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9. i) Express each of the complex numbers Z (2+3i)(1- 2i), Z2 = (3+ 5i)/2 - i in the form a + ib where a and b are real numbers. Find arg Z2 and arg( Z2 gi. your answer in degrees correct to one decimal place.

9. i) Express each of the complex numbers Z (2+3i)(1- 2i), Z2 = (3+ 5i)/2 - i in the form a + ib where a and b are real numbers. Find arg Z2 and arg( Z2 gi. your answer in degrees correct to one decimal place.

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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otherwise, find the roots of the equatiön 3. i) Given that the complex numbers z,and z, are the roots of the equation: x² - 4i – 3 = 0. Express z,and z, in the form a + bi where a and b are real. ii) Indicate by shaded areas on separate Argand diagrams the regions defined by: a) /Z + 3/<1_b) /Z+4/>/Z+1/ c) {"< ArgZ < "}n {Im Z < 4} %3D 3 2 T - isin -) in the form a + bi, where . :3 4. a) Given that z = 2- 3i, express i'z, and Z/Z* and Z (cos- 4 4 a and b are real. Z+1 b) Given that Fi, find z in the forms a + bi, where a and b are real. Z-1 c) Given that the complex number 2i represents the point A on the Argand diagram and that the point B represents the image of A when reflected in the line y = 2x, find the complex number which represents the point B. 5. i) Find two complex numbers Z, and Z, which satisfy the simultaneous equations Z, + Z2 = -i, Z, – iZ2= -4 +i ii) Given that Z= 1 + iv3 express Z* in the form | a) a+ bi where a and b are real. b) r(cosq + isinq), where r> 0 and - 1<q<I iii) By shading in three separate Argand diagrams show the regions in which the points representing z can lie when (a) Imz< 2 b) /Z-2i/ < 2. c) Z--2i/</Z - 2/. Shade in another Argand diagram the region in whichz can lie when ail the three inequalities apply 6. i) Given thati the complex numbers Z, and Z2 where Z, = (1+i), Z2 1 1 express Z 2+i 3-i and Za in the form a + ib, where a, b are real. ii) Solve the simultaneous equations Z; + Z4 = 6, 2Z; - 2iZ4 = 8+ 3i, expressing your %3D answer in the form a + ib, where a and b are real numbers. iii) A regular hexagon ABC DEF is drawn in the Argand diagram so that it centre is at the origin and the two adjacent vertices A and B are at the points represented by the complex numbers ZA and ZB = 1 and ZB = ½ + i½v3. Find, in the form a + bi where a and b are real numbers, the complex numbers which represent the other four vertices. 7. Given that /2Z - t/= /Z-3i/, show that the locus of the complex number Z is a circle, giving %3D the radius and the coordinates of the centre. 8. a) Find in the form a + bi, where a and b are real, the values of Z for which 3 i) /z/=3 and arg Z ii) + =1. = 1. 1-2i 3 b) Find the roots of the equation: Z* +6Z“ + 25 = 0. Represent these roots as points on the Arganddiagram., 9. i) Express each of the complex numbers Z= (2+3i)(1- 2i), Z2 = (3 + 5i)/2 - i in the form a + %3D %3D Z2 ib where a and b are real numbers. Find arg Z2 and arg gi.ng your answer in degrees Z1 correct to one decimal place. 76
Transcribed Image Text:otherwise, find the roots of the equatiön 3. i) Given that the complex numbers z,and z, are the roots of the equation: x² - 4i – 3 = 0. Express z,and z, in the form a + bi where a and b are real. ii) Indicate by shaded areas on separate Argand diagrams the regions defined by: a) /Z + 3/<1_b) /Z+4/>/Z+1/ c) {"< ArgZ < "}n {Im Z < 4} %3D 3 2 T - isin -) in the form a + bi, where . :3 4. a) Given that z = 2- 3i, express i'z, and Z/Z* and Z (cos- 4 4 a and b are real. Z+1 b) Given that Fi, find z in the forms a + bi, where a and b are real. Z-1 c) Given that the complex number 2i represents the point A on the Argand diagram and that the point B represents the image of A when reflected in the line y = 2x, find the complex number which represents the point B. 5. i) Find two complex numbers Z, and Z, which satisfy the simultaneous equations Z, + Z2 = -i, Z, – iZ2= -4 +i ii) Given that Z= 1 + iv3 express Z* in the form | a) a+ bi where a and b are real. b) r(cosq + isinq), where r> 0 and - 1<q<I iii) By shading in three separate Argand diagrams show the regions in which the points representing z can lie when (a) Imz< 2 b) /Z-2i/ < 2. c) Z--2i/</Z - 2/. Shade in another Argand diagram the region in whichz can lie when ail the three inequalities apply 6. i) Given thati the complex numbers Z, and Z2 where Z, = (1+i), Z2 1 1 express Z 2+i 3-i and Za in the form a + ib, where a, b are real. ii) Solve the simultaneous equations Z; + Z4 = 6, 2Z; - 2iZ4 = 8+ 3i, expressing your %3D answer in the form a + ib, where a and b are real numbers. iii) A regular hexagon ABC DEF is drawn in the Argand diagram so that it centre is at the origin and the two adjacent vertices A and B are at the points represented by the complex numbers ZA and ZB = 1 and ZB = ½ + i½v3. Find, in the form a + bi where a and b are real numbers, the complex numbers which represent the other four vertices. 7. Given that /2Z - t/= /Z-3i/, show that the locus of the complex number Z is a circle, giving %3D the radius and the coordinates of the centre. 8. a) Find in the form a + bi, where a and b are real, the values of Z for which 3 i) /z/=3 and arg Z ii) + =1. = 1. 1-2i 3 b) Find the roots of the equation: Z* +6Z“ + 25 = 0. Represent these roots as points on the Arganddiagram., 9. i) Express each of the complex numbers Z= (2+3i)(1- 2i), Z2 = (3 + 5i)/2 - i in the form a + %3D %3D Z2 ib where a and b are real numbers. Find arg Z2 and arg gi.ng your answer in degrees Z1 correct to one decimal place. 76
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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