Solve all Q14 explaining detailly each step 11) In an Argand diagram P represents the complex number {z:3/z+3/ = /z+4i/}. Show that Plies on a circle and find;a) the radius of the circle b) the complex number represented by its centre.Z210. i) Given that Z, = V3 + 1 and Z, = 1 – iv3, find a) /Z b) arg/Z/ c) argii) Obtain a polynomial equation of the fourth degree with real coefficients, given that two ofits roots are 2 + I and 1 – 3i. represents all the 4 roots of this equation in an Argand diagram.Z*where Z*__Z.Z11. Given that Z = 3(1+iV3), calculate a) the modulus and argument of- anddenotes the complex conjugate of Z. b) The two roots of Z in the form a + ib, where a and bare real.iz+112. If Re ()=2, show that the locus of the point representing Z in the Argand diagrams is az+1circle 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 that1+2i%3Db) Given that Z = x + iy, where x, y ER, find the locus of the point in the Argand diagram for1which the imaginary part of Z +- is zero.V3-icalculate /Z/ and arg Z.14. Given that ZV3+ i1+iv315. a) Write the complex numberin the form r (cose + sin0), where 0 is in radians.2-2ib) 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.169Find, also in degrees to one16. Find, in the form: x + iy, where x, y E R, the square root of5-12idecimal 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 and17. i) Given that Z2-ifind the values of /Z/ and arg /Z²/.ZZ-11ii) Given that1+2ifind real numbers p and q such that (p+iq)Z= 3+4i.1-2i18. i) Given that Z, = v3 + i and Z2 = 1 - iv3, find,%3D21in the form p + iq where p and q are real.Z2Z1b) argZ and argZ2ii) 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+iV3Z, = (-1+ iv3)(1 – iv3), Z2 =1-iV3Z3 = Z mark on the Argand diagram the pointsrepresenting Z1 and Z2ii) Prove that for any complex number Z, if /Z/ 0.

Question

Solve all Q14 explaining detailly each step 11) In an Argand diagram P represents the complex number {z:3/z+3/ = /z+4i/}. Show that Plies on a circle and find;a) the radius of the circle b) the complex number represented by its centre.Z210. i) Given that Z, = V3 + 1 and Z, = 1 – iv3, find a) /Z b) arg/Z/ c) argii) Obtain a polynomial equation of the fourth degree with real coefficients, given that two ofits roots are 2 + I and 1 – 3i. represents all the 4 roots of this equation in an Argand diagram.Z*where Z*__Z.Z11. Given that Z = 3(1+iV3), calculate a) the modulus and argument of- anddenotes the complex conjugate of Z. b) The two roots of Z in the form a + ib, where a and bare real.iz+112. If Re ()=2, show that the locus of the point representing Z in the Argand diagrams is az+1circle 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 that1+2i%3Db) Given that Z = x + iy, where x, y ER, find the locus of the point in the Argand diagram for1which the imaginary part of Z +- is zero.V3-icalculate /Z/ and arg Z.14. Given that ZV3+ i1+iv315. a) Write the complex numberin the form r (cose + sin0), where 0 is in radians.2-2ib) 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.169Find, also in degrees to one16. Find, in the form: x + iy, where x, y E R, the square root of5-12idecimal 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 and17. i) Given that Z2-ifind the values of /Z/ and arg /Z²/.ZZ-11ii) Given that1+2ifind real numbers p and q such that (p+iq)Z= 3+4i.1-2i18. i) Given that Z, = v3 + i and Z2 = 1 - iv3, find,%3D21in the form p + iq where p and q are real.Z2Z1b) argZ and argZ2ii) 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+iV3Z, = (-1+ iv3)(1 – iv3), Z2 =1-iV3Z3 = Z mark on the Argand diagram the pointsrepresenting Z1 and Z2ii) Prove that for any complex number Z, if /Z/<1 then Re(Z+1) > 0.

Accepted Answer

Argument and modules of complex number

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