 5 + a + b = 9 A N [ marks] Let a + b + c = 0, where 0 is the zero vector. Find c. 9c. [ marks] valid approach (M) c = (a + b), 5 + x = 0, + y = 0 5 c = ( ) [ marks] A N Consider f(x) = x sin x. Find f (x). 0a. [ marks] evidence of choosing product rule (M) u v + vu correct derivatives (must be seen in the product rule) cosx, x (A)(A) f (x) = x cosx + xsin x A N [ marks] 0b. Find the gradient of the curve of f at x =.
substituting into their f (x) (M) f ( ), correct values for both sin and cos seen in f (x) (A) f ( ) 0 + ( ) ( ) = A N cos( ) + ( )sin( ) Let f(x) = x 5, for x 5. Find f. a. () METHOD attempt to set up equation (M) = y 5, = x 5 correct working (A) = y 5, x = + 5 f () = 9 A N METHOD interchanging x and y (seen anywhere) (M) x = y 5 correct working (A) x = y 5, y = x + 5 f () = 9 A N Let g be a function such that g exists for all real numbers. Given that, find. b. g(0) = (f g )() recognizing g () = 0 (M) f(0) correct working (A) (f g )() = 0 5, 5 (f g )() = 5 A N Note: Award A0 for multiple values, ±5. a. Find the value of log 0 log 5.
evidence of correct formula (M) loga logb = log a, log( ), log8 + log5 log5 b 5 Note: Ignore missing or incorrect base. 0 correct working log 8, = 8 0 5 = log log (A) A N 5 Find the value of 8. b. log [ marks] attempt to write 8 as a power of (seen anywhere) (M) ( ) log 5, = 8, a multiplying powers (M) log 5, alog 5 correct working (A) log 5, log 5 5, ( ) log log 5 8 = 5 [ marks] A N Let log p = 6 and log q = 7. (a) Find log. a. p p q (b) Find log ( ). [7 marks] (c) Find log (9p).
(a) METHOD evidence of correct formula log u n = n logu, log p log ( p ) = METHOD A N valid method using p = 6 log ( 6 ), log, log log ( p ) = [ marks] A N (M) (M) (b) METHOD evidence of correct formula (M) p log( ) = logp logq, 6 7 q p q log ( ) = METHOD A N valid method using p = 6 and q = 7 (M) log ( 6 ), log, log 7 p q log ( ) = [ marks] A N (c) METHOD evidence of correct formula (M) log uv = log u + log v, log9 + logp log 9 = (may be seen in expression) A + logp log (9p) = 8 A N METHOD valid method using p = 6 (M) log (9 6 ), log ( 6 ) correct working A log 9 + log 6, log 8 log (9p) = 8 A N Total [7 marks] b. Find log p. [ marks]
METHOD evidence of correct formula log u n = n logu, log p log ( p ) = METHOD A N valid method using p = 6 log ( 6 ), log, log log ( p ) = [ marks] A N (M) (M) p q c. Find log ( ). [ marks] METHOD evidence of correct formula (M) p log( ) = logp logq, 6 7 q p q log ( ) = METHOD A N valid method using p = 6 and q = 7 (M) log ( 6 ), log, log 7 p q log ( ) = [ marks] A N d. Find log (9p).
METHOD evidence of correct formula (M) log uv = log u + log v, log9 + logp log 9 = (may be seen in expression) A + logp log (9p) = 8 A N METHOD valid method using p = 6 (M) log (9 6 ), log ( 6 ) correct working A log 9 + log 6, log 8 log (9p) = 8 A N Total [7 marks] In the following diagram, OP = p, OQ = q and PT = PQ. Express each of the following vectors in terms of p and q, a. QP; [ marks] appropriate approach (M) QP = QO + OP, P Q QP = p q A N [ marks] b. OT. recognizing correct vector for QT or PT (A) QT = (p q), PT = (q p) appropriate approach (M) OT = OP + PT, OQ + QT, + OP PQ p+q OT = (p + q) (accept ) A N
The following is a cumulative frequency diagram for the time t, in minutes, taken by 80 students to complete a task. 5a. Find the number of students who completed the task in less than 5 minutes. [ marks] attempt to find number who took less than 5 minutes (M) line on graph (vertical at approx 5, or horizontal at approx 70) 70 students (accept 69) A N [ marks] 5b. Find the number of students who took between 5 and 5 minutes to complete the task. 55 students completed task in less than 5 minutes (A) subtracting their values (M) 70 55 5 students A N
5c. Given that 50 students take less than k minutes to complete the task, find the value of k. [ marks] correct approach (A) line from y-axis on 50 k = A N [ marks] Consider a function f(x) such that f(x)dx = 8. 6 6a. Find 6 f(x)dx. [ marks] appropriate approach f(x), (8) 6 f(x)dx = 6 A [ marks] (M) N 6b. Find (f(x) + )dx. 6 [ marks] appropriate approach (M) f(x) +, 8 + dx = x (seen anywhere) (A) substituting limits into their intrated function and subtracting (in any order) (6) (), 8 + 6 (f(x) + )dx = 8 A [ marks] N (M) Let f(x) = sin(x + ) + k. The graph of f passes through the point (, 6). 7a. Find the value of k. METHOD attempt to substitute both coordinates (in any order) into f correct working (A) sin =, + k = 6 k = 5 A N METHOD recognizing shift of left means maximum at 6 R) recognizing k is difference of maximum and amplitude (A) k = 5 f ( ) = 6, = sin(6 + ) + k 6 A N (M)
7b. Find the minimum value of f(x). [ marks] evidence of appropriate approach (M) minimum value of sin x is, + k, f (x) = 0, (, ) minimum value is A N [ marks] 5 7c. Let g(x) = sin x. The graph of g is translated to the graph of f by the vector ( p ). q [ marks] Write down the value of p and of q. p =, q = 5 (accept ( )) 5 [ marks] AA N Let f(x) = e. The line is the tangent to the curve of at. 8. x L f (, e ) Find the equation of L in the form y = ax + b. [6 marks] recognising need to differentiate (seen anywhere) f, e x attempt to find the gradient when x = (A) (M) attempt to substitute coordinates (in any order) into equation of a straight line correct working f () f () = e [6 marks] y e = e (x ), e = e () + b (A) y e = e x e, b = e y = e x e A N R (M) 9a. Find (x )dx. 0 [ marks] correct intration AA x x 0 e.g. x, [ x], (x ) Notes: In the first examples, award A for each correct term. In the third example, award A for and A for (x ). substituting limits into their intrated function and subtracting (in any order) (M) 0 e.g. ( (0)) ( ()),0 ( 8), ( 0) 0 (x )dx = 8 A N 6
9b. Part of the graph of f(x) = x, for x, is shown below. The shaded rion R is enclosed by the graph of f, the line x = 0, and the x-axis. The rion R is rotated 60 about the x-axis. Find the volume of the solid formed. attempt to substitute either limits or the function into volume formula e.g. 0 f dx, b, a ( x ) 0 x Note: Do not penalise for missing or dx. (M) correct substitution (accept absence of dx and ) (A) e.g. 0 ( x ), 0 (x )dx, 0 (x )dx volume = 8 A N Part of the graph of f(x) = ax 6x is shown below. The point P lies on the graph of f. At P, x =. Find f (x). 0a. [ marks]
f (x) = ax x AA N Note: Award A for each correct term. [ marks] The graph of f has a gradient of at the point P. Find the value of a. 0b. [ marks] setting their derivative equal to (seen anywhere) e.g. f (x) = attempt to substitute x = into f (x) (M) e.g. a() () correct substitution into f (x) e.g. a, a = 5 a = 5 A [ marks] N (A) A (5,, 0) 5 The line L passes through the point and is parallel to the vector. a. Write down a vector equation for line L. [ marks] any correct equation in the form r = a + tb (accept any parameter for t) 5 where a is, and b is a scalar multiple of A N 0 5 5 e.g. r = + t, r = 5i j + 0k + t( 8i + j 0k) 0 5 Note: Award A for the form a + tb, A for L = a + tb, A0 for r = b + ta. [ marks] b. The line L intersects the x-axis at the point P. Find the x-coordinate of P. [6 marks]
recognizing that y = 0 or z = 0 at x-intercept (seen anywhere) (R) attempt to set up equation for x-intercept (must suggest x 0 ) (M) e.g. L = x,, 0 5 + t = x r = 0 0 0 one correct equation in one variable (A) e.g. t = 0, 0 + 5t = 0 finding t = A correct working (A) e.g. x = 5 + ( )() x = (accept (, 0, 0) ) A N [6 marks] Let f(x) = e 6x. Write down f (x). a. [ mark] f (x) = 6e 6x [ mark] A N The tangent to the graph of f at the point P(0, b) has gradient m. b. (i) Show that m = 6. (ii) Find b. [ marks] (i) evidence of valid approach e.g. f (0), 6e 6 0 correct manipulation A e.g. 6e 0, 6 m = 6 AG N0 (ii) evidence of finding f(0) e.g. y = e 6(0) b = A N [ marks] (M) (M) Hence, write down the equation of this tangent. c. [ mark]
y = 6x + A N [ mark] The diagram below shows the graph of a function f, for x. Write down the value of f(). a. [ mark] f() = A N [ mark] Write down the value of f. b. ( ) [ marks] f ( ) = 0 A N [ marks]
Sketch the graph of f on the grid below. c. EITHER attempt to draw y = x on grid (M) OR attempt to reverse x and y coordinates (M) writing or plotting at least two of the points (, ), (, 0), (0, ), (, ) THEN correct graph A N
The diagram below shows part of the graph of a function f. The graph has a maximum at A(, 5) and a minimum at B(, ). The function f can be written in the form f(x) = p sin(qx) + r. Find the value of (a) a. p (b) q (c) r. [6 marks] (a) valid approach to find p (M) amplitude, p = 6 p = A N [ marks] = max min (b) valid approach to find q (M) period =, q = q = [ marks] A N period (c) valid approach to find r (M) axis = r = [ marks] max+min A N, sketch of horizontal axis, f(0) Total [6 marks] b. p [ marks]
valid approach to find p (M) amplitude, p = 6 p = A N [ marks] = max min q c. [ marks] valid approach to find q period =, q = q = [ marks] A N period (M) r. d. [ marks] valid approach to find r axis = r = [ marks] max+min A N (M), sketch of horizontal axis, f(0) Total [6 marks] A rocket moving in a straight line has velocity v km s and displacement s km at time t seconds. The velocity v is given by 5. v(t) = 6 e t + t. When t = 0, s = 0. Find an expression for the displacement of the rocket in terms of t. [7 marks] evidence of anti-differentiation (6 e t + t) (M) s = e t t + + C AA Note: Award A for e t t, A for. attempt to substitute ( 0, 0) into their intrated expression (even if C is missing) (M) correct working (A) 0 = + C, C = 7 s = e t t + + 7 A N6 Note: Exception to the FT rule. If working shown, allow full FT on incorrect intration which must involve a power of e. [7 marks]
The random variable X has the following probability distribution, with P(X > ) = 0.5. Find the value of r. 6a. [ marks] attempt to substitute P(X > ) = 0.5 e.g. r + 0. = 0.5 r = 0. A N [ marks] (M) Given that E(X) =., find the value of p and of q. 6b. [6 marks] correct substitution into E(X) (seen anywhere) (A) e.g. 0 p + q + r + 0. correct equation A e.g. q + 0. + 0. =., q +. =. q = 0. A N evidence of choosing M e.g. p + 0. + 0. + 0. =, p + q = 0.5 correct working (A) p + 0.7 =, 0. 0. 0., p + 0. = 0.5 p = 0. A N Note: Exception to the FT rule. Award FT marks on an incorrect value of q, even if q is an inappropriate value. Do not award the final A mark for an inappropriate value of p. [6 marks] p i = Events A and B are such that P(A) = 0., P(B) = 0.6 and P(A B) = 0.7. The values q, r, s and t represent probabilities. Write down the value of t. 7a. [ mark]
t = 0. A N [ mark] 7b. (i) Show that r = 0.. (ii) Write down the value of q and of s. (i) correct values A e.g. 0. + 0.6 0.7, 0.9 0.7 r = 0. AG N0 (ii) q = 0., s = 0. AA N 7c. (i) Write down P( B ). (ii) Find P(A B ). (i) 0. A N (ii) P(A ) = A N B The diagram below shows part of the graph of f(x) = acos(b(x c)), where a > 0. The point P (,) is a maximum point and the point Q(, ) is a minimum point. Find the value of a. 8a. [ marks]
evidence of valid approach e.g. a = max y value min y value [ marks] A N (M), distance from y = (i) 8b. Show that the period of f is. (ii) Hence, find the value of b. [ marks] (i) evidence of valid approach (M) e.g. finding difference in x-coordinates, evidence of doubling e.g. ( ) period = AG N0 A (ii) evidence of valid approach e.g. b = (M) b = [ marks] A N 8c. Given that 0 < c <, write down the value of c. [ mark] c = A N [ mark] Let f(x) = (sin x + cosx). Show that f(x) can be expressed as + sin x. 9a. [ marks] attempt to expand (M) e.g. (sin x + cosx)(sin x + cosx) ; at least terms correct expansion A e.g. sin x + sin xcosx + cos x f(x) = + sin x AG N0 [ marks]
9b. The graph of f is shown below for 0 x. [ marks] Let g(x) = + cosx. On the same set of axes, sketch the graph of g for 0 x. AA N Note: Award A for correct sinusoidal shape with period and range [0, ], A for minimum in circle. 9c. The graph of g can be obtained from the graph of f under a horizontal stretch of scale factor p followed by a translation by the [ marks] vector ( k ). 0 Write down the value of p and a possible value of k. p =, k = AA N [ marks] A box contains six red marbles and two blue marbles. Anna selects a marble from the box. She replaces the marble and then selects a second marble. 0a. Write down the probability that the first marble Anna selects is red. [ mark]
Note: In this question, method marks may be awarded for selecting without replacement, as noted in the examples. P(R) = (= ) [ mark] 6 8 A N 0b. Find the probability that Anna selects two red marbles. [ marks] attempt to find P(Red) P(Red) e.g. P(R) P(R),, P(R) = (= ) [ marks] 6 6 9 6 A 6 8 N 5 7 (M) Find the probability that one marble is red and one marble is blue. 0c. METHOD attempt to find P(Red) P(Blue) e.g. P(R) P(B),, (M) recognizing two ways to get one red, one blue e.g. P(RB) + P(BR), ( ), P(R,B) = (= ) METHOD A N (M) recognizing that P(R,B) is P(B) P(R) (M) attempt to find P(R) and P(B) (M) e.g. P(R) =, ; P(B) =, 6 6 8 8 6 8 P(R,B) = (= ) 6 8 8 6 8 6 5 7 A 7 6 8 + N 7 8 6 7 8 7 Let sin θ =, where < θ <. Find cosθ. a.
METHOD evidence of choosing sin θ + cos θ = (M) correct working (A) e.g. cos 9 θ =, cosθ = ±, cosθ = 9 cosθ = A N Note: If no working shown, award N for. METHOD approach involving Pythagoras theorem e.g. + x =, (M) finding third side equals cosθ = A N (A) Note: If no working shown, award N for. b. Find tanθ. [5 marks] correct substitution into sin θ (seen anywhere) (A) e.g. ( ) ( ) correct substitution into cosθ (seen anywhere) (A) e.g. ( ) ( ), ( ), valid attempt to find tanθ e.g., correct working A ( ) e.g.,, A N (M) Note: If students find answers for cosθ which are not in the range [, ], award full FT in (b) for correct FT working shown. [5 marks] ( )( ) ( ) ( ) ()()( ) 9 tanθ = 5 8 ( ) ( ) 5 ( ) Let f(x) = 7 x and g(x) = x +. Find (g f)(x). a. [ marks]
attempt to form composite e.g. g(7 x), 7 x + (g f)(x) = 0 x A [ marks] (M) N Write down g. b. (x) [ mark] g (x) = x A N [ mark] c. Find (f g )(5). [ marks] METHOD valid approach (M) e.g. g (5),, f(5) f() = METHOD A N attempt to form composite of f and e.g. (f g )(x) = 7 (x ), x (f g )(5) = [ marks] A N g (M) The first three terms of an infinite geometric sequence are, 6 and 8. a. Write down the value of r. [ mark] 6 r = (= ) [ mark] A N b. Find u 6. [ marks] correct calculation or listing terms (A) 6 e.g. ( ), 8 ( ),,,, u 6 = [ marks] A N
Find the sum to infinity of this sequence. c. [ marks] evidence of correct substitution in e.g., S = 6 [ marks] A N S A Let g(x) = xsin x. Find g. a. (x) [ marks] evidence of choosing the product rule e.g. u v + vu correct derivatives cosx, (A)(A) g (x) = xcosx + sin x A N [ marks] (M) b. Find the gradient of the graph of g at x =. attempt to substitute into gradient function e.g. g () correct substitution (A) e.g. cos + sin gradient = A N (M) The diagram shows two concentric circles with centre O. The radius of the smaller circle is 8 cm and the radius of the larger circle is 0 cm. Points A, B and C are on the circumference of the larger circle such that AOB is radians. 5a. Find the length of the arc ACB. [ marks]
correct substitution in l = rθ e.g. 0, (A) arc length (= ) A N [ marks] 6 = 0 6 0 0 Find the area of the shaded rion. 5b. [ marks] area of large sector = 0 (= ) area of small sector = 8 (= ) evidence of valid approach (seen anywhere) e.g. subtracting areas of two sectors, (A) (A) area shaded = 6 (accept, etc.) A N [ marks] 6 6 00 6 6 6 M ( ) 0 8 Let g(x) = ln x, for x > 0. x 6a. Use the quotient rule to show that g ln x (x) =. x [ marks] d dx lnx = x d dx x, = x (seen anywhere) AA attempt to substitute into the quotient rule (do not accept product rule) M e.g. x ( ) x ln x x x correct manipulation that clearly leads to result A x x ln x x( ln x) x x e.g.,,, g (x) = [ marks] ln x x AG x x N0 x ln x x The graph of g has a maximum point at A. Find the x-coordinate of A. 6b. evidence of setting the derivative equal to zero e.g. g (x) = 0, lnx = 0 lnx = A (M) x = e A N
Consider f(x) = kx kx +, for k 0. The equation f(x) = 0 has two equal roots. Find the value of k. 7a. [5 marks] valid approach (M) e.g. b ac, Δ = 0, ( k) (k)() correct equation A e.g. ( k) (k)() = 0, 6 k = 8k, k k = 0 correct manipulation e.g. 8k(k ), k = [5 marks] A N 8± 6 A The line y = p intersects the graph of f. Find all possible values of p. 7b. [ marks] recognizing vertex is on the x-axis M e.g. (, 0), sketch of parabola opening upward from the x-axis p 0 A N [ marks] In a group of 6 students, take art and 8 take music. One student takes neither art nor music. The Venn diagram below shows the events art and music. The values p, q, r and s represent numbers of students. 8a. (i) Write down the value of s. (ii) Find the value of q. (iii) Write down the value of p and of r. [5 marks]
(i) s = A N (ii) evidence of appropriate approach e.g. 6, + 8 q = 5 q = 5 A N (iii) p = 7, r = AA N [5 marks] (M) 8b. (i) (ii) A student is selected at random. Given that the student takes music, write down the probability the student takes art. Hence, show that taking music and taking art are not independent events. [ marks] (i) P(art music) = 5 A N 8 (ii) METHOD P(art) = (= ) 6 A evidence of correct reasoning e.g. R the events are not independent AG N0 METHOD evidence of correct reasoning e.g. 5 8 P(art) P(music) = 96 (= ) 56 8 6 R A the events are not independent AG N0 [ marks] 8 6 5 6 8c. Two students are selected at random, one after the other. Find the probability that the first student takes only music and the second student takes only art. [ marks] P(first takes only music) = 6 P(second takes only art) = 7 5 evidence of valid approach e.g. 6 [ marks] 7 5 (M) P(music and art) = (= ) 0 80 7 (seen anywhere) (seen anywhere) A N A A
The Venn diagram below shows events A and B where P(A) = 0., P(A B) = 0.6 and P(A B) = 0.. The values m, n, p and q are probabilities. 9a. (i) Write down the value of n. (ii) Find the value of m, of p, and of q. [ marks] (i) n = 0. A N (ii) m = 0., p = 0., q = 0. AAA N [ marks] 9b. Find P( B ). [ marks] appropriate approach e.g. P( B ) = P(B), m + q, (n + p) (M) P( B ) = 0.6 [ marks] A N The following diagram shows quadrilateral ABCD, with AD = BC, AB = ( ), and AC = ( ). 0a. Find BC. [ marks]
evidence of appropriate approach e.g. AC AB, ( ) BC = ( ) A N [ marks] (M) 0b. Show that BD = ( ). [ marks] METHOD AD = ( ) correct approach (A) A e.g. AD AB, ( ) BD = ( ) METHOD AG recognizing CD = BA correct approach A N0 e.g. BC + CD, ( ) BD = ( ) [ marks] AG N0 (A) 0c. Show that vectors BD and AC are perpendicular.
METHOD evidence of scalar product correct substitution A e.g. ( )() + ()(), 8 + 8 A (M) e.g. BD AC, ( ) ( ) BD AC = 0 therefore vectors BD and AC are perpendicular AG N0 METHOD attempt to find angle between two vectors e.g. a b ab correct substitution ( )()+()() A e.g., cosθ = 0 θ = 90 A (M) therefore vectors BD and AC are perpendicular AG N0 8 International Baccalaureate Organization 06 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Printed for Atherton High School