Grade 12 - Exp. / Complex numbers and geometries 41 exercises (100% corrected)

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-3-2-101234567-4-3-2-112345uv 1. Module E.8594 In the complex plane provided with a O ; −→ u ; −→ v , consider the three points A , B , C having re-spective affixes a , b , c defined by: a = √ 3 − i · 3 ; b = − 1 − i ; c = 1 − 3 · i 1 Establish that : a − b c − b = 3 2 + 1 2 · i 2 Establish that the triangle ABC is isosceles at B . E.4099 In the complex plane provided with a O ; −→ u ; −→ v orthonormal direct, consider the three points A , B , C of affixes a , b , c defined by respectively: a = − 3 + 3 · i ; b = − 3 2 − i ; c = 5 2 − 5 2 · i Establish that the triangle ABC is isosceles at B . 2. Argument and angle measurement E.8593 In the complex plane provided with a O ; −→ u ; −→ v , consider the three points A , B , C having re-spective affixes a , b , c defined by: a = 2 3 + i · 3 ; b = 1 − 2 · i ; c = 3 − i Determine the measure of the angle ∠ ABC . E.5358 Consider the plane provided with the reference frame O ; −→ u ; −→ v orthonormal direct and the three points A , B and C of affixes : z A = 1 − 2 · i ; z B = − 3 − 4 · i ; z C = − 2 3 − i · 3 1 a Give the algebraic writing of the complex number Z defined by the quotient : Z = z C − z A z B − z A b Deduce the trigonometric writing of the complex num-ber Z . 2 a Deduce from the previous questions that the triangle ABC is an isosceles triangle at A . b Give the measure of the angle ∠ BAC . E.5535 We place ourselves in the complex plane provided with a direct orthonormal reference O ; −→ u ; −→ v . 1 Consider the points A , B and C with respective affixes : a =2 ; b =3+i √ 3 ; c =2i 3 . Determine a measure of the angle ∠ ABC . 2 Deduce that the affix ! of the center Ω of the circum-scribed circle of the triangle ABC is 1+i · √ 3 . E.5360 In the plane provided with a O ; −→ u ; −→ v orthonormal direct, consider the three points A , B and C of affixes : z A = − 2 + i ; z B = − 3 · i ; z C = 6 + 5 · i 1 Place these three points in the frame below : 2 Demonstrate that the triangle ABC is a right-angled tri-angle. 3 Consider the point D of affix z D où : z D =4 − i a Give the algebraic writing of the complex number Z defined by: Z = z B − z A z D − z A b Justify that the half-right [ AD ) is the bisector of the angle −−→ AB ; −→ AC . 3. Argument and orthogonality E.8592 In the plane provided with a O ; −→ u ; −→ v orthonormal direct, consider the points E , F , G and H of affixes : z E = 5 − 3 · i ; z F = 4 + i · − 3 + 3 z G = i ; z H = − 2 3 − i 1 Establish equality: z F − z E z H − z G = − 1 2 · i 2 Deduce that the straight lines ( EF ) and ( GH ) are per-pendicular. https://chingmath.fr chapExoCorrec/8594 sacados/8594 chapExoCorrec/4099 sacados/4099 chapExoCorrec/8593 sacados/8593 chapExoCorrec/5358 sacados/5358 chapExoCorrec/5535 sacados/5535 Extrait du Liban 2012 chapExoCorrec/5360 sacados/5360 -3-2-101234567-4-3-2-112345uv chapExoCorrec/8592 sacados/8592
E.4320 The complex plane is provided with a reference frame O ; −→ u ; −→ v orthonormal direct. We’ll take 2 cm as the graphic unit. The point with affix i is called J . 1 Consider the points A , B , C , H with respective affixes : a = − 3 − i ; b = − 2 + 4 · i ; c = 3 − i ; h = − 2 . Place these points on a figure, which will be completed as the exercise progresses. 2 Show that J is the center of the circle C circumscribed by the triangle ABC . Specify the radius of the circle C . 3 Give the algebraic form of the complex number b − c h − a . Deduce that the straight lines ( AH ) and ( BC ) are per- pendicular. E.3875 Consider the plane provided with a reference frame O ; −→ u ; −→ v orthonormal direct. Let A and B be the points with respective affixes : z A = i ; z B = 2 + 2 · i 1 Let I be a point of the plane whose affix z I is : z I = − 4 + 23 2 · i Show that point I belongs to the perpendicular bisector of segment [ AB ] . 2 Let M be a point in the plane with affix (2+i) . Show that the point M belongs to the circle of diameter [ AB ] . 4. Argument and parallelism E.3862 Consider the complex plane provided with a reference frame O ; −→ u ; −→ v orthonormal direct Consider the points A , B , C and D with affixes : z A = 4 − 3 · i ; z B = 4 − 3 + i · 1 − 3 z C = − 1 + 2 · i ; z D = 2 3 − 1 1 Establish equality: z B − z A z D − z C = − 1 2 2 Deduce that the straight lines ( AB ) and ( CD ) are paral-lel. E.3847 The complex plane is referred to a O ; −→ u ; −→ v direct orthonormal coordinate system. We pose : a =3 ; b =5 − 2 · i ; c =5+2 · i We denote by A , B and C the points with affixes a , b and c respectively. Let M be a point of affix z of the plane, distinct from the points A and B 1 Show that ABC is an isosceles right-angled triangle. 2 Give a geometric interpretation of the argument of the complex number: z − 3 z − 5 + 2 · i 3 Then determine the set of points M with affix z such that z − 3 z − 5+2 · i is a strictly negative real number. 5. Nature of a triangle E.5965 The plane is given a direct orthonor-mal coordinate system O ; −→ u ; −→ v . Consider the two points A and B of respective affixes : z A = 2 + 2 · i ; z B = 1 − i 1 Give the trigonometric writing of the complex number z A z B . 2 Deduce the nature of the triangle OAB . E.3809 The complex plane is pro-vided with a reference frame O ; −→ u ; −→ v orthonormal direct (graphic unit 4 cm ) . Let I be the point with affix 1 . We note C the circle of diameter [ OI ] and name its center Ω . We pose a 0 = 1 2 + 1 2 · i and note A 0 its image. 1 Show that the point A 0 belongs to the circle C . 2 Let B be the point with affix b , with b = − 1+2 · i , and B be the point with affix b such that b = a 0 · b . a Calculate b . b Demonstrate that the triangle OBB is rectangular at B . E.8597 Consider the complex plane provided with a reference frame O ; −→ u ; −→ v orthonormal direct. Consider the points A , B , C of respective affixes : z A = 1 + i ; z B = 2 + i · ( 3 + 1) ; z C = (1 − 3) + i · 2 1 Determine the algebraic writing of the quotient : z C − z A z B − z A 2 Determine the modulus and an argument of the previous quotient. 3 Establish that the triangle ABC is isosceles right-angled. https://chingmath.fr chapExoCorrec/4320 sacados/4320 Extrait d'Antilles Juin 2011 chapExoCorrec/3875 sacados/3875 chapExoCorrec/3862 sacados/3862 chapExoCorrec/3847 sacados/3847 Extrait Polynesie Septembre 2006 chapExoCorrec/5965 sacados/5965 chapExoCorrec/3809 sacados/3809 Extrait Amerique du Sud Novembre 2003 chapExoCorrec/8597 sacados/8597
E.3823 The complex plane is pro-vided with a reference frame O ; −→ u ; −→ v orthonormal direct of graphic unit 2 cm Consider the points A , B and C with respective affixes : z A = − 2 · i ; z B = − 3 + i ; z C = 3 + i 1 a Give the exponential writing of the numbers z A , z B and z C . b Deduce the center and radius of the circle Γ passing through the points A , B and C . c Make a figure and place the point A , draw the circle Γ , then place the points B and C . 2 a Give the algebraic writing of the complex z B − z A z C − z A , then its exponential writing. b Deduce the nature of the triangle ABC . E.3861 Consider the complex plane provided with a O ; −→ u ; −→ v orthonormal direct reference frame. Consider the points D , E , F of respective affixes : z D = 4 − 3 · i ; z E = 4 + 3 + i · − 1 − 3 z F = − 3 + 2 · i + 4 1 Determine the algebraic writing of the quotient : z E − z D z F − z D 2 Determine the modulus and an argument of the previous quotient. 3 Establish that the triangle DEF is equilateral. E.4237 The plane is provided with a O ; −→ u ; −→ v direct orthonormal coordinate system. Consider the points C , D , E of respective affixes : z C = 3 + 1 + 3 · i ; z D = 2 − i ; z E = i 1 Determine the algebraic writing of the number z C − z E z D − z E , then its trigonometric writing. 2 Deduce the nature of the triangle CDE . 6. Nature of a quadrilateral E.3854 In the complex plane provided with a O ; −→ u ; −→ v orthonormal direct, consider the points A , B , C , D with affixes : z A = − 3 − i ; z B = 1 − i · 3 z C = 3 + i ; z D = − 1 + i · 3 1 Give the modulus and one argument of each of the four complex numbers z A , z B , z C and z D . 2 Construct with ruler and compass the points A , B , C and D (we’ll take as graphic unit 2 cm ) . 3 Determine the midpoint of segment [ AC ] , that of segment [ BD ] . Calculate the quotient z A z B . Deduce the nature of the quadrilateral ABCD . E.5359 Consider the plane provided with the reference frame ( O ; −→ u ; −→ v orthonormal direct and the four points A , B , C and D with respective affixes : z A = − 2+i ; z B = − 3 · i ; z C =3 − 2 · i ; z D =3+2 · i 1 a Give the algebraic writing of the complex number Z defined by the quotient : Z = z C − z A z D − z B b Deduce the trigonometric writing of the complex Z . 2 a Justify that the quadrilateral ABCD has its diago-nals perpendicular and of the same length. b Is the quadrilateral ABCD a rhombus? E.8595 In the plane provided with a O ; −→ u ; −→ v orthonormal direct, we coniderate the four points A , B , C , D with respective affixes a , b , c , d defined by: a = 6 ; b = 2 + 2 · i ; c = 1 ; d = 5 − 2 · i Show that the quadrilateral ABCD is a rectangle but is not a square. 7. Using Cartesian equations E.8596 Consider the complex plane provided with a O ; −→ i ; −→ j orthonormal direct and the two points C and D of respective affixes c and d defined by: c = 1 + i ; d = 3 − 2 · i For any complex number z different from d , let z 0 be the com-plex number defined by: z 0 = z − c z − d 1 Noting a +i · b , where a; b ∈ R , the algebraic writing of the number z , determine the algebraic writing of the quotient z 0 . 2 We wish to characterize the set E of points in the plane with affix z , such that the number z 0 is a real number. a Determine a relationship on the real numbers a and b so that the number z 0 is a real number. b Justify that the set E is the set of points M of the plane distinct from D such that the points C , D , M are aligned https://chingmath.fr chapExoCorrec/3823 sacados/3823 chapExoCorrec/3861 sacados/3861 chapExoCorrec/4237 sacados/4237 chapExoCorrec/3854 sacados/3854 Pondichery Avril 2008 chapExoCorrec/5359 sacados/5359 chapExoCorrec/8595 sacados/8595 chapExoCorrec/8596 sacados/8596
-2-101-11uv E.8598 Consider the complex plane provided with a O ; −→ i ; −→ j orthonormal direct and the two points C and D of respective affixes c and d defined by: c = 2 − i ; d = 1 + 2 · i For any complex number z different from d , let z 0 be the com-plex number defined by: z 0 = z − c z − d 1 Noting a +i · b , where a; b ∈ R , the algebraic writing of the number z , determine the algebraic writing of the quotient z 0 . 2 We wish to characterize the set E of points in the plane with affix z , such that the number z 0 is a pure imaginary. a Determine a relationship on the real numbers a and b so that the number z 0 is a pure imaginary. b Justify that the set E is the circle of diameter [ CD ] deprived of the point D . 8. Plan transformation E.3841 The complex P plane is pro-vided with a O ; −→ u ; −→ v direct orthonormal coordinate sys-tem, graphic unit 2 cm . We call (Γ) the circle of center O and radius 1 . A figure will be made and completed throughout the exercise. 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 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 with respective af-fixes a and b . 1 Calculate a and b . 2 Place points A , A , B and B . 3 Demonstrate that : − b b − b = √ 3 3 · i 4 Deduce the nature of the triangle OBB . E.5532 In the complex plane referred to the O ; −→ u ; −→ v orthonormal direct, we denote by A and B the points of affixes 1 and − 1 respectively. Let f be the transformation of the plane which to any point M of affix z =1 , associates the point M of affix z such that : z = 1 − z z − 1 Let C be the point with affix : z C = − 2+i . 1 Calculate the affix z C of the point C image of C by the transformation f , and place the points C and C in the datum given below : 2 Show that the point C belongs to the circle C with cen-ter O and radius 1 . 3 Show that the points A , C and C are aligned. https://chingmath.fr chapExoCorrec/8598 sacados/8598 chapExoCorrec/3841 sacados/3841 chapExoCorrec/5532 sacados/5532 -2-101-11uv
E.3814 The complex plane P is re-ferred to the reference frame O ; −→ e 1 ; −→ e 2 orthonormal direct of graphic unit 1 cm . Let A be the point with affix 3 · i . We call f the application which, to any point M of affix z , distinct from A , associates the point M of affix z defined by: z = 3 · i · z − 7 z − 3 · i Search for points invariant by f . 1 Develop ( z − 7 · i)( z +i) . 2 Show that f admits two invariant numbers which are the affixes of two points that will be noted B and C . Definition: let f be a complex function (from C in C ) . The number z is said to be an invariant of the function f if f ( z )= z . E.5533 In the complex plane referred to the reference frame O ; −→ u ; −→ v orthonormal direct, we de-note by A the point of affixes 1 . Let f be the transformation of the plane which to any point M of affix z =1 , associates the point M of affix z such that : z = 1 − z z − 1 1 Show that, for any complex number z =1 , z − z z − 1 is real. 2 What can we deduce for the points A , M and M ? E.4239 In the plane provided with a O ; −→ u ; −→ v orthonormal direct, consider the points A and B of affixes 2 and ( − 2) respectively. We define the application f which to any point M of af-fix z and different from A associates the point M of affix : z = z · ( z − 2) z − 2 1 Show that for any complex number z , the number ( z − 2)( z − 2) is real. 2 Deduce that for any complex number distinct from 2 : z +2 z − 2 est réel. 3 Show that the straight lines ( AM ) and ( BM ) are paral-lel. 9. Plane transformation and image of a set E.3873 The complex plane P is provided with a reference frame O ; −→ u ; −→ v orthonormal direct. We call (Γ) the circle of center O and radius 1 . 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 af-fix z , associates the point M = F ( M ) of affix z defined by: z = z +i − 1 z Note A the point with affix i and „ a real number. 1 Let A be the image of the point A by the transformation F . Determine the affix of the point A . 2 Show that if z =e i · θ then z = 2 · sin „ +1 · i 3 Deduce that if M belongs to the circle (Γ) then M be- longs to the segment [ A C ] où C has affix − 1 . E.4078 In the complex plane referred to the O ; −→ u ; −→ v direct orthonormal coordinate system. Note A the affix point i . For any point M of affix z such that z =i , we define the point M whose affix z is defined by: z = 1 3 · ( z − i) 1 Establish the following proposition : If M is a point on the circle with center A of radius r , then M is a point on the circle with center O of radius 1 3 · r . 2 Demonstrate that : arg( z ) = − −→ u ; −−→ AM 10. Transformation of the plane and reciprocal image of a set E.3825 In the complex plane provided with a O ; −→ u ; −→ v orthonormal direct, consider the transformation f of the complex plane which to any point M of affix z asso-ciates the point M of affix z defined by: z = z 2 +i · z +1 − i 1 Consider the points A and B with respective affixes : 3 + 2 · i ; − 3 · i Determine the image of these two points by the transfor-mation f . 2 a Let M be a point of the plane. Let a +i · b denote the affix of the point M . Express as a function of a and b the real part and imaginary part of the point M . b Determine a condition on a and b so that the point M belongs to the axis of reals. c Determine a condition on a and b so that the point M belongs to the axis of the imaginary. https://chingmath.fr chapExoCorrec/3814 sacados/3814 Extrait Asie Juin 2004 chapExoCorrec/5533 sacados/5533 chapExoCorrec/4239 sacados/4239 chapExoCorrec/3873 sacados/3873 chapExoCorrec/4078 sacados/4078 chapExoCorrec/3825 sacados/3825
E.6086 Consider the plane provided with a O ; −→ u ; −→ v direct orthonormal coordinate system. Consider the points : A (0 ; 1) ; B (2 ; − 1) To any point of the plane M , different from A and B , of affix z , we associate a point M of affix z defined by the relation: z = z − 2 + i z − i 1 Geometrically interpret the length OM and the angle −→ u ; −−−→ OM using the points A , B and M . 2 Consider the point M (4 ; 1) . Determine the algebraic and exponential writing of the complex number z associated with z . 3 In this question, we wish to characterize the set of points M of the plane such that the point M belong to the ordinate axis: a Note z = x +i · y . Establish the following property: ˇ z is a pure imaginary if x 2 + y 2 − 2 x − 1=0 ı b Justify that the set sought is the circle of diameter [ AB ] . E.3801 The plane P is referred to a reference frame O ; −→ u ; −→ v orthonormal direct. Let f be the application which to any point M of P of non-zero affix z associates the point M of affix : z = 1 2 · z + 1 z We’ll make a figure that will be completed as we go along . 1 Let E be the point with affix z E = − i . Determine the affix of the point E , image of E by f . 2 Determine the set of points M such that M = M . 3 Let A and B be the points with affixes 1 and − 1 . respectively Let M be a point distinct from the points O , A and B . a Show that, for any complex number z different from 0 , 1 and − 1 , we have : z + 1 z − 1 = z + 1 z − 1 2 b Deduce an expression for M B M A as a function of MB MA , then an expression for the angle −−→ M A ; −−−→ M B as a function of the angle −−→ MA ; −−→ MB . 4 Let Δ be the perpendicular bisector of segment [ AB ] . Show that if M is a point of Δ distinct from the point O , then M is a point of Δ . 5 Let Γ be the circle of diameter [ AB ] . a Show that if the point M belongs to Γ then the point M belongs to the line ( AB ) . b Does any point on the line ( AB ) have an antecedent by f ? 11. Root n -th of unit E.6106 We are interested in the set U 4 defined by the set of solutions to the equation : z 4 = 1 1 a Factorize the expression z 4 − 1 into a product of prime factors. b Give the algebraic writing, then the exponential writ-ing of the set U 4 . 2 Consider the plane provided with a reference frame O ; −→ u ; −→ v orthonormal direct and, for j ∈ 0 ; 1 ; 2 ; 3 , the points M j of affixes : z j = e i · π 2 × j a Show that the points M j , for j ∈ 0 ; 1 ; 2 ; 3 , belong to the circle of center O and radius 1 . b Establish, for any j ∈ 0 ; 1 ; 2 , the equality: ∠ M j OM j +1 = ı 2 c Specify the nature of the quadrilateral: M 0 M 1 M 2 M 3 E.8599 Definition: The set U 7 is the set of 7 th roots of unity. That is, they are the roots of the equation z 7 − 1 . 1 Give the exponential notation of the elements of U 7 . 2 In the complex plane equipped with an orthonormal di-rect coordinate system O ; −→ u ; −→ v , show that the images of the numbers of U 7 are the vertices of a regular hep-tagon. 3 a Establish the factorization : z 7 − 1 = z − 1 1 + z + z 2 + z 3 + z 4 + z 5 + z 6 b Establish the equality: ω ∈ U 7 w =0 E.8600 Solve the following equations in C : a z + i 4 = 1 b z 3 = 8 Indication : we will give the algebraic writings of the solu-tions. 12. Courses E.3853 The following results are assumed to be known : 1 In the complex plane, we give by their affixes z A , z B and z C three points A , B and C then : https://chingmath.fr chapExoCorrec/6086 sacados/6086 chapExoCorrec/3801 sacados/3801 chapExoCorrec/6106 sacados/6106 chapExoCorrec/8599 sacados/8599 chapExoCorrec/8600 sacados/8600 chapExoCorrec/3853 sacados/3853 Pondichery Avril 2008
011uv ⏐ ⏐ ⏐ z B − z C z A − z C ⏐ ⏐ ⏐ = CB CA arg z B − z C z A − z C = −→ CA ; −−→ CB (2 ı ) 2 Let z be a complex number and let „ be a real number: z =e i · θ if, and only if, | z | =1 and arg( z )= „ +2 · k · ı où k is a relative integer. Course demonstration: demonstrate that the rotation r of angle ¸ and center Ω of affix ! is the transformation of the plane that associates any point m of affix z with the point M of affix z such that : z − ! = e i · ω · ( z − w ) E.3849 The complex plane is referenced to the origin O ; −→ u ; −→ v . Reminders: ˇ For any nonzero vector ~ w with affix z , we have : | z | = −→ w ; arg( z ) = −→ u ; −→ w ı. Let M , N , and P be three points on the plane, with respective affixes m , n , and p such that m = n and m = p . 1 Prove that : arg p − m n − m = −−→ MN ; −−→ MP 2 Interpret the number geometrically: ⏐ ⏐ ⏐ p − m n − m ⏐ ⏐ ⏐ . E.4105 In the complex plane provided with the O ; −→ u ; −→ v orthonormal direct, consider the points M and M distinct from O with affixes z and z respectively. We pose : z = x + i · y ; z = x + i · y où x , x , y , y are real numbers. Recall that z denotes the conjugate of z and that | z | denotes the modulus of z . 1 Express the complex z · z as a function of x , x , y , y . 2 a Show that the vectors −−→ OM and −−−→ OM are orthogonal if, et seulement si, Re z · z =0 . b Show that the points O , M and M are aligned if, and only if, Im ( z · z )=0 E.8601 Definition: For n a non-zero natural number, consider the set U n of n -th roots of unity. A root ! is said to be a root n -th primitive of unity if the successive powers of ! (i.e. ! , ! 2 , ! 3 ,. . . , ! n ) generate all the elements of U n . Show that if n is a prime integer then any element of U n different from 1 is a primitive root n -ième of U n . 13. Suite E.6384 The complex plane is provided with a reference frame O ; −→ u ; −→ v orthonormal direct. For any natural number n , let A n be the point with affix z n de-fined by: z 0 = 1 ; z n +1 = 3 4 + √ 3 4 · i · z n 1 Determine the affixes of the poitns A 1 and A 2 . 2 Demonstrate that the triangle OA n A n +1 is right-angled at A n +1 . 3 We admit that, for any natural number n : z n = ⏐ ⏐ z n ⏐ ⏐ · e i · n · π 6 Place the points A 0 , A 1 , A 2 , A 3 and A 4 in the frame below : 14. Unclassified financial years E.3815 In the plane provided with a O ; −→ u ; −→ v , consider the points B and C of affixes − i and 7 · i respectively. Show that any point M of affix z verifying: z = 3 · i + 4 · e i · θ où „ ∈ R belongs to the circle C of diameter [ BC ] . https://chingmath.fr chapExoCorrec/3849 sacados/3849 Extrait Amerique du Nord Novembre 2006 chapExoCorrec/4105 sacados/4105 chapExoCorrec/8601 sacados/8601 chapExoCorrec/6384 sacados/6384 011uv chapExoCorrec/3815 sacados/3815 Inspire Asie Juin 2004