Complex Analysis Questions and Answers – Complex Conjugates

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This set of Complex Analysis Multiple Choice Questions & Answers (MCQs) focuses on “Complex Conjugates”.

1. Consider two complex numbers, x and y satisfying |x|=|y| and Arg(x)+Arg(y)=π. What is x in terms of y?
a) y̅
b) -y̅
c) y
d) -y
View Answer

Answer: b
Explanation: Let |x|=|y|=r, therefore, x=re and y=re, where, α+β=π,
Hence, x=rei(π-β)=ree-iβ
Therefore, x=-re-iβ=-y̅.
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2. Let two complex numbers z and ω satisfy z > \(\overline{\omega}\). Find the value of the expression zIm(ω)Im(z)+zω, if Re(z) and Re(ω) are the roots of the equation x2–5x+6.
a) 2
b) 4
c) 6
d) 8
View Answer

Answer: d
Explanation: Since two complex numbers with non-zero complex parts cannot be compared, therefore, Im(z) and Im(ω) are zero. Also, since z and ω are the roots of x2–5x+6, zω=6. Hence the expression equals 1+1+6=8.

3. Consider the Argand Plane shown below.

If another complex number ω=z̅+4i, then find the area of the triangle having O, z and ω as its vertices.
a) 6
b) 12
c) 24
d) 36
View Answer

Answer: d
Explanation: Since ω=z̅+4i and z=6+8i (from image), therefore ω=6-8i+4i=6-4i and we get a triangle having vertices (0,0), (6,8) and (6,-4) in the Argand Plane shown. Therefore, the area is (1/2)×(base)×(height)=(1/2)×(8+4)×6=36.

4. For two complex numbers p and q, if Arg(p)-Arg(q)=π/2 as well as |pq|=1, what is the value of p̅q ?
a) -i
b) -1
c) i
d) 1
View Answer

Answer: a
Explanation: p̅q=|p|e-iarg p×|q|eiarg q
= e-i(arg p-arg q) (since |pq|=1)
= e-iπ/2=-i.

5. What is the area of the rectangle whose vertices are the roots of the equation zz̅3+z̅z3=350, given that Re(z) and Im(z) are integers ?
a) 12
b) 24
c) 36
d) 48
View Answer

Answer: d
Explanation: Taking zz̅ common, we have, (z2+z̅2) zz̅=350
Hence, (x2+y2)(x2-y2)=175=25×7
∴ x2+y2=25 and x2-y2=7
Hence, x=±4, y=±3, implying, Area=8×6=48.
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6. Consider the complex number z, for which a line segment A is drawn connecting origin and the point z. Also, consider the line segment B connecting origin and z̅. if z = x+iy, and the smaller angle between A and B is α, then select the incorrect option.
a) α=π/2 if x=y
b) α>π/2 if |y|>|x|
c) α=π/2 if x=-y
d) α=2×Arg(z)
View Answer

Answer: d
Explanation: if x=y or x=-y, |Arg(z)|=π/4 and α=2× π/4=π/2.
Now, if |y|>|x|, then |Arg(z)|>π/4 and α>2×π/4=π/2.
If z=i-1, then, Arg(z)=3π/4 but α=π/2, ∴ α≠2×Arg(z).

7. For a complex number z, if Re(z) and Im(z) are the roots of x2-7x+12=0 and z+z̅ is one of the roots of x2–10x+16=0, then, find Re(z)-Im(z).
a) 1
b) -1
c) 3
d) -3
View Answer

Answer: a
Explanation: Re(z) and Im(z) have values 3 and 4. Also 2Re(z) is either 2 or 8.
Merging, the two conditions, we get, Re(z)=4 and Im(z)=3.
∴ The required value is 4-3=1.

8. Let z=sinx+icos2x and ω=cosx-isin2x. Then for what values of x are z and ω conjugate of each other?
a) x = nπ
b) x = 0
c) x = (n+1/2)π
d) no value of x
View Answer

Answer: d
Explanation: z̅=ω
∴ sinx=cosx and cos2x=sin2x
This is not possible for any value of x.

9. If a complex number z, with integral real and imaginary parts, satisfies z2+ z̅2=16, then find the value of |z|.
a) 21/2
b) 4
c) 81/2
d) 101/2
View Answer

Answer: d
Explanation: Let z=x+iy, then, x2–y2=8 (by replacing z by x+iy in the given equation)
Since, x and y are integers, |x|=3 and |y|=1 are the only possible values.
Therefore, |z|=(32+12)1/2=101/2.
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10. If the complex numbers (x2-3x+2)+i(y2-11y+40) and (x2-6x+8)+i(y2-9x+10) are conjugates of each other, Then what is the value of |x+iy|?
a) 131/2
b) 191/2
c) 231/2
d) 291/2
View Answer

Answer: d
Explanation: We have, x2-3x+2=x2-6x+8 and y2-11y+40+y2-9x+10=0
Hence, x=2, y=5
∴ |2+5i|=291/2.

11. The hyperbola, x2–y2=1 can be represented on the Argand Plane by which of the following equations?
a) z2-z̅2=1
b) z2+z̅2=1
c) z2-z̅2=2
d) z2+z̅2=2
View Answer

Answer: d
Explanation: Put z = x+iy. Simplifying z2-z̅2=1 gives 4xyi=1(incorrect);
Simplifying z2+z̅2=1 gives 2x2–2y2=1(incorrect);
Simplifying z2-z̅2=2, we get 2xyi=1(incorrect);
Finally, simplifying z2+z̅2=2 gives x2–y2=1(correct).

12. If z and ω are the complex conjugates of each other, then find the value of (lnz+lnω)/ln|z|.
a) 1
b) 2
c) 3
d) 4
View Answer

Answer: b
Explanation: lnz+lnω=ln|z|+iargz+ln|ω|+iargω
Since, |z|=|ω| and arg z+arg ω=0, ∴ lnz+lnω=2ln|z|
∴ Required value=2ln|z|/ln|z|=2.

13. Let zz̅=64, and ω\(\overline{\omega}\)=36. Find the maximum possible value of |z+ω|.
a) 8
b) 10
c) 12
d) 14
View Answer

Answer: d
Explanation: We know that, zz̅=64=|z|2, ∴ |z|=8, similarly, |ω|=6.
Therefore, |z+ω| ≤ |z|+|ω|=8+6=14.
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14. Let z and ω be complex numbers such that ω has non-zero imaginary part and z≠1. If the expression (ω-\(\overline{\omega}\)z)/(1-z) is purely real, then find the set of values of z.
a) {z : |z|=1}
b) {z : z=z̅}
c) {z : z≠1}
d) {z : |z|=1, z≠1}
View Answer

Answer: d
Explanation: (ω-\(\overline{\omega}\)z)/(1-z) = (\(\overline{\omega}\)-ωz̅)/(1-z̅)⇒(zz̅-1)(\(\overline{\omega}\)-ω) = 0
⇒zz̅=1⇒|z|2=1⇒|z|=1.

15. Let z and ω be complex numbers satisfying zz̅+ω\(\overline{\omega}\)=100. If |z|,|ω|∈I, then find the minimum possible value of the expression |z+ω|.
a) 1
b) 2
c) 3
d) 4
View Answer

Answer: b
Explanation: zz̅+ω\(\overline{\omega}\)=100⇒|z|2+|ω|2=100, also, since |z|, |ω|∈I+, therefore, (|z|,|ω|)=(8,6) or (6,8). Hence, |z+ω|≥||z|-|ω||=|8-6|=2.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn