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Complex numbers 1 Complex numbers in algebraic, trigonometric and exponential form 9000034808 (level 1): Find the algebraic form of the complex number z = 2 (cos π + i sin π). 9000034801 (level 1): Given complex numbers z1 = 4 − i and z2 = 1 − 2i, find z1 − z2 . 3 − 3i 3+i 5 − 3i −2 3−i −3 − i 3+i 3−i −1 − 3i −1 + 3i √ 2 2 2 10 √ 2 π 2 9000034805 (level 1): Find the complex number z which satisfies 2z = 2 − 3i. 3 1− i 2 −3i 4 − 6i 5 π 9 − π 2 3 − π 2 3 −1 + i 2 3 2 A −1 1 i 1 −5 9000034807 (level 1): Find the polar form of the complex number z = 2i. 2 cos cos π π + i sin 2 2 π π + i sin 2 2 3 π 2 9000035701 (level 1): 5 Find the algebraic form of the complex number graphed in the complex plane. 4 Im 9000034806 (level 1): Simplify i15 . −i 3π 9000034810 (level 1): π π and Given complex numbers z1 = 2 cos + i sin 4 4 √ 7π 7π + i sin z2 = 2 cos , find the angle in the polar form 4 4 z1 of the quotient . z2 1 − 3i 9000034804 (level 1): Find the absolute value of the complex number z = 3 − i. √ 2 π 9 3π 2 9000034803 (level 1): Find the complex conjugate of z = 1 − 3i. 1 + 3i 2i 9000034809 (level 1): π π and Given complex numbers z1 = 2 cos + i sin 6 6 √ 4π 4π z2 = 3 cos + i sin , find the angle in the polar form 3 3 of the product z2 z2 . 9000034802 (level 1): Find the opposite number to the complex number z = 3 − i. −3 + i −2i 2 −4 −3 −2 −1 1 −1 √ π π 2 cos + i sin 2 2 −2 −3 2 (cos 0 + i sin 0) −4 1 −5 2 3 Re 4 5 5 z = −3 + 2i z = 2 − 3i 4 Im z = −3 − 2i z = 2 + 3i 3 5 9000035702 (level 1): 4 Find the absolute value of the complex number graphed in the complex plane. 3 Im 2 A 1 2 −5 −4 −3 −2 −1 1 −4 −3 −2 −1 1 2 Re 3 4 −4 √ √ π π 3π 3π z = 2 2 cos − i sin + i sin z = 2 2 cos 4 4 4 4 −5 −3 √ π π z = 2 2 − cos + i sin 4 4 −4 √ −5 3 5 5 −3 −2 5 Re 4 −2 5 −1 A 3 −1 1 −5 2 √ 5π 5π + i sin z = 2 2 cos 4 4 4 9000035705 (level 1): Find the absolute value of the complex number z = (1 − 2i)(2 + i). 9000035703 (level 1): Find the absolute value of the complex number graphed in the complex plane. Im 5 3 √ 10 √ 2 2 5 4 A 9000035706 (level 1): Find the absolute value of the complex number z = 3 √ 2 2 2 √ 2 5 2 2 + 6i . 1 − 2i √ 2 3 1 −5 −4 −3 −2 −1 1 2 3 Re 4 9000035707 (level 1): Find the real part of the complex number 2 + 2i2 + i3 − i4 + 2i5 . 5 −1 √ 2 5 √ 2 3 4 −2 −1 √ 6 1 5 −3 9000035708 (level 1): Find the imaginary part of the complex number 1 + 2i12 + 3i19 − i22 + 2i105 . −3 9000035704 (level 1): −4 Find the polar form of the complex number graphed in the complex plane. −5 −1 −5 9000035710 (level 1): 2 1 4 Find the complex conjugate of 2 − 4i 2 + 4i √ √ − 2+i 2 3+i + (i + 1)(2 + i). 2−i −2 − 4i −2 + 4i 13 + i −7 − i Find the imaginary part of the complex number z = 3 2 − 3 2 1 2 − 5π 5π 3 cos + i sin 4 4 2+i . 1−i 1 2 4 −0.8 Find the absolute value of the complex number z = 1 5 √ 7 5 2−i . 2+i √ − 3+i √ 5 5 −3 − i −3 + i Find the opposite number to the complex number z = 1 −1 −4 √ 3−i √ 3+i √ − 3−i i + 3i(2 − i)2 − 4(1 − i)3 20 − 18i 3+i 9000031206 (level 2): −i √ 7π 7π 3 2 cos + i sin 4 4 9000035801 (level 2): Find the complex conjugate of the following complex number. 9000031205 (level 2): Find the complex conjugate of z = i5 − 3i10 . 3−i 9000031210 (level 2): √ π π Given complex numbers z1 = 2 3 cos + i sin and 6 6 √ 4π 4π z1 + i sin , find the quotient . z2 = 3 cos 3 3 z2 1 9000031204 (level 2): √ √ − 2−i 2 π π 3 cos + i sin 4 4 −4i 4i 2−i Find the real part of the complex number z = . 2+i 0.8 √ 2−i 2 9000031209 (level 2): √ π π Given complex numbers z1 = 2 2 cos + i sin and 4 4 √ 7π 7π z2 = 2 cos + i sin , find the product z1 z2 . 4 4 9000031203 (level 2): 0.6 √ √ 3π 3π 3 2 cos + i sin 4 4 13 + 11i 9000031202 (level 2): √ 2+i 2 9000031208 (level 2): Find the polar form of the complex number z = −3 + 3i. 9000031201 (level 2): Given complex numbers z1 = 1 − 2i and z2 = 3 + 5i, find z1 z2 . 13 − i √ 20 − 24i 20 + 18i −8 + 26i 9000035802 (level 2): Solve the following equation for z ∈ C. 1+i . 1−i 3z − 2z = 8 − 10i i 8 − 2i 9000031207 (level 2): Find the of the complex number algebraic form 3π 3π z = 2 cos + i sin . 4 4 1 + 5i 8 − 10i 2 + 2i 9000035803 (level 2): Given the complex number z = −1 + 2i, find the imaginary 1 part of the complex number . z 3 − 1 2 2 5 2 5 − Evaluate i50 . 1 2 −1 9000035804 (level 2): Find the algebraic form of the following complex number. i 1 −i i 1 9000037402 (level 2): Evaluate i7 . (2 + i) (3 + 2i) −1 −i 4 + 7i 8 − 7i 8 + 7i 4 − 7i 9000037408 (level 2): Find the polar form of the complex number 9000035805 (level 2): Given the complex numbers √ 3π 2π 2π 3π a = 2 cos + i sin , b = 2 cos + i sin , 3 3 4 4 cos 4π 4π + i sin 3 3 cos 3 3 + i sin 2π 2π find the product ab. √ 17π 17π + i sin 2 2 cos 12 12 √ π π 2 2 cos + i sin 2 2 √ √ 5π 5π + i sin 2 2 cos 7 7 5π 5π + i sin 2 2 cos 12 12 2 3 cos 5π 5π + i sin 6 6 2 3 2 3 cos cos cos π + i sin 6 6 7π 7π + i sin 6 6 3 2− i 2 8 7 − i 5 5 1 . cos 7π + i sin 7π 6 6 π π + i sin 6 6 π π cos − + i sin − 6 6 9000037501 (level 2): Find the absolute value of the following complex number. √ 3 + 2i π √ 11 √ 3 13 √ 3 2 9000037502 (level 2): Find the total sum of the complex numbers a, b and c. √ √ a = 3 + 2i, b = 1 − 4i, c = 3 − 3i 9000035807 (level 2): Given the complex numbers a = 2 − 3i, b = 1 + 2i, find the a quotient . b 4 7 − − i 5 5 3 3 − i sin 2π 2π 5π 5π cos − + i sin − 6 6 5π 5π cos + i sin 6 6 a . b 2 11π 11π cos + i sin 3 6 6 cos Find the polar form of the complex number 9000035806 (level 2): Given the complex numbers 5π 5π 11π 11π a = 2 cos + i sin , b = 3 cos + i sin , 3 3 6 6 find the quotient 4π 4π cos − + i sin − 3 3 9000037409 (level 2): 1 . cos 2π + i sin 2π 3 3 4 7 + i 3 3 9000037401 (level 2): 4 4+ √ √ 3 + i( 2 − 7) 4+ √ √ 2 + i( 3 − 3) √ 4+i 3 4+ √ √ 3 − i( 2 − 7) 9000037503 (level 2): Given complex numbers √ √ a = 2 + 3i, b= √ 2− √ 9000037508 (level 2): Find the absolute value of the following complex number. √ π π 2 cos + i sin 3 3 3i, find the product ab. 5 √ 2 √ 2+i 3 √ √ √ 2−i 3 a = 5 + 2i, b = 3 − i, π π , a = 3 cos + i sin 3 3 c=i find the product abc. 1 − 17i −1 − 17i 1 + 17i √ 2 3−i 11 √ π π 3 2 cos + i sin 2 2 √ π π 3 2 cos − i sin 2 2 √ π π −3 2 cos + i sin 2 2 10i find the quotient a find the quotient . b b=2−i 7 1 +i 5 5 9000037507 (level 2): Given complex numbers √ a = 3 + 2i, find the quotient 2π 2π 2 cos + i sin 3 3 √ −3 2 π π a = cos + i sin , 3 3 a = 3 + 5i, 13 1 +i 3 3 b= √ 9000037510 (level 2): Given complex numbers 9000037506 (level 2): Given complex numbers 1 13 +i 5 5 2−2 find the product ab. 9000037505 (level 2): Find the complex conjugate of the following complex number. √ −2 3 − i √ −2 3 + i √ 2 2+2 9000037509 (level 2): Given complex numbers 9000037504 (level 2): Given complex numbers −1 + 17i √ 2 a . b b= √ 2π 2π 2 cos + i sin 3 3 a . b √ π π 2 cos − + i sin − 2 3 3 √ π π 2 cos − − i sin − 2 3 3 7 1 +i 3 3 √ π π 2 − cos − − i sin − 2 3 3 b= √ √ π π 2 − cos − + i sin − 2 3 3 2−i √ √ √ 6−2 2 2+ 3 +i 3 3 √ √ √ 6−2 2 2+ 3 −i 3 3 √ √ √ 6−3 2 2+ 3 +i 2 2 √ √ √ 6−2 2 2+ 3 −i 2 2 9000038601 (level 2): Find the polar form of the following complex number. √ 1 3 − +i 2 2 5 cos 2π 2π + i sin 3 3 π π + i sin 3 3 √ 2π 2π 5 cos + i sin 3 3 3π 3π + i sin 2 2 √ 2π 2π 5 cos + i sin 5 5 cos π π + i sin − cos − 3 3 cos 9000038602 (level 2): Find the polar form of the following complex number. √ 1 3 +i 2 2 cos cos π π + i sin 3 3 3π 3π + i sin 2 2 cos 4π 4π + i sin 2 cos 3 3 √ 3π 3π 5 cos + i sin 2 2 9000038606 (level 2): Find the algebraic form of the following complex number. cos 2π 2π + i sin 3 3 √ √ 2 2 +i 2 2 √ √ 2 2 −i 2 2 π π + i sin 4 4 √ √ √ 3 3 +i 2 2 √ 3 3 −i 2 2 π π + i sin − cos − 3 3 9000038607 (level 2): Find the algebraic form of the following complex number. π π 3 cos + i sin 2 2 9000038603 (level 2): Find the polar form of the following complex number. √ √ 2 6 +i 2 2 √ π π 2 cos + i sin 3 3 √ π π 5 cos + i sin 3 3 √ 3i 2π 2π 2 cos + i sin 3 3 −3i 8(cos π + i sin π) −8 9000038604 (level 2): Find the polar form of the following complex number. √ √ 3 3 √ + i√ 2 2 √ √ π π 2 cos + i sin 3 3 √ 3 − 3i 9000038608 (level 2): Find the algebraic form of the following complex number. 3π 3π + i sin 2 cos 2 2 √ π π 3 cos + i sin 4 4 3 + 3i 8 + 8i 8 8 − 8i 9000038609 (level 2): Find the algebraic form of the following complex number. 3π 3π 5 cos + i sin 4 4 3π 3π 3 cos + i sin 4 4 √ √ 5 2 5 2 − +i 2 2 2π 2π 2 cos + i sin 3 3 √ √ 5 2 5 2 −i 2 2 5 5 −i 2 2 9000038605 (level 2): Find the polar form of the following complex number. √ √ 5 15 +i − 2 2 9000038610 (level 2): 6 5 5 +i 2 2 Find the algebraic form of the following complex number. 3π 3π 2 cos + i sin 4 4 √ √ √ − 2+i 2 √ 1 1 − i 3 3 3 6 z=− + i 5 5 z =1+i 1 3 z=− − i 5 5 9000039109 (level 2): Assuming z ∈ C, solve the following equation. 2 − 2i 2 + 2i 2−i 2 z= 2z − iz = 1 − i 9000039101 (level 2): Find the polar form of the complex number √ 3π 3π + i sin 2 cos 4 4 √ π π 2 cos + i sin 4 4 √ i14 − 1 . i9 + 1 z =1−i √ 1 3 − i 2 2 3 4 − − i 5 5 √ 1 3 z=− − i 2 2 −i z=− x = 13, y = 8 √ 3 1 z= + i 2 2 1 1 z=− + i 3 3 √ 3 1 + i 2 2 x = 8, y = 13 −2i x = 2, y = 3 2 9000039104 (level 2): Assuming x ∈ R, y ∈ R, solve the following equation. (3 − 2i)x + (5 − 7i)y = 1 + 3i x = −1, y = 2 √ 1 3 z=− + i 2 2 9000070110 (level 2): 5 5 Given z1 = 4 cos π + i sin π and 3 3 1 1 z1 z2 = 2 cos π + i sin π , evaluate . 6 6 z2 (2 + 5i)x + (1 − i)y = 13i + 8 x = 2, y = −1 1 1 − i 3 3 9000039110 (level 2): Assuming z ∈ C, solve the following equation. √ √ 1+i 3 z =1−i 3 √ 7π 7π 2 cos + i sin 4 4 9000039103 (level 2): Assuming x ∈ R, y ∈ R, solve the following equation. x = 3, y = 2 z= 5π 5π + i sin 2 cos 4 4 9000039102 (level 2): Identify a complex number with absolute value different from 1. 1+i z =1+i 4i i 1 − i 2 Moivre’s theorem 9000035709 (level 1): Simplify (1 − i)−3 . x = −2, y = −1 1 1 − + i 4 4 x = −1, y = −2 1 + 3i −2 − 2i 1 1 + i 2 2 9000035808 (level 2): Evaluate (1 − i)10 . 9000039108 (level 2): Assuming z ∈ C, solve the following equation. −32i 2z − i z = 1 − i 7 32 32i −32 Evaluate the following complex number. π π 3 cos + i sin 4 4 9000035809 (level 2): Given the complex number z = −1 + i, find the angle in the polar form of the number z 6 . π 2 3π 2 3π 4 √ 2 2 − +i 2 2 √ 9 9i 3 + 2 2 3 2 − 3 2 √ 1 3 − −i 2 2 9000037404 2): √ (level π π Given z = 2 cos − i sin , find z 2 . 3 3 √ −1 − i 3 √ 1+i 3 √ −2 − i 2 7 − 7i √ 3 1 − − i 2 2 2+i 2 1+i −i 1 + 2i 1 −i i 11 i −i 1−i 1−i 1 i i13 1 cos 1+i −1 9000070105 (level 2): Evaluate the following complex number. 9000037410 (level 2): 3 Evaluate (1 − i) . 2 + 2i √ 1 3 − +i 2 2 3 1 − + i 2 2 (sin 2π + i cos 2π) cos −2 − 2i √ 9000070104 (level 2): Evaluate the following complex number. 9000037407 2): (level π π 13 Evaluate cos + i sin . 2 2 i √ 2 2 +i 2 2 9 9000037406 2): (level π 40 π Evaluate cos + i sin . 4 4 1 √ (cos π + i sin π) −1 1 √ 2 2 −i 2 2 9000070103 (level 2): Evaluate the following complex number. √ 9000037405 (level 2): 7 Evaluate (1 + i) . 8 − 8i √ √ 2 2 − −i 2 2 9000070102 (level 2): Evaluate the following complex number. π π 10 cos + i sin 3 3 9000037403 2): √ (level π π Given z = 3 cos + i sin , find z 4 . 3 3 √ 9 9i 3 − − 2 2 √ √ 7π 4 π π + i sin 2 2 cos π π − i sin 2 2 3π 3π + i sin 2 2 i 9000070106 (level 2): Evaluate the following complex number. 9000070101 (level 2): (1 − i)8 8 sin π π + i cos 2 2 −16i 16 −16 16i 9000035602 (level 2): Find the values of the parameter m ∈ C which guarantee that the following quadratic equation has a double solution. 9000070107 (level 2): Evaluate the following complex number. 1 π + cos + i cos 2π 2 3 −4 + 4i −4 − 4i 4 − 4i mx2 − 2x − 1 + i = 0 5 1 1 m=− − i 2 2 4 + 4i m = −1 m = −1 + i 1 1 m=− + i 2 2 9000070108 (level 2): Evaluate the following complex number. 9000035603 (level 2): Find the solution set of the following equation. √ !6 3 1 + i 2 2 −1 1 4x2 + 9 = 0 −i i 3 3 − i; i 2 2 2 2 − i; i 3 3 9 9 − i; i 4 4 3 3 − ; 2 2 9000070109 (level 2): Evaluate the following complex number. √ −8i 3 3−i x2 − 2ix + 3 = 0 −8 8 9000035604 (level 2): Solve the following equation. 3 8i {−i; 3i} Quadratic equations with complex roots 3 −∞; − 4 ∪ 3 ;∞ 4 3 ;∞ 4 9000035605 (level 2): 7 7 The number cos π + i sin π is a solution of a quadratic 6 6 equation with real valued coefficients. Find the second solution. 5 5 cos π + i sin π 6 6 3 3 − ; 4 4 3 3 − ; 4 4 {1 − 2i; 1 + 2i} {−3i; i} 9000035601 (level 2): Find the values of the parameter t ∈ R which guarantee that the following quadratic equation has solutions with nonzero imaginary part. tx2 − 3x + 4t = 0 {−4i; 4i} 7 7 cos π + i sin π 6 6 1 1 cos π + i sin π 6 6 cos 11 11 π + i sin π 6 6 9000035606 (level 2): Find the quadratic equation with √ real valued coefficients and one of the solutions x1 = −1 + i 3. 3 3 R\ − ; 4 4 9 x2 + 2x + 4 = 0 x2 − 2x + 4 = 0 x2 − 2x − 2 = 0 x2 + 2x + 2 = 0 Find the value of the parameter a which guarantees that the quadratic equation x2 + 2x − 4 = 0 x2 + 2ax + a = 0 has a pair of complex conjugate solutions with a nonzero imaginary part. 9000035607 (level 2): Identify the quadratic equation with solutions x1 = 2i, x2 = −i. x2 − ix + 2 = 0 x2 + ix + 2 = 0 x2 + ix − 2 = 0 x2 − ix − 2 = 0 x2 = 1 − 4i, q = 7 − 4i x2 = 1, q = −1 − 2i x2 + 4 = 0 √ x2 = 3 + i 2, p = −2i x2 − 4i = 0 2 5 x2 − 4 = 0 x2 + 4i = 0 3x2 + 4x + 2 = 0 3x2 − 4x + 2 = 0 9000064503 (level 2): Find the values of the real coefficients a, b and c such that the quadratic equation ax2 + bx + c = 0 √ x2 = −3 − i 2, p = −2i 2 has zeros x1,2 = ±i 9000039105 (level 2): Find the quadratic equation with real coefficients such that one of the solutions is the complex number x1 = 1 + 2i. x2 − 2x + 3 = 0 − x2 + 4x + 6 = 0 √ x2 − 2x + 5 = 0 2 4 + i 5 5 x2 − 4x + 6 = 0 √ with a parameter p ∈ C has a solution x1 = 3 − i 2. Find the second solution x2 and the parameter p. √ x2 = −3 − i 2, p = 6 2 5 9000064502 (level 2): √ Find the quadratic equation with the solution x1,2 = 2 ± i 2. 9000035609 (level 2): The equation x2 + px − 11 = 0 √ x2 = 3 + i 2, p = 6 such an a does not exist 9000064501 (level 2): Find the quadratic equation with the solution x1,2 = ±2i. x2 = −1, q = 1 + 2i √ √ x2 = −3 − i 2, p = 2i 2 a ∈ (−∞; 0) ∪ (1; ∞) 2 with a parameter q ∈ C has a solution x1 = 1 + 2i. Find the second solution x2 and the parameter q. x2 = −1 − 4i, q = 9 − 6i a ∈ [0; 1] 9000039107 (level 2): Find the sum of the reciprocal values of the solutions of 5x2 − 2x + 1 = 0. 9000035608 (level 2): The equation x2 − 2ix + q = 0 x2 = −1, q = −1 − 2i a ∈ (0; 1) x2 + 2x + 5 = 0 √ 5 . 3 a = 9, b = 0, c = 5 a = 5, b = 0, c = 9 a = 9, b = 0, c = −5 a = 5, b = 0, c = −9 9000064504 (level 2): Find the values of the real coefficients a, b and c such that the quadratic equation x2 + 2x − 3 = 0 ax2 + bx + c = 0 9000039106 (level 2): 10 i has solutions x1,2 = 1 ± . 2 Solve the following quadratic equation in the complex plane. 2x2 + x + 1 = 0 a = 4, b = −8, c = 5 a = 1, b = −4, c = 5 a = 4, b = 8, c = 5 a = 1, b = 4, c = 5 x1,2 9000064505 (level 2): Find the factorization of the following quadratic polynomial in the set of polynomial with complex valued coefficients. x1,2 2x2 + 32 2(x + 4i)(x − 4i) 2(x − 4i)2 √ −1 ± i 7 = 4 x1,2 √ −1 ± i 7 = 2 x1,2 √ 1±i 7 = 4 √ 1±i 7 = 2 9000069901 (level 2): Solve the following quadratic equation in the complex plane. x2 + 4x + 5 = 0 (x + 4i)(x − 4i) 2(x + 4i)2 9000064506 (level 2): Find the factorization of the following quadratic polynomial in the set of polynomial with complex valued coefficients. x1 = −2 + i, x2 = −2 − i x = −2 x1 = 2 + i, x2 = 2 − i x1 = −3, x2 = −1 9000069902 (level 2): Solve the following quadratic equation in the complex plane. 2 2x + 4x + 5 3x2 + 2x + 2 = 0 √ ! √ ! 6 6 2 x+1+ i x+1− i 2 2 √ √ 1 1 5 5 x1 = − + i, x2 = − − i 3 3 3 3 √ ! √ ! 6 6 i x−1− i 2 x−1+ 2 2 x1 = − √ √ 1 5 1 5 x1 = + , x2 = + 3 3 3 3 √ ! √ ! 6 6 x+1− i x+1+ i 2 2 √ ! √ ! 6 6 x−1− i x−1+ i 2 2 x1 = x2 + 2x + 2 4x2 + 12 = 0 x1,2 = ±3 x1,2 = ±3i √ √ 1 5 1 5 + i, x2 = − i 3 3 3 3 9000069903 (level 2): Find the factorization of the quadratic polynomial 9000064507 (level 2): Solve the following quadratic equation in the complex plane. √ x1,2 = ±i 3 1 3 in the set of polynomials with complex valued coefficients. √ x1,2 = ± 3 9000064508 (level 2): 11 (x + 1 + i)(x + 1 − i) (x − 1 + i)(x − 1 − i) (x − i)(x + i) (x − 1 + i)(x + 1 − i) 9000069904 (level 2): Find the factorization of the quadratic polynomial 9000069909 (level 2): One of the solutions of the quadratic equation x2 + 2x + 5 9x2 − 6x + p = 0 in the set of polynomials with complex valued coefficients. (x + 1 − 2i)(x + 1 + 2i) (x − 1 − 2i)(x − 1 + 2i) (x + 1 − 2i)(x − 1 + 2i) (x − 1 − 2i)(x + 1 + 2i) with a real parameter p is 1 + i. 3 Find the value of p. 9000069905 (level 2): Find the sum of all the complex solutions of the following quadratic equation. 4 5 − 24 i 5 4 10 −1 3 9000069910 (level 2): Find the values of the parameter p ∈ R which guarantee that the equation x2 + 2px + 16 = 0 5x2 + 4x + 8 = 0 − −10 10 0 has solution with a nonzero imaginary part. p ∈ (−4; 4) 9000069906 (level 2): Find the sum of all the complex solutions of the following quadratic equation. 4 p ∈ (−∞; 4) p ∈ (4; ∞) p ∈ {} Binomial equations 2 x − 8x + 17 = 0 8 4 9000034301 (level 2): Find the solution set of the following equation in the set of complex numbers. x3 − 1 = 0 0 4i 9000069907 (level 2): Find the quadratic equation with real valued coefficients and one of the solutions x1 = −5 + i. x2 + 10x + 26 = 0 x2 − 10x + 26 = 0 x2 − 10x − 24 = 0 2 x + 10x + 24 = 0 9000069908 (level 2): One of the solutions of the quadratic equation √ √ 1 3 3 1 {1; − + i ; − −i } 2 2 2 2 √ √ {1; −1 + i 3; −1 − i 3} √ 1 3 {1; − + i } 2 2 √ 1 3 {1; − − i } 2 2 9000034302 (level 2): Find the solution set of the following equation in the set of complex numbers. x3 + 8 = 0 2x2 + px + 5 = 0 with a real parameter p is x = −1 + √ √ √ {−2; 1 + i 3; 1 − i 3} 6 i. 2 Find the value of p. 4 −4 8 √ √ {−2; −1 + i 3; −1 − i 3} −8 12 √ √ 3 1 3 1 {−2; + i ; −i } 2 2 2 2 x1,2 = 2(1 ± i), x3,4 = −2(1 ± i) √ √ 1 3 3 1 {−2; − + i ; − −i } 2 2 2 2 x1,2 = x6 − 64 = 0 √ 3 1 3 1 − i; − − i} 2 2 2 2 √ √ √ x1,2 = ±2, x3,4 = 1 ± i 3, x5,6 = −1 ± i 3 √ 3 1 3 1 + i; − − i} {−1; − 2 2 2 2 √ {−1; {i; − √ 2 2 (1 ± i), x3,4 = − (1 ± i) 2 2 9000034306 (level 2): Solve the following equation in the set of complex numbers. 9000034303 (level 2): Find the solution set of the following equation in the set of complex numbers. x3 + i = 0 {i; √ x1,2 = ±2, x3,4 √ 3 1 3 1 − i; − − i} 2 2 2 2 √ √ √ x1,2 = ±4, x3,4 = 1 ± i 3, x5,6 = −1 ± i 3 √ √ √ x1,2 = ±8, x3,4 = 2 ± 2i 3, x5,6 = −2 ± 2i 3 √ 3 1 3 1 + i; − − i} 2 2 2 2 9000034307 (level 2): Let x be any of the complex solutions of the following equation. √ x5 + 3 − i = 0 9000034304 (level 2): Find the solution set of the following equation in the set of complex numbers. x4 − 1 = 0 Find the absolute value of x. √ 5 {1; −1; i; −i} {1 − i; −1 − i} are √ 10 2 x4 + 16 = 0 √ 6 5 5 2 cos π + i sin π , 12 12 √ 13 13 6 x2 = 2 cos π + i sin π . 12 12 x1 = 9000034305 (level 2): Solve the following equation in the set of complex numbers. 2(1 ± i), x3,4 4 x3 + 1 + i = 0 {1 + i; 1 − i; −1 + i; −1 − i} √ √ 5 2 2 9000034308 (level 2): Two solutions of the equation {1 + i; −1 + i} x1,2 = √ √ 1 3 3 1 = ±i , x5,6 = − ± i 2 2 2 2 Find the third solution. √ = − 2(1 ± i) x1,2 = 1 ± i, x3,4 = −1 ± i 13 x3 = √ 6 21 21 2 cos π + i sin π 12 12 x3 = √ 6 9 9 2 cos π + i sin π 12 12 x3 = √ 6 17 17 2 cos π + i sin π 12 12 x3 = √ 6 19 19 2 cos π + i sin π 12 12 9000034309 (level 2): Find the angle ϕ such that the angles in the polar form of any two solutions of the equation √ x5 − 1 + i 3 = 0 differ by a integer multiple of ϕ. ϕ= 2 π 5 ϕ= 3 π 5 ϕ= 4 π 5 ϕ=π 9000035810 (level 2): Given the complex number z = −2 + 2i, find all the roots of √ 3 z. √ π π 6 w0 = 8 cos + i sin 4 4 √ 11π 11π 6 + i sin w1 = 8 cos 12 12 √ 19π 19π 6 w2 = 8 cos + i sin 12 12 π π w0 = 2 cos + i sin 4 4 11π 11π + i sin w1 = 2 cos 12 12 19π 19π w2 = 2 cos + i sin 12 12 √ 3 −2 + √ 3 2 π π w0 = 2 cos + i sin 3 3 w1 = 2 (cos π + i sin π) 5π 5π w2 = 2 cos + i sin 3 3 14