Complex number excise, Exercises of Mathematics

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Problem 3.1: Find all values of z such that z? + 81 = 0. Problem 3.2: Let w = 5 — 3i,x = -1+ 4i, y = 2—i, and z = 1 + 2i. Compute each of the following: (a) xt+y+z (c) wz-2w-z (e) 2w?z + 3wz? (b) xyz (a) w? +2wz +z? Problem 3.3: Evaluate i2°°”. Problem 3.4: Let f(x) = (-2+3i)x* — (7 + 2i)x + 37 — 13i, where the domain of f is all complex numbers. Determine each of the following: (a) f() (6) f@ () f@+i) Problem 3.5: (a) Find all real values of k such that the product (3 + 2i)(3 + ki) is a real number. (b) Let z =a + bi, where a and b are real numbers, and let Z = a — bi. Prove that z -Z is real. Problem 3.6: Write as a complex number. In other words, find real numbers a and b such that 1 342i aan ath Problem 3.7: Let w = 2 - 5i and z = -3 + i. Express each of the following in the form a + bi, where a and b are real numbers. 1 Z Zz Zz cal pal —F @ 3 OD © Tati 1 Zz — d = e) 2 @ = ; 2z —3i Problem 3.8: Find all complex numbers z such that ae = -5 + 3.1.1 Find all z such that 4z* + 12 = 0. 3.1.2 Leta =3 + 4i and b = 12 —5i. Compute each of the following: (a) a-b (c) @+3a+2 (b) ab @ £25 b b 3.1.3 Write each of the following as a complex number: (a) 2i+7-2i (b) aT 24+3i+1+2i xo + x4 3.14 f(x)= sine Find each of the following: (a) f@ (b) f(-i) () fG@-1) 3.1.5 What is (i — i-!)~!? (Source: AHSME) 3.1.6 Solve each of the following equations for z: Z+3i 1+2i @) 735 =2 ) = Zz =445i — =10-4i +5i (c) +75 0 -4i 3.1.7, Let S = i" +i", where n is an integer. Find the total number of possible distinct values of S. (Source: AHSME) 3.2.1 Plot each of the following in the complex plane: (a) 4+7i (b) -6-2i (c) (8+i)(-2+5i) 3.2.2 Find the magnitude of each of the following complex numbers: (a) 24-7i (b) 24+2-V3i (ce) (1+ 2i)(2 +i) 3.2.3 Letw=3+5i and z = 12 +2i. Find the area of the convex quadrilateral in the complex plane that has vertices w, z, W, and Z. 3.2.4 Find the distance between the points 4 + 7i and -3 — 17i in the complex plane. 3.2.5 Show that the midpoint of the segment connecting z; and z2 on the complex plane is (z; + Z2)/2. 3.2.6 1+2i 2+i° 6+11i 11 + 63 (c) Notice anything interesting? Can you generalize your observations from the first two parts? (a) Find the magnitude of (b) Find the magnitude of . (You can use a calculator for this part.) 3.2.7* Four complex numbers lie at the vertices of a square in the complex plane. Three of the numbers are 1 + 2i, -2 +iand —1— 2i. What is the fourth number? (Source: AMC 12) Problem 3.13: Let z and w be complex numbers. (a) Letz=a+biand w =c + di, where a, b,c, and d are real. Show thatz+ Ww =Z+0. (b) Show that Zw =Z-w. Problem 3.14: (a) Show that Z =z for all complex numbers z. (b) Show that Z = z if and only if z is real. (c) Show that Z = -z if and only if z is imaginary. Problem 3.15: (a) Prove that 2Z = |z/? for all complex numbers z. (b) Prove that |zw| = |z||~| for all complex numbers z and w. Problem 3.16: Solve the equation z + 22 = 6 — 4i for z. Problem 3.17: In this problem, we find all complex numbers z such that 2 = 21-201. (a) Let z =a + biin the given equation. Find a system of equations involving a and b. (b) Solve for b in terms of a in one of the equations, and substitute the expression you found for b into the other equation. (c) Solve the equation you formed in part (b) for all possible values of a. Hints: 90 (d)_ Find all complex numbers z such that 22 = 21 -20i. 3.23 Let w = 2+3iand z = 4 —5i. Express each of the following as a complex number: 2 © —ytxi (a) 2w-3z (c) san e oi 1 w? + 2w*z + wz? a-i* ® 2 tw © arM 3.24 Solve the equation —3 — x? = 9 + x?. 3.25 Simplify u 7 hae — 1+i 3.26 Find the complex number z such that == =-1+i. 3.27 Four complex numbers are plotted in the plane, as shown. One of them is Im © z. The other three are —z, Z, and —Z. Which is which? 3.28 Let w and z be complex numbers. Re (a) Show that wz + @z is real. (b) Show that wz — Wz is imaginary. 3.29 Find two complex numbers whose squares equal 5 — 12i. 3.30 Find all real numbers c such that 7 + / is 5 units from 10 + ci on the complex plane. 3.31 Find the area of the region enclosed by the graph of |z — 4 + 5i] = 2v3. 3.32 The diagram to the right shows several numbers in the complex plane. The circle has radius 1 and is centered at the origin. One of these numbers is the reciprocal of F. Which one? (Source: AHSME) 3.33 Find all complex numbers z such that |z + 1 — i] = |z — 2}. 3.34 The product of the complex numbers a + bi and c + di is real, where a, b, c, and d are real numbers. Prove that the product of the complex numbers b+ ai and d + ciis also real. B 3.35 Describe the graph of each of the following: (a) z-Z=-8i (b) (4-iz-(4+)zZ=16i (Cc) |7+i-22|=4 3.36 Graph |z — 5 + 2i| < 4 on the complex plane. 3.37 Simplify (i + 1)°2 — (i — 1)32, 3.39 Evaluate i+2i +3? + 4i+---+ 641%. 3.40 Find all complex numbers z such that |z — 1| = |z + 3| = |z - i]. (Source: ARML) Hints: 163 3.41 Find the area of the region of points z in the complex plane such that lz+4-4i)<4V2 and |z-4-4i| > 42. 3.42 Let wand z be complex numbers and z # 0. Prove that = a a is a real number. 3.43 Show that the points w, x, y, and z are the vertices of a parallelogram in the complex plane if and only if the sum of some two of them is equal to the sum of the other two. Hints: 56 3.44x Define a sequence of complex numbers by z) = 0, Zn41 = 23 + iforn > 1. How far away from the origin is 2111? (Source: AHSME) Hints: 60 3.45*x A function f is defined on the complex numbers by f(z) = (a + bi)z, where a and b are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that |a+bil = 8, find the value of b?. (Source: AIME) Hints: 330, 146 3.46 For anonzero complex number z, let f(z) = 1/2. (a) Show that f(f(z)) =z for all z # 0. (b)x Let w = f(z). As z varies along the line (1 + 2i)z — (1 — 2i)2 = i, what curve does w trace? Hints: 59 +8b 8a-b =2andb+ poe 0 a2 + b2 a+y 3.47x Find all ordered pairs of real numbers (a,b) such that a + Hints: 143 3.48% Let u and v be complex numbers such that |u| = |v] = 1, u # v, and u # —v. Show that 4uv (ut+ov)? is real, and that ut+v u-v is imaginary. Hints: 334 28. 29. 30. Express the following in polar form (mod-Amplitude form) i) 1+iv3 ii) -1-iv3 iii) —J3 +43 (March 2014) (Aril-2001) iv) 1+cos6—isin8 Vv) =-1+i vi) —242iV3 vii) 1-i (March 2015) viii) —V7+iN21 — (Mar-2011, May 2015,2016) ix) -l-i xyIf (v3 + i) =r(cos@+isin@) find the value 9 in radian measure ans = 1) 25 2) -25 3) 25i 4) -25i Let z = x+iy and a point P represents z in the z-1 Argand plane. If the real part of ri isl, then a point that lies on the locus of P is (A.P. EAMCET-2018) 2) (-2016, 2017) 4) (2016, -2017) 1) (2016, 2017) 3) (-2016, -2017) 14+2i. . The complex number 1-i lies in the 2) 2™ quadrant 4) 4" quadrant 1) I quadrant 3) 3™ quadrant If three complex numbers are in A.P. then they lie on 1) straight line 2) a circle 3) a parabola 4) ellipse In the Argand plane, the points represented by the complex numbers 2 - 6i, 4 - 7i, 3 - Si and 1—4i form 1) parallelogram 2) rectangle 3) rhombus 4) square If z, =—1 and z, =i then find i) Arg .22 (March-2013) ii) Are{ 2 iii) are( =} Tt i) If Arey, =? and Arg z, =~ then find (May-2009) ot | (May-2013,2014,Mar-2016) Arg 2, +Argz) (March 2016, May2016) ii) If z#0, find Arg z +Arg = 142i - 1-d-ip then find Arg z iii) If == If z =x + iy and if the point P in the argand plane represents z then find the locus of P satisfying the following equations i) Im(2)=4 ii) o¢2+2)=3 iii) |<|=2 iv) |--2+3i]=4 vy) 12¢-31=7 vi) Iz? =4Re(z+2) vii) [z+ -1z-iP=2 viii) {ze C3|z Il} ix) le + ail =Ic-ail tT x) Amplitude of (z— 1) is > (May-2007, 2015) xi) |:-3+iJ=4 (March 2014) xii) |7-2-3i]=5 (May 2013) _ 7 T xiii) |z]=1 xiv) Amp (z) = = 39. Match the following Complex number Polar form Do 14+V3i a) 2 cis (n/3) I 1-V3i b) 2 cis (20/3) TH) 143i ©) 2 cis (20/3) 40. 14+ P+ P44 + 10 2G 3)i-1 4) -1 41. i+ 2? + 3+ 4 400. + 100j10 = 1) 25(1-i) 2) 50 (1-i) 3) 100 (1-i) 4)0 10. The least positive integer n for which (1+i)" a—i p> isa real number is I) 1 24 3) 6 4) 14 (1+i)° +i) = Ij2+7 2) 2-101 3) -2 - 101 4) -2 + 107 ee (344i) . The multiplicative inverse of 25 is 1) 443i 2)3-4i = 3) -4-31 4) -4437 The additive inverse of (1+2i) (3-4i) is 1) 1142 = 2) T1-2% = 3) -11+2% 4) -11-2i x(2)- mo\3 9+6i 9-61, ; ; 3 2) B 3)9+6i 4)9-6i ; 35 Express (##)-C ‘) in the form of 2-31 243i a+ib ) 13 “13 7) 13 , 13 The conjugate complex number of aon —2i 2 i. 2 1 I)>=t+> 2) Fe 7Fe 25 25 2 25 i, 2 = +— i )-—-— i 25 25 25 iran 13e 2=a+ib, then the ordered pair (a, b) = ({A.P. EAMCET-2018) 1) (12, 5) 2) (5, 12) 3) (24, 10) 4) (10, 24) Let A(3-4), B(2+i) be two points in the Argand plane. If the point P represents the complex number z = x+iy, which satisfies |z-3+il = \z-2-il, then the locus of the point P is (A.P. EAMCET-2018) 1) the circle with AB as diameter 2) the line passing through A and B 3) the perpendicular bisector of AB 4) the ellipse with AB as major axis 12. 13. 14. 16. 17. 18. 19. 20. 21. If Ze 1-2i, 2, = 1+i and z= 3+4i, then = (A.P. EAMCET-2018) 3) 13. 3. 1) 13-6i 2) 13-3i 3) 6->! 4) 3 a fea st it (V3+i)" =2"(a+ib) then a? +5*= 1) ¥2 2)4 3) 3 4)3 If ©,,@,,@, respectively denote the moduli of the complex number -i, 1/3(1+i) and -1+i then their increasing order is (EAMCET-2005) 1) o,,0,,0, 2) O,,,.0, 3) O,,,,0, 4) O,,0,,0, X t If 2 and rm are the arguments of z, and 7, z respectively, then ane( =] oa yprZ 3 , a> ) rm 2) rm 9 ) 3 If z, =-1 and z,= i the value of are( +} 2 5, = t 1) z 2) ay 3) 3 4) 3 The mod-amplitude form of ~/3 —i is ._{ 5x ani¢( ~% Ic = 2cis| —— yai($#) a(t) = .{ 3) 2i( = 4) 2eie{ =] z=1+iV¥3 31 Argzl+lArgZ l= (EAMCET-2010) 2n 2 — Ho 2) 3 4) 3 V-8—67 is equal to 1) +U-3i) 2 3) (143i) 4 The value of J15+8i + .J15—8i is equal to 1) 15 2)8 3) 23 4)7 If |3z—2|=5 then the locus of z is 1) 9x° +9y? +12x-21=0 2) 9x° +9y? -12x-21=0 3) Ox +9? + 12x +2150 4) 9x° -9y? +12x-21=0 us| a nt ) od +(1-2i) none of these