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Question Bank No: 2

1. The complex number Z which satisfy $| Z |$ <2 are

a)		on the x axis
b)		inside the circle with radius 2 and centre at origin
c)		on the circle with radius 2 and centre at origin
d)		none of these

2. The number of non zero integral solution of the equation ${| 1 - i |}^{x} = 2^{x}$ is

a)		infinite
b)		1
c)		2
d)		none of these

3. If Z is a complex number then $| Z + 1 |$ = $\sqrt{3}$ $| Z - 1 |$ represent

a)		straight line
b)		Ellipse
c)		Hyperbola
d)		Circle

4. If Z is a complex number then $| 3 Z - 1 |$ =3 $| Z - 2 |$ represents

a)		straight line
b)		Ellipse
c)		Hyperbola
d)		Circle

5. The number $\frac{{(1 + i)}^{n}}{{(1 + i)}^{n - 2}}$ is equal to

a)		${4 i}^{n - 2}$
b)		${2 i}^{n - 4}$
c)		${2 i}^{n - 1}$
d)		none of these

6. ${(1 + i)}^{2 n} + {(1 - i)}^{2 n}$ , n $\in N is$

a)		purely imaginary
b)		a purely real number
c)		a non real complex number
d)		a complex number in which real and imaginary parts are equal

7. The complex number Z= x+iy which satisfy the equation $| Z + 1 |$ lies on

a)		x- axis
b)		a circle with (-1,0) as center and radius 1
c)		y- axis
d)		none of these

8. The smallest positive integer n for which

8. ${(\frac{1 + i}{1 - i})}^{n} = 1$ is

a)		n=8
b)		n=12
c)		n=16
d)		none of these

9. The cube root of unity are 1, $ω, ω^{2}$ then the roots of the equation ${(x - 1)}^{3} = 8 = 0 are$

a)		-1,-1,-1
b)		-1,1+2 $ω, 1 + 2 ω^{2}$
c)		-1,1-2 $ω, 1 - 2 ω^{2}$
d)		1,1+2 $ω, 1 + 2 ω^{2}$

10. The points $Z_{1}$ , $Z_{2}, Z_{3}$ , $Z_{4}$ in a complex plane are the vertices of a parallelogram taken in order, then

a)		$Z_{1}$ $+ Z_{4} = Z_{2}$ $+ Z_{3}$
b)		$Z_{1}$ $+ Z_{3} = Z_{2}$ $+ Z_{4}$
c)		$Z_{1}$ $+ Z_{2} = Z_{3}$ $+ Z_{4}$
d)		$Z_{1}$ $- Z_{4} = Z_{3}$ $- Z_{4}$

11. If n is any positive integer, thewn the value of $\frac{i^{4 n = 1} - i^{4 n - 1}}{2}$ is equals

a)		1
b)		-1
c)		I
d)		-I

12. If $α$ is a complex number such that $α^{2} + α + 1 = 0$ then $α^{31}$ is

a)		$α$
b)		$α^{2}$
c)		0
d)		1

13. If n = 4m+3, m is an intergral, then $i^{n}$ is equal to

a)		I
b)		-I
c)		-1
d)		1

14. Arg Z +Arg $\bar{Z}$ (Z $\neq 0)$ is

a)		0
b)		$π$
c)		$\frac{π}{2}$
d)		none of these

15. If n is an integral then $i^{n}$ is

a)		I
b)		1,-1
c)		i,-I
d)		1,-1,i,-I

16. If $x_{n}$ =cos $(\frac{π}{2^{n}})$ +i sin $(\frac{π}{2^{n}})$ , then $x_{1}, x_{2}$ , $x_{3,}$ ------ $\propto is$

a)		-I
b)		-1
c)		I
d)		1

17. ${(1 + i)}^{6} + {(1 - i)}^{6}$ is

a)		0
b)		$2^{7}$
c)		$2^{6}$
d)		None of these

18. The conjugate of $\frac{1}{2 + i}$ is

a)		$\frac{2 + i}{5}$
b)		$\frac{2 - i}{5}$
c)		$\frac{5}{2 - i}$
d)		$\frac{5}{2 + i}$

19. The principal value of the argument of 1+i is

a)		$\frac{π}{2}$
b)		$\frac{π}{3}$
c)		$\frac{π}{4}$
d)		none of these

20. If $ω$ is a complex cube root of unity, then the value $\frac{a + b ω + c ω^{2}}{c + a ω + b ω^{2}}$ + $\frac{a + b ω + c ω^{2}}{b + c ω + a ω^{2}}$ ?

a)		1
b)		0
c)		-1
d)		2

21. If $| z + 4 | \leq$ 3, then the maximum value of $| z + 1 |$ is

a)		10
b)		6
c)		0
d)		4

22. If $| z |$ = 1 and z $\neq \pm 1$ , then all the values of $\frac{z}{1 - z^{2}}$ lie on

a)		a line not passing through the origin
b)		$\| z \| = \sqrt{2}$
c)		the x- axis
d)		the y- axis

23. A man walks a distance of 3 units from the origin towards the north- east (N $45^{\circ}$ E) direction. From there he walks a distance of 4 units towards the north- west (N $45^{\circ}$ W) direction to reach a point P. Then the position of P in the Argand plane is

a)		3 $e^{\frac{i π}{4}} + 4 i$
b)		$(3 - 4 i) e^{\frac{i π}{4}}$
c)		$(4 + 3 i) e^{\frac{i π}{4}}$
d)		$(3 + 4 i) e^{\frac{i π}{4}}$

24. If $z^{2} + z + 1$ = 0 , then $\sum_{r = 1}^{6} {(z^{r} + \frac{1}{z^{r}})}^{2} =$

a)		6
b)		12
c)		18
d)		24

25. The value of $\sum_{k = 1}^{10} (\sin \frac{2 k π}{11} + i \cos \frac{2 k π}{11})$ is

a)		-1
b)		-I
c)		I
d)		1

26. If w = $α + i β$ , where $β \neq 0$ and $z \neq 1$ such that $\frac{w - \bar{w} z}{1 - z}$ is real, then

a)		$\| z \| = 1$
b)		$z = \bar{z}$
c)		$z = - \bar{z}$
d)		$z = 0$

27. If x is a complex root of the equation $| \begin{matrix} 1 & x & x \\ x & 1 & x \\ x & x & 1 \end{matrix} | +$ $| \begin{matrix} 1 - x & 1 & 1 \\ 1 & 1 - x & 1 \\ 1 & 1 & 1 - x \end{matrix} |$ = 0 , then $x^{2005} + \frac{1}{x^{2005}} =$

a)		1
b)		-1
c)		I
d)		w

28. If w = $\frac{z}{z - \frac{i}{3}}$ and $| w |$ = 1, then z lies on

a)		a circle
b)		an ellipse
c)		a parabola
d)		a straight line

29. If the cube roots of unity are 1, w, $w^{2}$ , then the roots of the equation ${(x - 1)}^{3} + 8 = 0$ are

a)		-1 , -1, -1
b)		-1, -1+2w, -1 - 2 $w^{2}$
c)		-1, 1+2w, 1+ 2 $w^{2}$
d)		-1, 1 - 2w, 1 - 2 $w^{2}$

30. If $| z^{2} - 1 | = {| z |}^{2} + 1$ , then z lies on

a)		the real axis
b)		an ellipse
c)		a circle
d)		the imaginary axis

31. If $z = x - iy$ and $z^{\frac{1}{3}} = p + iq$ , then $(\frac{x}{p} + \frac{y}{q}) \div (p^{2} + q^{2}) =$

a)		1
b)		-2
c)		2
d)		-1

32. Let z, w be complex numbers such that $\bar{z} + i \bar{w} = 0$ and $\arg (zw)$ = $π$ . Then $\arg z$ =

a)		$\frac{π}{4}$
b)		$\frac{5 π}{4}$
c)		$\frac{3 π}{4}$
d)		$\frac{π}{2}$

33. If 1, w, $w^{2}$ are the cube roots of unity, then $| \begin{matrix} 1 & w^{n} & w^{2 n} \\ w^{n} & w^{2 n} & 1 \\ w^{2 n} & 1 & w^{n} \end{matrix} | =$

a)		0
b)		1
c)		w
d)		$w^{2}$

34. If ${(\frac{1 + i}{1 - i})}^{x} = 1$ , then $x$ =

a)		4n
b)		2n
c)		4n + 1
d)		2n + 1

35. If z and w are non -zerocomplex numbers such that $| zw | = 1$ and $\arg z - \arg w$ = $\frac{π}{2}$ , then $\bar{z} w$ is

a)		1
b)		-1
c)		I
d)		-I

36. Let $z_{1}$ and $z_{2}$ be the roots of $z^{2} + az + b = 0$ . If the origin, $z_{1}$ and $z_{2}$ form an equilateral triangle, then

a)		$a^{2} = b$
b)		$a^{2} = 2 b$
c)		$a^{2} = 3 b$
d)		$a^{2} = 4 b$

37. The locus of the centre of a circle which touches the circle $| z - z_{1} | = a$ and $| z - z_{2} | = b$ externally is

a)		an ellipse
b)		a hyperbola
c)		circle
d)		none

38. z and w are two non- zero complex numbers such that $| z |$ = $| w |$ and arg z +arg w = $π$ , then z =

a)		$\bar{w}$
b)		$- \bar{w}$
c)		w
d)		-w

39. The locus of z which lies in shaded region is represented by

a)		$z : \| z + 1 \| > 2$ , $\| \arg (z + 1) \|$ $< \frac{π}{4}$
b)		$z : \| z - 1 \| > 2$ , $\| \arg (z - 1) \|$ $< \frac{π}{4}$
c)		$z : \| z + 1 \|$ $< 2$ , $\| \arg (z + 1) \|$ $< \frac{π}{2}$
d)		$z : \| z - 1 \|$ $< 2$ , $\| \arg (z - 1) \|$ $< \frac{π}{2}$

40. If a, b, c are integers not all equal and $ω$ is a cube root of unity ( $ω \neq 1$ ), then the minimum value of $| a + b ω + c ω^{2} |$ is

a)		0
b)		1
c)		$\frac{\sqrt{3}}{2}$
d)		$\frac{1}{2}$

41. If $ω$ ( $\neq 1$ ) be a cube root of unity and ${(1 + ω^{2})}^{n} = {(1 + ω^{4})}^{n}$ , the least positive value of n is

a)		2
b)		3
c)		5
d)		6

42. If z is a complex number such that $| z | = 1$ , $z$ $\neq 1$ , then the real part of $\frac{z - 1}{z + 1}$ is

a)		$\frac{1}{{\| z + 1 \|}^{2}}$
b)		$\frac{- 1}{{\| z + 1 \|}^{2}}$
c)		$\frac{\sqrt{2}}{{\| z + 1 \|}^{2}}$
d)		0

43. The value of the determinant $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$ is

a)		3 $ω$
b)		3 $ω (ω - 1)$
c)		3 $ω^{2}$
d)		3 $ω (1 - ω)$

44. For all complex numbers $z_{1}$ , $z_{2}$ satisfying $| z_{1} | = 12$ and $| z_{2} - 3 - 4 i |$ = 5, the minimum value of $| z_{1} - z_{2} |$ is

a)		0
b)		2
c)		7
d)		17

45. If a, b, c and u, v, w are complex numbers representing the vertices of two triangles such that $c = (1 - r) a + r b$ , $w = (1 - r) u + r v$ , where r is a complex number, then the two triangles

a)		have the same area
b)		are similar
c)		are congruent
d)		none of these

46. The complex numbers $z_{1}$ , $z_{2}$ , $z_{3}$ satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}$ are the vertices of a triangle which is

a)		of area zero
b)		right- angled isosceles
c)		equilateral
d)		obtuse angled isosceles

47. Let $z_{1}$ and $z_{2}$ be nth roots of unity which subtend a right angle at the origin. Then n must be the form

a)		4k + 1
b)		4k + 2
c)		4k + 3
d)		4k

48. If $z_{1}$ , $z_{2}$ , $z_{3}$ are complex numbers such that $| z_{1} | = | z_{2} | = | z_{3} | = | \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} | = 1$ , then $| z_{1} + z_{2} + z_{3} |$ is

a)		1
b)		$< 1$
c)		$> 3$
d)		3

49. For positive integers $n_{1}$ and $n_{2}$ the value of the expression ${(1 + i)}^{n_{1}} + {(1 + i^{3})}^{n_{1}} + {(1 + i^{5})}^{n_{2}} + {(1 + i^{7})}^{n_{2}}$ is a real number if and only if

a)		$n_{1} = n_{2} + 1$
b)		$n_{1} = n_{2} - 1$
c)		$n_{1} = n_{2}$
d)		$n_{1} > 0$ , $n_{2} > 0$

50. 4 + 5 ${(- \frac{1}{2} + \frac{i \sqrt{3}}{2})}^{334} + 3 {(- \frac{1}{2} + \frac{i \sqrt{3}}{2})}^{365}$ =

a)		1- $i \sqrt{3}$
b)		-1 + $i \sqrt{3}$
c)		$i \sqrt{3}$
d)		$- i \sqrt{3}$