# NOUN TMA Questions and Answers: MTH210- Introduction to Complex Analysis

\[\frac{-17-19i}{10}\]

Q2 For any complex number and any integer k, \[i^{4k+2}=ô€³¦?ô€³¦.\]

-1

Q3 Simplify \[(4-5i)(2+3i)\]

\[23+2i\]

Q4 \[(a+bi)(c+di)=ô€³¦?ô€³¦.\]

\[(ac-bd)+(ad+bc)i\]

Q5 \[\frac{z^4-1}{z-i}=ô€³¦?ô€³¦ô€³¦?ô€³¦.\]

\[z^3+iz^2-z+i\]

Q6 The square of the imaginary number of a complex number has the value

-1

Q7 The Fundamental Theorem of Algebra states that

Every non-constant polynomial with coefficients in th e set of complex numbers, C (or set of real numbers,R) has a root in C

Q8 If in a complex number, \[z=x=iy, x=0\]then z is said to be ô€³¦?ô€³¦ô€³¦?ô€³¦ô€³¦?ô€³¦.

purely ima ginary

Q9 One of the following is not true about a complex number

The imaginary part of \[3+5i\;is\; 5i\]

Q10 The equation \[x^2+1=0\] has ô€³¦?ô€³¦ô€³¦?ô€³¦.. real solutions

no

Q11 \[If \;z_1 = 2(cos 15 + i sin 15)\; and \;\;z_2 = \frac{1}{2}(cos 30 + i sin 30)\] are complex numbers, then\[ z_1z_2\;= \]

\[cos 45 + i sin 45\]

Q12 Let f(z) =u + v be an analytic function, one of the following statements in not correct

a non constant analytic function can take only real or only p ure imaginary values

Q13 In a complex function \[f(z) =u(x,y) + iv(x,y),\; z + iy\] is analytic in a domain D iff

v is a harmonic conjugate to u in D

Q14 The argument of the cube of a complex number is same as

\[arg z + arg z^2\]

\[arg z + 2 arg z\]

\[3 arg z\]

---All of the options

Q15 One of the following is true about a continuous function

A function f(z) is continuous if it is continuous at all points w here it is defined.

A funcion is continuous if and only if its real and imaginary parts are continuou s

All polynomials P(z) are continuous

---all the options

Q16 All the following are true except

The differences of analytic functions are analytic

Q17 For any complex number z, the argument of its square is given by

\[arg z^2 = 2 arg z\]

Q18 Simplify using Euler's equation: \[(1+i)^24\]

\[2^{12}\]

Q19 Evaluate \[e^{-1}e^\frac{i\pi}{2}\] using Eulerô€³¦??s equation

\[\frac{i}{e}\]

Q20 Evaluate \[e^{i\pi}\] using Eulerô€³¦??s equation

-1

\[cos 45 + i sin 45\]

Q12 Let f(z) =u + v be an analytic function, one of the following statements in not correct

a non constant analytic function can take only real or only p ure imaginary values

Q13 In a complex function \[f(z) =u(x,y) + iv(x,y),\; z + iy\] is analytic in a domain D iff

v is a harmonic conjugate to u in D

Q14 The argument of the cube of a complex number is same as

\[arg z + arg z^2\]

\[arg z + 2 arg z\]

\[3 arg z\]

---All of the options

Q15 One of the following is true about a continuous function

A function f(z) is continuous if it is continuous at all points w here it is defined.

A funcion is continuous if and only if its real and imaginary parts are continuou s

All polynomials P(z) are continuous

---all the options

Q16 All the following are true except

The differences of analytic functions are analytic

Q17 For any complex number z, the argument of its square is given by

\[arg z^2 = 2 arg z\]

Q18 Simplify using Euler's equation: \[(1+i)^24\]

\[2^{12}\]

Q19 Evaluate \[e^{-1}e^\frac{i\pi}{2}\] using Eulerô€³¦??s equation

\[\frac{i}{e}\]

Q20 Evaluate \[e^{i\pi}\] using Eulerô€³¦??s equation

-1

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