Complex Numbers · NDA previous year questions

If $$z=\frac{-2(1+2i)}{3+i}$$ where $i=\sqrt{-1}$, then the argument $\theta (-\pi <\theta \le \pi )$ of $z$ is

1. A.$\frac{3\pi }{4}$
2. B.$\frac{\pi }{4}$
3. C.$\frac{5\pi }{6}$
4. D.$-\frac{3\pi }{4}$

If $1,\omega ,{\omega }^2$ are the cube roots of unity, then the value of $$(1+\omega )(1+{\omega }^2)(1+{\omega }^4)(1+{\omega }^8)$$ is

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

What is the square root of $i$, where $i=\sqrt{-1}$?

1. A.$\frac{1+i}{2}$
2. B.$\frac{1-i}{2}$
3. C.$\frac{1+i}{\sqrt{2}}$
4. D.None of the above

$(x^3-1)$ can be factorised as where $\omega$ is one of the cube roots of unity.

1. A.$(x-1)(x-\omega )(x+{\omega }^2)$
2. B.$(x-1)(x-\omega )(x-{\omega }^2)$
3. C.$(x-1)(x+\omega )(x+{\omega }^2)$
4. D.$(x-1)(x+\omega )(x-{\omega }^2)$

What is $${\left[\frac{\sin \frac{\pi }{6}+i(1-\cos \frac{\pi }{6})}{\sin \frac{\pi }{6}-i(1-\cos \frac{\pi }{6})}\right]}^3$$ where $i=\sqrt{-1}$, equal to?

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

What is the real part of $$(\sin x+i\cos x{)}^3$$ where $i=\sqrt{-1}$?

1. A.$-\cos 3x$
2. B.$-\sin 3x$
3. C.$\sin 3x$
4. D.$\cos 3x$

Consider the following statements in respect of the given equation : 1. All the roots of the equation are complex. 2. The sum of all the roots of the equation is 6. 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

Let z be a complex number such that $|z|=4$ and $\arg z=\frac{5\pi }{6}$. What is z equal to ? where $i=\sqrt{-1}$

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

If $P$ and $Q$ are two complex numbers, then the modulus of the quotient of $P$ and $Q$ is:

1. A.Greater than the quotient of their moduli
2. B.Less than the quotient of their moduli
3. C.Less than or equal to the quotient of their moduli
4. D.Equal to the quotient of their moduli

What is the argument of the complex number $\frac{(1+i)(2+i)}{3-i}$ where $i=\sqrt{-1}$?

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

The roots of the equation $2a^2{x}^2-2abx+b^2=0$ when $a<0$ and $b>0$ are:

1. A.Sometimes complex
2. B.Always irrational
3. C.Always complex
4. D.Always real

For the next three (03) items that follow: Given that $C(n,r):C(n,r+1)=1:2$ and $C(n,r+1):C(n,r+2)=2:3$. What is $n$ equal to?

1. A.11
2. B.12
3. C.13
4. D.14

For the next three (03) items that follow: Given that $C(n,r):C(n,r+1)=1:2$ and $C(n,r+1):C(n,r+2)=2:3$. What is $r$ equal to?

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

For the next three (03) items that follow: Given that $C(n,r):C(n,r+1)=1:2$ and $C(n,r+1):C(n,r+2)=2:3$. What is $P(n,r):C(n,r)$ equal to?

1. A.6
2. B.24
3. C.120
4. D.720

Out of 7 consonants and 4 vowels, words are to be formed by involving 3 consonants and 2 vowels. The number of such words formed is:

1. A.25200
2. B.22500
3. C.10080
4. D.5040

How many different words can be formed by taking four letters out of the letters of the word ‘AGAIN’ if each word has to start with A?

1. A.6
2. B.12
3. C.24
4. D.None of the above

The roots of the equation

1. A.are imaginary
2. B.are distinct and real
3. C.are equal and real
4. D.cannot be ascertained

What is ${\tan }^{-1}\left(\frac{1}{2}\right)+{\tan }^{-1}\left(\frac{1}{3}\right)$ equal to?

1. A.$\frac{\pi }{2}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{\pi }{4}$
4. D.$\frac{\pi }{6}$

What is $(1-{\sin }^2\theta )(1+{\tan }^2\theta )$ equal to?

1. A.${\sin }^2\theta$
2. B.${\cos }^2\theta$
3. C.${\tan }^2\theta$
4. D.1

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

1. A.$\frac{5\pi }{4}$
2. B.$-\frac{5\pi }{4}$
3. C.$\frac{3\pi }{4}$
4. D.None of the above

If $4^x-6\cdot 2^x+8=0$, then the values of x are

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

A bag contains balls of two colours, 3 black and 3 white. What is the smallest number of balls which must be drawn from the bag, without looking, so that among these there are two of the same colour ?

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

$(x+1{)}^2-1=0$ has

1. A.one real root
2. B.two real roots
3. C.two imaginary roots
4. D.four real roots

What is the modulus of $\frac{\sqrt{2}+i}{\sqrt{2}-i}$ where $i=\sqrt{-1}$?

1. A.3
2. B.1/2
3. C.1
4. D.None of the above

What is the number of diagonals which can be drawn by joining the angular points of a polygon of 100 sides?

1. A.4850
2. B.4950
3. C.5000
4. D.10000