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Practice QuestionsMCQ (Single Correct Answer)
1
If z1 and z2 are complex numbers with $$\left| {{z_1}} \right| = \left| {{z_2}} \right|$$, then which of the following is/are correct?
1. z1 = z2
2. Real part of z1 = Real part of z2
3. Imaginary part of z1 = Imaginary part of z2
Select the correct answer using the code given below.
1. z1 = z2
2. Real part of z1 = Real part of z2
3. Imaginary part of z1 = Imaginary part of z2
Select the correct answer using the code given below.
NDA 2015 Paper 2
2
If the point z1 = 1 + i, where $$i = \sqrt { - 1} $$ is the reflection of a point z2 = x + iy in the line $$i\overline z - iz = 5$$, then the point z2 is
NDA 2015 Paper 2
3
$$z\overline z + (3 - i)z + (3 + i)\overline z + 1 = 0$$ represents a circle with
NDA 2015 Paper 2
4
What is $$\sqrt {{{1 + {\omega ^2}} \over {1 + \omega }}} $$ equal to, where $$\omega$$ is the cube root of unity?
NDA 2016 Paper 2
5
What is $${\omega ^{100}} + {\omega ^{200}} + {\omega ^{300}}$$ equal to, where $$\omega$$ is the cube root of unity?
NDA 2016 Paper 2
6
If $${\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {z + 1}}} \right) = 0$$, where z = x + iy is a complex number, then which one of the following is correct?
NDA 2016 Paper 2
7
If $$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^{107}} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^{107}}$$, then what is the imaginary part of z equal to?
NDA 2016 Paper 2
8
What is the number of distinct solutions of the equation $${z^2} + |z|\, = \,0$$ (where, z is the complex number)?
NDA 2016 Paper 2
9
What is z1 + z2 + z3 equal to?
NDA 2016 Paper 1
10
Consider the following statements
1. $${z_1}{z_2}{z_3}$$ is purely imaginary.
2. $${z_1}{z_2} + {z_2}{z_3} + {z_3}{z_1}$$ is purely real.
Which of the above statements is/are correct?
1. $${z_1}{z_2}{z_3}$$ is purely imaginary.
2. $${z_1}{z_2} + {z_2}{z_3} + {z_3}{z_1}$$ is purely real.
Which of the above statements is/are correct?
NDA 2016 Paper 1
11
Suppose, $$\omega$$ is a cube root of unity with $$\omega$$ $$\ne$$ 1. Suppose, P and Q are the points on the complex plane defined by $$\omega$$ and $$\omega$$2. If O is the origin, then what is the angle between OP and OQ?
NDA 2016 Paper 1
12
Suppose, $$\omega$$1 and $$\omega$$2 are two distinct cube roots of unity different from 1. Then, what is $${({\omega _1} - {\omega _2})^2}$$ equal to?
NDA 2016 Paper 1
13
The smallest positive integer n for which $${\left( {{{1 + i} \over {1 - i}}} \right)^n} = 1$$, is
NDA 2017 Paper 2
14
If $$\left| {z - {4 \over z}} \right| = 2$$, then the maximum value of $$\left| z \right|$$ is equal to
NDA 2017 Paper 2
15
Geometrically $${\mathop{\rm Re}\nolimits} ({z^2} - i) = 2$$, where, $$i = \sqrt { - 1} $$ and Re is the real part, represents
NDA 2017 Paper 2
16
The value of $${i^{2n}} + {i^{2n + 1}} + {i^{2n + 2}} + {i^{2n + 3}}$$, where $$i = \sqrt { - 1} $$, is
NDA 2017 Paper 1
17
The value of $${\left( {{{ - 1 + i\sqrt 3 } \over 2}} \right)^n} + {\left( {{{ - 1 - i\sqrt 3 } \over 2}} \right)^n}$$, where n is not a multiple of 3 and $$i = \sqrt { - 1} $$, is
NDA 2017 Paper 1
18
If 1, $$\omega$$, $$\omega$$2 are the cube roots of unity, then (1 + $$\omega$$)(1 + $$\omega$$2)(1 + $$\omega$$3)(1 + $$\omega$$ + $$\omega$$2) is equal to
NDA 2017 Paper 1
19
The modulus and principal argument of the complex number $${{1 + 2i} \over {1 - {{(1 - i)}^2}}}$$ are
NDA 2017 Paper 1
20
If $$\left| {z + 4} \right| \le 3$$, then the maximum value of $$\left| {z + 1} \right|$$ is
NDA 2017 Paper 1
21
The number of roots of the equation $${z^2} = 2\overline z $$ is
NDA 2017 Paper 1
22
What is the value of $${\left( {{{ - 1 + i\sqrt 3 } \over 2}} \right)^{3n}} + {\left( {{{ - 1 - i\sqrt 3 } \over 2}} \right)^{3n}}$$, where $$i = \sqrt { - 1} $$?
NDA 2018 Paper 2
23
Which one of the following is correct in respect of the cube roots of unity?
NDA 2018 Paper 2
24
What is the principal argument of ($$-$$1 $$-$$i), where $$i = \sqrt { - 1} $$?
NDA 2018 Paper 1
25
The number of non-zero integral solution of the equation $$|1 - 2i{|^x} = {5^x}$$ is
NDA 2018 Paper 1
26
If $$\alpha$$ and $$\beta$$ are different complex numbers with $$|\alpha |\, = 1$$, then what is $$\left| {{{\alpha - \beta } \over {1 - \alpha \overline \beta }}} \right|$$ equal to?
NDA 2018 Paper 1
27
What is $${i^{1000}} + {i^{1001}} + {i^{1002}} + {i^{1003}}$$ equal to (where, $$i = \sqrt { - 1} $$)?
NDA 2018 Paper 1
28
The modulus-amplitude form of $$\sqrt 3 + i$$, where $$i = \sqrt { - 1} $$ is
NDA 2018 Paper 1
29
What is the value of the sum $$\sum\limits_{n = 2}^{11} {({i^n} + {i^{n + 1}})} $$, where $$i = \sqrt { - 1} $$?
NDA 2018 Paper 1
30
If $$x = 1 + i$$, then what is the value of $${x^6} + {x^4} + {x^2} + 1$$?
NDA 2019 Paper 2
31
What is the value of $${\left[ {{{i + \sqrt 3 } \over 2}} \right]^{2019}} + {\left[ {{{i - \sqrt 3 } \over 2}} \right]^{2019}}$$?
NDA 2019 Paper 2
32
The common roots of the equations $${z^3} + 2{z^2} + 2z + 1 = 0$$ and $${z^{2017}} + {z^{2018}} + 1 = 0$$ are
NDA 2019 Paper 1
33
What is the modulus of z?
NDA 2019 Paper 1
34
What is the principal argument of z?
NDA 2019 Paper 1
35
If ω ≠ 1 is a cube root of unity, then what is (1 + ω - ω2)100 + (1 - ω + ω2)100 equal to?
NDA Mathematics 1st September 2024
36
What is the value of the sum $\rm \sum\limits_{n=1}^{20}(i^{n-1}+i^n+i^{n+1})$ where i = √-1 ?
NDA Mathematics 1st September 2024
37
what is the value of $\rm \left|\frac{Z_1}{Z_2}\right|$,
NDA Mathematics 1st September 2024
38
what is the value of $\rm \frac{1}{2}+Re\left(\frac{Z_1}{Z_2}\right)?$
NDA Mathematics 1st September 2024
39
If $z$ is any complex number and $i z^3+z^2-z+i=0$, where $i=\sqrt{-1}$, then what is the value of $(|z|+1)^2$ ?
NDA Mathematics 21 April 2024
40
If $x, y$ and $z$ are the cube roots of unity, then what is the value of $xy + yz + zx$?
NDA Mathematics 21 April 2024
41
Let $z_1$ and $z_2$ be two complex numbers such that $\left|\frac{z_1 + z_2}{z_1 - z_2}\right| = 1$, then what is $\operatorname{Re} \left(\frac{z_1}{z_2}\right) + 1$ equal to?
NDA Mathematics 21 April 2024
42
If $\omega \neq 1$ is a cube root of unity, then what are the solutions of $(z-100)^3 + 1000 = 0$?
NDA Mathematics 21 April 2024
43
What is $(1 + i)^4 + (1 - i)^4$ equal to, where $i=\sqrt{-1}$?
NDA Mathematics 21 April 2024
44
If z z̅ = |z + z̅ |, where z = x + iy, i = $\sqrt{-1}$, then the locus of z is a pair of:
NDA Mathematics 3 September 2023
45
What is the value of $\sqrt{12+5 i}+\sqrt{12-5 i}$ where $i=\sqrt{-1}$ ?
NDA Mathematics 3 September 2023
46
Which one of the following is a square root of $-\sqrt{-1} $?
NDA Mathematics 3 September 2023
47
What are the roots of equation-I ?
NDA Mathematics 3 September 2023
48
Which one of the following is a root of equation-II?
NDA Mathematics 3 September 2023
49
What is the number of common roots of equation-I and equation-II?
NDA Mathematics 3 September 2023
50
If ω is a non-real cube root of 1, then what is the value of $\left|\frac{1-\omega}{\omega+\omega^2}\right| ?$
NDA Mathematics 16 April 2023
51
If α and β are the distinct roots of equation x2 - x + 1 = 0, then what is the value of $\left|\frac{\alpha^{100}+\beta^{100}}{\alpha^{100}-\beta^{100}}\right|$ ?
NDA Mathematics 16 April 2023
52
If 2 - i√3 where i = √-1 is a root of the equation x2 + ax + b = 0, then what is the value of (a + b)?
NDA Mathematics 16 April 2023
53
If $z=\frac{1+i √{3}}{1-i √{3}}$ where i = √-1 then what is the argument of z ?
NDA Mathematics 16 April 2023
54
If z is a complex number such that $\frac{z-1}{z+1}$ is purely imaginary, then what is |z| equal to ?
NDA Mathematics 16 April 2023
55
What is the modulus of z?
NDA Mathematics 4 September 2022
56
What is angle θ such that z is purely real ?
where n is an integer
NDA Mathematics 4 September 2022
57
What is angle θ such that z is purely imaginary ?
where n is an integer
NDA Mathematics 4 September 2022
58
What is the principal argument of $\frac{1}{1 + i}$ where $i = \sqrt{-1}?$
NDA Mathematics 10 April 2022
59
What is the modulus of $\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)^{200}?$
NDA Mathematics 10 April 2022
60
If x2 + x + 1 = 0, then what is the value of x199 + x200 + x201?
NDA Mathematics 14 November 2021
61
What is $\rm \sum\limits_{n=1}^{8n+7} i^n$ equal to, where i = √-1?
NDA Mathematics 14 November 2021
62
If z = x + iy, where i = √-1, then what does the equations zz̅ + ∣z∣2 + 4(z + z̅) - 48 = 0 represent?
NDA Mathematics 14 November 2021
63
Which one of the following is a square root of $\rm 2a+2\sqrt{a^2 + b^2}$, where a, b ∈ ℝ?
NDA Mathematics 14 November 2021
64
Consider the following statements in respect of the roots of the equation x3 - 8 = 0 :
1. The roots are non-collinear.
2. The roots lie on a circle of unit radius.
Which of the above statements is/are correct?
NDA Mathematics 14 November 2021
65
If i = √-1, then how many values does i-2n have for different n ∈ ℤ?
NDA Mathematics 14 November 2021
66
Consider the following in respect of a complex number z:
1. $\rm {\overline{\left(z^{-1}\right)}}=(\bar{z})^{-1}$
2. zz-1 = |z|2
Which of the above is/are correct?
NDA Mathematics 18 April 2021
67
1. The difference of Z and its conjugate is an imaginary number.
2. The sum of Z and its conjugate is a real number.
NDA Mathematics 18 April 2021
68
What is the modulus of the complex number i2n + 1(-i)2n - 1, where n ∈ N and i = √-1?
NDA Mathematics 18 April 2021
69
The smallest positive integer n for which
$\rm \left(\frac{1-i}{1+i}\right)^{n^2}=1$
where i = √-1, is
NDA Mathematics 18 April 2021
70
If Z = 1 + i, where i = √-1, then what is the modulus of $\rm z+\frac{2}{z}?$
NDA Mathematics 18 April 2021