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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.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter7: Distance And Approximation
Section7.1: Inner Product Spaces
Problem 7AEXP
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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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