Grade 12 - Exp. / Annals on complex numbers 41 exercises (including 34 corrected)

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1. Algebra E.3190 Part A. Organized knowledge transfer Prerequisites: The following two results are recalled: If z is a non-zero complex number, we have the follow-ing equivalence : | z | = r arg z = „ à 2 ı près ⇐⇒ z = r cos „ + i sin „ r > 0 For all real numbers a and b : cos a + b = cos a cos b − sin a sin b sin a + b = sin a cos b + sin b cos a Let z 1 and z 2 be two non-zero complex numbers. Prove the relations : | z 1 z 2 | = | z 1 |·| z 2 | ; arg( z 1 · z 2 )=arg( z 1 )+arg( z 2 ) to the near-est 2 ı . Part B For each proposition, indicate whether it is true or false and propose a demonstration for the indicated answer. In the case of a false proposition, the demonstration will consist of provid-ing a counter-example. An answer without a demonstration does not earn a point. Recall that if z is a complex number, z denotes the conjugate of z and z denotes the modulus of z : 1 If z = − 1 2 + 1 2 · i , then z 4 is a real number. 2 If z + z =0 , then z =0 . 3 If z + 1 z =0 , then z =i or z = − i . 4 If | z | =1 and if | z + z | =1 , then z =0 . E.6260 We denote by ( E ) the equation z 4 +4 z 2 +16=0 of complex unknown z . 1 Solve in C the equation Z 2 +4 · Z +16=0 . Write the solutions of this equation in exponential form. 2 We denote by a the complex number whose modulus is equal to 2 and whose one argument is equal to ı 3 . Calculate a 2 in algebraic form. Deduce the solutions in C of the equation : z 2 = − 2 + 2 · i · √ 3 . We’ll write the solutions in algebraic form. 3 Organized restitution of connaissances It is assumed known that for any complex number z = x +i · y où x ∈ R and y ∈ R , the conjugate of z is the com-plex number z defined by z = x − i · y . Show that : For all complex numbers z 1 and z 2 , z 1 · z 2 = z 1 · z 2 . For any complex number z and any non-zero natural number n : z n = z n . 4 Demonstrate that if z is a solution of the equation ( E ) then its conjugate z is also a solution of ( E ) . Deduce the solutions in C of the equation ( E ) . We’ll admit that ( E ) admits at most four solutions. E.3122 The plane ( P ) is provided with the direct orthonormal reference frame O ; I ; J (graphic unit : 2 cm ) . To any point M of the plane ( P ) is associated the complex number z , affix of the point M . 1 a Determine the modulus and one argument of each of the complex numbers : z 1 = − 1 ; z 2 = 1 − i 3 2 ; z 3 = − 1 − i 3 . b Determine the modulus and one argument of each of the cubes z 1 3 , z 2 3 , z 3 3 of the above complex numbers, then the real part and imaginary part of z 1 3 , of z 2 3 , of z 3 3 . 2 a If z = x +i · y = · e iθ is a complex number (with y and „ real and real greater than zero) , determine the real and imaginary parts of z 3 as a function of x and y , then the modulus and an argument of z 3 as a function of and „ . b Determine the set ( E ) of points M with affix z char-acterized by: z 3 is a real number. c Determine and plot the set ( E ) of points M with affix z , characterized by: z 3 is a real number and 1 z 3 8 . E.5949 The plane is referred to a direct orthonormal reference frame O ; −→ u ; −→ v . Note C the set of complex numbers. For each of the following propositions, say whether it is true or false, justifying the answer. 1 Proposition: For any natural number n : (1 + i) 4 n = ( − 4) n 2 Let ( E ) be the equation ( z − 4)( z 2 − 4 z +8)=0 où z denotes a complex number. Proposition: The points whose affixes are the solutions, in C , of ( E ) are the vertices of a triangle of area 8 . 3 Proposition: For any real number ¸ : 1+e 2i α =2 · e i α · cos( ¸ ) . 4 Let A be the point of affix z A = 1 2 · 1+i and M n the point with affix ( z A ) n où n denotes a natural number greater than or equal to 2 . Proposition: if n − 1 is divisible by 4 , then the points O , A and M n are aligned. 5 Let j be the complex number of modulus 1 and argument 2 ı 3 . Proposition: 1+ j + j 2 =0 2. Geometry without argument properties https://chingmath.fr chapExoCorrec/3190 sacados/3190 chapExoCorrec/6260 sacados/6260 chapExoCorrec/3122 sacados/3122 chapExoCorrec/5949 sacados/5949
E.5430 The complex plane is provided with a reference frame O ; −→ u ; −→ v orthonormal direct. Let i be the complex number such that : i 2 = − 1 . Let A be the point with affix z A =1 and B be the point with affix z B =i . To any point M of affix z M = x +i y , with x and y two reals such that y =0 , we associate the point M of affix : z M = − i · z M We denote by I the middle of segment [ AM ] . The aim of the exercise is to show that for any point M does not belong to ( OA ) , the median ( OI ) of triangle OAM is also a height of triangle OBM (property 1 ) and that BM =2 · OI (property 2) . 1 In this question and only in this question, we take : z M =2 · e − i π 3 . a Determine the algebraic form of z M . b Show that z M = − 3 − i . Determine the modulus and an argument of z M . c Place the points A , B , M , M and I in the frame O ; −→ u ; −→ v taking 2 cm as the graphic unit. Plot the line ( OI ) and quickly check the properties 1 and 2 using the graph. 2 We return to the general case, taking z M = x +i y with y =0 . a Determine the affix of point I as a function of x and y . b Determine the affix of the point M as a function of x and y . c Write the coordinates of the points I , B and M . d Show that the straight line ( OI ) is a height of the tri-angle OBM . e Show that : BM = 2 · OI . E.6262 Note C the set of complex num-bers. The complex plane is provided with an orthonormal reference frame O ; −→ u ; −→ v . We will take as unit 2 cm on each axis. The graph will be made on a sheet of graph paper and com-pleted as the questions are asked. Consider the function f which associates to any complex num-ber z : f ( z ) = z 2 + 2 · z + 9 1 Calculate the image of − 1+i · 3 . 2 Solve in C the equation : f ( z )=5 . Write the solutions of this equation in exponential form Then construct on the graph, with ruler and compass, the points A and B whose affixes are solutions of the equation ( A being the point whose affix has a positive imaginary part) . We’ll leave the construction lines visible. 3 Let – be a real number. Consider the equation f ( z )= – of unknown z . Determine the set of values of – for which the equation f ( z )= – admits two complex conjugate solutions. 4 Let ( F ) be the set of points in the complex plane whose affix z verifies : | f ( z ) − 8 | =3 Prove that ( F ) is the circle of center Ω( − 1 ; 0) and radius 3 . Plot ( F ) on the graph. 5 Let z be a complex number, such that z = x +i · y où x and y are real numbers. a Show that the algebraic form of f ( z ) is : x 2 − y 2 + 2 · x + 9 + i · 2 · x · y + 2 · y . b Let ( E ) be the set of points in the complex plane whose affix z is such that f ( z ) is a real number. Show that ( E ) is the union of two straight lines D 1 and D 2 whose equations will be specified. Complete the graph in the appendix by drawing these straight lines. 6 Determine the coordinates of the intersection points of the sets ( E ) and ( F ) . E.6778 The complex plane is referred to a reference frame O ; −→ u ; −→ v orthonormal direct. Consider the point A of affix 4 , the point of affix 4 · i and the points C and D such that ABCD is a square of center O . For any non-zero natural number n , we call M n the point with affix : z n = 1 + i n . 1 Write the number 1+i in exponential form. 2 Show that there exists a natural number n 0 , to be spec-ified, such that, for any integer n n 0 , the point M n is outside the square ABCD . https://chingmath.fr chapExoCorrec/5430 sacados/5430 chapExoCorrec/6262 sacados/6262 Antilles-Guyanne Septembre 2014 chapExoCorrec/6778 sacados/6778
E.6779 The complex plane is provided with a O ; −→ u ; −→ v direct orthonormal coordinate system. Note C the set of points M of the plane of affix z such that : ⏐ ⏐ z − 2 ⏐ ⏐ =1 1 Justify that C is a circle, whose center and radius are to be specified. 2 Let a be a real number. The line with equation y = a · x . is called D Determine the number of points of intersec-tion between C and D as a function of the values of a . E.3131 In the complex plane provided with the orthonormal reference frame O ; −→ u ; −→ v , consider the points M and M of affixes z and z respectively. We pose : z = x + i · y z = x + i · y where x , x , y , y are real numbers. Recall that z denotes the conjugate of z and that | z | denotes the modulus of z . 1 Show that the vectors −−→ OM and −−−→ OM are orthogonal if, and only if, Re ( z · z )=0 . 2 Show that the points O , M and M are aligned if, and only if, Im ( z · z )=0 . Applications 3 N is the point with affix z 2 − 1 . What is the set of points M such that the vectors −−→ OM and −−→ ON are orthogonal? 4 Assume z is non-zero. P is the point with affix 1 z 2 − 1 . We are looking for the set of points M with affix z such that the points O , N and P are aligned. a Show that : 1 z 2 − 1 z 2 − 1 = − z 2 · ⏐ ⏐ ⏐ 1 z 2 − 1 ⏐ ⏐ ⏐ 2 . b Using the equivalence demonstrated at the beginning of the exercise, conclude on the set sought. E.6777 The plane is provided with the direct orthonormal frame O ; −→ u ; −→ v . We give the complex number: j = − 1 2 +i · 3 2 The aim of this exercise is to study some properties of the number j and to highlight a link of this number with equilat-eral triangles. Part A : properties of the number j 1 a Solve in the set C of complex numbers the equation : z 2 + z + 1 = 0 b Verify that the complex number j is a solution of this equation. 2 Determine the modulus and one argument of the complex number j , then give its exponential form. 3 Demonstrate the following equalities: a j 3 = 1 b j 2 = − 1 − j 4 Note P , Q , R the respective images of the complex num-bers 1 , j and j 2 in the plane. What is the nature of the triangle PQR ? Justify the answer. Part B Let a , b , c be three complex numbers verifying the equality: a + j · b + j 2 · c = 0 We note A , B , C the respective images of the numbers a , b , c in the plane. 1 Using question A 3 b , prove equality: a − c = j · c − b 2 Deduce that : AC = BC 3 Demonstrate equality: a − b = j 2 · b − c 4 Deduce that the triangle ABC is equilateral. 3. Geometry with argument E.3157 For each of the 3 questions, only one of the three propositions is correct. The candidate will indicate on the copy the number of the question and the letter corresponding to the chosen answer. No justification is required. A correct answer earns 1 point ; an incorrect answer deducts 0.5 point ; the absence of an answer is counted 0 point. If the total is negative, the mark is reduced to zero. Throughout the exercise, the complex plane is referred to a direct orthonormal reference frame O ; −→ u ; −→ v 1 The point M lies on the circle with center A ( − 2 ; 5) and radius 3 . Its affix z verifies : a ⏐ ⏐ z − 2 + 5 · i ⏐ ⏐ 2 = 3 ; b ⏐ ⏐ z + 2 − 5 · i ⏐ ⏐ 2 = 3 ; c ⏐ ⏐ z − 2 + 5 · i ⏐ ⏐ 2 = 3 . 2 Consider three points A , B and C of affixes a , b and c re- spectively, two distinct and such that the triangle ABC is not equilateral. The point M is a point whose affix z is such that the complex numbers z − b c − a and z − c b − a are pure imaginary. a M is the center of the circle circumscribing the triangle ABC ; b M belongs to circles of respective diameters [ AC ] and [ AB ] ; c M is the orthocenter of the triangle ABC . 3 Let A and B be the points with affixes 1+i and 5+4 · i re-spectively, and C a point on the circle of diameter [ AB ] . We call G the isobarycenter of the points A , B and C and note z G its affix. a ⏐ ⏐ z G − 3 − 2.5 · i ⏐ ⏐ = 5 6 ; b z G − 1 + i = 1 3 · 4 + 3 · i ; https://chingmath.fr chapExoCorrec/6779 sacados/6779 Antilles-Guyane Juin 2016 chapExoCorrec/3131 sacados/3131 France Septembre 2006 5 points chapExoCorrec/6777 sacados/6777 Asie Juin 2015 chapExoCorrec/3157 sacados/3157
c z G − 3 + 2.5 · i = 1 3 · 4 + 3 · i E.3852 For each question, only one of the three answers is correct. The candidate must indicate on the copy the number of the question and the letter corresponding to the chosen answer. No justification is required. A correct answer earns 0.5 point ; an incorrect answer deducts 0.25 point ; the absence of an answer is counted 0 point. If the total is negative, the mark is reduced to zero. The complex plane is provided with a dorect orthonormal ref-erence frame of origin O . 1 A solution to the equation 2 · z + z =9+i is : a 3 b i c 3 + i 2 Let z be a complex number; | z +i | is equal to : a | z | + 1 b | z − 1 | c | i · z + 1 | 3 Let z be a non-zero complex number of argument „ . An argument of − 1+i · 3 z is : a − ı 3 + „ b 2 ı 3 + „ c 2 ı 3 − „ 4 Let n be a natural number. The complex 3+i n is a pure imaginary if and only if : a n = 3 b n = 6 · k + 3 , with k relatif c n = 6 k with k relatif 5 Let A and B be two points with respective affixes i and − 1 . The set of points M of affix z verifying | z − i | = | z +1 | is : a La droite ( AB ) b The circle of diameter [ AB ] c La straight line perpendicular to ( AB ) passing through O . 6 Let Ω be the point of affix 1 − i . The set of points M with affix z = x +i · y verifying | z − 1+i | = | 3 − 4 · i | has equation : a y = − x + 1 b ( x − 1) 2 + y 2 = 5 c z = 1 − i + 5 · e i · θ 7 Let A and B be the points with affixes 4 and 3 · i respec-tively. The affix of the point C such that the triangle ABC is isosceles with −−→ AB ; −→ AC = ı 2 is : a 1 − 4 · i b − 3 · i c 7 + 4 · i 8 The set of solutions in C of the equation z − 2 z − 1 = z is : a 1 − i b L’ensemble vide. c 1 − i ; 1+i E.5964 The plane is referred to the or-thonormal reference frame O ; −→ u ; −→ v . With any point M of affix z of the plane, we associate the point M of affix z by the application f which admits for complex writing: z = (3 + 4i) · z + 5 · z 6 1 Consider the points A , B , C with respective affixes : z A = 1 + 2i ; z B = 1 ; z C = 3i Determine the affixes of the points A , B , C respective images of A , B , C by f . Place points A , B , C , A , B , C . 2 We pose z = x +i · y (with x and y real) . Determine the real part and imaginary part of z as a function of x and y . 3 Show that the set of points M invariant by f is the straight line ( D ) of equation y = 1 2 x . Draw ( D ) . What remark can be made? 4 Let M be any point in the plane and M its image by f . Show that M belongs to the line ( D ) . 5 a Show that, for any complex number z : z − z z A = z + z 6 + i · z − z 3 Deduce that the number z − z z A is a real number. b Deduce that, if M = M , the straight lines ( OA ) and ( MM ) are parallel. 6 Given any point N , how do we construct its image N ? (two cases will be studied depending on whether N belongs to ( D ) ) . Perform the construction on the figure. E.6085 The plane is referred to the or-thonormal frame O ; −→ i ; −→ j . Let A (1 ; 0) and B ( − 1 ; 0) . To any point M of affix z =0 , we associate the point M of affix z defined by: z × z =1 1 a Build M when : z =2(1+i) b In the general case, show that the straight line ( AB ) is the bisector of the angle −−→ OM ; −−−→ OM and that : OM × OM = OA 2 . 2 a Check that : ∀ z ∈ C ∗ , z + z 2 − 1 z + z 2 + 1 = z − z 2 2 b Let I be the middle of [ MM ] . Using a show that : IA × IB = IM 2 and that for M = A and M = B the straight line ( MM ) is bisector of the angle −→ IA ; −→ IB https://chingmath.fr chapExoCorrec/3852 sacados/3852 Nouvelle Caledonie Novembre 2007 4 points chapExoCorrec/5964 sacados/5964 chapExoCorrec/6085 sacados/6085 Caen Juin 1987
A1A2A3A4A5A0O E.3857 The complex plane is referred to a O ; −→ u ; −→ v direct orthonormal coordinate system of graphic unit : 4 cm . Consider the point A of affix z A =2+i and the circle (Γ) of center A and radius 2 . 1 Make a figure that will be completed throughout the ex-ercise. 2 a Determine the affixes of the intersection points of (Γ) and the O ; −→ u axis. b We denote by B and C the points with respective af-fixes z B =1 and z C =3 . Determine the affix z D of the point D diametrically opposite the point B on the cir-cle (Γ) . 3 Let M be the point with affix 3 5 + 6 5 · i a Calculate the complex number: z D − z M z B − z M . b Geometrically interpret the argument of the number z D − z M z B − z M ; deduce that the point M belongs to the cir-cle (Γ) . 4 Note (Γ ) the circle of diameter [ AB ] . The straight line ( BM ) intersects the circle (Γ ) at a point N . a Show that the straight lines ( DM ) and ( AN ) are par-allel. b Determine the affix of the point N . 5 We denote by M the image of the complex number z verifying the relation: z − z B z M − z B = − i a Determine the algebraic writing of the complex num-ber z . b What is the nature of the triangle MM B ? c Show that the point M belongs to the circle (Γ ) . 4. Complex sequences E.6045 The complex plane is provided with an orthonormal reference frame O ; −→ u ; −→ v . For any natural number n , let A n be the point with affix z n defined by: z 0 = 1 ; z n +1 = 3 4 + √ 3 4 · i · z n pour tout n ∈ N We define the sequence r n by: r n = | z n | for any natural number n . 1 Give the exponential form of the complex number: 3 4 + √ 3 4 · i . 2 a Show that r n is geometric of reason √ 3 2 . b Deduce the expression of r n as a function of n . c What about the length OA n when n tends to + ∞ ? 3 Consider the function f of a following algorithm: Function f(p) r ← 1 n ← 0 As long as r>p n ← n+1 r ← √ 3 2 R End As long as Return n a What value is returned by calling the function f with the value 0.5 for the argument p ? b For p =0.01 , we get n =33 . What does this function do? 4 a Demonstrate that the triangle OA n A n +1 is right-angled at A n +1 . b On admet que : z n = r n · e i · nπ 6 Determine the values of n for which A n is a point on the y-axis. c Complete the figure given below by representing the points A 6 , A 7 , A 8 and A 9 . Construction lines will be apparent. https://chingmath.fr chapExoCorrec/3857 sacados/3857 Amerique du Nord Mai 2008 5 points chapExoCorrec/6045 sacados/6045 A1A2A3A4A5A0O
E.6900 Consider the complex numbers z n defined for any integer n 0 by the given z 0 , où z 0 is different from 0 and 1 , and the recurrence relation: z n +1 = 1 − 1 z n 1 a In this question, it is assumed that z 0 =2 . Deter-mine the numbers z 1 , z 2 , z 3 , z 4 , z 5 and z 6 . b In this question, it is assumed that z 0 =i . Determine the algebraic form of the complex numbers z 1 , z 2 , z 3 , z 4 , z 5 and z 6 . c In this question, we return to the general case où z 0 is a given complex number. What can we conjecture for the values taken by z 3 n depending on the values of the natural number n ? Prove this conjecture. 2 Determine z 2016 in the case où z 0 =1+i . 3 Are there values of z 0 such that z 0 = z 1 ? What can we say about the sequence z n in this case? E.6904 Consider the sequence z n of complex numbers defined for any natural number n by: z 0 = 0 z n +1 = 1 2 · i × z n + 5 In an orthonormal plane, note M n the point with affix z n . Consider the complex number z A =4+2 · i and A the point in the plane with affix z A . 1 Let u n be the sequence defined for any natural number n by: u n = z n − z A . a Show that, for any natural number n : u n +1 = 1 2 · i × u n b Demonstrate that, for any natural number n : u n = 1 2 · i n · − 4 − 2 · i 2 Demonstrate that, for any natural number n , the points A , M n and M n +4 are aligned. E.5948 Consider the sequence z n with complex terms defined by z 0 =1+i and, for any natural num-ber n , by: z n +1 = z n + | z n | 3 For any natural number n , we pose : z n = a n +i · b n , où a n is the real part of z n and b n is the imaginary part of z n . The aim of this exercise is to study the convergence of the sequences ( a n ) and ( b n ) . Part A 1 Donner a 0 and b 0 . 2 Calculate z 1 , then deduce that : a 1 = 1+ 2 3 ; b 1 = 1 3 . 3 Consider the function f below, taken from an algorithm, whose argument n is a non-zero natural number: Function f(n) A ← 1 B ← 1 For K varying 1 to n A ← A + A 2 + B 2 3 B ← B 3 End To Return A a We call this function with the value 2 for the argument n . Copy and complete the table below containing the state of the variables during the call to this function (we’ll round calculated values to 10 − 4 near) . K A B 1 2 b For a given number N , what does the value returned by the function f correspond to in relation to the situ-ation studied in this exercise? Part B 1 For any natural number n , express z n +1 as a function of a n and b n . Deduce the expression of a n +1 as a function of a n and b n , and the expression of b n +1 as a function of a n and b n . 2 What is the nature of the sequence b n ) ? Deduce the expression of b n as a function of n , and determine the limit of ( b n ) . 3 a Recall that for all complex numbers z and z : | z + z | | z | + | z | (inégalité triangulaire) Show that for any natural number n : | z n +1 | 2 | z n | 3 b For any natural number n , we pose u n = | z n | Show by recurrence that, for any natural number n , u n 2 3 n · 2 Deduce that the sequence u n converges to a limit to be determined. c Show that, for any natural number n , | a n | u n . De-duce that the sequence a n converges to a limit that we will determine. https://chingmath.fr chapExoCorrec/6900 sacados/6900 chapExoCorrec/6904 sacados/6904 Liban Mai 2016 3 points chapExoCorrec/5948 sacados/5948
-4-20246810121416-22468A0A3A4A5A6 E.3170 The complex plane is provided with a direct orthonormal reference frame O ; −→ u ; −→ v . The graphic unit will be 5 cm . We pose z 0 =2 and, for any natural number n , z n +1 = 1+i 2 z n . Note A n the point in the plane with affix z n . 1 Calculate z 1 , z 2 , z 3 , z 4 and check that z 4 is a real num-ber. Place the points A 0 , A 1 , A 2 , A 3 and A 4 on a figure. 2 For any natural number n , we pose : u n = | z n | . Justify that the sequence ( u n ) is a geometric sequence then establish that, for any natural number n : u n = 2 · 1 2 n 3 From what rank n 0 do all points A n belong to the disk with center O and radius 0.1 ? 4 a Establish that, for any natural number n : z n +1 − z n z n +1 = i . Deduce the nature of the triangle OA n A n +1 . b For any natural number n , note ‘ n the length of the broken line A 0 A 1 A 2 : : : A n − 1 A n . This gives : ‘ n = A 0 A 1 + A 1 A 2 + · · · + A n − 1 A n . Express ‘ n , as a function of n . What is the limit of the sequence ( ‘ n ) ? E.6255 We define, for any natural number n , the complex numbers z by: z 0 = 16 z n +1 = 1 + i 2 · z n for any natural number n . Let r n be the modulus of the complex number z n : r n = | z n | . In the plane provided with a direct orthonormal reference frame of origin O , consider the points A n of affixes z n . 1 a Calculate z 1 , z 2 and z 3 . b Place the points A 1 and A 2 on the graph given below. c Write the complex number 1+i 2 in trigonometric form. d Demonstrate that the triangle OA 0 A 1 is isosceles right-angled at A 1 . 2 Demonstrate that the sequence r n is geometric of rea-son √ 2 2 . Is the sequence r n convergent? Interpret the previous result geometrically. Let I n be the length of the broken line connecting the point A 0 to the point A n , passing successively through the points A 1 , A 2 , A 3 . . . Thus : L n = n − 1 i =0 A i A i +1 = A 0 A 1 + A 1 A 2 + · · · + A n − 1 A n 3 a Show that for any natural number n : A n A n +1 = r n +1 b Give an expression for L n as a function of n . c Determine the possible limit of the sequence L n . https://chingmath.fr chapExoCorrec/3170 sacados/3170 chapExoCorrec/6255 sacados/6255 -4-20246810121416-22468A0A3A4A5A6
0246810121416-8-6-4-22468 −u−vOA0A1A2A3A4A6 E.6349 The complex plane is referred to a direct orthonormal frame of reference. Consider the equation : ( E ) : z 2 − 2 · z · 3 + 4 = 0 1 Solve the equation ( E ) in the set C of complex numbers. 2 Consider the sequence M n of affix points defined for n 1 by: z n = 2 n · e i · ( − 1) n · π 6 . a Verify that z 1 is a solution of ( E ) . b Write z 2 and z 3 in algebraic form. c Place the points M 1 , M 2 , M 3 and M 4 on the figure given below and draw the segments [ M 1 M 2 ] , [ M 2 M 3 ] and [ M 3 M 4 ] . 3 Show that, for any integer n 1 : z n = 2 n · 3 2 + ( − 1) n · i 2 . 4 Calculate lengths M 1 M 2 and M 2 M 3 . For the remainder of the exercise, we assume that, for any integer n 1 : M n M n +1 = 2 n · 3 . 5 We note ‘ n = M 1 M 2 + M 2 M 3 + · · · + M n M n +1 . a Show that, for any natural number n 1 : ‘ n = 2 3 · 2 n − 1 . b Determine the smallest integer n such that : ‘ n 1000 . E.6937 Consider the complex numbers z n defined, for any natural number n , by: z 0 =1 ; z n +1 = 1 + i · √ 3 3 · z n Let A n be the point with affix z n in the O ; −→ u ; −→ v orthonor-mal frame given below : The aim of this exercise is to study the construction of points A n . 1 a Check that : 1+i · 3 3 = 2 3 · e i π 6 b Deduct z 1 and z 2 in exponential form. 2 a Show that for any natural number n : z n = 2 3 n · e i · n · π 6 b For what values of n , are the points O , A 0 and A n aligned? 3 For any natural number n , we pose : d n = ⏐ ⏐ z n +1 − z n ⏐ ⏐ . a Interpret geometrically d n . b Calculate d 0 . c Show that for any non-zero natural number n : z n +2 − z n +1 = 1 + i · 3 3 · z n +1 − z n . d Deduce that the sequence d n is geometric then that for any natural number n : d n = 3 3 · 2 3 n 4 a Show that for any natural number n : ⏐ ⏐ z n +1 ⏐ ⏐ 2 = ⏐ ⏐ z n ⏐ ⏐ 2 + d n 2 b Deduce that, for any natural number n , the triangle OA n A n +1 is rectangular at A n . c Construct, using a non-scale ruler and compass, the point A 5 on the figure above. d Justify this construction. https://chingmath.fr chapExoCorrec/6349 sacados/6349 0246810121416-8-6-4-22468 chapExoCorrec/6937 sacados/6937 −u−vOA0A1A2A3A4A6
-2-2-1-111220 -2-2-1-111220 -2-2-1-111220 E.6941 We want to model the shell of a nautilus in the plane using a broken line in the shape of a spiral. We’re interested in the area bounded by this line. The plane is given a direct orthonormal coordinate system O ; −→ u ; −→ v . Let n be an integer greater than or equal to 2 . For any integer k ranging from 0 to n , we define the complex numbers : z k = 1 + k n · e i 2 kπ n and note M k the point with affix z k . In this model, the circumference of the nautilus is the broken line connecting all points M k with 0 k n . For example, for the integers n =6 , n =10 and n =20 , we ob-tain the figures below : Part A : Broken line formed from seven points In this part, it is assumed that n =6 . Thus, for 0 k 6 , we have : z k = 1+ k 6 · e i 2 kπ 6 1 Determine the algebraic form of z 1 . 2 Check that z 0 and z 6 are integers to be determined. 3 Calculate the length of the height from M 1 in the trian-gle OM 0 M 1 then establish that the area of this triangle is equal to 7 3 24 . Part B: Broken line formed from n +1 points In this part, n is an integer greater than or equal to 2 . 1 For any integer k such that 0 k n , determine the length OM k . 2 For k integer such that 0 k n − 1 , determine a measure of the angles −→ u ; −−−→ OM k and −→ u ; −−−−−→ OM k +1 . Deduce a measure of the angle −−−→ OM k ; −−−−−→ OM k +1 . 3 For k integer such that 0 k n − 1 , demonstrate that the length of the height from M k +1 in the triangle OM k M k +1 is equal to 1+ k +1 n · sin 2 ı n . 4 We admit that the area of the triangle OM k M k +1 is equal to : a k = 1 2 · sin 2 ı n · 1 + k n 1 + k +1 n and the total area bounded by the broken line is equal to : A n = a 0 + a 1 + ··· + a n − 1 Consider the function f of an algorithm given below, tak-ing as argument an integer n and returning the value of the variable A corresponding to the area A n correspond-ing to the rank n : Function f(n) A ← 0 For k from 0 to n − 1 A ← A+ 1 2 sin 2 ı n 1+ k n 1+ k+1 n End To Return A We call the function f with the value 10 for the argument n . Copy and complete the table below, which collects all the values taken from the variables k and A when calling the function f . k 0 1 2 3 4 5 6 7 8 9 A 0.323 0.711 1.170 1.705 2.322 3.027 3.826 4.726 5 We admit that A 2 =0 and that the sequence A n con-verges and that : lim n ↦→ + ∞ A n = 7 ı 3 ≈ 7.3 Copy and complete the lines ‘ .3 so that at the end of its execution, the value of the variable n representing the smallest integer n such that A n 7.2 . We do not ask to determine n ‘ .1 n ← 2 ‘ .2 1 ← 0 ‘ .3 As long as ... ‘ .4 n ← n+1 ‘ .5 A ← 0 ‘ .6 For k ranging from 0 to n − 1 ‘ .7 A ← A+ 1 2 sin 2 ı n 1+ k n 1+ k+1 n ‘ .8 Fin Pour ‘ .9 End as long as 5. Plan transformation E.3141 The complex plane is referenced to the orthonormal frame O ; −→ u ; −→ v . The graphic unit will be 1 cm . 1 Question from cours Recall that : ˇ For any non-zero vector −→ w , of affix z , we have : | z | = −→ w et arg( z ) = −→ u ; −→ w ı. Let M , N and P be three points in the plane, with affixes m , n and p respectively, such that m = n and m = p . a Demonstrate that : arg p − m n − m = −−→ MN ; −−→ MP . b Geometrically interpret the number ⏐ ⏐ ⏐ p − m n − m ⏐ ⏐ ⏐ . 2 Consider the points A , B , C and D of affixes : z 1 = 4+i ; z B = 1+i ; z C = 5i ; z D = − 3 − i Place these points on a figure. 3 Let f be the application of the plane in itself which, to any point M of affix z associates the point M of affix z such that : z = 1 + 2i z − 2 − 4i a Specify the images of points A and B by f . b Show that f admits a single invariant point Ω whose https://chingmath.fr chapExoCorrec/6941 sacados/6941 -2-2-1-111220 -2-2-1-111220 -2-2-1-111220 sacados/3141
affix ! should be specified. 4 a Show that for any complex number z , we have : z − z = − 2i 2 − i − z b Deduce, for any point M different from the point Ω , the value of MM Ω M and a measure in radians of the angle −−→ M Ω ; −−−→ MM . c What is the nature of the triangle Ω MM ? d Let E be the point with affix z E = − 1 − i 3 . Write z E in exponential form and then place the point E on the figure. Then carry out the construction of the point E associated with the point E . E.3166 The complex is referred to a di-rect orthonormal reference frame O ; −→ u ; −→ v . The graphic unit will be 2 cm . Let f be the application which to any point M of the plane of non-zero affix z associates the point M of affix z such that z = 4 z , where z denotes the complex number conjugate of z . 1 Determine the set of points invariant by f . 2 Determine the set of points whose image by the applica-tion f is the point J of affix 1. 3 Let ¸ be a non-zero complex number. Show that the point A of affix ¸ admits a unique antecedent by f , whose affix is to be specified. 4 a Give a measure of the angle −−→ OM ; −−−→ OM . Interpret this result geometrically. b Express ⏐ ⏐ z ⏐ ⏐ as a function of ⏐ ⏐ z ⏐ ⏐ . If r denotes a strictly positive real, deduce the image by f of the circle of center O and radius r . c Choose a point P of the complex plane not lying on the coordinate axes and such that OP =3 , and geomet-rically construct its image P by f . 5 Consider the circle C 1 , of center J and radius 1. Show that the image by f of any point of C 1 , distinct from O , belongs to the line D of equation x =2 . E.3160 The plane is referenced to the orthonormal frame O ; −→ u ; −→ v (graphic unit 3 cm ) To any point M of affix z of the plane, we associate the point M of affix z by the application f which admits for complex writing: z = (3 + 4i) z + 5 z 6 1 Consider the points A , B , C with respective affixes : z A = 1 + 2i ; z B = 1 ; z C = 3i . Determine the affixes of the points A , B C respective images of A , B , C by f . Place points A , B , C , A , B , C . 2 We pose z = x +i y (with x and y real) . Determine the real part and imaginary part of z as a function of x and y . 3 Show that the set of points M invariant by f is the straight line ( D ) of equation y = 1 2 x . Draw ( D ) . What remark can be made? 4 Let M be any point in the plane and M its image by f . Show that M belongs to the line ( D ) . 5 a Show that, for any complex number z : z − z z A = z + z 6 + i z − z 3 Deduce that the number z − z z A is real. b Deduce that, if M = M , the straight lines ( OA ) and ( MM ) are parallel. 6 Given any point N , how do we construct its image N ? (two cases will be studied depending on whether N belongs to ( D ) ) . Perform the construction on the figure. https://chingmath.fr chapExoCorrec/3166 sacados/3166 chapExoCorrec/3160 sacados/3160
E.3185 The complex plane is pro-vided with a direct orthonormal reference frame O ; −→ u ; −→ v (graphic unit 2 cm ) Recall that for any non-zero vector −→ w , of affix z , we have : | z | = −→ w ; arg( z ) = −→ u ; −→ w à 2 ı près. Part A. Organized restitution of knowledge Prerequisite: we know that if z and z are two non-zero complex numbers, then : arg( z · z ) = arg( z ) + arg ( z ) Let z and z be two non-zero complex numbers. Show that : arg z z = arg( z ) − arg( z ) Part B Let A and B be the points with affixes − i and 3i . respectively We denote f the application which, to any point M of the plane, of affix z , distant from A , associates the point M of affix z such that : z = i z + 3 z + i 1 a Demonstrate that f admits two invariant points J and K belonging to the circle of diameter [ AB ] . Place these points on the drawing. b Note C the point with affix c = − 2+i . Show that the point C , image of C by f , belongs to the x-axis. 2 For any point M of the plane distinct from A and from B , show that : arg( z ) = −−→ MA ; −−→ MB + ı 2 à 2 ı près. 3 Study two sets of points. a Determine the set of points M with affix z such that z is a pure imaginary complex number. b Let M of affix z be a point on the circle of diameter [ AB ] deprived of the points A and B . To which set does the point M belong? E.3197 Consider the complex plane P referred to a direct orthonormal reference O ; −→ u ; −→ v . Throughout the exercise, P \ 0 denotes the plane P de-prived of the point of origin O . 1 Course question The following results are taken as prerequisites : If z and z are two non-zero complex numbers, then : arg( zz ) = arg( z ) + arg( z ) to the nearest 2 kı , with k relative integer. For any non-zero vector −→ w of affix z , we have : arg( z ) = −→ u ; −→ v à 2 kı près, with k relative integer. a Let z and z be non-zero complex numbers, show that : arg z z = arg z − arg z à 2 kı près, with k relative integer. b Show that if A , B , C are three points of the plane, two by two distinct, of respective affixes a , b , c , we have : arg c − a b − a = −−→ AB ; −→ AC à 2 kı près, with k relative integer. 2 Consider the application f of P \ 0 which, to the point M of the plane of affix z , associates the point M of affix z , associates the point M of affix z defined by: z = 1 z a Demonstrate that for z = 0 , we have : arg( z ) = arg( z ) to the nearest 2 kı , with k a relative integer. Deduce that, for any point M of P \ 0 the points M and M = f ( M ) belong to the same half-lineline of origin O . b Determine the set of points M of P \ 0 such that : f ( M )= M . c M is a point in the plane P distinct from O , U and V , we admit that M is also distinct from O , U and V . Etablir l’égalité: z − 1 z − i = 1 i z − 1 z + i = − i z − 1 z − i Deduce a relationship between arg z − 1 z − i and ( z − 1 z − i 3 a Let z be a complex number such that z = 1 and z = i and let M be the point of affix z . Show that M is on the line ( UV ) deprived of U and V if, and only if, z − 1 z − i is a non-zero real number. b Determine the image by f of the line ( UV ) deprived of U and V . https://chingmath.fr chapExoCorrec/3185 sacados/3185 Asie Juin 2006 4 points chapExoCorrec/3197 sacados/3197 France Juin 2006 5 points
E.3816 The plane is provided with a direct orthonormal reference frame O ; −→ u ; −→ v , graphic unit 2 cm . We call A the point with affix − 2i . To any point M of the plane of affix z , we associate the point M of affix : z = − 2 · z + 2 · i 1 Consider the point B of affix b =3 − 2 · i . Determine the algebraic form of the affixes a and b of the points A and B associated with the points A and B respectively. Place these points on the drawing. 2 Show that if M belongs to the line (Δ) of equation y = − 2 then M also belongs to (Δ) . 3 Demonstrate that for any point M of affix z , we have : | z + 2 · i | = 2 ·| z + 2 · i | E.3212 Let P be the complex plane refer-enced to O ; −→ u ; −→ v (graphic unit 4 cm ) . Let A be the point of affix 1 . We denote f the application of P deprived of A in P which, to any point M of affix z , associates the point M of affix z such that : z = 1 z − 1 1 a Let B be the point with affix b =4+i √ 3 . Determine the algebraic form and the exponential form of the af-fix b of B . b Determine the affixes of the points whose image by f is their symmetric with respect to O . 2 a Express | z | and arg( z ) in terms of | z − 1 | and arg( z − 1) . b Let C be the circle of center A and radius r . It is as-sumed that M is a point of C . Déterminer | z | . Deduce that M belongs to a circle C whose center and radius will be specified. c Place any point M on the circle of center A and ra-dius 1 2 and construct its image M . (we’ll leave the construction lines) . E.3874 In the complex plane ( P ) pro-vided with a direct orthonormal reference frame O ; −→ u ; −→ v of graphic unit 4 cm , consider the point A of affix A = − 1 and the application f , of plane ( P ) in itself, which to point M of affix z , distinct from A , associates the point M = f ( M ) of affix z such that : z = i · z z + 1 1 Determine the affix of points M such that M = M . 2 Demonstrate that for any point M distinct from A and from O , we have : OM = OM AM −→ u ; −−−→ OM = −−→ MA ; −−→ MO + ı 2 à 2 ı près 3 a Let B be the affix point b = − 1 2 +i . Place the point B and the perpendicular bisector (Δ) of the segment [ OA ] in the reference frame. b Calculate in algebraic form the affix b of the point B image of the point B by f . Establish that B belongs to the circle ( C ) of center O and radius 1 . Place the point B and draw the circle ( C ) in the ref-erence frame. c Using question 2 , demonstrate that, if a point M be-longs to the perpendicular bisector (Δ) , its image M by f belongs to the circle ( C ) . d Let C be the point such that the triangle AOC is direct equilateral. Using the results of question 2 , construct, using a ruler and compass, the image of point C by f (The construction lines should be left visible) . 4 In this question, we propose to determine, by two differ-ent methods, the set (Γ) of points M distinct from A and O whose image M by f belongs to the x-axis. Questions a and b can be treated independently. a We pose z = x +i · y with x and y real such that : ( x ; y ) = ( − 1 ; 0) ; ( x ; y ) = (0 ; 0) Show that the imaginary part of z is equal to : Im ( z ) = x 2 + y 2 + x ( x + 1) 2 + y 2 Deduce the nature and characteristic elements of the set (Γ) and plot it in the reference frame. b Using question 2 , geometrically find the nature of the set (Γ) . https://chingmath.fr sacados/3816 Centres etrangers Juin 2004 chapExoCorrec/3212 sacados/3212 Antilles-Guyane Septembre 2005 4 points chapExoCorrec/3874 sacados/3874
DAOI E.4072 The attached sheet will show the constructions requested during the exercise. In the complex plane referred to the direct orthonormal ref-erence frame O ; −→ u ; −→ v , the point A has affix i . We denote f the application which, to any point M of affix z with z =i associates the point M of affix z such that : z = − z 2 z − i The aim of the exercise is to geometrically construct the point M knowing the point M 1 An example Consider the point K of affix 1+i . a Place the point K . b Determine the affix of the point K image of K by f . c Place point K . 2 Points for which the problem does not arise a Consider the point L of affix i 2 . Determine its image L by f . What do we notice? b A point is said to be invariant by f if it is confused with its image. Show that there are two points invariant by f whose affixes we will determine. 3 A construction process We name G the isobarycenter of the points A , M and M and g the affix of G . a Check equality: g = 1 3 · ( z − i) . b Deduce that : If M is a point on the circle with center A radius r , then G is a point on the circle with center O radius 1 3 · r . c Demonstrate that : arg( g )= − −→ u ; −−→ AM . d On the attached sheet, we have marked a point D on the circle of center A and radius 1 2 . We name D the image of D by f . From the previous questions, deduce the construction of the point D and perform it on the appended figure to be returned with the copy) . On the figure below the segment OI such that : −→ u = −→ OI is divided into six segments of equal length. E.4030 The complex plane P is pro-vided with a direct orthonormal reference frame O ; −→ u ; −→ v , graphic unit 2 cm . The circle with center O and radius 1 is called Γ . We’ll make a figure that we’ll complete throughout the exer-cise. We call F the application of the plane P deprived of the point O in P which, at any point M different from O , of affix z , associates the point M = F ( M ) of affix z defined by: z = z + i − 1 z 1 Consider the points A and B of respective affixes a =i and b =e i · π 6 and their images A and B by F of affixes a and b respectively. a Calculate a and b . b Place points A , A , B and B . c Démontrer que : − b b − b = 3 3 · i . d Deduce the nature of the triangle OBB . 2 We search for the set ( E ) of points in the plane P de-prived of the point O which have as their image by F , the point O . a Demonstrate that, for any complex number z : z 2 + i · z − 1 = z + 3 2 + 1 2 · z − 3 2 + 1 2 · i b Deduce the affixes of the points of the set ( E ) . c Show that the points of ( E ) belong to (Γ) . 3 Let „ be a real. a Show that if z =e i · θ then z = 2 · sin „ +1 · i b Deduce that if M belongs to the circle (Γ) then M belongs to the segment [ A C ] où C has affix − i . https://chingmath.fr chapExoCorrec/4072 sacados/4072 Antilles-Guyane Juin 2008 5 points DAOI sacados/4030 Polynesie Septembre 2009 5 points
6. Unclassified financial years E.8135 1 Give the exponential and trigonometric forms of the com-plex numbers 1+ i and 1 − i . 2 For any natural number n , we have pose : S n = 1 + i n + 1 − i n a Determine the trigonometric form of S n . b For each of the following two statements, say whether it is true or false, justifying the answer. An unjustified answer will not be taken into account and the absence of an answer is not penalized Assertion A : For any natural number n , the com-plex number S n is a real number. Assertion B: There are infinitely many natural numbers n such that S n =0 . E.8145 Asia June 2018 E.8148 Antilles guyanbes September 2018 E.3151 For each question, only one of the four proposed answers is correct. The candidate will indicate on the copy the number of the question and the letter corre-sponding to the chosen answer. Each correct answer scores 1 point, each wrong answer deducts 0.5 point. An absence of response is counted as 0 points. If the total is negative, the mark is reduced to zero. No justification is required. 1 Let z be the complex number of module 2 and argu-ment ı 3 . Then we have : a z 14 = − 128 · 3 − 128 · i b z 14 = 64 − 64 · i c z 14 = − 64 + 64 · i · 3 d z 14 = − 128 + 128 · i · 3 2 Consider, in the complex plane referred to an orthonor-mal reference frame, the point S of affix 3 and the point T of affix 4i . Let ( E ) be the set of points M of affix z such that : ⏐ ⏐ z − 3 ⏐ ⏐ = ⏐ ⏐ 3 − 4i ⏐ ⏐ . a ( E ) is the perpendicular bisector of the segment [ ST ] ; b ( E ) is the straight line ( ST ) ; c ( E ) is the circle with center Ω of affix 3 − 4i and radius 3 ; d ( E ) is the circle of center S and radius 5 3 Consider a regular hexagon ABCDEF , whose sides are of length 1 . The scalar product −→ AC · −−→ CF is equal to : a 3 b − 3 c − 3 d 3 2 4 A function g is defined on the interval −∞ ; 0 by g ( x )= x 2 − 2 x x − 3 ; let Γ be its representative curve in a plane reference frame. a Γ admits an asymptote of equation y = − 1 . b Γ admits no asymptote. c Γ admits an asymptote of equation y = x . d Γ admits an asymptote of equation y =1 . 5 Let the function f be defined on R by f ( x )= x 0 e − t 2 dt . The function f  , second derivative of the function f on R , is defined by: a f  ( x ) = x 0 − 2 t · e − t 2 d t b f  ( x ) = x 0 − 2 x · e − x 2 d x c f  ( x ) = − 2 x · e − x 2 d f  ( x ) = e − x 2 E.6048 This exercise is a multiple-choice questionnaire. No justification is required. For each question, only one of the four propositions is correct. Each correct an-swer earns 1 point. A wrong answer or an absence of an an-swer does not take away a point. The candidate must indicate on the copy the question number and the chosen answer. 1 Soit z 1 = 6 · e i π 4 et z 2 = 2 · e − i π 3 . The exponential form of i · z 1 z 2 is : a 3e i 19 π 12 b 12e − i π 12 c 3 · e i 7 π 12 d 3e i 13 π 12 2 The equation − z = z , of complex unknown z , admits : a a solution b two solutions c an infinite number of solutions whose image points in the complex plane lie on a straight line. d an infinity of solutions whose image points in the com-plex plane lie on a circle. 3 In a space frame, consider the three points A 1 ; 2 ; 3 , B − 1 ; 5 ; 4 et C − 1 ; 0 ; 4 . The line parallel to the line ( AB ) passing through the point C has the paramet-ric representation : a x = − 2 t − 1 y = 3 t z = t + 4 , t ∈ R b x = − 1 y = 7 t z = 7 t + 4 , t ∈ R c x = − 1 − 2 t y = 5 + 3 t z = 4 + t , t ∈ R d x = 2 t y = − 3 t z = − t , t ∈ R 4 In an orthonormal space frame, consider the plane P passing through the point D − 1 ; 2 ; 3 and of normal vector −→ n 3 ; − 5 ; 1 , and the straight line Δ of paramet-ric representation : x = t − 7 y = t + 3 z = 2 t + 5 , for all t ∈ R a The line Δ is perpendicular to the plane P . b The line Δ is parallel to the plane P and has no com-mon point with the plane P . c The line Δ and the plane P are secant. d The line Δ is included in the plane P . https://chingmath.fr sacados/8135 Liban Mai 2018 3 points sacados/8145 Asie Juin 2018 5 points sacados/8148 chapExoCorrec/3151 sacados/3151 chapExoCorrec/6048 sacados/6048
TABCDEFGH E.6051 The four questions in this exercise are independent. For each question, an assertion is proposed. Indicate whether each is true or false, justifying the answer. An unjustified answer earns no points. In questions 1 and 2 , the plane is referred to the direct orthonormal frame O ; −→ u ; −→ v . Consider the points A , B , C , D and E of affixes : a = 2 + 2i ; b = − 3 + i ; c = 1 + i 3 d = − 1 + 3 2 i ; e = − 1 + 2 + 3 i 1 Assertion 1: points A , B and C are aligned. 2 Assertion 2: the points B , C and D belong to the same circle with center E . 3 In this question, space is provided with a reference frame O ; −→ i ; −→ j ; −→ k . Consider the points : 1 ; 0 ; 0 ; J 0 ; 1 ; 0 ; K 0 ; 0 ; 1 Assertion 3: the line D of parametric representation : x = 2 − t y = 6 − 2 t z = − 2 + t où t ∈ R cuts plane ( IJK ) at point E − 1 2 ; 1 ; 1 2 4 In the cube ABCDEFGH , the point T is the midpoint of the segment [ HF ] . Assertion 4: the straight lines ( AT ) and ( EC ) are or- thogonal. E.8138 For each of the following four state-ments, indicate whether it is true or false, justifying your answer. One point is awarded for each correct answer cor-rectly justified. An inaccurate or unjustified answer does not earn or lose any points. 1 A type of oscilloscope has a lifetime, expressed in years, which can be modeled by a random variable D that fol-lows an exponential law with parameter – . We know that the average lifetime of this type of oscillo-scope is 8 years. Assertion 1: for a randomly chosen oscilloscope of this type that has already operated 3 years, the probability that the lifetime is greater than or equal to 10 years, rounded to the hundredth, is equal to 0.42 . Recall that if X is a random variable that follows an ex-ponential law of parameter – , we have for any positive real t : P X t = 1 − e − λ · t 2 In 2016 , in France, law enforcement agencies carried out 9.8 million alcohol screening tests on motorists, and 3.1 % of these tests were positive. Source OFDT (Observatoire Français des Drogues et des Toxicomanies) In a given region, on 15 June 2016 , a gendarmerie brigade screened 200 motorists. Assertion 2: rounding to the hundredth, the probabil-ity that, out of 200 screenings, there were strictly more 5 positive screenings, is equal to 0.59 . 3 Consider in R the equation : ln 6 x − 2 + ln 2 x − 1 = ln x Assertion 3: the equation admits two solutions in the interval 1 2 ; + ∞ . 4 Consider in C the equation : 4 · z 2 − 20 · z + 37 2 · z − 7 + 2 · i = 0 Assertion 4: the solutions of the equation are the af-fixes of points belonging to the same circle of center point P of affix 2 . https://chingmath.fr chapExoCorrec/6051 sacados/6051 Asie Juin 2013 TABCDEFGH sacados/8138