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Question Bank No: 3

1. If $| \begin{matrix} 6 i & - 3 i & 1 \\ 4 & 3 i & - 1 \\ 20 & 3 & i \end{matrix} |$ = x + iy, then

a)		x = 3, y = 1
b)		x = 1, y = 3
c)		x = 0, y = 3
d)		x = y = 0

2. If $ω \neq 1$ is a cube root of unity, then ${(1 + ω - ω^{2})}^{7}$ =

a)		128 $ω$
b)		-128 $ω$
c)		128 $ω^{2}$
d)		-128 $ω^{2}$

3. $\sum_{n = 1}^{13} (i^{n} + i^{n + 1})$ =

a)		I
b)		i - 1
c)		-I
d)		0

4. If z $\neq$ 0 and arg z = $\frac{π}{4}$ , then

a)		Re $z^{2}$ = 0
b)		Im $z^{2}$ = 0
c)		Re $z^{2}$ = Im $z^{2}$
d)		none

5. If z $\neq$ 0 and Re z = 0, then

a)		Re $z^{2} = 0$
b)		Im $z^{2} = 0$
c)		Re $z^{2} = Im z^{2}$
d)		none

6. The complex numbers $\sin x + i \cos 2 x$ and $\cos x - i \sin 2 x$ are conjugate to each other, for x =

a)		0
b)		n $π$
c)		$(n + \frac{1}{2})$ $π$
d)		none

7. The value of $\sum_{r = 1}^{6} (\sin \frac{2 π r}{7} - i \cos \frac{2 π r}{7})$

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

8. If $z_{1}$ and $z_{2}$ are two non- zero complex numbers such that $| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$ , then arg $z_{1 -}$ arg $z_{2}$ =

a)		0
b)		$\frac{π}{2}$
c)		$- \frac{π}{2}$
d)		$π$

9. Let $z_{1}$ and $z_{2}$ be complex numbers such that $z_{1}$ $\neq z_{2}$ , $| z_{1} |$ = $| z_{2} |$ . If $z_{1}$ has positive real part and $z_{2}$ has negative imaginary part, then $\frac{z_{1} + z_{2}}{z_{1} - z_{2}}$ may be

a)		real and positive
b)		real and negative
c)		pure imaginary
d)		none of these

10. If $z_{1} = a + i b$ and $z_{2} = c + i d$ are complex numbers such that $| z_{1} | = | z_{2} | = 1$ and Re ( $z_{1} {\bar{z}}_{2}$ ) = 0 , then the complex numbers $w_{1} = a + i c$ and $w_{2} = b + i d$ satisfy

a)		$\| w_{1} \| = 0$
b)		$\| w_{2} \| = 1$
c)		Re ( $w_{1} {\bar{w}}_{2}$ ) = 1
d)		none of these

11. If z = x + iy and w = $\frac{1 - iz}{z - i}$ , then $| w |$ = 1 implies that

a)		z lies on the imaginary axis
b)		z lies onthe unit circle
c)		z lies on the real axis
d)		none of these

12. The points $z_{1}$ , $z_{2}$ , $z_{3}$ , $z_{4}$ in the complex plane are the vertices of a parallelogram taken in order if and only if

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)		none of these

13. The inequality $| z - 4 | < | z - 2 |$ represents the region given by

a)		Re z $>$ 0
b)		Re z $<$ 0
c)		Re z $>$ 2
d)		none of these

14. If z = ${(\frac{\sqrt{3}}{2} + \frac{i}{2})}^{5} + {(\frac{\sqrt{3}}{2} - \frac{i}{2})}^{5}$ , then

a)		Re z = 0
b)		Im z = 0
c)		Re z $>$ 0 , Im z $<$ 0
d)		Re z $>$ 0, Im z $<$ 0

15. The complex numbers z = x + iy which satisfy the equation $| \frac{z - 5 i}{z + 5 i} |$ = 1 lie on

a)		the x- axis
b)		straight line y = 5
c)		a circle through the origin
d)		none of these

16. If w is a complex cube root of unity, then the value of

16. $\frac{a + bw + {cw}^{2}}{c + aw + {cw}^{2}}$ $\begin{matrix} + \\ + \end{matrix}$ $\frac{a + bw + {cw}^{2}}{c + aw + {cw}^{2}}$ is

a)		1
b)		0
c)		– 1
d)		2

17. If x = a + b, y = aw + b $w^{2}$ and z = a $w^{2}$ + bw, then which one of the following is true

a)		x + y + z $\neq$ 0
b)		$x^{2}$ + $y^{2}$ + $z^{2}$ = $a^{2}$ + $b^{2}$
c)		$x^{3}$ + $y^{3}$ + $y^{3}$ = 3 ( $a^{3}$ + $b^{3}$ )
d)		xyz = 2 ( $a^{3}$ + $b^{3}$ )

18. The principal value of the amplitude of ( 1 + i ) is

a)		$\frac{π}{12}$ 
b)		$\frac{π}{4}$ 
c)		$π$ 
d)		$\frac{3 π}{4}$ 

19. Let x, y $ϵ$ R, then x + iy is a non-real complex number if

a)		x = 0
b)		y = 0
c)		x ≠ 0
d)		y ≠ 0

20. If n is any integer, the $i^{n}$ is

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

21. a + ib > c + id; a, b, c, d $ϵ$ R is meaningful only when

a)		a = 0, d = 0
b)		a = 0, c = 0
c)		b = 0, c = 0
d)		b = 0, d = 0

22. If (x + iy) (p + iq) = ( $x^{2}$ + $y^{2}$ ) i, then

a)		p = x, q = y
b)		p = y, q = x
c)		p = $x^{2}$ , q = $y^{2}$
d)		none of these

23. If b + ic = (1 + a) z and $a^{2}$ + $b^{2}$ + $c^{2}$ = 1, then $\frac{1 + iz}{1 - iz}$ is equal to

a)		$\frac{a - ib}{1 + c}$
b)		$\frac{a - ib}{1 + c}$
c)		$\frac{a + ib}{1 - c}$
d)		$\frac{a + ib}{1 + c}$

24. Let a and b be two distinct complex numbers such that │a │= │b │. If real part of a is positive and imaginary part of b is negative, then the complex number $\frac{a + b}{a - b}$ may be

a)		Zero
b)		purely imaginary
c)		real and +ve
d)		real and –ve

25. If z1 and z2 are two non-zero complex numbers such that │ $z_{1}$ + $z_{2}$ │= │ $z_{1}$ │+ │ $z_{2}$ │, then Arg $z_{1}$ – Arg $z_{2}$ is equal to

a)		- $π$ 
b)		- $\frac{π}{2}$ 
c)		0
d)		$\frac{π}{2}$

26. The points $z_{1}$ , $z_{2}$ , $z_{3}$ , $z_{4}$ in complex plane are the vertices of a parallelogram taken in order iff

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)		$z_{1}$ – $z_{2}$ = $z_{3}$ – $z_{4}$

27. If the cube roots of unity are 1, w, $w^{2}$ , then 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 $w^{2}$

28. Let $z_{1}$ and $z_{2}$ be two complex numbers such that $z_{1}$ ≠ $z_{2}$ and │ $z_{1}$ │≠│ $z_{2}$ │. If $z_{1}$ has a positive real part and $z_{2}$ has negative imaginary part, then $z_{1}$ + $z_{2}$ may be

28. $z_{1}$ – $z_{2}$

a)		zero or purely imaginary
b)		real and positive
c)		real and negative
d)		none of these

29. The complex number z = x + iy which satisfy the equation │z│+ 1│= 1 lie on

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

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

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

31. If z is a complex number then │z + 1│= $\sqrt{3}$ │z - 1│ represents

a)		St. line
b)		Ellipse
c)		Hyperbola
d)		circle

32. If w and $w^{2}$ are complex cube roots of unity, the ${(1 - w + w^{2})}^{5}$  + ${(1 - w^{2} + w)}^{5}$ is equal to

a)		8
b)		16
c)		32
d)		64

33. 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 the origin
d)		none of these

34. If a,b are non real cube roots of unity, then ab + $a^{5}$  + $b^{5}$ is equal to

a)		1
b)		0
c)		-1
d)		3

35. If $p^{2}$ - p + 1 = 0, then the value of $p^{3 n}$ is

a)		± 1
b)		1
c)		– 1
d)		0

36. Which of the following is correct

a)		2 + 3i > 1 + 4i
b)		6 + 2i > 3 + 3i
c)		2 + 8i > 5 + 7i
d)		none of these

37. If z is any complex number such that │z + 4│ $\leq$ 3, then least value of │z +1│is

a)		– 6
b)		0
c)		3
d)		2

38. Given that │z│= 4 and arg z = 5 $π$ , then z =

a)		-2 $\sqrt{3}$ + 2i
b)		2 $\sqrt{3}$ + 2i
c)		2 $\sqrt{3}$ – 2i
d)		- $\sqrt{3}$ + I

39. If the point represented by the complex number $z_{1}$ = a + ib, $z_{2}$ = a' + ib' and $z_{1}$ – $z_{2}$ are collinear, then

a)		ab' + a'b = 0
b)		ab' – a'b = 0
c)		ab + a'b' = 0
d)		ab – a'b' = 0

40. If n1, n2 are positive integers, then ${(1 + i)}^{n 1}$ + ${(1 + i^{3})}^{n}$ + ${(1 + i^{5})}^{n 2}$ + ${(1 + i^{7})}^{n 2}$ is a real number iff

a)		n1 = n2 +1
b)		n1 + 1 = n2
c)		n1 = n2
d)		n1, n2 are any two +ve integers

41. If │z1│=│z2│=│z3│=│z4│, then the points representing $z_{1}$ , $z_{2}$ , $z_{3}$ , $z_{4}$ are

a)		concyclic
b)		vertices of a square
c)		vertices of a rhombus
d)		none

42. Which of the following is not applicable for a complex number?

a)		Addition
b)		Subtraction
c)		division
d)		inequality

43. If z = -1, then the principal value of the arg ( $z^{\frac{2}{3}}$ ) is equal to

a)		$\frac{π}{3}$
b)		2 $π / 3$ or 2 $π$
c)		10 $π / 3$
d)		$π$ 

44. If w is a complex root of unity, then

a)		$w^{4}$ = 1
b)		$w^{14}$ = $w^{2}$
c)		w6 = w
d)		$w^{5}$ = 1

45. If a complex number lies in the IIIrd quadrant then its conjugate lies in quadrant number

a)		I
b)		II
c)		III
d)		IV