Complex Numbers · NDA previous year questions

If $z=-\bar{z}$, then which one of the following is correct?

1. A.The real part of z is zero.
2. B.The imaginary part of z is zero.
3. C.The real part of z is equal to imaginary part of z.
4. D.The sum of real and imaginary parts of z is z.

If $\alpha$ and $\beta$ are the complex cube roots of unity, then what is the value of $(1+\alpha )(1+\beta )(1+{\alpha }^2)(1+{\beta }^2)$ ?

1. A.$-1$
2. B.0
3. C.1
4. D.4

If p, q, r are positive integers and $\omega$ is the cube root of unity and $f(x)=x^{3p}+x^{3q+1}+x^{3r+2}$, then what is $f(\omega )$ equal to?

1. A.$\omega$
2. B.$-{\omega }^2$
3. C.$-\omega$
4. D.0

What is the argument of $(1-\sin \theta )+i\cos \theta$? $(i=\sqrt{-1})$

1. A.$\frac{\pi }{2}-\frac{\theta }{2}$
2. B.$\frac{\pi }{2}+\frac{\theta }{2}$
3. C.$\frac{\pi }{4}-\frac{\theta }{2}$
4. D.$\frac{\pi }{4}+\frac{\theta }{2}$

If $\omega$ is the imaginary cube root of unity, then what is $(2-\omega +2{\omega }^2{)}^{27}$ equal to?

1. A.$3^{27}\omega$
2. B.$-3^{27}{\omega }^2$
3. C.$3^{27}$
4. D.$-3^{27}$

What is the modulus of $\frac{1}{1+3i}-\frac{1}{1-3i}$?

1. A.$\frac{3}{5}$
2. B.$\frac{9}{25}$
3. C.$\frac{3}{25}$
4. D.$\frac{5}{3}$

If $p,q,r$ are rational numbers, then the roots of the equation $x^2-2px+p^2-q^2+2qr-r^2=0$ are

1. A.complex
2. B.pure imaginary
3. C.irrational
4. D.rational

If $A=\{1,2,5,6\}$ and $B=\{1,2,3\}$, then what is $(A\times B)\cap (B\times A)$ equal to?

1. A.$\{(1,1),(2,1),(6,1),(3,2)\}$
2. B.$\{(1,1),(1,2),(2,1),(2,2)\}$
3. C.$\{(1,1),(2,2)\}$
4. D.$\{(1,1),(1,2),(2,5),(2,6)\}$

If $\omega$ is a complex cube root of unity, then what is ${\omega }^{10}+{\omega }^{-10}$ equal to?

1. A.2
2. B.-1
3. C.-2
4. D.1

1. A.$0$
2. B.$1$
3. C.$2$
4. D.$3$

If $\omega$ is a complex cube root of unity and $x={\omega }^2-\omega -2$, then what is the value of $x^2+4x+7$?

1. A.-2
2. B.-1
3. C.0
4. D.1

If $\cos x\ne -1$, then what is $\frac{\sin x}{1+\cos x}$ equal to?

1. A.$-\cot \frac{x}{2}$
2. B.$\cot \frac{x}{2}$
3. C.$\tan \frac{x}{2}$
4. D.$-\tan \frac{x}{2}$

What is the modulus of $\frac{1+2i}{1-(1-i{)}^2}$?

1. A.1
2. B.$\sqrt{5}$
3. C.$\sqrt{3}$
4. D.5

If the line through the points A (k, 1, -1) and B (2k, 0, 2) is perpendicular to the line through the points B and C (2 + 2k, k, 1), then what is the value of k?

1. A.-1
2. B.1
3. C.-3
4. D.3

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

1. A.$-1$
2. B.1
3. C.2
4. D.0

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

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

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.$3\omega -2i$

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

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

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

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

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

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.$\sqrt{3}/2$