Complex Analysis Questions and Answers – Roots of Complex Numbers

«
»

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

1. The nth roots of any number are in ____________
a) arithmetic progression
b) geometric progression
c) harmonic progression
d) no specific pattern
View Answer

Answer: b
Explanation: We know that re=rei(α+2π). Hence, if reiα/n is a nth root of a number, then, re i(α/n+2π/n) is also a nth root of that number. Hence, the roots are in G.P. with common ratio= ei2π/n.
advertisement

2. In the Argand Plane shown below, a,b,c,d are the 4-th roots of 16. Find the area of the closed Polygon having a,b,c,d as its vertices.

a) 2 sq. units
b) 4 sq. units
c) 8 sq. units
d) 16 sq. units
View Answer

Answer: c
Explanation: The complex numbers a,b,c,d are in G.P. with common ratio eiπ/2. Therefore, a square is formed having side of 81/2(since O-a length is 2, therefore, b-a length is 81/2) . Hence the required area=(81/2)2=8 sq. units.

3. For k=1,2,…9, if we define zk=cos(3kπ/10)+isin(2kπ/10), then is it true that for each zk, there exists zj satisfying zk× zj=1?
a) True
b) False
View Answer

Answer: a
Explanation: zk= ei(2kπ/10)⇒ zk× zj=ei(2π/10)(k+j), zk is 10th root of unity.
⇒\(\overline{z_k}\) is also 10th root of unity. Taking zj as \(\overline{z_k}\), we have zk×zj=1.

4. For k=1,2,…9, if we define zk=cos(3kπ/10)+isin(2kπ/10), then is it possible that z1×z=zk has no Solution z?
a) True
b) False
View Answer

Answer: b
Explanation: z=zk/z1=ei(2kπ/10-2π/10)=eiπ/5(k-1)
For k = 2; z=eiπ/5, therefore a solution always exists.

5. Find the value of the expression (-1/2+i√3/2)637+(-1/2-i√3/2)337.
a) -1
b) 0
c) 1
d) i
View Answer

Answer: a
Explanation: (-1/2+i√3/2)=ω and (-1/2-i√3/2)=ω2, ω being the complex cube root of unity.
Therefore, expression has the value ω637337=ω+ω2=-1.
advertisement

6. Let a and b be complex cube roots of unity. If x=7a+2b and y=2a+7b, then evaluate xy.
a) 9
b) 39
c) 45
d) 53
View Answer

Answer: d
Explanation: Write a=ω and b=ω2.
Now, xy=(7a+2b)(2a+7b)=(7ω+2ω2)(2ω+7ω2)=14ω2+14ω+53=39 (since, 1+ω+ω2=0).

7. For integral x,y,z, find the range of |x+yω+zω2| if it is not true that x=y=z.
a) [1, ∞)
b) [√3, ∞)
c) (0, √3)
d) (0, ∞)
View Answer

Answer: a
Explanation: Put a=∞ and b=c=0 to show that maximum value tends to infinity.
Now, z=|x+yω+zω2|, hence, z2=|x+yω+zω2|2=(x2+y2+z2-xy-yz-zx)=1/2{(x-y)2+(y-z)2+(z-x)2}
Now if x=y, then y≠z and x≠z (given that x,y,z are not all equal)⇒(y-z)2≥1, also (z-x)2≥1 and
(x-y)2=0. Therefore, z2≥½(0+1+1)=1, hence |z|≥1.

8. Find the value of (1+ω)(1+ω2)(1+ω4)(1+ω8)…to 2n factors.
a) 1
b) 2
c) 3
d) 4
View Answer

Answer: a
Explanation: (1+ω)(1+ω2)(1+ω4)(1+ω8)…up to 2n factors
= (-ω2)(-ω)(1+ω)(1+ω2) …up to 2n factors
= 1×1×1×L up to n factors = 1.

9. If α,β,ȣ are the roots of equation x3–3x2+3x+7=0 and ω is cube root of unity, then evaluate
(α-1)/(β-1)+(β-1)/(ȣ-1)+(ȣ-1)/(α-1).
a) ω
b) ω2
c) 3ω
d) 3ω2
View Answer

Answer: d
Explanation: equation can be simplified to (x-1)3=-8⇒(x-1)/(-2)=(1)1/3⇒roots: 1,ω,ω2
⇒ α=-1, β=1-2ω, ȣ=1-2ω2. Therefore, required value=-2/(-2ω)+(-2ω)/(-2ω2)+(-2ω2)/(-2)
= 1/ω+1/ω+1/ω=3/ω=3ω2.
advertisement

10. Find the possible value(s) of Re(i1/2)+|Im(i1/2)|.
a) -1, 1
b) 0, √2
c) 0, 1
d) 1, √2
View Answer

Answer: b
Explanation: We can write: i=eiπ/2⇒i1/2=eiπ/4=1/√2+i/√2, also i1/2=ei3π/4=-1/√2-i/√2
Hence, Re(i1/2)+|Im(i1/2)|=1/√2+1/√2=√2 OR -1/√2+1/√2=0.

11. Let ω and ω2 be the non-real cube roots of unity and 1/(a+ω)+1/(b+ω)+1/(c+ω)=2ω2 and 1/(a+ω2)+1/(b+ω2)+1/(c+ω2)=2ω, then calculate 1/(a+1)+1/(b+1)+1/(c+1).
a) 1
b) 2
c) 3
d) 4
View Answer

Answer: b
Explanation: Given relations can be written as: 1/(a+ω)+1/(b+ω)+1/(c+ω)=2/ω, and
1/(a+ω2)+1/(b+ω2)+1/(c+ω2)=2/ω2⇒ω and ω2 are roots of 1/(a+x)+1/(b+x)+1/(c+x)=2/x.
⇒[3x2+2(a+b+c)x+bc+ca+ab]/[(a+x)(b+x)(c+x)]=2/x⇒x3+(bc+ca+ab)x-2abc=0.
Let 3rd root be α (apart from ω and ω2). Then, α+ ω+ ω2=coefficient of x2=0⇒α=1. Hence, 1/(a+1)+1/(b+1)+1/(c+1)=2/1=2.

12. Find the cube root of 8i lying in the first quadrant of the complex plane.
a) i-√3
b) 2i+√3
c) i+2√3
d) i+√3
View Answer

Answer: d
Explanation: Let z3=8i⇒z3=-8i3⇒[z/(-2i)]3=1.
⇒z/(-2i)=1 or ω or ω2⇒z=-2i or -2iω or -2iω2
⇒z=-2i or i+√3 or i-√3 out of which i+√3 lies in the first quadrant.

13. If ω is the complex cube root of unity, then which among the following is a factor of the polynomial x6+ 4x5+3x4+2x3+x+1?
a) x+ω
b) x+ω2
c) (x+ω)(x+ω2)
d) (x–ω)(x–ω2)
View Answer

Answer: d
Explanation: Let f(x)=x6+ 4x5+3x4+2x3+x+1. Therefore, f(ω)= ω6+4ω5+3ω4+2ω3+ω+1=0
Since, ω2 is the complex conjugate of ω, therefore ω2 is also a root (roots occur in conjugate pairs). Therefore (x–ω)(x–ω2) is a root. Also, f(-ω)=(-ω)6+4(-ω)5+3(-ω)4+2(-ω)3+(-ω)+1=1-4ω2+3ω-2-ω+1≠0⇒-ω and hence -ω2 are not the roots.
advertisement

14. Find ∑r=1(ar+b) ωr-1 if ω is a complex nth root of unity.
a) n(n+1)a/2
b) nb/(1-n)
c) na/(ω-1)
d) n(n+1)a/(ω-1)
View Answer

Answer: c
Explanation: (a+b)+(2a+b)ω+(3a+b)ω2+…+(na+b)ωn-1=b(1+ω+ω2+….+ωn-1)+a(1+2ω+3ω2+…+nωn-1)
Let S=1+2ω+3ω2+…+nωn-1⇒Sω= ω+2ω2+…+(n-1)ωn-1+nωn⇒S(1-ω)=1+ω+ ω2+…+ωn-1–nωn=-n
⇒S=n/(ω-1)⇒E=na/(ω-1).

15. Which of the following is not equal to (-1)1/3?
a) -1
b) (-√3+i)/(2i)
c) (√3+i)/(2i)
d) (√3–i)/(2i)
View Answer

Answer: d
Explanation: Let x=(-1)1/3⇒x3=-1⇒(-x)3=1⇒-x=1, ω, ω2
Therefore, x=-1,- ω,- ω2=-1, (-√3+i)/(2i), (√3+i)/(2i).

Sanfoundry Global Education & Learning Series – Complex Analysis.

To practice all areas of Complex Analysis, here is complete set of 1000+ Multiple Choice Questions and Answers.

Participate in the Sanfoundry Certification contest to get free Certificate of Merit. Join our social networks below and stay updated with latest contests, videos, internships and jobs!

advertisement
advertisement
advertisement
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