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Question Bank No: 1

1. The complex number z satisfying the equation $| z - 1 |$ = $| z - 3 |$ = $| z - i |$ is

a)		2 + i
b)		$\frac{3}{2} + \frac{1}{2} i$
c)		2 + 2i
d)		None of these

2. The cube roots of unity

a)		are collinear
b)		lie on a circle of radius $\sqrt{3}$
c)		form an equilateral triangle
d)		None of these

3. If $z_{1}$ , $z_{2}$ are two complex numbers such that Im $(z_{1} + z_{2})$ = 0, Im( $z_{1} z_{2}$ ) = 0, then

a)		$z_{1} = z_{2}$
b)		$z_{1} {= - z}_{2}$
c)		$z_{1} = \bar{z_{2}}$
d)		None of these

4. If z = ( $α$ + 3) + i $\sqrt{5 - α^{2}}$ , then the locus of z is

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

5. If x < 0, then arg (x) is equal to

a)		0
b)		$π$ /2
c)		$π$
d)		None of these

6. The curve represented by $| z |$ = Re(z) + 2 is

a)		A straight line
b)		a circle
c)		an ellipse
d)		None of these

7. If the complex number z lies on the boundary of the circle of radius 3 and centre at -4, then the greatest value of $| z + 1 |$ is

a)		4
b)		5
c)		6
d)		9

8. The complex numbers z = x + iy which satisfies the equation $| z + 1 |$ =1 lies on

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

9. If (2 + i) (2 + 2i) (2 + 3i) .... (2 + ni) = x + iy, then 5.8.13 ........ (4 + $n^{2}$ ) is equal to

a)		$x^{2}$ - $y^{2}$
b)		$x^{2}$ + $y^{2}$
c)		$x^{4}$ - $y^{4}$
d)		$x^{4}$ + $y^{4}$

10. If the amplitude of z - 2 - 3i = $\frac{π}{4}$ , then the locus of z is

a)		x - y = 2
b)		x + y = 0
c)		x + y = 1
d)		x - y + 1 = 0

11. If $α$ , $β$ are non-real cube roots of unity, then $α$ $β$ + $α^{2}$ + $β^{2}$ is equal to

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

12. If $a^{2}$ + $b^{2}$ = 1, then $\frac{1 + a + ib}{1 + a - ib}$ is equal to

a)		a + ib
b)		a - ib
c)		b + ia
d)		b - ia

13. 1 + $i^{2}$ + $i^{4}$ + ...... + $i^{2 n}$ is

a)		positive
b)		negative
c)		zero
d)		dependent on n

14. A square root of 3 + 4i is

a)		$\sqrt{3}$ + i
b)		2 - i
c)		2 + I
d)		None of these

15. If Re $(\frac{z - 8 i}{z + 6})$ = 0, then z lies on the curve

a)		$x^{2}$ + $y^{2}$ + 6x - 8y = 0
b)		4x - 3y + 24 = 0
c)		$x^{2}$ + $y^{2}$ - 8 = 0
d)		None of these

16. Let z be a purely imaginary number such that Im(z) > 0. Then arg(z) is equal to

a)		$π$
b)		$π$ /2
c)		0
d)		– $π$ /2

17. If z ( $\neq$ - 1) is a complex number such that $\frac{z - 1}{z + 1}$ is purely imaginary, then $| z |$ is equal to

a)		1
b)		2
c)		3
d)		5

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

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

19. The square root of 5 + 12i is

a)		3 + 2i
b)		3 - 2i
c)		$\pm$ (3 + 2i)
d)		None of these

20. ${(\frac{2 i}{1 + i})}^{2}$ is equal to

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

21. If the lines x+2ay+a=0, x+3by+b=0 and x+4cy+c=0 are concurrent, then a, b, c are in

a)		AP
b)		GP
c)		HP
d)		none of these

22. The medians of a triangle meet at (0,-3) and two vertices are at (-1,4) and (5,2). Then the third vertex is at

a)		(4,15)
b)		(-4,-15)
c)		(-4,15)
d)		(4,-15)

23. The equation of the straight line which is $⊥$ r to y = x and passes through (3,2) is

a)		x-y=5
b)		x+y=5
c)		x+y=1
d)		x-y=1

24. The inclination of the straight line passes through the point (-3,6) and mid point of the line joining the point (4,-5) and (-2,9) is

a)		$\frac{π}{2}$
b)		$\frac{π}{6}$
c)		$\frac{π}{3}$
d)		$\frac{3 π}{4}$

25. A triangle with vertices (4,0) (-1,-1) (3,5) is

a)		isosceles and right angled
b)		isosceles and not right angled
c)		right angled but not isosceles
d)		neither right angled nor isosceles

26. The points (-2,-5), (2,-2), (8,a) are collinear, then the value of a

a)		- $\frac{5}{2}$
b)		$\frac{5}{2}$
c)		$\frac{3}{2}$
d)		$\frac{1}{2}$

27. If 2p is the length of the $⊥$ r from the origin to the line $\frac{x}{a} + \frac{y}{b}$ =1, then

a)		$a^{2}, 8 p^{2}, b^{2}$ are in AP
b)		$a, 8 p^{2}, b^{2}$ are in GP
c)		$a^{2}, 8 p^{2}, b^{2}$ are in HP
d)		None of these

28. The acute angle between the lines ax+by=c=0 and (a+b)x = (a-b) y, a $\neq b$ is

a)		$15^{\circ}$
b)		$30^{\circ}$
c)		$45^{\circ}$
d)		$60^{\circ}$

29. The angles of the triangle formed by the lines x+y=0, x-y=0 and x=7 are

a)		$30^{\circ}, 60^{\circ}, 90^{\circ}$
b)		$60^{\circ}, 60^{\circ}, 60^{\circ}$
c)		$45^{\circ}, 45^{\circ}, 90^{\circ}$
d)		none of these

30. If the distance between the straight lines y = mx+ $C_{1}$ and y = mx+ $C_{2}$ is $| C_{1} - C_{2} |$ then

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

31. Distance between two parallel lines 2x+3y-2=0 and 2x+3y-4 is

a)		$\frac{1}{\sqrt{13}}$
b)		$\sqrt{13}$
c)		$\frac{2}{\sqrt{13}}$
d)		$\frac{3}{\sqrt{13}}$

32. The circumcentre of the triangle formed by the vertices of the triangle (3,7), (5,7) and (3,-2) is

a)		(3,7)
b)		(3,-2)
c)		(5,7)
d)		(4, $\frac{5}{2}$ )

33. If the centroid and circumcentre of a triangle are (3,3) (6,2) then the ortho centre is

a)		(9,5)
b)		(3,-1)
c)		(-3,1)
d)		(-3,5)

34. If the ortho centre and centroid of a traingle are (-3,5), (3,3) then the circumcentre is

a)		(6,2)
b)		(0,8)
c)		(6,-2)
d)		(0,4)

35. The mid points of AB, BC, CA of a $Δ ABC$ are (6,-1), (-4,-3), (2,-5) respectively. Centroid of the $Δ ABC$ is

a)		(4,1)
b)		( $\frac{4}{3}$ ,-3)
c)		( $\frac{4}{3}$ ,3)
d)		( $\frac{- 4}{3}$ ,3)

36. The fourth vertex of the parallelogram formed by the points (5,-1), (-3,-2), (9,12) is

a)		(17,13)
b)		(17,10)
c)		(15,13)
d)		(-15,13)

37. The y- axis divides the line joining the points (3,6) and (12,-3) in the ratio

a)		1:4
b)		1: -4
c)		2:5
d)		2:-5

38. The x- axis divides the line joining the points (5,7) and (-1,3) in the ratio

a)		7:3
b)		7:-3
c)		6:5
d)		6: -5

39. The area of the triangle formed by (-4,-1) (1,2) and (4,-3) is

a)		17
b)		16
c)		15
d)		none of these

40. The circumcentre of a triangle formed by A (1,2) B(-2,2) C(1,5) is

a)		(1,2)
b)		(-2,2)
c)		(1,5)
d)		(- $\frac{1}{2}$ , $\frac{7}{2}$ )

41. ${(\frac{- 1 + i \sqrt{3}}{2})}^{3 n} + {(\frac{- 1 - i \sqrt{3}}{2})}^{3 n}$ =

a)		0
b)		1
c)		2
d)		3

42. ${(\frac{1 + i}{1 - i})}^{3} - {(\frac{1 - i}{1 + i})}^{3}$ =x+iy then (x,y) =

a)		(0,2)
b)		(-2,0)
c)		(0,-2)
d)		none of these

43. The smallest positive integer n for which ${(1 + i)}^{2 n}$ is

a)		4
b)		8
c)		2
d)		12

44. ${Sin}^{- 1} (\frac{Z - 1}{i})$ where Z is non real can be the angles of a triangle if

a)		Re (Z)= 1 Im(z) = 2
b)		Re (Z)= 1 -1<Im(z) = 2 $\leq 1$
c)		Re (Z) + Im(z) = 0
d)		none of these

45. If Re $(\frac{Z + 4}{2 Z - i})$ = $\frac{1}{2}$ , then Z is represented by a point lying on

a)		a circle
b)		an ellipse
c)		a straight line
d)		none of these

46. If $Z_{1}, Z_{2}, Z_{3}$ are the vertices of an equilateral triangle circumscribing the circle $| Z | = 1$ if $Z_{1} = 1 + \sqrt{3} i$ and $Z_{1}, Z_{2}, Z_{3}$ are in the anticlockwise sense, then $Z_{2}$ is

a)		$1 - \sqrt{3} i$
b)		2
c)		$\frac{1}{2} (1 - \sqrt{3} i$ )
d)		none of these

47. If the fourth roots of unity are $Z_{1}, Z_{2}, Z_{3}, Z_{4}$ then ${Z_{1}}^{2} + {Z_{2}}^{2}$ + ${Z_{3}}^{2} + {Z_{4}}^{2}$ is equal to

a)		1
b)		0
c)		I
d)		None of these

48. If $e^{i θ} = \cos θ + i \sin θ$ , then for the $Δ$ ABC $e^{iA} \times e^{iB} \times e^{iC}$ is

a)		-I
b)		1
c)		-1
d)		none of these

49. If Z is a complex number such that $| Z + 1 | = Z + 2 (1 + i)$ then Z is

a)		$\frac{1}{2} (1 + 4 i)$
b)		$\frac{1}{2} (3 + 4 i)$
c)		$\frac{1}{2} (1 - 4 i)$
d)		$\frac{1}{2} (3 - 4 i)$

50. If ${(a + ib)}^{5} = α + i β$ then ${(b + ia)}^{5}$ is equals

a)		$β + i α$
b)		$α - i β$
c)		$β - i α$
d)		$- α - i β$