Complex Numbers · NDA previous year questions

2016 12 questions

1. Q52016 · 18 Sept 2016

   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 $i=\sqrt{-1}$
   Show answer

   Answer: B

2. Q62016 · 18 Sept 2016

   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
   Show answer

   Answer: D

3. Q72016 · 18 Sept 2016

   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.$|z|=1$
   Show answer

   Answer: D

4. Q82016 · 18 Sept 2016

   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
   Show answer

   Answer: A

5. Q92016 · 18 Sept 2016

   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
   Show answer

   Answer: C

6. Q102016 · 18 Sept 2016

   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
   Show answer

   Answer: C

7. Q42016 · 17 Apr 2016

   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.$1$
   Show answer

   Answer: C

8. Q52016 · 17 Apr 2016

   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
   Show answer

   Answer: C

9. Q1132016 · 17 Apr 2016

   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
   Show answer

   Answer: A

10. Q1142016 · 17 Apr 2016

    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
    Show answer

    Answer: D

11. Q1162016 · 17 Apr 2016

    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
    Show answer

    Answer: C

12. Q1172016 · 17 Apr 2016

    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
    Show answer

    Answer: B