Karnataka 1st PUC Mathematics Complex Numbers and Quadratic Equations MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
18TH CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – I PUC
NAME OF THE CHAPTER: COMPLEX NUMBERS AND QUADRATIC EQUATIONS

 1 
Number system consists of real numbers  -5, 7, , 3..............etc  and imaginary numbers
 3 

 -5, -9....etc.
If we combine these two numbers by some mathematical operations, the resulting
number is known as Complex Number i.e.,
“Complex Number is the combination of real and imaginary numbers”.
Complex Numbers
A number of the form x  iy, where x , y  R and i = -1 is called a complex number

so the quantity  1 is denoted by 'i' called iota thus i   1 .
A complex number is usually denoted by z and the set of complex number is denoted by c
i.e., c  {x  i y : x R, y  R, i  1}
For example, 5  3i,  1  i, 0  4i, 4  0i etc. are complex numbers.
 Euler was the first mathematician to introduce the symbol i (iota) for the square root
of – 1 with property i 2  1 . He also called this symbol as the imaginary unit.
 Iota (i) is neither 0, nor greater than 0, nor less than 0.
 The square root of a negative real number is called an imaginary unit.

 For any positive real number a, we have -a = -1×a = -1 a = i a

 i -a = - a.

 The property a b = ab is valid only if at least one of a and b is non-negative.

If a and b are both negative then a b = - ab .

 If a  0 then a = |a|i .

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Integral powers of iota (i):
Since i = -1 hence we have i 2 = -1 , i 3 = -i and i 4 = 1 .
To find the value of i n (n  4 ), first divide n by 4. Let q be the quotient and r be the
remainder.
i.e., n = 4q + r where 0  r  3

i n = i 4q+r = (i 4 )q .(i)r = (1)q .(i)r = i r

In general we have the following results i 4n = 1, i 4n+1 = i, i 4n+2 = -1, i 4n+3 = -i , where n is any
integer.
In other words, i n = (-1)n/ 2 if n is even integer and i n = (-1)n-1/ 2 i if n is odd integer.
The value of the negative integral powers of i are found as given below:
1 i3 1 1 1 i i 1 1
i -1 = = 4 = i 3 = -i, i -2 = 2 = = -1, i -3 = 3 = 4 = = i, i -4 = 4 = = 1
i i i -1 i i 1 i 1
Real and Imaginary Parts of a Complex Number.
If x and y are two real numbers, then a number of the form z  x  iy is called a complex
number.
Here ‘x’ is called the real part of z and ‘y’ is known as the imaginary part of z.
The real part of z is denoted by Re(z) and the imaginary part by Im(z).
If z = 3 – 4i, then Re(z) = 3 and Im(z) = – 4.
 A complex number z is purely real if its imaginary part is zero
i.e., Im(z) = 0 and purely imaginary if its real part is zero i.e., Re(z) = 0.
 i can be denoted by the ordered pair (0,1).
 The complex number (a, b) can also be split as (a, 0) + (0, 1) (b, 0).
Algebraic Operations with Complex Numbers
Let two complex numbers z1  a  ib and z2  c  id

Addition : (a + ib) + (c + id) = (a + c) + i(b + d)
Subtraction : (a + ib) - (c + id) = (a - c) + i(b - d)
Multiplication : (a + ib)(c + id) = (ac - bd) + i(ad + bc)
a + ib
Division : (when at least one of c and d is non-zero)
c + id
a  ib (a  ib) (c  id)
 . (Rationalization)
c  id (c  id) (c  id)

a  ib (ac  bd ) i(bc  ad )
  2 .
c  id c 2  d 2 c d2

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Properties of algebraic operations with complex numbers:
Let z 1 , z 2 and z 3 are any complex numbers then their algebraic operation satisfy following
operations:
(i) Addition of complex numbers satisfies the commutative and associative properties
i.e., z1 + z 2 = z 2 + z1 and (z1 + z 2 ) + z 3 = z 1 + (z 2 + z 3 ).

(ii) Multiplication of complex number satisfies the commutative and associative properties.
i.e., z 1 z 2 = z 2 z 1 and (z1 z 2 )z 3 = z 1 (z 2 z 3 ).

(iii) Multiplication of complex numbers is distributive over addition
i.e., z1 (z 2 + z 3 ) = z 1z 2 + z 1z 3 and (z 2 + z 3 )z 1 = z 2 z 1 + z 3 z 1 .

Note :
 0  0  0 i is the identity element for addition.

 1  1  0 i is the identity element for multiplication.

 The additive inverse of a complex number z  a  ib is  z (i.e. – a – ib).

1
 For every non-zero complex number z, the multiplicative inverse of z is .
z

Equality of Two Complex Numbers
Two complex numbers z 1 = x1 + iy 1 and z 2 = x 2 + iy 2 are said to be equal if and only

if their real parts and imaginary parts are separately equal.
i.e., z1  z 2  x1  iy1  x2  iy2  x1  x2 and y1  y 2 .

Thus, one complex equation is equivalent to two real equations.
 A complex number z  x  iy  0 iff x  0, y  0.
 The complex number do not possess the property of order
i.e., (a  ib)  (or )  (c  id) is not defined.
For example, the statement 9  6 i  3  2i makes no sense.
Conjugate of a Complex Number
If there exists a complex number z = a + i b, (a ,b)  R, then its conjugate is defined as
z = a-ib .
z+z z-z
Hence, we have Re(z) = and Im (z) = .
2 2i

Geometrically, the conjugate of z is the reflection or point image of z in the real axis.

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Properties of conjugate:
If z, z1 and z 2 are existing complex numbers, then we have the following results:

(i) (z)  z (ii) z1  z 2  z1  z 2

(iii) z1  z 2  z1  z 2 (iv) z1z 2  z1 z 2 , In general z1 .z 2 .z 3 .....zn  z1 .z 2 .z 3 .....zn

z  z
(v)  1   1 , z 2  0 (vi) (z)n  (z n )
 z2  z2
(vii) z  z  2Re(z)  2Re(z)  purely real (viii) z  z  2iIm(z)  purely imaginary

(ix) z z  purely real (x) z1 z2 + z1z 2 = 2Re(z 1 z 2 ) = 2Re(z1z 2 )

(xi) z - z = 0 i.e., z  z  z is purely real i.e., Im(z) = 0
(xii) z + z = 0 i.e., z = -z  either z = 0 or z is purely imaginary i.e., Re(z) = 0

(xiii) z1 = z 2  z1 = z2 (xiv) z = 0  z = 0

(xv) zz = 0  z = 0 (xvi) If w = f(z) then w = f(z) (xvii) reiθ = re-iθ

Reciprocal of a complex number:
For an existing non-zero complex number z = a + ib ,
1 z 1 a - ib Re(z) i[-Im(z)] z
the reciprocal is given by z -1 = = 2
i.e., z -1 =  2 2
= 2
+ 2
= 2.
z |z| a + ib a +b |z| |z| |z|

Modulus of a Complex Number
Modulus of a complex number z  a  ib is defined by a positive real number given by

| z |= a 2 + b 2 , where a, b real numbers. Geometrically |z| represents the distance of point
P (represented by z) from the origin,
i.e. |z| = OP. Y
P(z)
If |z| = 0, then z is known as zero modular complex number
and is used to represent the origin of reference plane. M

If |z| = 1 the corresponding complex number is known as X
O
unimodular complex number.
Clearly z lies on a circle of unit radius having centre (0, 0).
 In the set C of all complex numbers, the order relation is not defined. As such
z 1 > z 2 > or z 1 < z 2 has no meaning. But | z 1 |>| z 2 |or | z 1 |<| z 2 | has got its meaning

since | z1 |and| z 2 | are real

numbers.

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Properties of modulus
(i) z  0  z  0 iff z  0 and |z| 0 iff z  0 (ii)  z  Re ( z )  z and  z  Im ( z )  z

(iii) z  z   z   z | zi | (iv) z z  z | z |2
2

(v) z1 z2  z1 z2 . In general z1 z2 z3 ......zn  z1 z2 z3 .... zn

z1 z1
(vi)  , (z 2  0 ) (vii) | z n |  | z |n , n  N
z2 z2
2
(viii) z1  z2 2  ( z1  z2 )( z1  z2 )  z1  z2  ( z1 z2  z1 z2 ) or | z1 |2  | z2 |2  2 Re( z1 z2 )
2

z1 z 
(ix) z 1  z 2 is purely imaginary or Re 1   0
2 2 2
 z1  z2 
z2  z2 

(x) z1  z2
2
 z1  z2
2
2 z z  1
2
2
2
(Law of parallelogram)

(xi) az1  bz2
2
 bz1  az2
2
 ( a 2  b2 )  z  z  , where a, b  R.
1
2
2
2

Argument of a Complex Number.
Let z  a  ib be any complex number. If this complex number is represented geometrically
by a point P, then the angle made by the line OP with real axis is known as argument or
amplitude of z and is expressed as
b
arg (z )    tan 1  ,   POM . Also, argument of a complex number is not unique,
a
since if  be a value of the argument, so also is 2n   , where n I .
Principal value of arg (z):
The value  of the argument, which satisfies the inequality Y

      is called the principal value of – 
(–,+) (+,+)
argument. Principal values of argument z will be  ,   ,     X'
(–,–) O (+,–)
X

– (  – ) –
and   according as the point z lies in the 1st , 2nd , 3rd and 4th
Y'
1 b
quadrants respectively, where   tan  (acute angle).
a

Principal value of argument of any complex number lies between       .

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Complex Numbers
1. If 4x+i(3x-y)=3+i(-6), where x and y are real numbers, then the values of x and y are
(Easy)
(a) x=3/5 and y=33/4 (b) x=3/4 and y=22/3
(c) x=3/4 and y=33/4 (d) x=3/4 and y=33/5
2. For a complex number 𝒛 = −𝟒𝒊 + 𝟕, what is the 𝐑𝐞⁡(𝒛) and Im (z) respectively? (Easy)
(a) -4,7 (b) 7,-4 (c) 4,-7 (d) -7,4
3. If (𝒙 + 𝒊𝒚)(𝟐 − 𝟑𝒊) = 𝟒 + 𝒊, then (Average)
(a) 𝑥 = −14/13, 𝑦 = 5/13 (b) 𝑥 = 5/13, 𝑦 = 14/13
(c) 𝑥 = 14/13, 𝑦 = 5/13 (d) 𝑥 = 5/13, 𝑦 = −14/13
4. If 𝒙, 𝒚 ∈ 𝑹, then 𝒙 + iy is a non-real complex number, if (Average)
(a) x=0 (b) y=0 (c) x≠0 (d) y≠0
5. If 𝒂 + 𝒊𝒃 = 𝒄 + 𝒊𝒅, then (Easy)
(a) 𝑎2 + 𝑐 2 = 0 (b) 𝑏 2 + 𝑐 2 = 0 (c) 𝑏 2 + 𝑑2 = 0 (d) 𝑎2 + 𝑏 2 = 𝑐 2 + 𝑑 2
Complex Numbers
1 2 3 4 5
C B B D D
Algebra Of Complex Numbers
−𝟑
1. If 𝒛 = 𝟓𝐢 ( 𝒊), then 𝐳 is equal to 𝟑 + 𝐛𝐢. The value of ' 𝐛 ' is (Easy)
𝟓

(a) 1 (b) 2 (c) 0 (d) 3
2. If 𝒛 = 𝒊𝟗 + 𝒊𝟏𝟗 , then 𝒛 is equal to 𝒂 + 𝒂. The value of ' 𝒂 ' is (Easy)
(a) 0 (b) 1 (c) 2 (d) 3
𝟐𝒊 𝟐
3. Find the Value of ( ) (Easy)
𝟏+𝒊

(a) i (b) 2i (c) 1-i (d) 1-2i
𝟏+𝒊 𝒙
4. If ( ) = 𝟏, then (Average)
𝟏−𝒊

(a) 𝑥 = 2𝑛 + 1, 𝑛 ∈ 𝑁 (b) 𝑥 = 4𝑛, 𝑛 ∈ 𝑁
(c) 𝑥 = 2𝑛 , 𝑛 ∈ 𝑁 (d) 𝑥 = 4𝑛 + 1 , 𝑛 ∈ 𝑁
𝟏−𝒊 𝟏𝟎𝟎
5. If (𝟏+𝒊) = 𝒂 + 𝒊𝒃 then (Average)

(a) a=2,b=-1 (b) a=1,b=0 (c) a=0,b=1 (d) a=-1,b=2
𝟏
6. 𝒊𝟓𝟕 + 𝒊𝟐𝟓 , when simplified has the value (Average)

(a) 0 (b) 2𝑖 (c) −2𝑖 (d) 2
7. 𝟏 + 𝒊𝟐 + 𝒊𝟒 + 𝒊𝟔 + ⋯ + 𝒊𝟐𝒏 (Average)
(a) positive (b) negative (c) 0 (d) cannot be determined
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𝐢𝟓𝟗𝟐 +𝐢𝟓𝟗𝟎 +𝐢𝟓𝟖𝟖 +𝐢𝟓𝟖𝟔 +𝐢𝟓𝟖𝟒
8. Find the Value of 𝐢𝟓𝟖𝟐 +𝐢𝟓𝟖𝟎 +𝐢𝟓𝟕𝟖 +𝐢𝟓𝟕𝟔 +𝐢𝟓𝟕𝟒 − 𝟏 (Average)

(a) -2 (b) 0 (c) -1 (d) 1
𝟏
𝒙 𝒚
9. If 𝒛 = 𝒙 − 𝒊𝒚 and 𝒛𝟑 = 𝒑 + 𝒊𝒒, then (𝒑 + 𝒒) /(𝒑𝟐 + 𝒒𝟐 ) is equal to (Difficult)

(a) -2 (b) -1 (c) 2 (d) 1
10. If 𝒛 = 𝟏 + 𝒊, then the multiplicative inverse of 𝒛𝟐 is (where, 𝒊 = √−𝟏 ) (Average)
i i
(a) 2i (b) 1-i (c) – 2 (d) 2
𝟑+𝟒𝒊
11. The multiplicative inverse of 𝟒−𝟓𝒊 is (Average)
8 31 8 31 8 31 8 31
(a) 25 − 25 𝑖 (b) − 25 − 25 𝑖 (c) − 25 + 25 𝑖 (d) ) 25 + 25 𝑖

12. If 𝒛(𝟐 − 𝒊) = (𝟑 + 𝒊), then 𝒛𝟐𝟎 is equal to (Average)
(a) 210 (b) −210 (c) 220 (d) −220
𝟏
𝒙 𝒚
13. If (𝒙 + 𝒊𝒚)𝟑 = 𝒂 + 𝒊𝒃, where 𝒙, 𝒚, 𝒂, 𝒃 ∈ 𝑹, then 𝒂 − 𝒃 = (Difficult)

(a) 𝑎2 − 𝑏 2 (b) −2(𝑎2 + 𝑏 2 ) (c) 2(𝑎2 − 𝑏 2 ) (d) 𝑎2 + 𝑏 2
14. If 𝒛 = 𝟐 − 𝟑𝒊, then value of 𝒛𝟐 − 𝟒𝒛 + 𝟏𝟑 is (Difficult)
(a) 0 (b) 1 (c) 2 (d) 3
15. If 𝐳 = 𝐢−𝟑𝟗 , then simplest form of 𝐳 is equal to 𝒂 + 𝒊. The value of ' 𝒂 ' is (Average)
(a) 0 (b) 1 (c) 2 (d) 3
16. If 𝒛𝟏 = 𝟐 + 𝟑𝒊 and 𝒛𝟐 = 𝟑 + 𝟐𝒊, then 𝒛𝟏 + 𝒛𝟐 equals to a+ai then value of 'a' is equal to
(Average)
(a) 3 (b) 4 (c) 5 (d) 2
17. If 𝒛𝟏 = 𝟐 + 𝟑𝒊 and 𝒛𝟐 = 𝟑 − 𝟐𝒊, then 𝒛𝟏 − 𝒛𝟐 equals to −𝟏 + 𝒃𝒊. The value of ' 𝒃 ' is
(Average)
(a) 1 (b) 2 (c) 3 (d) 5
𝟏 𝟒
18. The value of (𝟏 + 𝐢)𝟒 (𝟏 + 𝐢 ) is (Average)

(a) 12 (b) 2 (c) 8 (d) 16
19. Evaluate: (𝟏 + 𝐢)𝟔 + (𝟏 − 𝐢)𝟑 (Average)
(a) −2 − 10i (b) 2 − 10i (c) −2 + 10i (d) 2 + 10i
𝐢𝟒𝐧+𝟏 −𝐢𝟒𝐧−𝟏
20. The value of is (Average)
𝟐

(a) 𝑖 (b) 2i (c) – i (d) −2i
21. Value of 𝐢𝟒𝐤 + 𝐢𝟒𝐤+𝟏 + 𝐢𝟒𝐤+𝟐 + 𝐢𝟒𝐤+𝟑 is (Average)
(a) 0 (b) 1 (c) 2 (d) 3

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22. The value of (𝟏 + 𝐢)𝟓 × (𝟏 − 𝐢)𝟓 is (Average)
(a) -8 (b) 8i (c) 8 (d) 32
𝟏 𝟑 𝟑+𝟒𝐢
23. (𝟏−𝟐𝐢 + 𝟏+𝐢) (𝟐−𝟒𝐢) is equal to: (Average)
1 9 1 9 1 9 1 9
(a) 2 + 2 i (b) 2 − 2 i (c) 4 − 4 i (d) 4 + 4 i
(𝟏+𝒊)𝟐
24. The real part of (𝟑−𝒊) is (Average)
1 1 1 1
(a) 3 (b) 5 (c) − 3 (d) − 5

Algebra Of Complex Numbers
1 2 3 4 5 6 7 8 9 10
c a b b b a d a a c
11 12 13 14 15 16 17 18 19 20
b b b a a c d d a a
21 22 23 24
a d d d
Modulus And Conjugate Of A Complex Number
5  12i  5  12i
1. What is the conjugate of (Average)
5  12i  5  12i
3 3
(a) −3i (b) 3i (c) 2 i (d) − 2 i
(𝟏+𝒊√𝟑)(𝟐+𝟐𝒊)
2. The modulus of is (Average)
(√𝟑−𝒊)

(a) 2 (b) 4 (c) 3√2 (d) 2√2
𝒛 𝟏
3. If 𝒛𝟏 = 𝟔 + 𝟑𝒊 and 𝒛𝟐 = 𝟐 − 𝒊, then 𝒛𝟏 is equal to 𝒂 (𝟗 + 𝟏𝟐𝒊). The value of ' 𝒂 ' is (Difficult)
𝟐

(a) 1 (b) 2 (c) 4 (d) 5
𝟕−𝒊
4. If 𝒛 = 𝟑−𝟒𝒊, then |𝒛|𝟏𝟒 = (Average)

(a) 27 (b) 27 i (c) −27 (d) −27 𝐢
𝒛
5. If 𝒛𝟏 = 𝟔 + 𝟑𝒊 and 𝒛𝟐 = 𝟐 − 𝒊, then 𝒛𝟏 is equal to (Average)
𝟐

1 1
(a) 5 (9 + 12i) (b) 9 + 12i (c) 3 + 2i (d) 5 (12 + 9i)

6. If (𝟏 − 𝒊)𝐧 = 𝟐𝐧 , then the value of 𝐧 is (Average)
(a) 1 (b) 2 (c) 0 (d) -1
7. If |𝒛 − 𝟒| < |𝒛 − 𝟐|, its solution is given by (Difficult)
(a). Re(𝑧) > 0 (b) Re(𝑧) < 0 (c) Re(𝑧) > 3 (d) Re(𝑧) > 2
(𝟏+𝒊√𝟑)(𝐜𝐨𝐬⁡𝜽+𝒊𝐬𝐢𝐧⁡𝜽)
8. Modulus of 𝒛 = 𝟐(𝟏−𝒊)(𝐜𝐨𝐬⁡𝜽−𝒊𝐬𝐢𝐧⁡𝜽) is (Average)

1 1 1
(a) (b) − (c) (d) 1
√3 √2 √2

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𝒛 +𝒛 +𝟏
9. If 𝒛𝟏 = 𝟐 − 𝒊 and 𝒛𝟐 = 𝟏 + 𝒊, then value of |𝒛𝟏 −𝒛𝟐+𝟏| is (Difficult)
𝟏 𝟐

(a) 2 (b) 2i (c) √2 (d) √2i
(𝟏+𝒊)𝟑 (𝟏−𝒊)𝟑
10. If (𝟏−𝒊)𝟑 − (𝟏+𝒊)𝟑 = 𝒙 + 𝒊𝒚 (Average)

(a) 𝑥 = 0, 𝑦 = −2 (b) x = 1, y = 1 (c) 𝑥 = −2, 𝑦 = 0 (d) 𝑥 = −1, 𝑦 = 1
11. Additive inverse of 𝟏 − 𝒊 is (Average)
(a) 0 + 0i (b) −1 − 𝑖 (c) −1 + i (d) 1+i
12. If 𝒛 is a complex number such that 𝒛𝟐 = (𝒛‾)𝟐 , then (Average)
(a) z is purely real (b) z is purely imaginary
(c) either z is purely real or purely imaginary (d) None of these
𝒂+𝒊𝒃
13. If 𝒙 + 𝒊𝒚 = √ 𝒄+𝒊𝒅, then (𝒙𝟐 + 𝒚𝟐 )𝟐 = (Average)

2
𝑎2 +𝑏2 𝑎+𝑏 𝑐 2 +𝑑2 𝑎2 +𝑏2
(a) 𝑐 2 +𝑑2 (b) 𝑐+𝑑 (c) 𝑎2 +𝑏2 (d) (𝑐 2 +𝑑2 )
𝒂+𝒊𝒃
14. If 𝒙 + 𝒊𝒚 = 𝒂−𝒊𝒃, then 𝒙𝟐 + 𝒚𝟐 = (Average)

(a) 1 (b) 2 (c) 0 (d) 4
𝟐+𝟓𝒊
15. The conjugate of the complex number 𝟒−𝟑𝒊 is equal to : (Average)
7−26i −7−26𝑖 −7+26𝑖 7+26i
(a) (b) (c) (d)
25 25 25 25
𝟑−𝟒𝒊𝒙
16. A real value of 𝒙 satisfies the equation ( ) = 𝜶 − 𝒊𝜷(𝜶, 𝜷 ∈ 𝑹), if 𝜶𝟐 + 𝜷𝟐 is equal to
𝟑+𝟒𝒊𝒙

(Difficult)
(a) 1 (b) -1 (c) 2 (d) -2
17. If 𝐳 = 𝟐 + 𝐢, then (𝐳 − 𝟏)(𝐳̅ − 𝟓) + (𝐳̅ − 𝟏)(𝐳 − 𝟓) is equal to (Difficult)
(a) 2 (b) 7 (c) -1 (d) -4
𝒛−𝟏
18. If |𝒛| = 𝟏, (𝒛 ≠ −𝟏) and 𝒛 = 𝒙 + 𝒊𝒚, then (𝒛+𝟏) is (Difficult)

(a) purely real (b) purely imaginary
(c) zero (d) undefined
19. If 𝒛‾ be the conjugate of the complex number 𝒛, then which of the following
relations is false? (Difficult)
(a) |𝑧| = |𝑧‾| (b) 𝑧 ⋅ 𝑧‾ = |𝑧‾|2 (c) ̅̅̅̅̅̅̅̅̅
𝑧1 + 𝑧2 = 𝑧‾1 + 𝑧‾2 (d) arg⁡𝑧 = arg⁡𝑧‾
20. (𝒙 − 𝒊𝒚)(𝟑 + 𝟓𝒊) is the conjugate of (−𝟔 − 𝟐𝟒𝒊), then 𝒙 and 𝒚 are (Average)
(a) 𝑥 = 3, 𝑦 = −3 (b) 𝑥 = −3, 𝑦 = 3
(c) 𝑥 = −3, 𝑦 = −3 (d) 𝑥 = 3, 𝑦 = 3

2026-27 MATHEMATICS CET MATERIAL Page 10 of 19

𝒛−𝟏
21. If 𝒛 is a complex number such that 𝒛+𝟏 is purely imaginary, then (Average)

(a) |𝑧| = 0 (b) |𝑧| = 1 (c) |𝑧| > 1 (d) |𝑧| < 1
𝒊+𝒛
22. The complex number 𝒛 which satisfies the condition |𝒊−𝒛| = 𝟏 lies on (Average)

(a) circle 𝑥 2 + 𝑦 2 = 1 (b) the 𝑥-axis
(c) the 𝑦-axis (d) the line 𝑥 + 𝑦 = 1
Modulus And Conjugate Of A Complex Number
1 2 3 4 5 6 7 8 9 10
c d d a a c c c c a
11 12 13 14 15 16 17 18 19 20
c c a a b a d b d a
21 22
b b
Diagram-Based, Matching & Statement-Type Questions
1. The graph drawn below depicts (Easy)

(A) 𝑍 = 3 + 4𝑖 (B) 𝑍 = 3 − 4𝑖 (C) 𝑍 = 4 + 3𝑖 (D) 𝑍 = 4 − 3𝑖
2. For the figure given below, (Easy)

1
(A) 𝑍 ∗ ⁡ = 𝑍 (B) 𝑍 ∗ = −𝑍 (C) 𝑍 ∗ = 𝑍̅ (D) 𝑍 ∗ = 𝑍
3. For the figure given below,|𝒛|⁡𝒊𝒔 (Average)

(A) 4 (B) 1 (C) √17 (D) 17
4. Match List I with List II (Average)

List I List II
a) Additive inverse of 1+ i 1
i)⁡1+⁡i
b) conjugate of 1+ i ii) 1 − ⁡i
c) multiplicative inverse of 1+ i iii) −1 − ⁡i
Choose the correct answer from the options given below:
A) a-i, b-ii, c-iii B) a-iii, b-ii, c-i C) a-ii, b-i, c-iii D) a-iii, b-i, c-ii
2026-27 MATHEMATICS CET MATERIAL Page 11 of 19

5. If z = x+iy, then match List I with List II (Easy)

List I List II
a) z 𝑧̅ i)⁡𝑥 2 + 𝑦 2
b) 𝑧 − 𝑧̅ ii) 2x
c)⁡𝑧 + 𝑧̅ iii) 𝑖2𝑦
Choose the correct answer from the options given below:
A) a-i, b-ii, c-iii B) a-iii, b-ii, c-i C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii.
6. If 𝒛𝒊 = 𝒙𝒊 +i𝒚𝒊 ,then match List I with List II (Easy)

List I List II
a) 𝑧1 𝑧2 ⁡ i)⁡(𝑥1 − 𝑥2 ) + 𝑖(𝑦1 − 𝑦2 )
b) 𝑧1+ 𝑧2 ⁡ ii) (𝑥1 𝑥2 − 𝑦1 𝑦2 ) + 𝑖(𝑥1 𝑦2 + 𝑦1 𝑥2 )
c)⁡𝑧1− 𝑧2 ⁡ iii) (𝑥1 + 𝑥2 ) + 𝑖(𝑦1 + 𝑦2 )
Choose the correct answer from the options given below:
A) a-ii, b-iii, c-i B) a-iii, b-ii, c-i C) a-ii, b-i, c-iii D) a-i, b-iii, c-ii
2⁡+⁡⁡2𝑖
7. Assertion (A): The multiplicative inverse of 2⁡ + ⁡⁡2𝑖 is
√8
𝑍̅
Reason (R): The⁡⁡multiplicative⁡inverse⁡of⁡⁡Z⁡⁡is⁡ |𝑍|2 (Average)
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation
of the Assertion (A)
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct
explanation of the Assertion A)
(C) Assertion (A) is true and Reason (R) is false
(D) Assertion (A) is false and Reason (R) is true
8. STATEMENT 1: The modulus of 3 − 4𝑖 is 5 (Easy)
STATEMENT 2: The⁡⁡modulus⁡of⁡⁡𝑎 + 𝑏𝑖⁡⁡is⁡√a + b
2 2

(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct
explanation of the STATEMENT 1
(C) STATEMENT 1 is true and STATEMENT 2 is false
(D) STATEMENT 1 is false and STATEMENT 1 is true
9. STATEMENT 1: 𝑇ℎ𝑒⁡𝑣𝑎𝑙𝑢𝑒⁡𝑜𝑓⁡𝑖⁡(𝑖𝑜𝑡𝑎)⁡𝑖𝑠⁡(−1)1/2⁡ (Easy)
STATEMENT 2: 𝒊⁡. 𝒊⁡ = √−1√−1= 1.
(A) Both STATEMENT 1 and 2 are true
(B) Both STATEMENT 1 and 2 are false.
(C) STATEMENT 1 is true and STATEMENT 2 is false.
(D) STATEMENT 1 is false and STATEMENT 1 is true.
1 3
10. STATEMENT 1: If 𝑧 = 3 + 𝑖,then the real part of ⁡⁡𝑖𝑠
𝑧 10
1 𝑥−𝑖𝑦
STATEMENT 2: If 𝑧 = 𝑥 + 𝑖𝑦, then 𝑧 = 𝑥 2 +𝑦 2. (Average)
(A) Both STATEMENT 1 and 2 are true and STATEMENT 2 is the correct explanation of
the STATEMENT 1
(B) Both STATEMENT 1 and 2 are true and STATEMENT 2 is not the correct explanation
of the STATEMENT 1
(C) STATEMENT 1 is true and STATEMENT 2 is false.
(D) STATEMENT 1 is false and STATEMENT 1 is true.

2026-27 MATHEMATICS CET MATERIAL Page 12 of 19

11. Assertion (A): The value of 1 + i2 + i4 + i6 + … + i20 is 0 (Easy)
Reason (R): If n is multiple of 4, then⁡i = −1⁡and⁡n⁡is⁡not⁡a⁡multiple⁡⁡of⁡4⁡then⁡i = 1⁡
𝑛 𝑛

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation
of the Assertion (A)
(B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct
explanation of the Assertion A)
(C) Assertion (A) is false and Reason (R) is false
(D) Assertion (A) is false and Reason (R) is true
Diagram-Based, Matching & Statement-Type Questions
1 2 3 4 5 6 7 8 9 10
C C C B D A D A C A
11
Practice Questions
𝟐𝟒𝒛‾
1. Let 𝒛 = 𝟏 + 𝒊, where 𝒊 = √−𝟏. If 𝒛 − 𝒛𝟐 = 𝝀𝒛, then the value of 𝝀 is equal to (Average)

A) 12 B) 13 C) 18 D) 23
𝐑𝐞(𝒛) 𝐈𝐦(𝒛) 𝟑
2. The complex number 𝒛 satisfying the equation 𝟐+𝒊 + 𝟏+𝟐𝒊 = 𝟏−𝟐𝒊, is (Average)

A) ⁡6 − 5i B) ⁡−4 + 5𝑖 C) ⁡5 − 4i D) ⁡4 − 5𝑖
𝟏 𝒊 𝟏𝟔
3. If 𝜶 is a real number satisfying 𝜶𝟐 − 𝜶𝟐 = 𝟐, then the value of (𝜶 + 𝜶) is equal to

(Difficult)
A) 2048 B) 4096 C) 2024 D) 4048
𝒊
𝒂−
4. Let 𝒛 = 𝒊−𝟐𝟐, where 𝒂 is a real number and 𝒊 = √−𝟏. If 𝐈𝐦(𝐳) = 𝟎, then the value of 𝒂 is

equal to (Difficult)
A) 1 B) 2 C) 3 D) 4
5. In a complex plane, if two vertices of an equilateral triangle are at −𝟑(𝟏 + 𝒊) and
𝟑(𝟏 − 𝒊), then the area of the triangle (in sq.units) is equal to (Difficult)

A) 18 B) 9√3 C) 6√3 D) 3√3
𝟓𝒊 𝟐𝟎𝟐𝟔
6. The value of [(𝟑+𝒊)(𝟑−𝒊)] is equal to (Average)
1 1 −1 −1
A) 22026 B) 21013 C) 21013 D) 22026

7. If 𝒁|𝒁| = 𝟐𝟒 + 𝟕𝒊, where 𝒁 is a complex number, then the value of |𝐳| is equal to
(Average)
A) 5 B) 7 C) 12 D) 15
8. Given that 𝒊𝟐 = −𝟏. Then 𝒊𝟏𝟑 + 𝒊𝟏𝟒 + 𝒊𝟏𝟓 + ⋯ + 𝒊𝟐𝟎𝟐𝟔 is equal to (Easy)
A) 𝑖 − 2 B) 𝑖 + 2 C) 2𝑖 + 1 D) 𝑖 − 1

2026-27 MATHEMATICS CET MATERIAL Page 13 of 19

9. Let 𝒙 and 𝒚 be real numbers. If (𝟑 + 𝒊)𝒙 + 𝒚 + (𝟏 − 𝒊)𝒚 + 𝟑𝒊 − 𝟒 = (𝟐𝒙 + 𝟏)𝒊 + (⁡𝒙 − 𝒚 + 𝟐)𝒊,
where 𝒊 = √−𝟏, then the pair 𝒙, 𝒚 is equal to (Average)
A) (1,2) B) (0,2) C) (0, −2) D) (3,2)
𝟓+𝟕𝒊 𝟑+𝟐𝒊 𝟏+𝟏𝟏𝒊
10. Let 𝒛𝟏 = 𝟕−𝟓𝒊 , 𝒛𝟐 = 𝟑−𝟐𝒊 and 𝒛𝟑 = 𝟏𝟏−𝒊 . Then 𝒛𝟏 𝒛𝟏 + 𝒛𝟐 𝒛𝟐 + 𝒛𝟑 𝒛𝟑 is equal to
(Difficult)
A) 2 B) 1 + 2𝑖 C) 1 D) 3
(𝟏+𝒊)𝒏
11. The value of (𝟏−𝒊)𝒏−𝟒 , where 𝒊 = √−𝟏 and 𝒏 is an integer, (Average)

𝑖𝑛
A) 4 B) 4𝑖 𝑛 C) −4𝑖 𝑛 D) -1

12. Given that 𝒊𝟐 = −𝟏. Then (𝒊𝟏 )(𝒊𝟐 )(𝒊𝟑 ) … (𝒊𝟐𝟎𝟐𝟔 ) is equal to (Average)

A) -1 B) 1 C) 2𝑖 D) – 𝑖
13. Given that 𝒊𝟐 = −𝟏. If 𝒛𝟏 = (𝟕 + 𝒊√𝟓)𝟐 + (𝟕 − 𝒊√𝟓)𝟐 and 𝒛𝟐 = (𝟑 + 𝟐𝒊)𝟑 − (𝟑 − 𝟐𝒊)𝟑, then
(Average)
A) 𝑍1 is a purely imaginary number and 𝑍2 is a purely real number
B) ⁡𝑍1 is a purely real number and 𝑍2 is a purely imaginary number
C) both 𝑍1 and 𝑍2 are purely imaginary numbers
D) both 𝑧1 and 𝑧2 are purely real numbers

14. If 𝒛𝟏 = 𝟏 + 𝟑𝒊, 𝒛𝟐 = −𝟑𝒊 + 𝟓, then (𝒛𝟏 𝒛𝟐 + 𝒛𝟐 𝒛𝟏 ) + ((𝒛𝟏 𝒛𝟐 + 𝒛𝟐 𝒛𝟏 ) is equal to (Difficult)

A) -16 B) 1 + 𝑖 C) 1 D) 1 − 𝑖
𝐜𝐨𝐬⁡𝜶+𝐢𝐬𝐢𝐧⁡𝜶 𝟏𝟎𝟎𝟎 𝐬𝐢𝐧⁡𝜶+𝐢𝐜𝐨𝐬⁡𝜶 𝟐𝟎𝟎𝟎
15. |𝐬𝐢𝐧⁡𝜶−𝐢𝐜𝐨𝐬⁡𝜶| + |𝐜𝐨𝐬⁡𝜶−𝐢𝐬𝐢𝐧⁡𝜶| ⁡ is equal to (Average)

A) 2 B) 5 C) 4 D) 16
16. If |𝒛 + 𝟒| = 𝟐|𝒛 + 𝟏|, where ⁡𝒛 is a complex number, then |𝒛| is equal to (Average)
A) 0 B) 2 C) 4 D) 8
17. If 𝒛(𝟑 − 𝒊) = 𝟐 + 𝒊, then 𝒛𝟐 = (Average)
𝑖 −𝑖 1 −1
A) 2 B) 2 C) 2 D) 2
𝟏−𝒊√𝟑
18. The imaginary part of is (Average)
𝟏+𝒊√𝟑

−1 1 √3 −√3
A) 2 B) 2 C) 2 D) 2

19. The sum of 𝒊 + 𝒊 + ⋯ upto 25 terms is equal to
𝟐 𝟒
(Easy)

A) 0 B) 𝑖 C) – 𝑖 D) -1
20. The equation 𝐥𝐦(𝟏 − 𝒊)𝒛 = 𝟏 represents the line (Average)
A) 𝑦 = 𝑥 + 1 B) 𝑦 = 1 − 𝑥 C) 𝑦 = 𝑥 − 1 D) 𝑦 = 𝑥 + 2

2026-27 MATHEMATICS CET MATERIAL Page 14 of 19

𝐜𝐨𝐬⁡𝟓𝟎∘ +𝒊𝐬𝐢𝐧⁡𝟓𝟎∘
21. The imaginary part of 𝐜𝐨𝐬⁡𝟓𝟎∘ −𝒊𝐬𝐢𝐧⁡𝟓𝟎∘ is equal to (Average)

A) cos⁡10∘ B) ⁡sin⁡80∘ C) ⁡cos⁡50∘ D) sin⁡40∘
𝟑 𝟑
√𝟑 𝒊 √𝟑 𝒊
22. The value of ( 𝟐 + 𝟐) + ( 𝟐 − 𝟐) is equal to (Easy)

A) 3√3 B) 2√3 C) 2 D) 0
23. If the complex numbers (𝟐 + 𝒊)𝒙 + (𝟏 − 𝒊)𝒚 + 𝟐𝒊 − 𝟑 and 𝒙 + (−𝟏 + 𝟐𝒊)𝒚 + 𝟏 + 𝒊 are equal,
then (𝒙, 𝒚) is (Average)
A) (1, −2) B) (−1,2) C) (2, −1) D) (2,1)
𝟑+𝟒𝒊
24. If 𝒙 + 𝒊𝒚 = 𝟓−𝟏𝟐𝒊, then 𝒙 + 𝒚 is equal to (Easy)
23 56 15 15
A) 169 B) 169 C) − 169 D) 169
𝒛−𝟓𝒊
25. If |𝒛+𝟓𝒊| = 𝟏, then (Average)

A) Re(𝑧) = 0 B) ⁡|𝑧| = 10 C) |𝑧| = 25 D) ⁡Im(𝑧) = 0
𝟐+𝒊
26. If the complex number 𝝀+𝒊 lies on the line 𝒚 = 𝒙 of the first quadrant, then the value

of 𝝀 is equal to (Difficult)
A) 3 B) -3 C) 2 D) -2
27. Let 𝒛 = 𝒙 + 𝒊𝒚, where 𝒚 > 𝟎. If 𝒛 + 𝒛‾ = 𝟔 and |𝒛| + |𝒛‾| = 𝟏𝟎, then 𝒛 = (Difficult)
A) 3 + 2𝑖 B) 3 + 5𝑖 C) 3 + 3𝑖 D) 3 + 4𝑖
28. If the complex number 𝟐 + 𝒊 is rotated through an angle 𝟗𝟎∘ in the anti-clockwise
direction about the origin in the complex plane, then the resulting complex
number is (Difficult)
A) 2 − 𝑖 B) 1 + 2𝑖 C) −1 + 2𝑖 D) −2 + 𝑖
𝟏𝟎𝒊 𝟐𝟎𝟐𝟒
29. The value of ((𝟐−𝒊)(𝟑−𝒊)) is equal to (Average)

1 2024
A) 22024 B) 21012 C) 42024 D) (2)
𝟐−𝜶𝒊
30. The value of 𝜶 for which the complex number 𝜶−𝒊 is purely imaginary, is (Easy)

A) 2 B) -2 C) 1 D) 0
31. The centre of a square is at the origin of the complex plane. If one of the vertices is
at −𝟑𝒊, then the area of the square is (Difficult)
A) 9 B) 12 C) 18 D) 24
(𝟏+𝒊)𝟏𝟎 (𝟐−𝒊)𝟔
32. The modulus of the complex number is equal to (Average)
(𝟐𝒊−𝟒)𝟒

A) 8 B) 10 C) 16 D) 30

2026-27 MATHEMATICS CET MATERIAL Page 15 of 19

𝟏+𝒊 𝟐+𝒊
33. Real part of (𝟏−𝒊) (𝟐−𝒊) is (Easy)
3 3 4 4
A) 5 B) − 5 C) 5 D) − 5
𝟏𝟔
34. Let 𝒛 be a non-zero complex number such that 𝒛 = 𝒛‾ . Then the locus of 𝒛 is
(Average)
A) a straight line B) a parabola
C) an ellipse D) a circle with centre at the origin
𝟏+(𝒂−𝒊𝒃)
35. If 𝒂𝟐 + 𝒃𝟐 = 𝟏, then 𝟏+(𝒂+𝒊𝒃) = is equal to (Average)

A) 𝑎 − 𝑖𝑏 B) 𝑎 + 𝑖𝑏 C) −𝑎 + 𝑖𝑏 D) −𝑎 − 𝑖𝑏
𝟏+𝒊 𝟐𝟎𝟐𝟒
36. |( ) |= (Easy)
√𝟐

A) 4 B) 21012 C) 1 D) √2

37.   i n + i n+1  =
24
(Easy)
n=1

A) 1 + 𝑖 B) 𝑖 C) 1 − 𝑖 D) 0
𝟏+𝒛
38. If 𝒛 is a complex number of unit modulus, then |𝟏+𝒛‾| equals (Average)
1
A) 2 B) 1 C) 2 D) 4

39. Let 𝒛 be a complex number satisfying |𝒛 + 𝟏𝟔| = 𝟒|𝒛 + 𝟏|. Then (Average)
A) ⁡|𝑧| = 2 B) |𝑧| = 4 C) |𝑧| = 8 D) |𝑧| = 10
40. If 𝟐𝒛 = 𝟕 + 𝒊√𝟑, then the value of 𝒛𝟐 − 𝟕𝒛 + 𝟒 is (Average)
39 39
A) − 4 B) 4 C) -9 D) 17

𝟏−𝒊 𝟏𝟎
41. If (𝟏+𝒊) = 𝒂 + 𝒊𝒃, then the values of 𝒂 and 𝒃 are, respectively, (Easy)

A) 1 and 0 B) 0 and 1 C) -1 and 0 D) 0 and - 1
𝒛 −𝒛
42. If 𝒛𝟏 and 𝒛𝟐 are two complex numbers with |𝒛𝟏 | = 𝟏, then |𝟏−𝒛
𝟏 𝟐
| is equal to (Average)
𝒛‾ 𝟏 𝟐

1 1
A) 0 B) 4 C) 2 D) 1

 -1  = ______
4
43. 
2n
(Easy)
n=1

A) 2 B) −i C) 0 D) 1
44. If Z1 and Z 2 are two non-zero complex numbers, then which of the following is not

true? (Average)

A) Z1  Z 2  Z1  Z 2 B) Z1Z 2  Z1 Z 2

C) Z1 Z 2  Z1.Z 2 D) Z1  Z 2  Z1  Z 2

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𝟏
45. Let 𝒛 = 𝟏 + 𝒊 . Then the value of 𝒛𝟒 is equal to (Average)

A) 4 B) -4 C) 1 − 𝑖 D) 1 + 𝑖
46. The modulus of the complex number (𝟐√𝟐 + 𝒊𝟐√𝟐)𝟐 is equal to (Average)
A) 16 B) 4 C) 32 D) 8
47. If 𝒛 + 𝒛‾ = 𝟔 and 𝒛 − 𝒛‾ = 𝟒𝒊, then |𝒛|𝟐 = (Average)
A) 36 B) 16 C) 15 D) 13
𝟐−𝒊
48. Let 𝐳 = 𝜶+𝒊, where 𝜶 is a real number. If 𝟒𝐑𝐞(𝒛) = 𝟑𝐈𝐦(𝒛‾) then the value of 𝜶 is
(Difficult)
A) 5 B) -5 C) 3 D) 2
49. Let 𝒔, 𝒕, 𝒓 be non-zero distinct positive real numbers. If the complex number
𝒛 = 𝒙 + 𝒊𝒚 satisfies 𝒔𝒛 + 𝒕𝒛‾ + 𝒓 = 𝟎, then z lies on (Difficult)
A) imaginary axis B) real axis C) 𝑦 = 𝑥 D) 𝑦 = 2𝑥
50. Let 𝒛 = 𝒙 + 𝒊𝒚 be a complex number, where 𝒊 = √−𝟏 is the complex unit.
Then ∣ 𝒛 − 𝟏 + 𝒊 ∣= 𝟓 is a circle with (Difficult)
A) centre at (-1,1) and radius 5 B) centre at (1,1) and radius √5
C) centre at (−1, −1) and radius √5 D) centre at (1, −1) and radius 5
51. Let 𝒛 be a complex number such that 𝒛𝟑 + 𝒊𝒛𝟐 − 𝒊𝒛 + 𝟏 = 𝟎 where 𝒊𝟐 = −𝟏.
Then |𝐳| = (Average)
1 1
A) 2 B) 2 C) 1 D) 4
𝟐𝝅 𝟐𝝅
𝟏+𝐬𝐢𝐧⁡ −𝒊𝐜𝐨𝐬⁡
52. Real part of 𝟐𝟕
𝟐𝝅
𝟐𝟕
𝟐𝝅 is equal to (Average)
𝟏+𝐬𝐢𝐧⁡ +𝒊𝐜𝐨𝐬⁡
𝟐𝟕 𝟐𝟕

2𝜋 2𝜋 2𝜋 2𝜋
A) ⁡cos⁡27 B) ⁡sin⁡27 C) 1 + sin⁡27 D) 1 + cos⁡27
𝟑𝝅
53. If 𝒛 = 𝟏 + 𝐢𝐭𝐚𝐧⁡𝜽, where 𝝅 < 𝜽 < 𝟐 , then |𝒛| is equal to (Difficult)

A) 1 + tan⁡𝜃 B) 2tan⁡𝜃 C) sec⁡𝜃 D) −sec⁡𝜃
𝟑+𝒊
54. If 𝒁 = 𝟐−𝒊 then 𝒁−𝟏 is equal to (Average)
1+𝑖 1−𝑖
A) 1 + 𝑖 B) C) D) 2(1 − 𝑖)
2 2

55. If 𝒛 is a complex number, then the minimum value of |𝒛 − 𝟐| + |𝒛 − 𝟒| is (Average)

A) √20 B) 2 C) 6 D) √6
𝟏
56. The point 𝒛 = (𝟏 + 𝒊) in the complex plane is rotated about the origin through an
√𝟐
𝝅
angle 𝟒 in the clockwise direction, then the new position of 𝒁 is (Difficult)
1 1
A) 2 B) 1 C) (1 − 𝑖) D) 2 (1 − 𝑖)
√2

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57. If 𝒛 = 𝟐 + 𝒊, 𝒊𝟐 = −𝟏, then the value of 𝒛𝟐 − 𝟒𝒛 + 𝟏𝟓 (Difficult)
A) 2 B) 6 C) 10 D) 12
𝒊 𝟐
58. The modulus of the complex number ( − ) is equal to (Average)
𝟐 𝒊
2 5 2 3
A) 5 B) 2 C) 3 D) 2

59. If the complex number 𝒛 varies so that the real and imaginary parts of 𝒛 − 𝟐 − 𝟑𝒊 are
equal, then the locus of 𝒁 is (Difficult)
A) a circle B) a straight line C) a parabola D) an ellipse
60. If 𝒌 = 𝟒𝒏 + 𝟑, where 𝒏 is an integer and 𝒊𝟐 = −𝟏 then 𝒊𝒌 is equal to (Average)
A) 0 B) 1 C) -1 D) −𝑖
61. The value of (𝟏 + 𝒊)𝟏𝟎 is equal to (Average)
A) 16 B) 16𝑖 C) 32 D) 32𝑖
62. The value of 𝒊𝟑 + 𝒊𝟒 + 𝒊𝟓 + ⋯ 𝒊𝟗𝟑 , where 𝒊 = √−𝟏, is equal to (Average)

A) 𝑖 B) 1 C) – 𝑖 D) -1
𝟑−𝟐𝒊 𝟐
63. Imaginary parts of ( 𝟐𝒊 ) is equal to (Easy)
5 −5
A) 4 B) 4 C) 3 D) -3
𝟏 √𝟑 𝟏 √𝟑
64. Let 𝒛𝟏 = 𝟐 + 𝒊 𝟐 and 𝒛𝟐 = − 𝟐 − 𝒊 𝟐 . If 𝒘 = 𝒛𝟏 + 𝒛𝟐 , then 𝒘
‾ = (Average)

A) 1 B) √3 C) 𝑖√3 D) −𝑖√3
𝝀+𝒊
65. All the points in 𝑨 = {𝝀−𝒊 ; 𝝀 ∈ ℝ} lie on (Difficult)

A) a circle with radius √2 B) a circle with radius 2
1
C) a circle with radius 2 D) a circle with radius 1
2027
66.  i n 1 + i , i 2 = -1 , is equal to (Easy)
n=1

A) ⁡𝑖 + 1 B) 𝑖 − 1 C) −𝑖 − 1 D) ⁡−𝑖 + 1
67. If 𝒙, 𝒚 ∈ ℝ and 𝒙 + 𝒊𝒚 = −(𝟔 + 𝒊)𝟑 , 𝒊𝟐 = −𝟏, then 𝒙 − 𝒚 is equal to (Easy)
A) 93 B) -93 C) 91 D) -91
68. Let 𝒛 = 𝒙 + 𝒊𝒚, where 𝒙, 𝒚 ∈ ℝ and 𝒊𝟐 = −𝟏. If |𝒛 − 𝒊| = |𝒛 − 𝟏|, then 𝒚 = (Easy)
A) 𝑥 B) 𝑥 + 1 C) −𝑥 − 1 D) 𝑥 + 2

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Practice Questions
1 2 3 4 5 6 7 8 9 10
B D B A B D A D B D
11 12 13 14 15 16 17 18 19 20
C D B A A B A D D A
21 22 23 24 25 26 27 28 29 30
B D D A D B D C B D
31 32 33 34 35 36 37 38 39 40
C B D D A C D B B C
41 42 43 44 45 46 47 48 49 50
C D C D B A D D B D
51 52 53 54 55 56 57 58 59 60
C B D C B B C B B D
61 62 63 64 65 66 67 68
D B C D D C D A

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