Grade 12 - Exp. / Complex numbers, modules and arguments 37 exercises corrected

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1. Trigonometry reminder E.3792 Let ¸ be a real number. Simplify the following entries : a cos ı 2 + ¸ b sin ¸ +3 · ı c cos ¸ − ı 2 d sin ı 2 − ¸ E.3793 Solve the following equations : a sin x + ı 4 = √ 3 2 b cos 2 x + ı 3 = 2 2 2. Graphical representation and algebraic writing E.3784 Consider the complex plane with a reference frame O ; −→ u ; −→ v orthonormal direct and the five points shown below : 1 Determine the algebraic writings of the affixes of the points A , B , C , D , E . 2 Place in the plane the points F , G , H and I with respec-tive affixes z 1 , z 2 , z 3 and z 4 defined by: z 1 = 3 − i ; z 2 = 3 2 · i z 3 = − 7 4 + 2 · i ; z 4 = 3 2 + 9 4 · i 3 Determine the affix of the middle of segment [ EF ] . E.6792 We equip the complex plane with an orthonormal direct reference frame O ; −→ u ; −→ v . We de- fine the sequence z n of complex numbers by: z 0 = 1 ; z n +1 = 3 4 + √ 3 4 · i · z n 1 Determine the algebraic expression of the first four terms of the sequence z n . 2 Using your calculator, give the algebraic expression of the terms z 4 and z 5 , rounding the real and imaginary parts to the nearest 10 − 2 . 3 For any natural number n , let M n be the images of the complex number z n . a Place the points M 0 ,. . . , M 5 on the coordinate plane below : b What conjecture can be made about the nature of tri-angles OM 0 M 1 , OM 1 M 2 , and OM 2 M 3 ? 3. Modulus of a complex number E.5345 Determine the modulus of the fol-lowing complex numbers : a 1 − 2 · i b − 5 · i c (3 − 2 · i)(2 + i) E.8568 Determine the modulus of the fol-lowing complex numbers : a − i · (1 − 2 · i) b 1 + i · √ 3 3 · i c 3 − i · √ 3 − 3 + i 4. Geometry and module https://chingmath.fr chapExoCorrec/3792 sacados/3792 chapExoCorrec/3793 sacados/3793 chapExoCorrec/3784 sacados/3784 -4-3-2-101234-2-1123uvABCDE chapExoCorrec/6792 sacados/6792 -0,6-0,4-0,20,20,40,60,81,2I-0,20,20,40,60,8JO chapExoCorrec/5345 sacados/5345 chapExoCorrec/8568 sacados/8568
E.3843 In the complex plane with a ref-erence frame O ; −→ u ; −→ v , consider the points A , B , C with respective affixes : z A =1+3 · i ; z B = − 2+2 · i ; z C = 2 2 +i · 6 2 +2 Let I be the point of the plane with affix z I =2 · i Show that the points A , B , C belong to the same circle with center I . Specify the radius of this circle. 5. Relationship ⏐ ⏐ z ⏐ ⏐ 2 = z · z E.8569 For any complex number z , we de-fine the complex number z by: z = (3 + 4 · i) · z + 5 · z 6 1 For any complex number z , establish the equality: z − z 1 + 2 · i = z + z 6 + i · z − z 3 2 Deduce that the number z − z 1+2 · i is a real number. E.8570 Establish the equations below : a ⏐ ⏐ ⏐ 5 · i 5 − 5 · i ⏐ ⏐ ⏐ = 2 2 b ⏐ ⏐ ⏐ 1 + 2 · i 5 + 5 · i ⏐ ⏐ ⏐ = 10 10 6. Module: algebraic properties E.3846 Establish the following equality for any non-zero complex number z : 1 z 2 − 1 z 2 − 1 = − z 2 · ⏐ ⏐ ⏐ 1 z 2 − 1 ⏐ ⏐ ⏐ 2 7. Modules and Cartesian equations E.3844 Consider the complex plane provided with a reference frame O ; −→ u ; −→ v orthonormal direct : 1 Let z be a complex number solution of the equation ( E ) : ( E ) : | z − 2 + i | = 5 Note M the image of the complex number z a Translate equation ( E ) in terms of distance. b Justify that the set of points M of affix z verifying the relation ( E ) is a circle. Specify its center and radius. c Let a +i · b be the algebraic writing of the affix of the point M ; determine a relation on a and b characteriz-ing the relation ( E ) . 2 Let z be a complex number solution of the equation ( F ) : ( F ) : | z + i | = | z + 1 − 2 · i | We note M the image of the complex number z a Translate equation ( F ) in terms of distance. b Justify that the set of points M of affix z verifying the relation ( F ) is a straight line. Specify its nature. c Let a +i · b be the algebraic writing of the affix of the point M ; determine a relation on a and b characteriz-ing the relation ( E ) . E.3813 The plane is referred to the O ; −→ u ; −→ v direct orthornormal reference frame. Consider the points A and B of affixes : z A = − 1 + i · 3 ; z B = − 1 − i · 3 1 Establish that the set Γ 2 of points M of affix z that ver-ify: 2( z + z ) + z · z = 0 is a circle with center Ω and affix − 2 . Specify its radius. Construct Γ 2 . 2 Check that points A and B are Γ 2 elements 8. Set U E.8573 Definition: note U the set of complex numbers z verifying ⏐ ⏐ z ⏐ ⏐ = 1 . Note: U = z ∈ C ⏐ ⏐ | z | =1 ; U ⊂ C 1 Justify that the complex numbers below belong to U : a z 1 = − i b z 2 = 3 2 − 1 2 · i c z 3 = − 2 2 + 2 2 · i 2 Let z and z be two complex numbers belonging to U , show that the product z · z belongs to U . 3 Let z be an element of U , show that the three expressions below define a complex number belonging to U : https://chingmath.fr chapExoCorrec/3843 sacados/3843 chapExoCorrec/8569 sacados/8569 chapExoCorrec/8570 sacados/8570 chapExoCorrec/3846 sacados/3846 chapExoCorrec/3844 sacados/3844 chapExoCorrec/3813 sacados/3813 chapExoCorrec/8573 sacados/8573
a − z b 1 z c z 2 4 Show that, for any complex number z ( z ∈ C ) , the com-plex number z ⏐ ⏐ z ⏐ ⏐ belongs to U . E.8574 Establish that the complex numbers below belong to U : a z 1 = 1 − 7 · i − 5 + 5 · i b z 2 = 2 − 9 · i − 6 + 7 · i c z 3 = 4 + 7 · i 8 − i 9. Argument on the whole U E.8575 definition : Consider the complex plane provided with a reference frame O ; −→ u ; −→ v orthonormal direct. For any complex number z belonging to U , we call argu-ment of z the measure of the oriented angle −→ u ; −−→ OM where M is the image of the point z . 1 In the representation below of the complex plane pro-vided with a reference frame O ; −→ u ; −→ v orthonormal direct, place the points M 1 , M 2 , M 3 belonging to the trigonometric circle and whose respective arguments of the affixes of these points have values : a „ 1 = ı 4 b „ 2 = 5 ı 6 c „ 3 − 2 ı 3 Proposition: in the complex plane provided with a ref-erence frame O ; −→ u ; −→ v orthonormal direct. The point M belonging to the trigonometric circle whose affix z has arg-ment „ admits as algebraic writing: z = cos „ + i · sin „ 2 Give the algebraic writings of the affixes of the points M 1 , M 2 , M 3 defined in question 1 . 10. Remarkable arguments and angles E.5565 Definition: consider the complex plane provided with a reference frame O ; −→ u ; −→ v orthonormal direct. Let z be a non-zero complex number, we call argument of z the value of the argument of the complex number z ⏐ ⏐ z ⏐ ⏐ Determine the arguments of the complex numbers below : a z 1 = 1 + i b z 2 = − 3 + 3 · 3 · i c z 3 = 3 − i E.3795 Geometric interpretation : in the plane provided with a reference frame O ; −→ u ; −→ v orthonormal direct, for any point M different from O . The argument of the affix of the point M has as its value the measure of the angle oriented −→ u ; −−→ OM . Consider the plane pro-vided with a reference frame O ; −→ u ; −→ v or-thogonal direct repre-sented below : The circle C with center O and radius 2 is shown dotted ; points C and D belong to the circle C . 1 Determine the moduli and arguments of the affixes of the points A , B , C , D and E . 2 Place the points F and G of affixes z and z respectively, https://chingmath.fr chapExoCorrec/8574 sacados/8574 chapExoCorrec/8575 sacados/8575 -3-2-10123-2-112uv chapExoCorrec/5565 sacados/5565 chapExoCorrec/3795 sacados/3795 -2-1012-2-112uvABCDE
verifying: | z | = 2 arg z = − 3 ı 4 ; | z | = 2 arg z = − ı 2 11. Argument and approximate value E.8572 For each of the complex numbers below, determine the approximate value of their argument to the nearest hundredth of a radian. a z 1 = 2 + i b z 2 = − 3 − 2 · i c z 3 = 1 − 2 · i E.5346 Consider the complex numbers be-low : z 1 = 1 + 2 · i ; z 2 = 5 − 2 · i ; z 3 = − 1 − 2 · i 1 Determine the modulus of each of the above complex numbers. 2 Determine, to the nearest hundredth of a radian, the value of the argument of each of the above complex num-bers. 12. Trigonometric writing E.5566 Determine the algebraic writing of the complex numbers z 5 and z 6 non-zero defined by: a | z 5 | = 5 ; arg( z 5 ) = − ı 3 b | z 6 | = 2 ; arg( z 6 ) = − ı 2 E.3797 Determine the trigonometric form of each of the complex numbers below : a z 1 = − 1 2 + i · 3 2 b z 2 = 3 + i c z 3 = − 1 − i d z 4 = 2 2 + i · 2 2 13. Moduli, arguments and geometric loci E.5380 Consider the plane provided with a reference frame O ; −→ u ; −→ v orthonormal and direct. For each of the parts of the plane shown below, describe this subset of C using the algebraic writing, modulus, argument of the affixes of their points. E.3796 Consider the plane provided with a reference frame O ; −→ u ; −→ v orthonormal direct. Describe the set of points M of the plane whose affix z verifies the following conditions : a | z | = 1 b | z | 3 c 1 | z | 2 d arg( z ) = ı 2 e ⏐ ⏐ arg( z ) ⏐ ⏐ = ı 4 14. Geometric locations E.6806 Consider the three sets, defined below, of pairs of reals : E 1 = ( x ; y ) | 2 · x + y − 1=0 E 2 = ( x ; y ) | x 2 + y 2 − 2 · x +4 · y − 4=0 E 3 = ( x ; y ) | x · y +2 · y =0 Consider the plane provided with a O ; I ; J direct orthonor-mal coordinate system. Give for each of the three sets above : the nature of their representation in the plane, elements characterizing their representation. E.6776 To any complex number z , we associate a complex number z defined by: z = z 2 + 4 · z + 3 Determine the set E of complex numbers z such that z is a real number. https://chingmath.fr chapExoCorrec/8572 sacados/8572 chapExoCorrec/5346 sacados/5346 chapExoCorrec/5566 sacados/5566 chapExoCorrec/3797 sacados/3797 chapExoCorrec/5380 sacados/5380 uv2 uv2 uvπ6 chapExoCorrec/3796 sacados/3796 chapExoCorrec/6806 sacados/6806 chapExoCorrec/6776 sacados/6776
E.6782 Note C the set of complex num-bers. The complex plane is provided with a reference frame O ; −→ u ; −→ v orthornormal direct. Consider the function f which associates : f ( z ) = z 2 +2 · z +9 Let z be a complex number, such that z = x +i · y where x and y are real numbers. 1 Show that the algebraic form of f ( z ) is : f ( z ) = x 2 − y 2 + 2 · x + 9 + i · 2 · x · y + 2 · y 2 Note ( E ) 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. E.6353 Of the answers given, only one is correct. Which one? Justify your answer. Consider the set E of complex numbers z verifying: 1 z 2 +1 be a real. The set E is : 1 the set of real numbers ; 2 the set of pure imaginary numbers deprived of i and − i ; 3 the union of the set of real numbers and the set of pure imaginary numbers deprived of i and − i ; 4 the number 0 . 15. Plan transformation E.3785 Consider the coplex plane provided with a O ; −→ u ; −→ v orthonormal direct and the points A and B shown below : Let z A and z B be the respective affixes of the points A and B . 1 Give the algebraic writings of the affixes of the points A , B . 2 a Place the point C of affix − z A . b Which geometric transformation transforms the point A into C ? 3 a Place the point D of affix z A . b What geometric transformation transforms the point A into D ? 4 a Place the point E of affix − z A . b Which geometric transformation transforms the point A into E ? 5 a Place the point F of affix 1 2 · z A . b What can be said about the position of the point F ? 6 a Place the point G of affix z A + − 2 − 2 · i . b Which geometric transformation transforms the point A into G ? 7 Consider the point H of affix z H verifying equality: z H − z B = 1 2 · z A − z B a By solving the equation, determine the algebraic writ-ing of the affix z H of the point H . b What can be the position of the point H ? E.6796 Consider the complex plane provided with a O ; −→ u ; −→ v orthonormal direct reference frame. 1 Consider the points A , B , C with respective affixes : z 1 = 0.5 + 0.5 · i ; z 2 = 1 + i ; z 3 = 2 + 2 · i Place the points A , B and C . 2 Consider the function f defined on the set of non-zero complex numbers by: f ( z ) = − 1 z 2 a Determine the complex numbers z 1 , z 2 , z 3 respective images by the function f of z 1 , z 2 , z 3 . b Place the points A , B , C respective images of the complex numbers z 1 , z 2 , z 3 . 3 Can we say that the transformation of the plane obtained using the complex function f is an isometry? Justify your answer. https://chingmath.fr chapExoCorrec/6782 sacados/6782 Extrait Antilles-Guyanes Septembre 2014 chapExoCorrec/6353 sacados/6353 fichierPlus/6353/ chapExoCorrec/3785 sacados/3785 -4-3-2-101234-2-1123uvAB chapExoCorrec/6796 sacados/6796 01212uv
16. Plane transformation and geometric locus E.6020 To any complex number z such as z =2 , we associate the complex number z defined by: z = z + 3 z − 1 1 We note x +i · y , where x ∈ R and y ∈ R , the algebraic writ-ing of the complex number z . Give the algebraic writing of the number z , associated with z , as a function of x and y . 2 In the complex plane with a reference frame O ; −→ u ; −→ v orthonormal direct : a Consider the Cartesian equation : ( E ) : x 2 + y 2 + 2 x − 3 = 0 Justify that, in the plane, the set of solutions of ( E ) is the circle C of center ( − 1 ; 0) and radius 2 . b Determine the set of complex numbers z such that the associated complex number z is a pure imaginary. E.6797 Consider the function f defined on C by the relation: f ( z ) = z + z 2 1 Let’s note a and b the real numbers such that z admit as algebraic writing z = a +i · b . Give the algebraic writing of the complex number f ( z ) . 2 In this question, we seek to determine the set E of com-plex numbers z such that f ( z ) is a real. That is : E = z ∈ C ⏐ ⏐ Im f ( z ) =0 a Let z be a complex number belonging to E . Determine the set E . b In the complex plane shown below fitted with a refer-ence frame O ; ; −→ u ; −→ v orthonormal direct, represent in red the set E . E.6383 Note C the set of complex numbers and consider the complex plane is fitted with a ref-erence frame O ; −→ u ; −→ v orthonormé direct. Consider the function f which associates : f ( z ) = z 2 +2 · z +9 1 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. 2 Let z be a complex number, such that z = x +i · y where 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 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 should be specified. Complete the graph by drawing these straight lines. 3 Determine the coordinates of the intersection points of the sets ( E ) and ( F ) . https://chingmath.fr chapExoCorrec/6020 sacados/6020 chapExoCorrec/6797 sacados/6797 -4-3-2-101234-4-3-2-11234uv chapExoCorrec/6383 sacados/6383 Extrait d'Antilles-Guyane Septembre 2014 -2-1012-2-112uv
E.6798 In the complex plane provided with a O ; −→ u ; −→ v orthonormal reference frame. Consider the set E defined by: E = a +i · a ⏐ ⏐ a ∈ R The image of this set in the plane is the first bisector of the plane. Consider the complex function defined on C by the relation: f ( z ) = 1 z 2 + 1 The aim of the exercise is to determine characteristics of the image of the set E in the plane where : E = f ( z ) | z ∈E That is, the representation in the plane of all the images of the set E by the function f . 1 Let z be an element of E ; then there exists a real a such that : z = a + a · i Establish that : f ( z )= 1 1+4 · a 4 − 2 · a 2 1+4 · a 4 · i 2 a Demonstrate that : ⏐ ⏐ f ( z ) − 0.5 ⏐ ⏐ =0.5 b What can we say about the image of the set E in the plane? c Show that the point A of affix 0.5+0.5 · i does not be-long to the set E . Refine, if necessary, the answer to question b . 17. Suites E.6381 Consider the sequence of com-plex numbers z n defined by: z 0 = 3 − i ; z n +1 = 1 + i · z n For any natural number n , we pose : u n = ⏐ ⏐ z n ⏐ ⏐ . 1 Calculate u 0 . 2 Demonstrate that u n is the geometric sequence of rea-son 2 and first term 2 . 3 For any natural number n , express u n as a function of n . 4 Determine the limit of the sequence u n . E.6092 The complex plane is provided with a O ; −→ u ; −→ v orthonormal reference frame. 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 num-ber n . 1 Give the trigonometric form of the complex number: 3 4 + 3 4 · i . 2 Show that the sequence r n is geometric of reason 3 2 3 Determine, as a function of n , the measure of the segment A n A n +1 . E.6382 Consider the sequence z n of complex numbers defined by: z 0 = 4 ; z n +1 = 1 + i 2 · z n for all n ∈ N In the plane with a direct orthonormal reference point of ori-gin O , note A n the image point of the complex number z n . 1 a Calculate z 1 , z 2 and z 3 . b Place the points A 1 , A 2 and A 3 in the frame below. 2 Note ‘ n the length of the broken line connecting point A 0 to point A n , passing successively through points A 1 , A 2 , A 3 . . . Thus : ‘ n = n k =1 A k − 1 A k = A 0 A 1 + A 1 A 2 + ··· + A n − 1 A n a Establish the relationship below for any non-zero nat-ural number n : A n − 1 A n = ⏐ ⏐ z n ⏐ ⏐ b Using reasoning by recurrence, establish the following equality for any non-zero natural number n : A n − 1 A n = 2 2 · 2 2 n − 1 c Give an expression for ‘ n as a function of n . d Determine the possible limit of the sequence ‘ n . 18. Unclassified financial years E.3845 ⏐ ⏐ ⏐ z + 3 · i z − 6 ⏐ ⏐ ⏐ = 1 2 https://chingmath.fr chapExoCorrec/6798 sacados/6798 chapExoCorrec/6381 sacados/6381 Extrait de Liban Mai 2014 chapExoCorrec/6092 sacados/6092 chapExoCorrec/6382 sacados/6382 -2-101234-1123uv sacados/3845
E.4314 The complex plane is referred to a O ; −→ u ; −→ v direct orthonormal coordinate system. We de-note by P , Q and R the points with respective affixes : z P = 3 2 · 1 + i ; z Q = 3 2 · (1 − i) ; z R = − 2 · i · 3 Let S be the symmetric of the point R with respect to the point Q . Check that the affix z S of the point S is : z S =3+i · 2 · 3 − 3 https://chingmath.fr chapExoCorrec/4314 sacados/4314