Complex Numbers · NDA previous year questions

Let $Z_1$ and $Z_2$ be complex numbers such that $\frac{3Z_1}{4Z_2}$ is purely imaginary. What is $\left|\frac{Z_1+Z_2}{Z_1-Z_2}\right|$ equal to ?

1. A.2
2. B.$\frac{3}{2}$
3. C.$\frac{5}{4}$
4. D.1

Consider the following inequalities : I. $1+4i>3+2i$ II. $2+3i<3+4i$ III. $4+3i>3+4i$ where $i=\sqrt{-1}$ How many of the above are valid ?

1. A.None
2. B.One
3. C.Two
4. D.All the three

Consider the following statements : I. $\sqrt{x}+x+1=0$ has two irrational roots. II. $5\sqrt{x}-x-4=0$ has two rational roots. Which of the statements given above is/are correct ?

1. A.I only
2. B.II only
3. C.Both I and II
4. D.Neither I nor II

Let $1,\omega ,{\omega }^2$ be three cube roots of unity. If x = a + b $y=a\omega +b{\omega }^2,z=a{\omega }^2+b\omega$ then what is $x^2+y^2+z^2$ equal to?

1. A.6ab
2. B.3ab
3. C.$a^2+b^2$
4. D.1

How many 4-digit numbers that are divisible by 4 can be formed using the digits 1, 2, 3 and 4 (repetition of digits is not allowed)?

1. A.3
2. B.6
3. C.9
4. D.12

If $\omega$ is a non-real cube root of unity, then what is a root of the following equation? $$\left|\begin{array}{ccc}x+1& \omega & {\omega }^2\\ \omega & x+{\omega }^2& 1\\ {\omega }^2& 1& x+\omega \end{array}\right|=0$$

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

What is ${\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)}^3$ equal to?

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

If $x^2-x+1=0$, then what is $(x-\frac{1}{x}{)}^2+(x-\frac{1}{x}{)}^4+(x-\frac{1}{x}{)}^8$ equal to?

1. A.81
2. B.85
3. C.87
4. D.90

How many 7-letter words (with or without meaning) can be constructed using all the letters of the word CAPITAL so that all the consonants come together in each word?

1. A.360
2. B.300
3. C.288
4. D.240

If $z\ne 0$ is a complex number, then what is $\mathrm{amp}(z)+\mathrm{amp}(\bar{z})$ equal to?

1. A.0
2. B.$\pi /2$
3. C.$\pi$
4. D.$2\pi$

How many sides are there in a polygon that has 20 diagonals?

1. A.6
2. B.7
3. C.8
4. D.10

If $\omega \ne 1$ is a cube root of unity, then what is $(1+\omega -{\omega }^2{)}^{100}+(1-\omega +{\omega }^2{)}^{100}$ equal to?

1. A.$2^{100}{\omega }^2$
2. B.$2^{100}\omega$
3. C.$2^{100}$
4. D.$-2^{100}$

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

What is the value of the sum ${\sum }_{n=1}^{20}({\mathrm{i}}^{n-1}+{\mathrm{i}}^n+{\mathrm{i}}^{n+1})$ where $\mathrm{i}=\sqrt{-1}$ ?

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

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

Direction : Consider the following for the items that follow : Let $Z_1$ and $Z_2$ be any two complex numbers such that $Z_1^2+Z_2^2+Z_1{Z}_2=0$ what is the value of $\left|\frac{Z_1}{Z_2}\right|$,

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

Direction : Consider the following for the items that follow : Let $Z_1$ and $Z_2$ be any two complex numbers such that $Z_1^2+Z_2^2+Z_1{Z}_2=0$ what is the value of $\frac{1}{2}+\mathrm{Re}\left(\frac{Z_1}{Z_2}\right)$?

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

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 $(|z|+1{)}^2$ ?

1. A.1
2. B.4
3. C.81
4. D.121

What is the sum of all four-digit numbers formed by using all digits 0, 1, 4, 5 without repetition of digits?

1. A.44440
2. B.46460
3. C.46440
4. D.64440

If $x$, $y$ and $z$ are the cube roots of unity, then what is the value of $xy+yz+zx$?

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

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 $\mathrm{Re}\left(\frac{z_1}{z_2}\right)+1$ equal to?

1. A.$-1$
2. B.0
3. C.1
4. D.5

If $\omega \ne 1$ is a cube root of unity, then what are the solutions of $(z-100{)}^3+1000=0$?

1. A.$10(1-\omega ),10(10-{\omega }^2),100$
2. B.$10(10-\omega ),10(10-{\omega }^2),90$
3. C.$10(1-\omega ),10(10-{\omega }^2),1000$
4. D.$(1+\omega ),(10+{\omega }^2),-1$

What is $(1+i{)}^4+(1-i{)}^4$ equal to, where $i=\sqrt{-1}$?

1. A.4
2. B.0
3. C.-4
4. D.-8

If $a,b,$ and $c$ ($a>0,c>0$) are in GP, then consider the following in respect of the equation $a{x}^2+bx+c=0$: 1. The equation has imaginary roots. 2. The ratio of the roots of the equation is $1:\omega$ where $\omega$ is a cube root of unity. 3. The product of roots of the equation is $\left(\frac{b^2}{a^2}\right)$. Which of the statements given above are correct?

1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3

Consider the following for the next items that follow: Consider equation-I : $z^3+2z^2+2z+1=0$ and equation-II : $z^{1985}+z^{100}+1=0$. What are the roots of equation-I ?

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