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Complex Number

1

Exercise-1

1. If 1 2 1 2

z z z z where z1 and z2 are different non zero complex numbers, then

a. 1 2Re z / 0z b. 1 2

Im / 0z z

c. 1 2

0z z d. None of these

2. Total number of complex numbers ‘z’ satisfying Re(z2)=0, 3z , is equal to

a. 2 b. 4

c. 6 d. None of these

3. Area of the triangle formed by the points represented by z,z+iz and iz is equal to

a. 21

4z b.

2z

c. 21

2z d. None of these

4. If 2 2

1 2 1 2z z z z where

1z and

2z are non zero complex numbers, then

a. 1 2Re / 0z z b. 1 2

Im z / 0z

c. 1 2Re 0z z d. None of these

5. For any three complex members z1, z2 and z3 1 2 3Im zz z is always equal to

a. 1 2 3

z z z b. 1 2 3

z z z

c. 1 2 3

z z z d. None of these

6. If 1

1,zz

then

a.max

1 5

2z

b.

min

1 5

2z

c. max

1 5

2z

d. None of these

7. 1 2 3 4, , ,z z z z are non-zero complex numbers. If 1 2 3 4

Im z Im z 0,z z then the

value of arg 1 2/z z +arg 3 4

/z z is always equal to

a. 0 b.

c. /2 d. None of these

Complex Number

2

8. If 1 2 3

1 2 3

1 1 11, then

zz z z

z z is always equal to

a. 1 2 3

z z z b. 1

2 3

z

z z

c. 1 2 3

z z z d. None of these

9. 1 2 3 1

1 , , ,..................n

z z z z

are the nth roots of unity, then 1 2 11 1 ......... 1

nz z z

is

equal to

a.n b. 1 .n

n

c. zero d. None of these

10. If a and are the roots of x2-2x+4=0 then n na is equal to

a. 2 .cos3

n nx b.

12 .cos

3

nn

c. 12 .cos

3

n n d.

11

2 .cos3

nn

11. If z= (i)i Where 1,i then Re (z) is

a. / 2e

b. Zero

c. e d. None of these

12. If

,i

iz i where 1,i then z is equal to

a. 1 b. / 2e

c. e d. None of these

13. Consider a ‘n’ sided regular polygon A1 A2…………………An having it’s center at origin. If ‘zi

‘represents the vertex A, then complex number representing the vertex that is adjacent

to A1, is

a.

2

1

i

nz e

b. 1

i

nz e

c.

2

1

i

nz e

d.

2

1

i

nz e

Complex Number

3

14. If A (z1), B (z2) and C (z3) are collinear points, then

a. 3 1 2 1 3 1 2 1z z z z z z z z b.

3 1 2 1z z z z

c. 3 1 3 1 2 1 2 1z z z z z z z z d. None of these

15. Slope of line 0,za za b b R is

a.

Re

Im a

a b.

Im a

Re a

c.

Re a

Im a d.

Im a

Re a

16. z,z2,z3 are three non-zero complex numbers such that 3 1 21z z z , where

0 ,R then points corresponding to z1,z2 and z3 are

a. Lie on a circle b. Vertices of a triangle

c. Collinear d. None of these

17. If .z r Area of the triangle whose vertices are z, z and z z, where is the non-

real cube root of unity, is 4 3 sq.units, then the value of ‘r’ is

a. 1 b. 2


18. If +

1 2 1 2, R , ,z z z z z z are fixed complex numbers, represents a ellipse, then

a. 1 2

2z z

b.

1 2z z

c. 1 22

z z

d. 1 2z z

19. z1 and z2 are the roots of 23 3 0z z b

If O (0), A (z1), B (z2) is an equilateral triangle then

a. b=-1 b. b=-3

c. b=1 d. b=3

20. If ‘z’ be any complex number such that 3 2 3 2 4z z , then locus of ‘z’ is

a. A circle b. An ellipse

c. A line segment d. None of these

Complex Number

4

21. If iz3+z2-z+i=0, where i= -1, then z is equal to

a. 1 b. 1/ 2

c. 1/4 d. None of these

22. z, and z2 are two non zero complex number such that 1 2 1 2

z z z z , then

a. 1

2

zRe 0

z

b. 1

2

zIm 0

z

c. 2

1

zIm 0

z

d. 2

1

zRe 0

z

23. If 4.z then the maximum value of iz+3-4i is equal to

a.2 b. 4

c. 3 d. 9

24. If 1

z and 2

z are two non-zero complex numbers satisfying 1 2 1

1 2 2

z1, then

z iz

z iz z

is

a. Purely imaginary b. Purely real

c. Of unit modulus d. None of these

25. If 4 2 , thenz z

a. Re z 0,2 b. Re z 3,

c. Re z 2, d. Re 2,3z

Complex Number

5

Exercise-2

1. For any two complex numbers z1 and z2 2 2

1 2 1 27 3 3 7z z z z is always equal to

a. 2 2

1 24 z z b. 2 2

1 28 z z

c. 2 2

1 22 z z d. None of these

2. Let z1 and z2 are non-zero complex numbers such that 1 2

1 2

1,z z

z z

then 1

2

z

z is always

a. Purely positive real number b. Purely imaginary

c. Equal to 1 d. None of these

3. If 1,z then the point representing the complex number-1+3z will lie on

a. A circle

b. A straight line

c. A parabola

d. A hyperbola

4. If 1 2

15 and z 3 4 5,z i then minimum value of 1 2

z z is

a.10 b.15


5. If the points A (z), B(-z) and C (1-z) are the vertices of an equilateral triangle ABC,

then Re (z) is equal to

a. 1

4 b.

3

2

c. 1

2 d. None of these

6. If z,z2= 1 2cosθz z where 1 2

,A z B z and AOB (‘0’ being the origin), then

1 2

z z is also equal to

a. 1 2

z z b. Re 1 2z z

c. Im 21z z d.

1 2z z

Complex Number

6

7. 1 2 3

If z 1, 2, 3z z and 1 2 1 3 2 3

9 4 12,z z z z z z then the value of is 1 2 3

z z z is

equal to

a. 2 b. 3

c. 4 d. 6

8. If 2

2

3,z z

z z

where

1 2 and zz are fixed complex numbers and z is a various complex

number, then ‘z’ lies on a

a. Circle with ‘z1 as it’s interior point.

b. Circle with ‘z2’ as its interior point.

c. Circle with ‘z1’ and ‘z2’ as it’s interior points

d. Circle with ‘z1’ and ‘z2’ as it’s exterior points

9. z1, z2, z3 are three points lying on the circle 1.z maximum value of

2 2 2

1 2 2 3 3 1,z z z z z z is equal to

a. 6 b. 9

c.12 d. None of these

10. Triangle ABC, A (z1), B (z2) and C (z3) is inscribed in the circle 5.z if H (zH) be the

orthocentre of triangle ABC, then ZH is equal to

a. 1 2 3

2

3z z z b. 1 2 3

4

3z z z

c. 1 2 3z z z d. 1 2 3

3 z z z

11. P (z) be a variable point in the argand plane such that minimum 1 , 1 ,z z z

then z z will be equal to

a. -1 or 1 b. 1 but not equal to-1

c. -1 but not equal to 1 d. None of these

12. 1, z1, z2, z3, ……………………..zn-1 are nth roots of unity then the value of

1 2 1

1 1 1.............

3 3 3n

z z z

is equal to

a.

13 1

3 1 2

n

n

n

b.

1

1

31

3

n

n

n

c.

131

3 1

n

n

n

d. None of these

Complex Number

7

13. Let z1=a+i,z2=-1+ib where a,b R+ if triangle OAB, where A(z1), B (z2) and ‘O’ is

origin, is equilateral, then

a. a=2- 3, b=2- 3 b. 2 3, 2 3a b

c. 2 3, 2 3a b d. None of these

14. 2 2

2 3 4 3z i z i represents the equation of a circle, then value of can’t be

a. 8 b. 2

c. 6 d. 9

15. If 1 2 23 5 71 1 1 1i

n n nni i i i is real, where n1, n2 are positive integers, then

a.1 2

n n b. 1 2

1n n

c. 2 1

1n n d. 1 2.n n N

16. A (z1), B (z2), C (z3) are the vertices of the triangle ABC (in anticlockwise order). If

4

ABC

and 2AB (BC), then

a. 2 3 1 2z z i z z b. 2 3 1 3

z z i z z

c. 2 3 1 3z z i z z d. None of these

17. If z1, and z2 are two distinct non-zero complex numbers such that 1 2

1 2

1 2

z, then

zz z

z z

is always

a. Purely real b. purely imaginary

c. Equal to zero d. None of these

18. tan ,a ib

i na ib

where a,b R+, is always equal to

a. 2 2

2ab

a b b.

2 2

2ab

a b c.

2 2

ab

a b d.

2 2

ab

a b

19. If z is non real complex number lying on z =1, then z is always equal to

a.

arg1 tan

2

arg1 tan

2

zi

z

b.

arg1 tan

2

arg1 tan

2

zi

z

Complex Number

8

c.

1 tan arg

1 tan arg

i z

z

d.

1 tan arg

1 tan arg

i z

z

20. if 1 2

1 2

21

2

z z

z z

and

21,z then value of

1z is equal to

a. 2 b. 1


21. If the equation 0,0 ,eax bx c a b c has non complex roots z1 and z2 then

a. 2

1, 1z z b.2

1, 1z z

c. 2

1, 1z z d. 2

1, 1z z

22. If z1,z2,z3 are distinct non zero complex numbers and a,b,c+R such that

2 2 2

1 2 2 3 3 1 1 2 2 3 3 1

athen

a b c b c

z z z z z z z z z z z z

is always equal to

a. Re (z1,z2, z3) b. Im(z1,z2, z3)

c. zero d. None of these

23 A (z1), B (z2), C (z3), are the vertices of an equilateral triangle ABC, whose

circumcentre is D (z0), then 2 2 2

1 2 3z z z is always equal to

a. 2

0z b.

2

03z

c. zero d. 1 2 3

0

z z z

z

24. If z =x+iy,z1/3=a-ib; a.b a.b0 and 2 2 ,x y

a ba b

then is equal to

a. 2 b. 3

c. 4 d. 1

25. Equation of tangent drawn to circle [z]=r at the point A (z0) is

a.

0

zRe 1

z

b.

0

zRe 1

z

c.

0

zIm 1

z

d. 0

zIm 1

z

Complex Number

9

Exercise-3

1. If ‘z’ lies on the circle 2 2 2,z i then the value of 2

arg2

z

z

is equal to

a. 3

b.

4

c.

6

d.

2

2. Let A1(z1), A2 ( 1z are the adjacent vertices of a regular polygon. If

1

1

Im z1 2,

Re z

then number of sides of the polygon is equal to

a. 6 b.8 c. 16 d. 12

3. If Re Im , thenz i z z z

a. Im(z)=2 b. Re (z)=2 c. Re (z)+Im (z)=2 d. None of these

4. If exactly one root of z2+az+b=0 where a, b c is purely imaginary, then

a. 2

b b ab a b a a b. 2

b b ab ab a a

c. 2

b b ab ab a a d. 2

b b ab ab a a

5. ABCD is a rhombus. It’s diagonals AC and BD intersect at P and satisfy 2BD=AC. If

D=1 i, P 2-i, then the complex number representing the point ‘A’ can be

a. 6+1 b. 6-i c. 3+3i d. -3+3i

6. Total number of complex numbers ‘z’ satisfying 3,Re 3 2 and z+1-i 2z z is

equal to

a. 2 b. 1


7. The minimum value of ‘a’ so that the line arg6

z

intersect the circle 2 3i ,z a

is equal to

a. 3 b. 2

c. 1/2 d. None of these

8. z1 and z2 are the roots of z2+az+b=0 where a,b are non zero complex numbers. It is

known that the line joining the points A (z1) and B (z2) pass through then origin, then

a2/b is

Complex Number

10

a. Purely real b. Purely imaginary

c. on the form x+iy, where x,y0 d. None of these

9. Center of the arc represented by 3i

argz-2i+4 4

z

is

a. 1

5i+52

b. 1

5i-52

c. 1

9i+52

d. 1

9i-52

10. If one vertex of the triangle having maximum area that can be inscribed in the circle

5z i is 3 3i, then another vertex of the triangle can be

a. 13 4 3 3 3 2

2i b. 1

3 4 3 3 3 22

i

c. 13 4 3 3 3 2

2i d. 1

3 4 3 3 3 22

i

11. a,b,c,a,b,c, are non zero complex numbers satisfying

1 1 1

1a b c

ia b c and

2 2 2

1 1 1

2 2 2

1 1 1

a0, then

a

a b c b c

a b c b c is equal to

a. 2i b. 2+2i


12. z1 and z2 are the roots of z2-az+b=0, where 1 2

z z =1 and a,b are non-zero complex

numbers, then

a. arg (A)=2 arg (B) b. 2 arg (A)=arg (B)

c. arg (A)=arg (B) d. None of these

13. A (z1), B (z2) are the points on the circles 1z and 3z respectively, then

a. 1 2 max1z z b. 1 2 min

4z z

c. 1 2 max4z z d. 1 2 min

1z z

14. A (z1),B (z2), C (z3) are vertices of a triangle ABC inscribed in the circle 2.z Internal

angle bisector of the angle A, meet the circumcircle again at D (z4), then

a. 2

4 2 3z z z b. 2 3

4

1

z zz

z

Complex Number

11

c. 1 2

4

3

z zz

z d. 1 3

4

2

z zz

z

15. In Q. (14), 4

2 3

argz

z z

is equal to

a. 4

b.

3

c. 2

d.

2

3

16. In Q. (14), altitude drawn from A to side BC meet the circumcircle at 5E z , then

a. 2 3

5

1

z zz

z b. 2 3

5

1

z zz

z

c. 1 3

5

5

z zz

z d. 1 3

5

5

z zz

z

17. A rectangle of maximum area is inscribed in the circle 3 4i 1.z If one vertex of

the rectangle is 4 +4i, then another adjacent vertex of this rectangle can be

a. 3+4i or 4-3i b. 3+5i or 3+3i

c. 3-5i or 3+3i d. 3-4i or 4+3i

18. If z1 is one of the roots of the equation 1

0 1 1........ 3,n n

n na z a z a z a

where

12a

for i=0,1,………………..n. then

a. 1

1

3z b. 1

1

3z c. 1

1

4z d.

1

3z

19. If A (z1), B (z2), C (z3) are the vertices of an equilateral triangle ABC, then value

2 3 1

3 2

2arg

z z z

z z

is equal to

a. 4

b.

2

c.

3

d.

6

20. z0 is one of the roots of the equation 1

0 1 1cos cos cos cos 2n n

n nz z z

where

1

R, then

a. 0

1

2z b. 0

1

2z

Complex Number

12

c. 0

1

2z d. None of these

21. Consider a square OABC, where ‘O’ is the origin, A (z0) and vertices are given in

anticlockwise order. The equation of the circle that can be inscribed in the square is

a. 0 01z z i z b.

0

0

1

2

z iz z

c. 0

0

12

2

z iz z

d. 0 0

2 1z z i z

22. qQ z is the foot of perpendicular drawn from P (zp) to the line 0,az za b b R

and ‘a’ is complex number, then

a. 0pq p

z a az z a b b. 2 0pq p

z a az z a b

c. 2 0pq p

z a az z a b d. 2 0pq p

z a az z a b

23. 1 2 3, ,A z B z C z are the vertices of a right angled isosceles triangle ABC. If

,2

C

then

a. 2

1 2 1 3 3 2z z z z z z b.

2

1 2 1 3 3 22z z z z z z

c. 2

1 2 1 3 3 22 z z z z z z d. None of these

24. If z1, z2 are the non zero complex root of z2-az+b=0 such that 1 2

,z z where a,b are

complex numbers.

If A (z1), B (z2) and AOB , ‘O’ being the origin, then

a. b2=4a cos2

2

b.

2 2a 4 cos2

b

c. 2 22 cos

2b a

d.

2 22 cos2

a b

25. If tangents drawn to circle 2z at 1A z and 2

B z meet at ,pP z then

a. 1 2

2p

z zz

b. 1 2

1 2

2p

z zz

z z

c. 1 2

1 2

2p

z zz

z z

d.

2 1

2z z z

p

Complex Number

13

Answers Exercise-1 Complex Number

1. b 2. b 3. c 4. a 5. d 6. a 7. d 8. c 9. a 10. c

11. a 12. a 13. d 14.a 15. a 16.c 17. d 18. b 19. c 20. c

21. a 22. d 23. d 24. b 25. b


1. d 2. b 3. a 4. c 5. a 6. b 7. a 8. b 9. b 10. c

11. a 12. d 13. d 14.b 15. d 16.c 17. b 18. b 19. b 20. a

21. d 22. c 23. b 24. c 25. a


1. b 2. b 3. d 4. a 5. a 6. d 7. a 8. a 9. d 10. a

11. a 12. b 13. c 14.a 15. c 16. b 17. b 18. a 19. b 20. b

21. c 22. c 23. b 24. b 25. c

Complex Number

14

Complex Number

15

Complex Number

16

Complex Number

17

Answers Exercise-1 Quadratic Equation and Expressions

1. a 2. a 3. b 4. a 5. b 6. d 7. d 8. b 9. a 10. b

11. b 12. c 13. b 14.d 15. d 16.b 17. d 18. b 19. d 20. a

21. c 22. a 23. c 24. b 25. c


1. d 2. a 3. d 4. b 5. c 6. b 7. a 8. a 9. b 10. b

11. d 12. c 13. c 14.b 15. c 16.b 17. b 18. d 19. b 20. a

21. a 22. a 23. d 24. c 25. c


1. d 2. b 3. a 4. b 5. c 6. a 7. c 8. d 9. a 10. b

11. c 12. a 13. a 14.a 15. c 16. a 17. b 18. c 19. a 20. d

21. d 22. a 23. b 24. b 25. a

complex number - wordpress.com · complex number 1 exercise-1 1. if z z z z 1 2 1 2 where z 1 and z...

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