Complex Numbers · NDA previous year questions

2025 8 questions

1. Q172025 · 14 Sept 2025

   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
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   Answer: A

2. Q182025 · 14 Sept 2025

   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
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   Answer: B

3. Q122025 · 13 Apr 2025

   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.$x={\omega }^2$
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   Answer: A

4. Q132025 · 13 Apr 2025

   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
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   Answer: A

5. Q142025 · 13 Apr 2025

   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
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   Answer: C

6. Q152025 · 13 Apr 2025

   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
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   Answer: C

7. Q162025 · 13 Apr 2025

   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.$2\pi$
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   Answer: A

8. Q172025 · 13 Apr 2025

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

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