Complex Numbers · NDA previous year questions

2009 13 questions

1. Q52009 · 30 Aug 2009

   If $\alpha$ is a complex number such that ${\alpha }^2+\alpha +1=0$, then what is ${\alpha }^{31}$ equal to ?

   1. A.$\alpha$
   2. B.${\alpha }^2$
   3. C.0
   4. D.1
   Show answer

   Answer: A

2. Q182009 · 30 Aug 2009
   
   1. A.$-1$
   2. B.1
   3. C.2
   4. D.0
   Show answer

   Answer: D

3. Q312009 · 30 Aug 2009

   The roots of the equation $(x-p)(x-q)=r^2$, where $p,q,r$ are real are

   1. A.always complex
   2. B.always real
   3. C.always purely imaginary
   4. D.None of the above
   Show answer

   Answer: B

4. Q322009 · 30 Aug 2009

   If $(\sin x+\mathrm{cosec}x{)}^2+(\cos x+\sec x{)}^2=k+{\tan }^{2}x+{\cot }^{2}x$, then what is the value of $k$ ?

   1. A.8
   2. B.7
   3. C.4
   4. D.3
   Show answer

   Answer: B

5. Q752009 · 30 Aug 2009

   If $\omega$ is the cube root of unity, then what is the conjugate of $2{\omega }^2+3i$ ?

   1. A.$2\omega -3i$
   2. B.$3\omega +2i$
   3. C.$2\omega +3i$
   4. D.$3\omega -2i$
   Show answer

   Answer: A

6. Q762009 · 30 Aug 2009

   Consider the following statements : 1. If $\theta =1200^{\circ }$, then $(\sec \theta +\tan \theta {)}^{-1}$ is positive. 2. If $\theta =1200^{\circ }$, then $(\mathrm{cosec}\theta -\cot \theta )$ is negative. Which of the statements given above 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: D

7. Q962009 · 30 Aug 2009

   If $a,b$ and $c$ are real numbers then the roots of the equation $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ are always

   1. A.real
   2. B.imaginary
   3. C.positive
   4. D.negative
   Show answer

   Answer: A

8. Q972009 · 30 Aug 2009

   If $({\log }_{x}x)({\log }_{3}2x)({\log }_{2x}y)={\log }_x{x}^2$, then what is the value of $y$ ?

   1. A.$9/2$
   2. B.9
   3. C.18
   4. D.27
   Show answer

   Answer: B

9. Q982009 · 30 Aug 2009

   If ${\log }_{k}x{\log }_{5}k=3$ then what is $x$ equal to ?

   1. A.$k^5$
   2. B.$5k^3$
   3. C.243
   4. D.125
   Show answer

   Answer: D

10. Q992009 · 30 Aug 2009

    Which term of the sequence $20,19\frac{1}{4},18\frac{1}{2},17\frac{3}{4},\dots$ is the first negative term ?

    1. A.$27^{\text{th}}$
    2. B.$28^{\text{th}}$
    3. C.$29^{\text{th}}$
    4. D.No such term exists
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    Answer: B

11. Q1002009 · 30 Aug 2009

    If ${\sin }^{-1}x+{\cot }^{-1}(1/2)=\pi /2$, then what is the value of $x$ ?

    1. A.0
    2. B.$1/\sqrt{5}$
    3. C.$2/\sqrt{5}$
    4. D.$\sqrt{3}/2$
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    Answer: B

12. Q1012009 · 30 Aug 2009

    For the two equations $x^2+mx+1=0$ and $x^2+x+m=0$, what is/are the value/values of $m$ for which these equations have at least one common root ?

    1. A.$-2$ only
    2. B.1 only
    3. C.$-2$ and 1
    4. D.$-2$ and $-1$
    Show answer

    Answer: C

13. Q1092009 · 30 Aug 2009

    If $z$ is a complex number such that $z+z^{-1}=1$, then what is the value of $z^{99}+z^{-99}$ ?

    1. A.1
    2. B.$-1$
    3. C.2
    4. D.$-2$
    Show answer

    Answer: D