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14. Given that Z calculate /Z/ and arg Z. %3D 3+ i'

14. Given that Z calculate /Z/ and arg Z. %3D 3+ i'

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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11) In an Argand diagram P represents the complex number {z:3/z+3/ = /z+4i/}. Show that P lies on a circle and find; a) the radius of the circle b) the complex number represented by its centre. Z2 10. i) Given that Z, = V3 + 1 and Z, = 1 – iv3, find a) /Z b) arg/Z/ c) arg ii) Obtain a polynomial equation of the fourth degree with real coefficients, given that two of its roots are 2 + I and 1 – 3i. represents all the 4 roots of this equation in an Argand diagram. Z* where Z* __ Z. Z 11. Given that Z = 3(1+iV3), calculate a) the modulus and argument of- and denotes the complex conjugate of Z. b) The two roots of Z in the form a + ib, where a and b are real. iz+1 12. If Re ()=2, show that the locus of the point representing Z in the Argand diagrams is a z+1 circle giving its centre and radius. (2-i)z = 0. 13. a) Find, in the form a + bi, where a, b E R the complex number Z such that 1+2i %3D b) Given that Z = x + iy, where x, y ER, find the locus of the point in the Argand diagram for 1 which the imaginary part of Z +- is zero. V3-i calculate /Z/ and arg Z. 14. Given that Z V3+ i 1+iv3 15. a) Write the complex number in the form r (cose + sin0), where 0 is in radians. 2-2i b) P1 and P2 are the points representing the complex numbers 3 +i and -1 + 3i respectively. Show that OP1 is perpendicular to OP2 where O is the origin. 169 Find, also in degrees to one 16. Find, in the form: x + iy, where x, y E R, the square root of 5-12i decimal place, the principal value of the argument of each of the square roots. 5 (3+2i), express Z in the form x + iy, where x and y are real numbers and 17. i) Given that Z 2-i find the values of /Z/ and arg /Z²/. Z Z-1 1 ii) Given that 1+2i find real numbers p and q such that (p+iq)Z= 3+4i. 1-2i 18. i) Given that Z, = v3 + i and Z2 = 1 - iv3, find, %3D 21in the form p + iq where p and q are real. Z2 Z1 b) argZ and arg Z2 ii) Show that the roots of the equation Z3 -1 = 0 lie at the vertices of an equilateral triangle. 19. i) Find the modulus and argument of each the complex numbers Z1, Z2 and Z3 where -1+iV3 Z, = (-1+ iv3)(1 – iv3), Z2 = 1-iV3 Z3 = Z mark on the Argand diagram the points representing Z1 and Z2 ii) Prove that for any complex number Z, if /Z/<1 then Re(Z+1) > 0.
Transcribed Image Text:11) In an Argand diagram P represents the complex number {z:3/z+3/ = /z+4i/}. Show that P lies on a circle and find; a) the radius of the circle b) the complex number represented by its centre. Z2 10. i) Given that Z, = V3 + 1 and Z, = 1 – iv3, find a) /Z b) arg/Z/ c) arg ii) Obtain a polynomial equation of the fourth degree with real coefficients, given that two of its roots are 2 + I and 1 – 3i. represents all the 4 roots of this equation in an Argand diagram. Z* where Z* __ Z. Z 11. Given that Z = 3(1+iV3), calculate a) the modulus and argument of- and denotes the complex conjugate of Z. b) The two roots of Z in the form a + ib, where a and b are real. iz+1 12. If Re ()=2, show that the locus of the point representing Z in the Argand diagrams is a z+1 circle giving its centre and radius. (2-i)z = 0. 13. a) Find, in the form a + bi, where a, b E R the complex number Z such that 1+2i %3D b) Given that Z = x + iy, where x, y ER, find the locus of the point in the Argand diagram for 1 which the imaginary part of Z +- is zero. V3-i calculate /Z/ and arg Z. 14. Given that Z V3+ i 1+iv3 15. a) Write the complex number in the form r (cose + sin0), where 0 is in radians. 2-2i b) P1 and P2 are the points representing the complex numbers 3 +i and -1 + 3i respectively. Show that OP1 is perpendicular to OP2 where O is the origin. 169 Find, also in degrees to one 16. Find, in the form: x + iy, where x, y E R, the square root of 5-12i decimal place, the principal value of the argument of each of the square roots. 5 (3+2i), express Z in the form x + iy, where x and y are real numbers and 17. i) Given that Z 2-i find the values of /Z/ and arg /Z²/. Z Z-1 1 ii) Given that 1+2i find real numbers p and q such that (p+iq)Z= 3+4i. 1-2i 18. i) Given that Z, = v3 + i and Z2 = 1 - iv3, find, %3D 21in the form p + iq where p and q are real. Z2 Z1 b) argZ and arg Z2 ii) Show that the roots of the equation Z3 -1 = 0 lie at the vertices of an equilateral triangle. 19. i) Find the modulus and argument of each the complex numbers Z1, Z2 and Z3 where -1+iV3 Z, = (-1+ iv3)(1 – iv3), Z2 = 1-iV3 Z3 = Z mark on the Argand diagram the points representing Z1 and Z2 ii) Prove that for any complex number Z, if /Z/<1 then Re(Z+1) > 0.
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