Grade 12 - Exp. / Complex numbers 40 exercises (100% corrected)

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1. Algebraic writing of a complex number E.6788 Knowing that i 2 = − 1 , simplify the writing of the following expressions : a i 2 b i 3 c i 4 d i 5 e i 14 f i 100 g i − 1 h i − 3 E.3782 Determine the algebraic form of each of the complex numbers below : a z 1 = 3 · 2 − i + i · 3 + 2 · i b z 2 = 5 + 2 · i · 1 − i E.8649 Determine the algebraic form of each of the complex numbers below : a z 3 = − 5 · i · 5 − 4 · i − 3 · i b z 4 = 5 + 2 · i · 5 − 2 · i E.6107 Determine the algebraic form of each of the complex numbers below : a z 1 = 5 + 2 · i 2 b z 5 = 2 − i 2 − 2 · 1 + 3 · i 2 E.3788 Let x be a real number. Consider the complex number z defined by the equality: z = x + 2 · i · 1 − x · i 1 Determine the algebraic writing of the complex number z . 2 a For quelle (s) valeur (s) of x , is z a real number? b For quelle (s) valeur (s) of x , z is a pure imaginary? 2. Calculation Using an Expression E.5310 1 Consider the polynomial P with coefficient in R defined on C by: P =2 · z 3 + z 2 +2 · z +1 Check that the three numbers below are roots of the poly-nomial P : z 1 = − i ; z 2 = i ; z 3 = − 1 2 2 Consider the polynomial Q with coefficient in C defined on C by: Q =i · z 2 + 2+i · z +2 Verify that the complex number z 4 =2 · i is a root of the polynomial Q . E.5311 1 Consider the polynomial P defined on C by: P = 2 · z 3 + 3 · z 2 + 2 · z − 2 Verify that the two complexes z 1 and z 2 are roots of the polynomial P where : z 1 = − 1 + i ; z 2 = − 1 − i 2 Consider the polynomial Q defined on C by: Q = z 3 + z 2 + 3 · z − 5 Verify that the two complexes z 3 and z 4 are roots of the polynomial Q where : z 3 = − 1 − 2 · i ; z 4 = − 1 + 2 · i 3. Equations E.5324 1 In C , consider the equation ( E ) defined by: ( E ) : z + 2 − i = (1 + i) · z Show that the complex number − 1 − 2i is a solution of the equation ( E ) . 2 In C , consider the equation ( F ) defined by: ( F ) : ( z − 1 + 2 · i)( z + 2 · i) = z 2 − i Show that the complex number − i is a solution of the equation ( F ) . E.3799 Reminder : A complex number is zero if, and only if, its real part and imaginary part are zero. Consider the equation ( E ) : i · z + 2 = i 1 Let a +i · b be the algebraic expression of a solution z to the equation ( E ) . Show that : 2 − b +i · a − 1 =0 2 Deduce the algebraic expression of the unique solution to the equation ( E ) . E.8650 In C , solve the following equations : a z + 2 · i · z = i b z + 2 − i · ( z + 1) = 0 E.8560 Solve, in C , the following equations : (3 + i) · z − 3 · (5 · z − 2) = 0 E.6250 Solve, in C , the equations : a z − 5 z − i = i b 2 − 3 · z z + i = − 2 · i + z Hint for the second equation: the set C of complex numbers is said integrates . That is, it verifies the prop-erty for any complex number a and b : a · b =0 = ⇒ a =0 or b =0 4. Conjugated: definitions https://chingmath.fr chapExoCorrec/6788 sacados/6788 chapExoCorrec/3782 sacados/3782 chapExoCorrec/8649 sacados/8649 chapExoCorrec/6107 sacados/6107 chapExoCorrec/3788 sacados/3788 chapExoCorrec/5310 sacados/5310 chapExoCorrec/5311 sacados/5311 chapExoCorrec/5324 sacados/5324 chapExoCorrec/3799 sacados/3799 chapExoCorrec/8650 sacados/8650 chapExoCorrec/8560 sacados/8560 chapExoCorrec/6250 sacados/6250
E.5326 For any complex number z admitting the algebraic writing a +i · b , with a ∈ R and b ∈ R , we call conjugate of the com-plex number z , the complex number, denoted z defined by: z = a − i · b Give the algebraic writing of the conjugate of each of the following complex numbers : a z = 1 + i b z = 2 · i − 3 c z = i · (1 + 2 · i) E.3789 Let z be a complex number. 1 Demonstrate that the following two numbers are real: z + z ; z · z 2 Demonstrate that the complex number z − z is a pure imaginary. 5. Conjugate: quotient simplification E.8561 Method : consider the number defined by z z , where z and z are two complex numbers. To obtain its algebraic writ-ing, consider the quotient z · z z · z whose denominator is a real number. Give the algebraic writing of each of the following complex numbers : a z = 1 i b z = 2 2 − i c z = 3 1 + 2 · i E.3786 Determine the algebraic form of each of the complex numbers below : a z 1 = 2 i b z 2 = 3 2 − 4 · i c z 3 = − 2 1 + i E.5133 Give the algebraic form of the com-plex numbers below : a z 1 = 1 + i i b z 2 = 1 1 − i c z 3 = − 2 + i 2 + i E.5309 Consider the two complex numbers z 1 and z 2 defined by: z 1 = 1 + i ; z 2 = 5 − 2 · i Determine the algebraic writing of the following numbers : a z 1 + z 2 b z 1 − z 2 c z 1 − 2 · z 2 d z 1 · z 2 e z 1 z 2 f z 2 z 1 − z 2 E.5327 Determine the algebraic form of each of the complex numbers below : a z = 2 + 2 · i − 1 + i b z = 3 − 4 · i 1 + i c z = 3 − i 1 + 3 · i E.8562 Determine the algebraic form of each of the complex numbers below : a z 4 = 3 − 2 · i 5 + 3 · i b z 5 = 2 + 3 · i 1 − i c z 6 = 2 2 + 1 1 + i E.4249 Let z be a complex number distinct from 0 and i . Consider the complex numbers n , p and q whose affixes are defined by: n =i · z +1+i ; p = − z +1+i ; q = − i · z Establish the following equality: z − n p − n =i+ 1 z 6. Equations of the form a · z = b E.8564 Solve the following equations and give the algebraic form of their solutions : a i · z = 1 + 2 · i b 1 + i · z = 5 c 4 · z = i − 2 E.8565 Solve the following equations : a 3 · z + 2i · z = 3 − i b 5 − i z + 3 = i 7. First-degree equations E.8566 Solve the following equations : a 2 − 3 · z z + i = − 2 · i + z b (1 + 2 · i)( z − 3 · i) = z + 2 + 3 · i 8. Conjugate: algebraic properties https://chingmath.fr chapExoCorrec/5326 sacados/5326 chapExoCorrec/3789 sacados/3789 chapExoCorrec/8561 sacados/8561 chapExoCorrec/3786 sacados/3786 chapExoCorrec/5133 sacados/5133 chapExoCorrec/5309 sacados/5309 chapExoCorrec/5327 sacados/5327 chapExoCorrec/8562 sacados/8562 chapExoCorrec/4249 sacados/4249 Extrait Amerique du Nord Juin 2011 chapExoCorrec/8564 sacados/8564 chapExoCorrec/8565 sacados/8565 chapExoCorrec/8566 sacados/8566
E.6789 Without performing any calcula-tions, justify that the complex numbers z 1 and z 2 are con-jugate complex numbers. a z 1 = (1+i) · (2 − i) ; z 2 = (1 − i ) · (2 + i ) b z 1 = i − 3 2 + 2 · i ; z 2 = − i − 3 2 − 2 · i c z 1 = (1 − i) 5 ; z 2 = (1 + i) 5 E.3824 For each of the complex numbers below : without calculation, give an expression for the complex number conjugate of the number z ; then, give the algebraic writing of the complex number z . a z = (2 − i)(5 + 3 · i) b z = i · (3 + 2 · i) − 3 + i E.8563 For each complex number, determine the algebraic form of its conjugate : a z = 5 − 2 · i i − 2 b z = (3 − 2 · i)(1 − i) 1 + i E.5333 Consider the two complex numbers z 1 and z 2 : z 1 = 3 − 2 · i 1 + i ; z 2 = 3 + 2 · i 1 − i 1 What can we say about the complex numbers z 1 and z 2 ? 2 a Determine the algebraic writing of the number z 1 . b Deduce the expression for z 1 + z 2 and z 1 − z 2 . 9. Equations with z and z E.6790 Reminders: A complex number is zero if, and only if, its real part is zero and its imaginary part is zero. Thus, for z ∈ C , this property translates to : z =0 ⇐⇒ ( z )=0 et ( z )=0 Determine the values of the real numbers a and b that satisfy the following equality: 2 · a + b · 2 + i − 3 · i = a · i − 2 + b · 1 − 2 · i E.3800 Solve the following equations : a z + z = 6 b z + z = i c z + 2 · z = 8 + i d i · z + 2 · ( z − 5) = 0 E.6194 Solve the following equations : a 3 + i · z + 2 · i = z b (3 + i) · z − 3 · (5 · z − 2) = 0 E.6791 Solve the following equations : a z z + 1 = 1 b 1 − z i · z + 2 = 2 10. Pair formula E.8567 Establish equality: 2+i 5 = − 38+ 41 · i E.6108 Give the algebraic form of the num- ber: z = 4 k =0 1 + i k 11. Suite E.3783 1 Simplify the writing of the following expression : A = 1 + i + i 2 + i 3 2 Determine the algebraic writing of the complex number: B = 1 + i + i 2 + · · · + i 99 12. Lesson - Complex numbers E.4101 Prerequisites Let z be a complex number such that z = a +i · b where a and b are two real numbers. Let z be the complex number defined by: z = a − i · b Questions 1 Prove that, for all complex numbers z and z : z × z = z × z 2 Prove that, for any nonzero natural number n and any complex number z : https://chingmath.fr chapExoCorrec/6789 sacados/6789 chapExoCorrec/3824 sacados/3824 chapExoCorrec/8563 sacados/8563 chapExoCorrec/5333 sacados/5333 chapExoCorrec/6790 sacados/6790 chapExoCorrec/3800 sacados/3800 chapExoCorrec/6194 sacados/6194 chapExoCorrec/6791 sacados/6791 chapExoCorrec/8567 sacados/8567 chapExoCorrec/6108 sacados/6108 chapExoCorrec/3783 sacados/3783 chapExoCorrec/4101 sacados/4101
z n = z n E.6812 Organized knowledge re-trieval Prerequisite: any complex number z can be written as : z = x +i · y where x ∈ R and y ∈ R Le conjugate of z is the complex number defined by: z = x − i · y Prove 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: z n = z n . E.3300 Prerequisite : the modulus of any complex number z , de-noted | z | , verifies | z | 2 = z z where z is the conjugate of z . Show that : for all complex numbers z 1 and z 2 : ⏐ ⏐ z 1 × z 2 ⏐ ⏐ = ⏐ ⏐ z 1 ⏐ ⏐ × ⏐ ⏐ z 2 ⏐ ⏐ . for any non-zero complex number z : ⏐ ⏐ ⏐ 1 z ⏐ ⏐ ⏐ = 1 | z | 13. Unclassified financial years E.8580 Determine the algebraic form of the com-plex number z defined by: z = 1 − i 1+i E.8587 Consider the complex number: j = − 1 2 +i · 3 2 . Demonstrate, using reasoning by recurrence, the following equality for any natural number n : (1 + j ) 2 n +1 = − j n +2 https://chingmath.fr chapExoCorrec/6812 sacados/6812 chapExoCorrec/3300 sacados/3300 chapExoCorrec/8580 sacados/8580 chapExoCorrec/8587 sacados/8587