Grade 12 - Exp. / Complex numbers and polynomial equations 29 exercises (100% corrected)

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1. Introduction to second-degree equations in C E.8588 1 In C , consider the equation z 2 +1=0 . a Verify that the complex number z 1 = i is a solution of this equation. b Propose another solution to this equation. c What can be said about these two roots? 2 In C , consider the equation 2 · z 2 − 2 · z +1=0 . a Verify that the two complex numbers : z 3 = 1 2 + 1 2 · i ; z 4 = 1 2 − 1 2 · i b What can we say about these two roots of the equa-tion? E.3787 1 Solve in R the following equation : x 2 + x + 1 = 0 2 Consider the complex number j = − 1 2 +i · 3 2 . a Determine the algebraic writing of the two complex numbers : j 2 ; j 3 b Determine the algebraic writing of the following com-plex numbers : 1+ j + j 2 ; 1+ j + j 2 c In C , of which quadratic equation are the numbers j and j solutions? E.8589 In C , consider the equation : z 2 + z +1=0 1 Let z be a solution of this equation admitting as algebraic writing a +i · b where a;b ∈ R . Show that the two real numbers a and b verify the sys-tem of equations : a 2 + a + 1 − b 2 = 0 2 · a · b + b = 0 2 a Deduce that this equation admits two solutions whose algebraic writings will be given. b What can be said about the two solutions of this equa-tion? 2. Second-degree equations in C E.3802 Solve the following quadratic equa-tions in C : a z 2 − 3 · z + 4 = 0 b z 2 − 4 · z + 4 = 0 c 3 · z 2 + 3 · z + 2 = 0 d z 2 − 4 · z − 1 = 0 E.5332 Solve the following equations in C : a 2 z 2 + 2 z + 1 = 0 b − z 2 + 2 z − 3 = 0 c z 2 + 3 z + 3 = 0 d − z 2 − 3 z − 2 = 0 E.8586 Solve the equation : 2 · z 2 − 6 · z +9= 0 E.6101 Solve in C the equation : Z 2 +4 · Z +16=0 . Write the solutions of this equation in exponential form. E.3810 1 Solve in the set of complex numbers the equation : z 2 − 2 z + 4 = 0 The solutions will be denoted z and z , z denoting the solution whose imaginary part is positive. Write the solutions of this equation algebraically and then exponentially. 2 Give the exact exponential writing of the complex num-ber z 2 004 , then its algebraic writing. E.6781 In C , we consider the equation E λ depending on the parameter – ∈ R ; x 2 − 3 · x + 4 = – For what values of – , the equation E λ admits two distinct conjugate solutions. E.6811 Let C be the set of complex numbers and ( E ) the equation with complex unknown z : ( E ) : z 2 + 2 · a · z + a 2 + 1 = 0 où a denotes any real number. Only one of the following propositions is correct. Copy the chosen answer and justify this proposition a For any value of a , ( E ) has no solution in C . b For any value of a , the solutions of ( E ) in C are not real and they are conjugate. c For any value of a , the solutions of ( E ) in C are not real and they are not conjugate. d There exists a value of a for which ( E ) admits at least one real solution. E.3803 Consider the complex function de-fined on C \{ 2 } by the relation: f : z ↦−→ z − 4 z − 2 Determine the set of complex numbers invariant by this func-tion. https://chingmath.fr chapExoCorrec/8588 sacados/8588 chapExoCorrec/3787 sacados/3787 chapExoCorrec/8589 sacados/8589 chapExoCorrec/3802 sacados/3802 chapExoCorrec/5332 sacados/5332 chapExoCorrec/8586 sacados/8586 chapExoCorrec/6101 sacados/6101 chapExoCorrec/3810 sacados/3810 Extrait Pondichery Avril 2004 chapExoCorrec/6781 sacados/6781 chapExoCorrec/6811 sacados/6811 Extrait d'Antilles-Guyane Septembre 2016 chapExoCorrec/3803 sacados/3803
E.6794 We give the complex number: j = − 1 2 +i · 3 2 1 Solve in the set C of complex numbers the equation : z 2 + z + 1 = 0 2 Demonstrate the following equalities: a j 3 = 1 b j 2 = − 1 − j 3 We assume the existence of three complex numbers a , b and c verifying the equality: a + j · b + j 2 · c = 0 a Demonstrate equality: a − c = j · c − b b Demonstrate equality: a − b = j 2 · b − c 3. Second-degree equation and factorization E.3818 Solve in C the equation : z − 2 · i z 2 − 2 · z +2 =0 Give the algebraic and exponential writing of the solutions of this equation (justify answers) . E.6780 In C , consider the equation : ( E ) : z 3 + z 2 − 2 = 0 1 Show that the complex number z 1 define by z 1 = − 1 − i is solution of the equation E . 2 Justify that the complex number z 2 defined by z 2 = z 1 is also a solution of the equation ( E ) . 3 Noting that 1 is also a solution of ( E ) , propose a factor-ized form in C of the polynomial z 3 + z 2 − 2 . This form will be developed to establish the factorization. E.8590 We want to solve in C the equation : ( E ) : z 3 + 4 · z 2 + 2 · z − 28 = 0 1 Determine two reals a and b such that the equation ( E ) is written as : z − 2 z 2 + a · z + b = 0 2 Solve ( E ) . E.5331 Consider in C the equation : ( E ) : z 3 + 4 · z 2 + 2 · z − 28 = 0 1 Determine two reals a and b such that the equation ( E ) is written : ( E ) : z − 2 z 2 + a · z + b = 0 2 Solve equation ( E ) . E.3850 Consider the equation ( E ) : z 3 − (4 + i) · z 2 + (7 + i) · z − 4 = 0 où z denotes a complex number. 1 Show that ( E ) admits a real solution, denoted z 1 . 2 Determine the two complex numbers a and b such that, for any complex number z we have : z 3 − (4+i) · z 2 +(7+i) · z − 4 = ( z − z 1 )( z − 2 − 2 · i)( a · z + b ) 3 Solve ( E ) . E.5958 For any number z , we pose : P ( z )= z 4 − 1 . 1 Factorize P ( z ) into a product of factors of the first de-gree. 2 Deduce the solutions in the set C of complex numbers of the equation P ( z )=0 , of unknown z . 3 Deduce from the previous question the solutions in C of the equation of unknown z : 2 · z + 1 z − 1 4 = 1 4. Root of a complex number E.8591 We denote by a the complex number whose modulus is equal to 2 and whose one argument is equal to ı 3 . 1 Calculate a 2 in algebraic form. 2 Deduce the two solutions in C of the equation : z 2 = − 2 + 2 · i 3 . We’ll write the solutions in algebraic form. E.5534 The complex plane is referred to a O ; −→ u ; −→ v direct orthonormal reference frame. Consider the application f of the plane into itself which, to any point M of affix z , associates the point M of affix z such that : z = z 2 . 1 Determine the set Γ 1 of points M of the plane such that : f ( M )= M . 2 Let A be the point of affix : a = 2 − i 2 a Express a in exponential form. b Deduce the affixes of the two antecedents of A by f . 3 Determine the set Γ 2 of points M of affix z such that the affix z of the point M is a pure imaginary number. 5. Equations with change of variables https://chingmath.fr chapExoCorrec/6794 sacados/6794 Extrait d'Asie Juin 2015 chapExoCorrec/3818 sacados/3818 Extrait de Liban Juin 2004 chapExoCorrec/6780 sacados/6780 chapExoCorrec/8590 sacados/8590 chapExoCorrec/5331 sacados/5331 chapExoCorrec/3850 sacados/3850 Extrait Nouvelle-Caledonie Novembre 2006 chapExoCorrec/5958 sacados/5958 Extrait d'Asie Juin 1999 chapExoCorrec/8591 sacados/8591 chapExoCorrec/5534 sacados/5534
E.3817 In the set C of complex num-bers : 1 Show that : (1 + i) 6 = − 8 · i . 2 Consider the equation ( E ) : z 2 = − 8 · i a Deduce from 1 a solution to the equation ( E ) . b The equation ( E ) has another solution ; write this so-lution in algebraic script. 3 Also deduce from 1 a solution to the equation : ( E ) : z 3 = − 8 · i E.6204 1 In C , consider the polynomial: z 2 +6 z +25 . Determine its roots. 2 a Give the algebraic writing of the complex number a and b defined by: a = (1 + 2 · i) 2 ; b = (1 − 2 · i) 2 b Deduce the solutions of the equation : z 4 + 6 z 2 + 25 = 0 E.6205 1 In C , consider the polynomial: 4 z 2 − 16 z +25 . Determine its roots. 2 a Give the algebraic writing of the complex number a and b defined by: a = 3 2 + 1 2 · i 2 ; b = 3 2 − 1 2 · i 2 b Deduce the solutions of the equation : 4 z 4 − 16 z 2 + 25 = 0 6. Non-polynomial equations or equations with non-real coefficients E.3835 1 Determine the set of roots of the polynomial: P = i · z 2 − 2 · i · z + 3 · i 2 Consider the polynomial: Q =i · z 2 +2 · z − 10 · i a Verify that the complex number z 1 is a root of the polynomial Q where : z 1 = − 3 + i b Determine the factored form of the polynomial Q . Hint: We can look for the two complex numbers a and b that satisfy the factorization : Q = z +3 − i a · z + b E.4238 The complex plane P is pro-vided 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 plane P deprived of 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 We are looking 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 . 1 Demonstrate that, for any complex number z : z 2 + i · z − 1 = z + 3 2 + 1 2 · i z − √ 3 2 + 1 2 · i 2 Deduce the affixes of the points of the set ( E ) . 3 Show that the points ( E ) belong to (Γ) . E.3812 In the complex plane pro-vided with a O ; −→ u ; −→ v orthonormal direct, note ( H ) the set of points M of affize z verifying: z 2 − 4 = 4 − z 2 1 Note x and y the real and imaginary parts of the affix z of a point M . Show that : M belongs to ( H ) if, and only if, x 2 − y 2 =4 2 Let A , B and C be the points with respective affixes : 2 ; − 3 − i · 5 ; − 3 + i · 5 Check that A , B and C belong to ( H ) . E.3851 1 Determine the complex number ¸ such that : ¸ · (1 + i) = 1 + 3 · i i · ¸ 2 = − 4 + 3 · i 2 For any complex number z , we pose : f ( z ) = z 2 − (1 + 3 · i) · z + ( − 4 + 3 · i) Show that f ( z ) admits as expression z − ¸ z − i · ¸ . Deduce the algebraic writings of the solutions of the equa-tion f ( z )=0 . 7. Factorization of z n − a n by z − a E.6105 Let a be a complex number other than 1 and z any complex number. 1 Consider the geometric sequence u n with first term 1 and common ratio z a . Let n be a non-zero natural number. Establish the equal-ity: u 0 + u 1 + u 2 + ··· + u n − 1 = 1 − z n a n 1 − z a 2 Deduce the factorization : z n − a n = z − a z n − 1 + z n − 2 a + z n − 3 a 2 + ··· + z 2 a n − 3 + za n − 2 + a n − 1 https://chingmath.fr chapExoCorrec/3817 sacados/3817 Extrait France Juin 2004 chapExoCorrec/6204 sacados/6204 chapExoCorrec/6205 sacados/6205 chapExoCorrec/3835 sacados/3835 chapExoCorrec/4238 sacados/4238 chapExoCorrec/3812 sacados/3812 Extrait Amerique du Nord Mai 2004 chapExoCorrec/3851 sacados/3851 Antilles Guyane Septembre 2007 chapExoCorrec/6105 sacados/6105
Note: this factorization can also be written as : z n − a n = z − a · n − 1 k =0 a k · z n − k − 1 8. Unclassified financial years E.6251 For each of the following four statements, indicate whether it is true or false and justify the answer. An unjustified answer is not taken into account. No answer is penalized. 1 The equation z − 4 z 2 − 4 · z +8 =0 admits 3 solutions in C one of which is real and the other 2 are conjugate to each other. 2 In the set of complex numbers, the equation : ( E ) : z − z + 2 − 4 · i = 0 admits a unique solution. 3 Consider the sequence z n of complex numbers defined by: z 0 = 2 ; z n +1 = 1 + i · z n The fifth term of the sequence is a real number. 4 To any complex number z , we associate a complex num-ber z defined by: z = z 2 + 2 z + 9 The set of complex numbers z such that z is a real is the set of complex numbers z = x +i · y où y =0 . https://chingmath.fr chapExoCorrec/6251 sacados/6251