Complex Numbers · NDA previous year questions

What is the principal argument of $(-1-i)$, where $i=\sqrt{-1}$ ?

1. A.$\pi /4$
2. B.$-\pi /4$
3. C.$-3\pi /4$
4. D.$3\pi /4$

If $\alpha$ and $\beta$ are different complex numbers with $|\alpha |=1$, then what is $\left|\frac{\alpha -\beta }{1-\alpha \bar{\beta }}\right|$ equal to ?

1. A.$|\beta |$
2. B.2
3. C.1
4. D.0

What is $i^{1000}+i^{1001}+i^{1002}+i^{1003}$ equal to (where $i=\sqrt{-1}$) ?

1. A.0
2. B.$i$
3. C.$-i$
4. D.1

The modulus-amplitude form of $\sqrt{3}+i$, where $i=\sqrt{-1}$ is

1. A.$2\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)$
2. B.$2\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)$
3. C.$4\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)$
4. D.$4\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)$

What is the value of the sum ${\sum }_{n=2}^{11}(i^n+i^{n+1})$, where $i=\sqrt{-1}$ ?

1. A.$i$
2. B.$2i$
3. C.$-2i$
4. D.$1+i$

If the ratio of AM to GM of two positive numbers a and b is 5 : 3, then a : b is equal to

1. A.3 : 5
2. B.2 : 9
3. C.9 : 1
4. D.5 : 3

If $x=1-y+y^2-y^3+\dots$ up to infinite terms, where $|y|<1$, then which one of the following is correct ?

1. A.$x=\frac{1}{1+y}$
2. B.$x=\frac{1}{1-y}$
3. C.$x=\frac{y}{1+y}$
4. D.$x=\frac{y}{1-y}$

What is the sum of all two-digit numbers which when divided by 3 leave 2 as the remainder ?

1. A.1565
2. B.1585
3. C.1635
4. D.1655

Let $\alpha$ and $\beta$ be real numbers and z be a complex number. If $z^2+\alpha z+\beta =0$ has two distinct non-real roots with $\text{Re}(z)=1$, then it is necessary that

1. A.$\beta \in (-1,0)$
2. B.$|\beta |=1$
3. C.$\beta \in (1,\infty )$
4. D.$\beta \in (0,1)$

The modulus and principal argument of the complex number $\frac{1+2i}{1-(1-i{)}^2}$ are respectively

1. A.1, 0
2. B.1, 1
3. C.2, 0
4. D.2, 1

If the graph of a quadratic polynomial lies entirely above $x$-axis, then which one of the following is correct?

1. A.Both the roots are real
2. B.One root is real and the other is complex
3. C.Both the roots are complex
4. D.Cannot say

What is $\sqrt{\frac{1+{\omega }^2}{1+\omega }}$ equal to, where $\omega$ is the cube root of unity ?

1. A.1
2. B.$\omega$
3. C.${\omega }^2$
4. D.$i\omega$, where $i=\sqrt{-1}$

What is ${\omega }^{100}+{\omega }^{200}+{\omega }^{300}$ equal to, where $\omega$ is the cube root of unity ?

1. A.1
2. B.$3\omega$
3. C.$3{\omega }^2$
4. D.0

If $\mathrm{Re}\left(\frac{z-1}{z+1}\right)=0$, where $z=x+iy$ is a complex number, then which one of the following is correct ?

1. A.$z=1+i$
2. B.$|z|=2$
3. C.$z=1-i$
4. D.$|z|=1$

If $z={\left(\frac{\sqrt{3}}{2}+\frac{i}{2}\right)}^{107}+{\left(\frac{\sqrt{3}}{2}-\frac{i}{2}\right)}^{107}$, then what is the imaginary part of z equal to ?

1. A.0
2. B.$\frac{1}{2}$
3. C.$\frac{\sqrt{3}}{2}$
4. D.1

What is the number of distinct solutions of the equation $z^2+|z|=0$ (where z is a complex number) ?

1. A.One
2. B.Two
3. C.Three
4. D.Five

What is the greatest value of the positive integer n satisfying the condition $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots +\frac{1}{2^{n-1}}<2-\frac{1}{1000}$ ?

1. A.8
2. B.9
3. C.10
4. D.11

For the next two (2) items that follow : Let $z_1,z_2$ and $z_3$ be non-zero complex numbers satisfying $z^2=i\bar{z}$, where $i=\sqrt{-1}$. What is $z_1+z_2+z_3$ equal to?

1. A.$i$
2. B.$-i$
3. C.$0$
4. D.$1$

For the next two (2) items that follow : Let $z_1,z_2$ and $z_3$ be non-zero complex numbers satisfying $z^2=i\bar{z}$, where $i=\sqrt{-1}$. 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. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

For the next two (2) items that follow : Let $z$ be a complex number satisfying $\left|\frac{z-4}{z-8}\right|=1$ and $\left|\frac{z}{z-2}\right|=\frac{3}{2}$ What is $|z|$ equal to?

1. A.6
2. B.12
3. C.18
4. D.36

For the next two (2) items that follow : Let $z$ be a complex number satisfying $\left|\frac{z-4}{z-8}\right|=1$ and $\left|\frac{z}{z-2}\right|=\frac{3}{2}$ What is $\left|\frac{z-6}{z+6}\right|$ equal to?

1. A.3
2. B.2
3. C.1
4. D.0

For the next two (2) items that follow: Let $f(x)$ be the greatest integer function and $g(x)$ be the modulus function. What is $(g\circ f)\left(-\frac{5}{3}\right)-(f\circ g)\left(-\frac{5}{3}\right)$ equal to?

1. A.$-1$
2. B.0
3. C.1
4. D.2

For the next two (2) items that follow: Let $f(x)$ be the greatest integer function and $g(x)$ be the modulus function. What is $(f\circ f)\left(-\frac{9}{5}\right)+(g\circ g)(-2)$ equal to?

1. A.$-1$
2. B.0
3. C.1
4. D.2

If $z_1$ and $z_2$ are complex numbers with $|z_1|=|z_2|$, then which of the following is/are correct? 1. $z_1=z_2$ 2. Real part of $z_1$ = Real part of $z_2$ 3. Imaginary part of $z_1$ = Imaginary part of $z_2$ Select the correct answer using the code given below:

1. A.1 only
2. B.2 only
3. C.3 only
4. D.None

If the point $z_1=1+i$ where $i=\sqrt{-1}$ is the reflection of a point $z_2=x+iy$ in the line $i\bar{z}-iz=5$, then the point $z_2$ is

1. A.$1+4i$
2. B.$4+i$
3. C.$1-i$
4. D.$-1-i$