Complex Numbers · NDA previous year questions

Consider the following for the next items that follow: Consider equation-I : $z^3+2z^2+2z+1=0$ and equation-II : $z^{1985}+z^{100}+1=0$. Which one of the following is a root of equation-II?

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

What is the number of common roots of equation-I and equation-II?

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

Consider the following for the next items that follow: A quadratic equation is given by $(a+b)x^2-(a+b+c)x+k=0$, where a, b, c are real. If $k=\frac{c}{2}$, $(c\ne 0)$, then the roots of the equation are :

1. A.Real and equal
2. B.Real and unequal
3. C.Real iff a > c
4. D.Complex but not real

If $\omega$ is a non-real cube root of 1, then what is the value of $\left|\frac{1-\omega }{\omega +{\omega }^2}\right|$ ?

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

What is the number of 6-digit numbers that can be formed only by using 0, 1, 2, 3, 4 and 5 (each once); and divisible by 6 ?

1. A.96
2. B.120
3. C.192
4. D.312

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

If z is a complex number such that $\frac{z-1}{z+1}$ is purely imaginary, then what is |z| equal to ?

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

How many real numbers satisfy the equation |x - 4| + |x - 7| = 15 ?

1. A.Only one
2. B.Only two
3. C.Only three
4. D.Infinitely many

A mapping $f:A\to B$ defined as $f(x)=\frac{2x+3}{3x+5},x\in A$. If f is to be onto, then what are A and B equal to ?

1. A.$A=R\setminus \{-\frac{5}{3}\}$ and $B=R\setminus \{-\frac{2}{3}\}$
2. B.$A=R$ and $B=R\setminus \{-\frac{5}{3}\}$
3. C.$A=R\setminus \{-\frac{3}{2}\}$ and $B=R\setminus \{0\}$
4. D.$A=R\setminus \{-\frac{5}{3}\}$ and $B=R\setminus \{\frac{2}{3}\}$

Let $z=\frac{1+i\sin \theta }{1-i\sin \theta }$ where $i=\sqrt{-1}$ What is the modulus of z?

1. A.1
2. B.$\sqrt{2}$
3. C.$1+{\sin }^2\theta$
4. D.$\frac{1+{\sin }^2\theta }{1-{\sin }^2\theta }$

Let $z=\frac{1+i\sin \theta }{1-i\sin \theta }$ where $i=\sqrt{-1}$ What is angle $\theta$ such that z is purely real ? where n is an integer

1. A.$\frac{n\pi }{2}$
2. B.$\frac{(2n+1)\pi }{2}$
3. C.$n\pi$
4. D.$2n\pi \text{ only}$

Let $z=\frac{1+i\sin \theta }{1-i\sin \theta }$ where $i=\sqrt{-1}$ What is angle $\theta$ such that z is purely imaginary ? where n is an integer

1. A.$\frac{n\pi }{2}$
2. B.$\frac{(2n+1)\pi }{2}$
3. C.$n\pi$
4. D.$2n\pi$

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

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

What is the modulus of ${\left(\frac{\sqrt{-3}}{2}-\frac{1}{2}\right)}^{200}$?

1. A.$\frac{1}{4}$
2. B.$\frac{1}{2}$
3. C.1
4. D.$2^{200}$

Consider the following statements: 1. $\frac{n!}{3!}$ is divisible by 6, where $n>3$ 2. $\frac{n!}{3!}+3$ is divisible by 7, where $n>3$ 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

Consider the following in respect of a complex number $z$: 1. $\overline{(z^{-1})}=(\bar{z}{)}^{-1}$ 2. $z{z}^{-1}=|z{|}^2$ Which of the above is/are correct?

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

1. The difference of Z and its conjugate is an imaginary number. 2. The sum of Z and its conjugate is a real number.

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

What is the modulus of the complex number $i^{2n+1}(-i{)}^{2n-1}$, where $n\in \mathbf{N}$ and $i=\sqrt{-1}$?

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

If $Z=1+i$, where $i=\sqrt{-1}$, then what is the modulus of $z+\frac{2}{z}$?

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

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

1. A.$240^{\circ }$
2. B.$210^{\circ }$
3. C.$120^{\circ }$
4. D.$60^{\circ }$

What is the modulus of the complex number $\frac{\cos \theta +i\sin \theta }{\cos \theta -i\sin \theta }$, where $i=\sqrt{-1}$ ?

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

If $A=\{x\in Z:x^3-1=0\}$ and $B=\{x\in Z:x^2+x+1=0\}$, where Z is set of complex numbers, then what is $A\cap B$ equal to?

1. A.Null set
2. B.$\left\{\frac{-1+\sqrt{3}i}{2},\frac{-1-\sqrt{3}i}{2}\right\}$
3. C.$\left\{\frac{-1+\sqrt{3}i}{4},\frac{-1-\sqrt{3}i}{4}\right\}$
4. D.$\left\{\frac{1+\sqrt{3}i}{2},\frac{1-\sqrt{3}i}{2}\right\}$

Consider the following for the next 02 (two) items: A complex number is given by $z=\frac{1+2i}{1-(1-i{)}^2}$. What is the modulus of z?

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

Consider the following for the next 02 (two) items: A complex number is given by $z=\frac{1+2i}{1-(1-i{)}^2}$. What is the principal argument of z?

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

What is the $n^{th}$ term of the sequence $25,-125,625,-3125,\dots$?

1. A.$(-5{)}^{2n-1}$
2. B.$(-1{)}^{2n}{5}^{n+1}$
3. C.$(-1{)}^{2n-1}{5}^{n+1}$
4. D.$(-1{)}^{n-1}{5}^{n+1}$