Complex Numbers · NDA previous year questions

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2024 13 questions

Q42024 · 1 Sept 2024
If $ω \neq = 1$ is a cube root of unity, then what is $(1 + ω - ω^{2})^{100} + (1 - ω + ω^{2})^{100}$ equal to?
1. A. $2^{100} ω^{2}$
2. B. $2^{100} ω$
3. C. $2^{100}$
4. D. $- 2^{100}$
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Answer: D
Q92024 · 1 Sept 2024
1. A.1
2. B. $2$
3. C.2
4. D. $3$
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Answer: B
Q102024 · 1 Sept 2024
What is the value of the sum $\sum_{n = 1}^{20} (i^{n - 1} + i^{n} + i^{n + 1})$ where $i = - 1$ ?
1. A.-2i
2. B.0
3. C.1
4. D.2i
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Answer: B
Q122024 · 1 Sept 2024
1. A.0
2. B. $ω$
3. C. $ω^{2}$
4. D. $1 - ω^{2}$
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Answer: A
Q412024 · 1 Sept 2024
Direction : Consider the following for the items that follow : Let $Z_{1}$ and $Z_{2}$ be any two complex numbers such that $Z_{1}^{2} + Z_{2}^{2} + Z_{1} Z_{2} = 0$ what is the value of $\frac{Z _{1}}{Z _{2}}$ ,
1. A.1
2. B.2
3. C.3
4. D.4
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Answer: A
Q422024 · 1 Sept 2024
Direction : Consider the following for the items that follow : Let $Z_{1}$ and $Z_{2}$ be any two complex numbers such that $Z_{1}^{2} + Z_{2}^{2} + Z_{1} Z_{2} = 0$ what is the value of $\frac{1}{2} + Re (\frac{Z _{1}}{Z _{2}})$ ?
1. A.-1
2. B.0
3. C.1
4. D.2
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Answer: B
Q22024 · 21 Apr 2024
If $z$ is any complex number and $i z^{3} + z^{2} - z + i = 0$ , where $i = - 1$ , then what is the value of $(∣ z ∣ + 1)^{2}$ ?
1. A.1
2. B.4
3. C.81
4. D.121
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Answer: B
Q32024 · 21 Apr 2024
What is the sum of all four-digit numbers formed by using all digits 0, 1, 4, 5 without repetition of digits?
1. A.44440
2. B.46460
3. C.46440
4. D.64440
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Answer: D
Q42024 · 21 Apr 2024
If $x$ , $y$ and $z$ are the cube roots of unity, then what is the value of $x y + y z + z x$ ?
1. A.0
2. B.1
3. C.2
4. D.3
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Answer: A
Q92024 · 21 Apr 2024
Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\frac{z _{1} + z _{2}}{z _{1} - z _{2}} = 1$ , then what is $Re (\frac{z _{1}}{z _{2}}) + 1$ equal to?
1. A. $- 1$
2. B.0
3. C.1
4. D.5
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Answer: C
Q142024 · 21 Apr 2024
If $ω \neq = 1$ is a cube root of unity, then what are the solutions of $(z - 100)^{3} + 1000 = 0$ ?
1. A. $10 (1 - ω), 10 (10 - ω^{2}), 100$
2. B. $10 (10 - ω), 10 (10 - ω^{2}), 90$
3. C. $10 (1 - ω), 10 (10 - ω^{2}), 1000$
4. D. $(1 + ω), (10 + ω^{2}), - 1$
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Answer: B
Q152024 · 21 Apr 2024
What is $(1 + i)^{4} + (1 - i)^{4}$ equal to, where $i = - 1$ ?
1. A.4
2. B.0
3. C.-4
4. D.-8
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Answer: D
Q322024 · 21 Apr 2024
If $a, b,$ and $c$ ( $a > 0, c > 0$ ) are in GP, then consider the following in respect of the equation $a x^{2} + b x + c = 0$ : 1. The equation has imaginary roots. 2. The ratio of the roots of the equation is $1 : ω$ where $ω$ is a cube root of unity. 3. The product of roots of the equation is $(\frac{b ^{2}}{a ^{2}})$ . Which of the statements given above are correct?
1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3
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Answer: D