Download Complex numbers (Chap 16)

Complex numbers (Chap 16)
Topics
a. Definition of complex numbers.
b. Conjugate and modulus of a complex number and their relation.
c. Addition and Subtraction of complex numbers.
d. Division and multiplication of complex numbers.
e. Solving equations
Exercise 16A, 16B
Q.1
If 𝑟 = 3 + 𝑖 and 𝑠 = 1 − 2𝑖 Then express the following in the form of 𝑎 − 𝑏𝑖
a. (1 + i)r
s
b. 1+i
c.
d.
Q.2
Evaluate the followings.
a. 𝑅e( 3 + 4i)
b. Im(3 – i)2
3−4i
c. Re ( 1+i )
d.
Q.3
a.
b.
c.
Q.4
a.
b.
c.
d.
Q.5
Q.6
1−i
s
s2
r
Im(1 + 2i)(3 − i)2
Solve the following quadratic equations.
𝑧2 + 9 = 0
𝑧 2 − 6𝑧 + 25 = 0
2𝑧 2 + 2𝑧 + 13 = 0
Write down the following polynomial as product of linear factors.
9𝑧 2 − 6𝑧 + 5
𝑧 4 − 16
𝑧 4 − 8𝑧 2 − 9
𝑧 3 − 3𝑧 2 + 𝑧 + 5
Prove that 1 + 𝑖 is the root of the equation 𝑧 4 + 𝑧 2 − 6𝑧 + 10 = 0. Find all other roots
Prove that −2 + 𝑖 is the root of the equation 𝑧 4 + 24𝑧 + 55 = 0. Find all other roots.
Topics
a. Geometrical representation of complex numbers( Argand diagram and vector diagram)
b. Solution of equation with real coefficients.
c. Locus of the points of the graph of complex equation.
d. Rules of modulus and conjugate.
Exercise 16C
Q.7
Draw Argand diagram showing the roots of the following equations.
a. 𝑧 4 − 1 = 0
b. 𝑧 3 + 6𝑧 + 20 = 0
c. 𝑧 4 + 4𝑧 3 + 4𝑧 2 − 9 = 0
Q.8
Identify in an Argand diagram the points corresponding to the following equations.
a. |𝑧| = 5
b. |𝑧 − 2| = 0
c. |𝑧 + 2𝑖| = |𝑧 + 4|
d. |𝑧 + 4| = 3|𝑧|
Q.9 Identify in an Argand diagram the points corresponding to the following inequalities.
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a. |𝑧| > 2
b. |𝑧 − 3𝑖| ≤ 1
c. |𝑧 + 1| ≤ |𝑧 − 𝑖|
d. |𝑧 − 3| > 2|𝑧|
Topics
a. Solving equations with complex coefficients.
b. To find the square root of a complex number.
c. To find the Least and greatest value of two loci.
Exercise 16D
Q.10 Find the square root of the following
a. −3 + 4𝑖
b.
5 + 12𝑖
c.
8 − 6𝑖
Q.11 Solve the following equations.
a. 𝑧 2 + 𝑧 + (1 − 𝑖) = 0
b. (1 + 𝑖)𝑧 2 + 2𝑖 𝑧 + 4𝑖 = 0
c. 𝑧 2 + (1 − 𝑖)𝑧 + (−6 + 2𝑖) = 0
Q.12 If(𝑥 + 𝑖𝑦)2 = 8𝑖, Prove that either 𝑥 = 0 or 𝑥 = ±√3𝑦, Hence find all the cube roots of
the 8𝑖,Show all roots on the Argand diagram.
Miscellaneous Exercise 16
Q.13 Give that 𝑧 is complex number such that 𝑧 + 3𝑧 ∗ = 12 + 8𝑖, find𝑧.
Q.14 Given that 3𝑖 is the root of the equation 3𝑧 3 − 5𝑧 2 + 27𝑧 − 45 = 0, find the other two roots.
Q.15 Two of the roots of the cubic equation, in which all the coefficients are real, are 2 and 1 + 3𝑖.
State the third root.
Q.16 It is given that 3 − 𝑖 is the root of the quadratic equation 𝑧 2 − (𝑎 + 𝑏𝑖)𝑧 + 4(1 + 3𝑖) = 0,
where 𝑎 and 𝑏 are real. In either order,
a. Find the values of 𝑎 and 𝑏.
b. Find the other root of the equation, giving that it is of the form 𝑘𝑖 , where 𝑘 is real.
Q.17 Find the roots of the equation 𝑧 2 = 21 − 20𝑖
Q.18 Verify that (3 − 2𝑖)2 = 5 − 12𝑖 .Find the roots of the equation (𝑧 − 𝑖)2 = 5 − 12𝑖
Q.19 Two complex numbers 𝑧 and 𝑤 , satisfy the inequality |𝑧 − 3 − 2𝑖| ≤ 2 and |𝑧 − 7 − 5𝑖| ≤ 1.
By drawing an Argand diagram. Find the Least possible value of |𝑧 − 𝑤|.
Exercise 17A, 17B
Q.20
Write the following complex numbers in the form of 𝑎 + 𝑖𝑏. Where appropriate leaves surds in
your answer, or give answer correct to 2 decimal places.
𝜋
3
𝜋
3
a. 2 (𝑐𝑜𝑠 + 𝑖𝑠𝑖𝑛 )
𝜋
𝜋
b. 5 (𝑐𝑜𝑠 (− 2 ) + 𝑖𝑠𝑖𝑛 (− 2 ))
c.
Q.21
(𝑐𝑜𝑠(−3) + 𝑖𝑠𝑖𝑛(−3))
Write the complex numbers in modulus-argument form. Where appropriate express the
argument as a rational multiple of 𝜋.
a. 1 + 2𝑖
b. 3 − 4𝑖
c. √2 − √2𝑖
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d. −1 + √3𝑖
Q.22 Show in an argand diagram the set of points satisfying the following equations.
𝜋
a. 𝑎𝑟𝑔𝑧 =
5
b. 𝑎𝑟𝑔𝑧 = −
c.
Q.23
a.
b.
c.
d.
2𝜋
5
3
𝑎𝑟𝑔(𝑧 − 1 − 2𝑖) = 4 𝜋
Show in an argand diagram the sets of points satisfying the following identities.
0
𝜋
< 𝑎𝑟𝑔𝑧 < 6
1
2
𝜋 < 𝑎𝑟𝑔(𝑧 − 1) < 3 𝜋
3
1
1
− 4 𝜋 < 𝑎𝑟𝑔(𝑧 + 1𝑖) < 4 𝜋
2
2
1
1
Q.24 If 𝑠 = 𝑐𝑜𝑠 3 𝜋 + 𝑖𝑠𝑖𝑛 3 𝜋 and 𝑡 = 𝑐𝑜𝑠 4 𝜋 + 𝑖𝑠𝑖𝑛 4 𝜋. Find
a. 𝑠𝑡 3
𝑠
b. 𝑡 2
Q.25
4
Write 1 + √3𝑖 and1 − 𝑖 in modulus argument form. Hence express
(1+√3𝑖)
(1−𝑖)6
in the form of
𝑎 + 𝑖𝑏.
Q.26
Q.27 Identify the set of points in an argand diagram for which
Miscellaneous Exercise 17
Q.28
a.
b.
Q.29
𝑧−3
𝜋
)=
𝑧−4𝑖
2
𝑧−𝑖
𝜋
arg ( ) =
𝑧+𝑖
4
Identify the set of points in an argand diagram for which arg (
𝜋
𝜋
Given that 𝑧 = 𝑡𝑎𝑛𝛼 + 𝑖 and 𝜔 = 4 (𝑐𝑜𝑠 + 𝑖𝑠𝑖𝑛 ), find in the simplest form
10
10
|𝑧|
𝑧
𝑎𝑟𝑔 𝜔
Given that (5 + 12𝑖)𝑧 = 63 + 16𝑖, find |𝑧| and 𝑎𝑟𝑔𝑧, giving this answer in radians correct to 3
𝜋
3
𝜋
3
significant figures. Given also that 𝜔 = 3 (𝑐𝑜𝑠 + 𝑖𝑠𝑖𝑛 ), find
𝑧
a. |𝑤|
b. 𝑎𝑟𝑔(𝑧𝑤)
Q.30 A complex number 𝑧 satisfies |𝑧 − 3 − 4𝑖| = 2. Describe in geometrical terms, with the aid of a
sketch, the locus of the points which represents 𝑧 in an argand diagram. Find
a. The greatest value of |𝑧|
b. The difference between the greatest and least value of arg 𝑧
Q.31
Given that |𝑧 − 5| = |𝑧 − 2 − 3𝑖|, show that on an argand diagrm the locus of the point which
represents 𝑧. Using your diagram, show that there is no value of 𝑧 satisfying both
𝜋
|𝑧 − 5| = |𝑧 − 2 − 3𝑖|, and 𝑎𝑟𝑔𝑧 = .
4
A complex number 𝑧 satisfies the inequality|𝑧 + 2 − 2√3𝑖| ≤ 2. Describe, in geometrical terms,
with the aid of sketch, the corresponding region in argand diagram. Find
a. The least value of |𝑧|
b. the greatest possible value of arg 𝑧
Past papers Questions:
Q.32
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Q.33 The complex number 1 + i√3 is denoted by u.
a. Express u in the form r(cos 𝜃 + 𝑖 𝑠𝑖𝑛𝜃), where r> 0 and -𝜋 < 𝜃 ≤ 𝜋. Hence or otherwise, find
the modulus and argument of 𝑢2 and 𝑢3
b. Show that u is a root of the equation 𝑧 2 − 2𝑧 + 4 = 0, and state the other root of this
equation.
c. Sketch an Argand diagram showing the points representing the complex numbers iand u. Shade
the region whose points represent every complex number z satisfying both the inequalities.
i. |𝑧 − 𝑖| ≤ 1
and
arg z ≥ arg u
Q34. (a)
Find the two square roots of the complex number -3 +4i, giving your answer in the form
𝑥 + 𝑖𝑦 , where x and y are real.
−1+3𝑖
(b)
The complex number z is given by z = 2+𝑖
c. Express z in the form x + iy, where x and y are real.
d. Show on a sketch of an Argand diagram, with origin O, the points A, B and C representing
the complex numbers -1 + 3i, 2 + i and z respectively.
e. State an equation relating the length OA, OB and OC.
2
Q35. The complex number 2i is denoted by u. The complex number with modulus 1 and argument 𝜋
3
is denoted by w.
𝑢
a.
Find in the form x + iy, where x and y are real, the complex numbers w, uw and 𝑤.
b.
Sketch an Argand diagram showing the points U, A and B representing the complex
𝑢
numbers u, uw and 𝑤 respectively.
c.
Prove that triangle UAB is equilateral.
7+4𝑖
Q36. The complex number u is given by u =
3−2𝑖
a. Express u in the form x + iy, where x and y are real.
b. Sketch an Argand diagram showing the point representing the complex number u.
Show on the same diagram the locus of the complex number z such that |𝑧 − 𝑢| = 2
c. Find the greatest value of arg z for points on this locus.
Q37.
a. Find the roots of the equation 𝑧 2 − 𝑧 + 1 = 0, giving your answer in the form 𝑥 + 𝑖𝑦,
where x and y are real.
b. Obtain the modulus and argument of each root.
c. Show that each root also satisfies the equation 𝑧 3 = −1.
Q38. The complex numbers 1 + 3i and 4 +2i are denoted by u and v respectively.
𝑢
a. Find in the form 𝑥 + 𝑖𝑦 , where x and y are real, the complex numbers 𝑢 – 𝑣 𝑎𝑛𝑑 𝑣 .
𝑢
State the argument of .In an Argand diagram, with origin O, the points A, B and C represent
𝑣
the numbers u, v and u-v respectively.
b. State fully the geometrical relationship between OC and BA.
1
c. Prove that angle AOB = 𝜋 radians.
4
Q39.
a. Solve the equation 𝑧 2 − 2𝑖𝑧 − 5 = 0, giving your answers in the form 𝑥 + 𝑖𝑦 where x
and y are real.
b. Find the modulus and argument of each root.
c. Sketch an Argand diagram showing the points representing the roots.
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Q40.
Q41.
The equation 2𝑥 2 + 𝑥 2 + 25 = 0 has one real root and two complex roots.
a.
Verify that 1 + 2i is one of the complex roots.
b.
Write down the other complex root of the equation.
c.
Sketch an Argand diagram showing the point representing the complex number 1 + 2i.
Show on the same diagram the set of points representing the complex numbers z which
satisfy |𝑧| = |𝑧 − 1 − 2𝑖|.
The complex number 2 + I is denoted by u. Its complex conjugate is denoted by 𝑢∗ .
a.
Show, on a sketch of an Argand diagram with origin O, the points A, B and C
representing the complex numbers u, 𝑢∗ and u + 𝑢∗ respectively. Describe in
geometrical terms the relationship between the four points O,A, B and C.
𝑢
b.
Express 𝑢∗in the form x + iy, where x and y are real.
c.
𝑢
4
1
By considering the argument of 𝑢∗ or otherwise, prove that 𝑡𝑎𝑛−1 [3] = 2𝑡𝑎𝑛−1 [2]
3+𝑖
Q42.
a.
b.
c.
The complex number u is given by
u=
2−𝑖
Express u in the form x + iy, where x and y are real.
Find the modulus and argument of u.
Sketch an Argand diagram showing the point representing the complex number u. Show on the
same diagram the locus of the point representing the complex number z such that
|𝑧 − 𝑢| = 1
Using your diagram, calculate the least value of |𝑧| for points on this locus.
2
Q43. The complex number −1+𝑖is denoted by u.
a. Find the modulus and argument of u and 𝑢2 .
b. Sketch an Argand diagram showing the points representing the complex numbers
u and 𝑢2 . Shade the region whose points represent the complex numbers z which satisfy both
the inequalities |𝑧| < 2 𝑎𝑛𝑑 |𝑧 − 𝑢2 | < |𝑧 − 𝑢|
Q44.
Q45.
4−3𝑖
(a)
(i)
(ii)
(b)
The complex number z is given by z = 1−2𝑖
Express z in the form x + iy, where x and y are real.
Find the modulus and argument of z.
Find the two square roots of the complex number 5 – 12i, giving your answer in the
for 𝑥 + 𝑖𝑦, where x and y are real.
The variable complex number z is given by
z = 2cos𝜃 + 𝑖(1 − 2𝑠𝑖𝑛𝜃)
Where 𝜃 takes all values in the interval –𝜋 < 𝜃 ≤ 𝜋.
a.
Show that |𝑧 − 𝑖| = 2, for all values of 𝜃. Hence sketch, in an Argand diagram, the locus
of the point representing z.
1
b.
Prove that the real part of 𝑧+2−𝑖 is constant for –𝜋 < 𝜃 ≤ 𝜋.
1
√3
Q46. The complex number w is given by w = − 2 + 𝑖 2
a. Find the modulus and argument of w.
1
1
b. The complex number z has modulus R and argument 𝜃, where –3 𝜋 < 𝜃 < 3 𝜋.
𝑧
State the modulus and argument of 𝑤𝑧 and the modulus and argument of 𝑤.
𝑧
c. Hence explain why, in an Argand diagram, the points representing 𝑧, 𝑤𝑧 𝑎𝑛𝑑 𝑤.
Are the vertices of an equilateral triangle?
d. In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the
origin. One of the vertices represents the complex number 4 + 2i. Find the complex numbers
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represented by the other two vertices. Give your answer in the form x + iy, where x and y are
real and exact.
Answers:
1.
a. 2 + 4𝑖
1
b. (−1 − 3𝑖)
c.
12.
13.
14.
15.
16.
17.
18.
19.
20.
2
1
(3 + 𝑖)
5
1
(5 + 9𝑖)
10
d.
2. 3, −3
3.
a. ±3𝑖
b. 3 ± 4𝑖
1
c. 2 (−1 ± 5𝑖)
4.
a. (3𝑧 − 1 − 2𝑖)(3𝑧 − 1 + 2𝑖)
b. (𝑧 − 2)(𝑧 + 2)(𝑧 − 2𝑖)(𝑧 + 2𝑖)
c. (𝑧 − 3)(𝑧 + 3)(𝑧 − 𝑖)(𝑧 + 𝑖)
d. (𝑧 + 1)(𝑧 − 2 − 𝑖)(𝑧 − 2 + 𝑖)
5. 1 − 𝑖, −1 + 2𝑖, −1 − 2𝑖
6. −2 − 𝑖, 2 + √7𝑖, 2 − √7𝑖
7.
a. ±1, ±𝑖
b. −2, 1 ± 3𝑖
c. −3, 1, −1 ± √2𝑖
8.
a. Circle, center O radius 5;𝑥 2 + 𝑦 2 =
25
b. Circle, center 2+0i radius 5;𝑥 2 +
𝑦 2 − 4𝑥 = 0
c. Line 𝑦 = 2𝑥 + 3
1
d. Circle, center 2 + 0𝑖 radius
21.
4
𝜋
𝜋
1
25. 2 (𝑐𝑜𝑠 3 + 𝑖𝑠𝑖𝑛 3 ) , √2 (𝑐𝑜𝑠 (− 4 𝜋) +
1
𝑖𝑠𝑖𝑛 (− 4 𝜋)) , −√3 + 𝑖
28. ;
a. sec 𝛼
2
b. 5 𝜋 − 𝛼
5
29. 5, −0.927, , 0.120
3
30. Circle, center 3 + 4𝑖 radius 2;
32. Interior and boundary of the Circle,
center −2 + 2√3𝑖 radius 2
33. A
1
1 2; 𝑥 2 + 𝑦 2 − 𝑥 = 2
9.
a.
b.
c.
d.
10. ,
a.
b.
c.
11. .
a.
b.
c. 1, −2 − 𝑖
−2𝑖, √3 + 𝑖, −√3 + 𝑖
3 − 4𝑖
5
3𝑖,
3
1 − 3𝑖 ; 𝑧 3 − 4𝑧 2 + 14𝑧 − 20 = 0
3, 3 ; 4𝑖
5 − 2𝑖, −5 + 2𝑖
3 − 𝐼, −3 + 𝑖
2
,
a. 1 + √3𝑖
b. 0 − 5𝑖
c. −0.9 − 0.14𝑖
,
a. 𝑟 = 2.24, 𝜃 = 1.11
b. 𝑟 = 5, 𝜃 = −0.93
c. 𝑟 = 2.24, 𝜃 = 1.11
1
d. 𝑟 = 2, 𝜃 = − 𝜋
𝜋
𝑥2 + 𝑦2 > 4
𝑥 2 + 𝑦 2 − 6𝑦 + 8 ≤ 0
𝑥+𝑦 ≤0
𝑥 2 + 𝑦 2 + 2𝑥 − 3 <0
𝜋
a. u = 2 (𝑐𝑜𝑠 3 + 𝑖 𝑠𝑖𝑛 3 )
b.
c.
34. .
a.
b.
c.
d.
35. .
a.
b.
1 + 2𝑖, −1 − 2𝑖
3 + 2𝑖, −3 − 2𝑖
3 − 𝑖, −3 + 𝑖
– 𝑖, −1 − 𝑖
−2𝑖, −1 + 𝑖
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other root is z = 1 - i√3
Nil
± (1 + 2𝑖)
1
7
+ 5𝑖
5
Nil
|𝑂𝐴| = |𝑂𝐵| × |𝑂𝐶|
√3 − 𝑖
Nil
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c. UAB is an equilateral triangle.
36. .
a. = 1+ 2i
b. |𝑧 − 𝑢| = 2 is a circle with centre at
u and radius 2 units.
c. R𝑂̂𝑋 = 2(𝑢𝑂̂X) = 2(1.07) = 2.21
radians.
37. .
a.
𝜋
1
2
−𝑖
origin and the point (1, 2). Therefore
z lies on the locus shown in the
diagram.
41. ,
a. Points O, A, C and B are vertices of
a rhombus.
3 4
b. 5+ 5 i
42. ,
a. u = 1 + i
𝜋
b. 4 radians
c. Nil
d. 0.414
43. M
1
a. 2 𝜋
b. 𝑁𝑖𝑙
44. a
a. 2 + i
b. 0.464 radians
c. ±(3 − 2𝑖)
45. A
a. 2
1
b. 4
46. C
3
a. 𝜋 radians
√3
2
b. –
3
c. Two roots obtained in part (i) are
also the roots of the equation 𝑧 3 =
−1
38.
a.
1
1
+ 2𝑖
2
𝜋
4
b.
c. OC and BA are parallel and OC and
BA are equal in length.
1
d. A𝑂̂𝐵 = 𝜋 r
4
39.
a.
b.
c.
40. ,
a.
b.
𝑧 = 2 + 𝑖, −2 + 𝑖
2.68 radians
2 + 𝑖, −2 + 𝑖
2
Other root = 1 – 2i
The locus is the perpendicular
bisector of the line segment joining
2
c. (-2 -√3 ) + 𝑖(√3 − 1)
2
3
b. 𝜃 − 𝜋
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