Complex Numbers · NDA previous year questions

2022 6 questions

1. Q162022 · 4 Sept 2022

   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.$\frac{1+{\sin }^2\theta }{1-{\sin }^2\theta }$
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   Answer: A

2. Q172022 · 4 Sept 2022

   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.$2n\pi \text{ only}$
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   Answer: C

3. Q182022 · 4 Sept 2022

   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.$2n\pi$
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   Answer: B

4. Q462022 · 10 Apr 2022

   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.$\frac{3\pi }{4}$
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   Answer: B

5. Q472022 · 10 Apr 2022

   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.$2^{200}$
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   Answer: C

6. Q482022 · 10 Apr 2022

   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
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   Answer: D