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- Question : 53SP - Perform each of the indicated operations: (a) (4 3i)
- Question : 54SP - Suppose z1
- Question : 55SP - Prove that (a) (z1z2)
- Question : 56SP - Prove that (a) (z1=z2)
- Question : 57SP - Find real numbers x and y such that 2x 3iy
- Question : 58SP - Prove that (a) Refzg
- Question : 59SP - Suppose the product of two complex numbers is zero. Prove that at least one of the numbers must be zero.
- Question : 60SP - Let w
- Question : 61SP - Perform the indicated operations both analytically and graphically. (a) (2
- Question : 62SP - Let z1, z2, and z3 be the vectors indicated in Fig. 1-40. Construct graphically: (a) 2z1
- Question : 63SP - Let z1
- Question : 64SP - The position vectors of points A, B, and C of triangle ABC are given by z1
- Question : 65SP - Let z1, z2, z3, z4 be the position vectors of the vertices for quadrilateral ABCD. Prove that ABCD is a parallelogram if and only if z1 z2 z3
- Question : 66SP - Suppose the diagonals of a quadrilateral bisect each other. Prove that the quadrilateral is a parallelogram.
- Question : 67SP - Prove that the medians of a triangle meet in a point.
- Question : 68SP - Let ABCD be a quadrilateral and E, F, G, H the midpoints of the sides. Prove that EFGH is a parallelogram.
- Question : 69SP - In parallelogram ABCD, point E bisects side AD. Prove that the point where BE meets AC trisects AC.
- Question : 70SP - The position vectors of points A and B are 2
- Question : 71SP - Describe and graph the locus represented by each of the following: (a) jz ij
- Question : 72SP - Find an equation for (a) a circle of radius 2 with center at ( 3, 4), (b) an ellipse with foci at (0, 2) and (0, 2) whose major axis has length 10.
- Question : 73SP - Describe graphically the region represented by each of the following: (a) 1 , jz
- Question : 74SP - Show that the ellipse jz
- Question : 75SP - Use the definition of a complex number as an ordered pair of real numbers to prove that if the product of two complex numbers is zero, then at least one of the numbers must be zero.
- Question : 76SP - Prove the commutative laws with respect to (a) addition, (b) multiplication.
- Question : 77SP - Prove the associative laws with respect to (a) addition, (b) multiplication.
- Question : 78SP - (a) Find real numbers x and y such that (c, d) (x, y)
- Question : 79SP - Prove that (cos u1, sinu1)(cos u2, sin u2) (cos un, sinun)
- Question : 80SP - (a) How would you define (a, b)1=n where n is a positive integer? (b) Determine (a, b)1=2 in terms of a and b.
- Question : 81SP - Express each of the following complex numbers in polar form: (a) 2 2i, (b) 1
- Question : 82SP - Show that 2
- Question : 83SP - Express in polar form: (a) 3 4i, (b) 1 2i.
- Question : 84SP - Graph each of the following and express in rectangular form: (a) 6(cos 1358
- Question : 85SP - An airplane travels 150 miles southeast, 100 miles due west, 225 miles 308 north of east, and then 200 miles northeast. Determine (a) analytically and (b) graphically how far and in what direction it is from its starting point.
- Question : 86SP - Three forces as shown in Fig. 1-41 act in a plane on an object placed at O. Determine (a) graphically and (b) analytically what force is needed to prevent the object from moving. [This force is sometimes called the equilibrant.]
- Question : 87SP - Prove that on the circle z
- Question : 88SP - (a) Prove that r1eiu1
- Question : 89SP - Evaluate each of the following: (a) (5 cis 208)(3 cis 408) (b) (2 cis 508)6 (c) (8 cis 408)3 (2 cis 608)4 (d) (3epi=6)(2e 5pi=4)(6e5pi=3) (4e2pi=3)2 (e) ffiffiffi3p i ffiffiffi3p
- Question : 90SP - Prove that (a) sin 3u
- Question : 91SP - Prove that the solutions of z4 3z2
- Question : 92SP - Show that (a) cos 368
- Question : 93SP - Prove that (a) sin 4u=sin u
- Question : 94SP - Prove De Moivre
- Question : 95SP - Find each of the indicated roots and locate them graphically. (a) (2 ffiffiffi 3 p 2i)1=2, (b) ( 4
- Question : 96SP - Find all the indicated roots and locate them in the complex plane. (a) Cube roots of 8, (b) square roots of 4 ffiffiffi 2 p
- Question : 97SP - Solve the equations (a) z4
- Question : 98SP - Find the square roots of (a) 5 12i, (b) 8
- Question : 99SP - Find the cube roots of 11 2i.
- Question : 100SP - Solve the following equations, obtaining all roots: (a) 5z2
- Question : 101SP - Solve z5 2z4 z3
- Question : 102SP - (a) Find all the roots of z4
- Question : 103SP - Prove that the sum of the roots of a0zn
- Question : 104SP - Find two numbers whose sum is 4 and whose product is 8.
- Question : 105SP - Find all the (a) fourth roots, (b) seventh roots of unity, and exhibit them graphically.
- Question : 106SP - a) Prove that 1
- Question : 107SP - Prove that cos 368
- Question : 108SP - Prove that the sum of the products of all the nth roots of unity taken 2, 3, 4, . . . , (n 1) at a time is zero
- Question : 109SP - Find all roots of (1
- Question : 110SP - Given z1
- Question : 111SP - Prove that z1 z2
- Question : 112SP - Suppose z1
- Question : 113SP - Prove that z1 (z2
- Question : 114SP - Find the area of a triangle having vertices at 4 i, 1
- Question : 115SP - Find the area of a quadrilateral having vertices at (2, 1), (4, 3), ( 1, 2); and ( 3, 2).
- Question : 116SP - Describe each of the following loci expressed in terms of conjugate coordinates z, z. (a) z z
- Question : 117SP - Write each of the following equations in terms of conjugate coordinates. (a) (x 3)2
- Question : 118SP - Let S be the set of all points a
- Question : 119SP - Answer Problem 1.118 if S is the set of all points inside the square.
- Question : 120SP - Answer Problem 1.118 if S is the set of all points inside or on the square.
- Question : 121SP - Given the point sets A
- Question : 122SP - Suppose A, B, C, and D are any point sets. Prove that (a) A < B
- Question : 123SP - Suppose A, B, and C are the point sets defined by jz
- Question : 124SP - Prove that the complement of a set S is open or closed according as S is closed or open.
- Question : 125SP - Suppose S1, S2, . . . , Sn are open sets. Prove that S1 < S2 < < Sn is open.
- Question : 126SP - Suppose a limit point of a set does not belong to the set. Prove that it must be a boundary point of the set.
- Question : 127SP - Let ABCD be a parallelogram. Prove that (AC)2
- Question : 128SP - Explain the fallacy: 1
- Question : 129SP - (a) Show that the equation z4
- Question : 130SP - (a) Prove that cosn f
- Question : 131SP - Let z
- Question : 132SP - Show that for any real numbers p and m, e2mi cot 1 p pi
- Question : 133SP - Let P(z) be any polynomial in z with real coefficients. Prove that P(z)
- Question : 134SP - Suppose z1, z2, and z3 are collinear. Prove that there exist real constants a, b, g, not all zero, such that az1
- Question : 135SP - Given the complex number z, represent geometrically (a) z, (b) z, (c) 1/z, (d) z2.
- Question : 136SP - Consider any two complex numbers z1 and z2 not equal to zero. Show how to represent graphically using only ruler and compass (a) z1z2, (b) z1=z2, (c) z21
- Question : 137SP - Prove that an equation for a line passing through the points z1 and z2 is given by argf(z z1)=(z2 z1)g
- Question : 138SP - Suppose z
- Question : 139SP - Is the converse to Problem 1.51 true? Justify your answer.
- Question : 140SP - Find an equation for the circle passing through the points 1 i, 2i, 1
- Question : 141SP - Show that the locus of z such that jz ajjz
- Question : 142SP - Let pn
- Question : 143SP - Prove that: (a) cosu
- Question : 144SP - Prove that (a) Refzg . 0 and (b) jz 1j , jz
- Question : 145SP - A wheel of radius 4 feet [Fig. 1-44] is rotating counterclockwise about an axis through its center at 30 revolutions per minute. (a) Show that the position and velocity of any point P on the wheel are given, respectively, by 4eipt and 4pieipt , where t is the time in seconds measured from the instant when P was on the positive x axis. (b) Find the position and velocity when t
- Question : 146SP - Prove that for any integer m . 1, (z
- Question : 147SP - Suppose points P1 and P2, represented by z1 and z2 respectively, are such that jz1
- Question : 148SP - Prove that for any integer m . 1, cot p 2m cot 2p 2m cot 3p 2m cot (m 1)p 2m
- Question : 149SP - Prove and generalize: (a) csc2(p=7)
- Question : 150SP - Let masses m1, m2, m3 be located at points z1, z2, z3, respectively. Prove that the center of mass is given by ^z
- Question : 151SP - Find the point on the line joining points z1 and z2 which divides it in the ratio p : q.
- Question : 152SP - Show that an equation for a circle passing through three points z1, z2, z3 is given by z z1 z z2 = z3 z1 z3 z2
- Question : 153SP - Prove that the medians of a triangle with vertices at z1, z2, z3 intersect at the point 1 3 (z1
- Question : 154SP - Prove that the rational numbers between 0 and 1 are countable. [Hint. Arrange the numbers as 0, 1 2 , 1 3 , 2 3 , 1 4 , 3 4 , 1 5 , 2 5 , 3 5 , . . . .]
- Question : 155SP - Prove that all the real rational numbers are countable.
- Question : 156SP - Prove that the irrational numbers between 0 and 1 are not countable.
- Question : 157SP - Represent graphically the set of values of z for which (a) jzj . jz 1j, (b) jz
- Question : 158SP - Show that (a) ffiffiffi p3 2
- Question : 159SP - Prove that ffiffiffi 2 p
- Question : 160SP - Let ABCD PQ represent a regular polygon of n sides inscribed in a circle of unit radius. Prove that the product of the lengths of the diagonals AC, AD, . . . , AP is 1 4 n csc2(p=n).
- Question : 161SP - Suppose sinu=0. Prove that (a) sin nu sin u
- Question : 162SP - Prove cos 2nu
- Question : 163SP - Suppose the product of two complex numbers z1 and z2 is real and different from zero. Prove that there exists a real number p such that z1
- Question : 164SP - Let z be any point on the circle jz 1j
- Question : 165SP - Prove that under suitable restrictions (a) zmzn
- Question : 166SP - Prove (a) Refz1z2g
- Question : 167SP - Find the area of the polygon with vertices at 2
- Question : 168SP - Let a1, a2, . . . , an and b1, b2, . . . , bn be any complex numbers. Prove Schwarz

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