Q: Let R denote the set of all real numbers. Let z_1=1+2 i and z_2=3 i be two complex numbers, where i= -1 . Let S= \ (x, y) R R : |x+i y-z_1 |=2 |x+i y-z_2 | \ . Then which of the following statements i

aligned & aligned & |x+i y-1-2 i|=2|x+i y-3 i| \\ & (x-1)^2+(y-2)^2=4 (x^2+(y-3)^2 ) \\ & 3 x^2+3 y^2+2 x-20 y+31=0 \\ & x^2+y^2+ 2 x 3 - 20 y 3 + 31 3 =0 aligned \\ & S is a circle with centre (- 1 3 , 10 3 ) and radius = 1 9 + 100 9 - 31 3 = 8 9 = 2 2 3 aligned

Q: For a non-zero complex number z, let (z) denote the principal argument of z, with - < (z) . Let be the cube root of unity for which 0 < ( ) < . Let = ( _ n=1 ^ 2025 (- )^n ). Then the value of 3 is __

aligned & = (- + ^2- ^3+ +(- )^ 2025 ) \\ & = ( - ((- )^ 2025 -1 ) - -1 ) \\ & = ( - - -1 (-2) ) \\ & = ( -2 +1 ) \\ & = ( -2 - ^2 ) \\ & = ( 2 ) \\ & = (2 ^2 ) \\ & = -2 3 \\ & 3 =-2 aligned

Q: Let S=\ a+b 2 : a, b Z \ , T_1= \ (-1+ 2 )^n: n N \ , and T_2= \ (1+ 2 )^n: n N \ . Then which of the following statements is (are) TRUE?

(1) aligned & (-1+ 2 )^ n = m + 2 n , m , where, n Z \\ & (1+ 2 )^ n = m _1+ 2 n _1, where, ~m _1, n _1 Z \\& Z T _1 T _2 S aligned but b 2 S for negative b Z . So Z T _1 T _2 S (2) aligned & ( 2 -1)^ n = 1 ( 2 +1)^ n 2024, where, n N \\ & T _2 (2024, ) aligned (4) ( (a+b 2 )=0) b=0, where, a Z Options (1), (3), (4) are Correct.

Q: Let A 1 , A 2 , A 3 , … , A 8 be the vertices of a regular octagon that lie on a circle of radius 2 . Let P be a point on the circle and let P A i denote the distance between the points P

Given, A 1 , A 2 , A 3 , … , A 8 vertices of a regular octagon lying on a circle of radius 2 . Now using the concept of n t h root of unity, Let any point P be, Z = ( 2 ) ( 1 ) 1 / 8 ⇒ Z 8 = 2 8 × 1 ⇒ Z 8 − 2 8 = 0 ⇒ Z = 2 , 2 α , 2 α 2 , 2 α 3 , … , 2 α 7 ; here α = e i 2 π 8 ⇒ Z 8 − 2 8 = ( Z − 2 ) ( Z − 2 α ) Z − 2 α 2 Z − 2 α 3 &#

Q: Let A = 1967 + 1686 i sin θ 7 - 3 i cos θ : θ ∈ R . If A contains exactly one positive integer n , then the value of n is

Let z = 1967 + 1686 i sin θ 7 - 3 i cos θ is a positive integer. ⇒ z = 1967 + 1686 i sin θ 7 + 3 i cos θ 7 - 3 i cos θ 7 + 3 i cos θ ⇒ z = 1967 × 7 - 1686 × 3 sin θ cos θ + i 1686 × 7 sin θ + 1967 × 3 cos θ 49 + 9 cos 2 θ Now taking imaginary part as zero we get, 1686 × 7 sin θ + 1967 × 3 cos &

Q: Let z be a complex number satisfying z 3 + 2 z 2 + 4 z ¯ - 8 = 0 , where z ¯ denotes the complex conjugate of z . Let the imaginary part of z be nonzero. Match each entry in List-I to the co

Given, z 3 + 2 z 2 + 4 z ¯ - 8 = 0 . . . . . . . 1 Now on taking conjugate both side we get, z 3 + 2 z ¯ 2 + 4 z - 8 = 0 . . . . . . . . 2 Note z ¯ = z Now subtracting both equations we get, 2 z 2 - z ¯ 2 + 4 z ¯ - z = 0 ⇒ z - z ¯ 2 z + z ¯ - 4 = 0 ∵ z = z ¯ (not possible) or 4 x = 4 ⇒ x = 1 as z + z ¯ = 2 x So, z = 1 + λ i ⇒ z = 1 + λ 2 & z &#17

Q: Let z ¯ denote the complex conjugate of a complex number z . If z is a non-zero complex number for which both real and imaginary parts of z ¯ 2 + 1 z 2 are integers, then which of the follow

Let z = r . e i θ So, z ¯ 2 + 1 z 2 = r 2 + 1 r 2 e - 2 i θ = a + i b (say), given a , b ∈ Z Now taking mod both side and squaring we get, r 2 + 1 r 2 2 = a 2 + b 2 ⇒ r 8 - a 2 + b 2 - 2 r 4 + 1 = 0 ⇒ r 4 = a 2 + b 2 - 2 ± a 2 + b 2 - 2 2 - 4 2 Now comparing from option we get, for a 2 + b 2 = 45 (i.e a , b = ± 6 , ± 3 or ± 3 , ± 6 We get r = 43 + 3 205 2 1 4

Q: Let z be a complex number with non-zero imaginary part. If 2 + 3 z + 4 z 2 2 - 3 z + 4 z 2 is a real number, then the value of z 2 is ______.

Let α = 4 z 2 + 3 z + 2 4 z 2 - 3 z + 2 = 1 + 6 z 4 z 2 - 3 z + 2 ⇒ α = 1 + 6 2 2 z + 1 z - 3 ∵ α ∈ R then 2 z + 1 z ∈ R ⇒ 2 z + 1 z = 2 z ¯ + 1 z ¯ ⇒ 2 z - z ¯ - z - z ¯ z 2 = 0 ⇒ z - z ¯ 2 - 1 z 2 = 0 ∵ z ≠ z ¯ (given) So z 2 = 1 2

Q: Let z ¯ denote the complex conjugate of a complex number z and let i = - 1 . In the set of complex numbers, the number of distinct roots of the equation z ¯ - z 2 = i z

Given z ¯ 1 - i = z 2 1 + i So z ¯ 1 - i = z 2 1 + i ⇒ z = z 2 ⇒ z = 0 or z = 1 Also, z ¯ 1 - i = z 2 1 + i ⇒ z ¯ 1 - i 1 + i = z 2 ⇒ - i z ¯ = z 2 Let z = x + i y ⇒ x 2 - y 2 - 2 x y i = y - i x i.e. x 2 - y 2 = y and - 2 x y = - x From - 2 x y = - x , we get x = 0 or y = - 1 2 i.e. the points are 0 , 0 , 0 , 1 , 3 2 , - 1 2 and - 3 2 , -

Q: Let θ 1 , θ 2 , … , θ 10 be positive valued angles (in radian) such that θ 1 + θ 2 + ⋯ + θ 10 = 2 π . Define the complex numbers z 1 = e i θ 1 , z k

Given, θ 1 + θ 2 + . . . . . + θ 10 = 2 π , z 1 = e i θ 1 , z k = z k - 1 e i θ k Now, z k = z k - 1 e i θ k ⇒ z 2 = z 1 e i θ 1 , z 3 = z 2 e i θ 2 , … … As, z 1 = e i θ 1 ⇒ z 1 = 1 z 2 = z 1 e i θ ⇒ z 2 = z 1 = 1 Similarly z 1 = z 2 … = z k = 1 ⇒ z 1 , z 2 , … , z k lies on a circle of unit radius. We know, z k - z k - 1 represent

Question 1

Let R denote the set of all real numbers. Let z_1=1+2 i and z_2=3 i be two complex numbers, where i= -1 . Let S= \ (x, y) R R : |x+i y-z_1 |=2 |x+i y-z_2 | \ . Then which of the following statements i

Accepted Answer

aligned & aligned & |x+i y-1-2 i|=2|x+i y-3 i| \ & (x-1)^2+(y-2)^2=4 (x^2+(y-3)^2 ) \ & 3 x^2+3 y^2+2 x-20 y+31=0 \ & x^2+y^2+ 2 x 3 - 20 y 3 + 31 3 =0 aligned \ & S is a circle with centre (- 1 3 , 10 3 ) and radius = 1 9 + 100 9 - 31 3 = 8 9 = 2 2 3 aligned

Question 2

For a non-zero complex number z, let (z) denote the principal argument of z, with - < (z) . Let be the cube root of unity for which 0 < ( ) < . Let = ( _ n=1 ^ 2025 (- )^n ). Then the value of 3 is __

Accepted Answer

aligned & = (- + ^2- ^3+ +(- )^ 2025 ) \ & = ( - ((- )^ 2025 -1 ) - -1 ) \ & = ( - - -1 (-2) ) \ & = ( -2 +1 ) \ & = ( -2 - ^2 ) \ & = ( 2 ) \ & = (2 ^2 ) \ & = -2 3 \ & 3 =-2 aligned

Question 3

Let S=\ a+b 2 : a, b Z \ , T_1= \ (-1+ 2 )^n: n N \ , and T_2= \ (1+ 2 )^n: n N \ . Then which of the following statements is (are) TRUE?

Accepted Answer

(1) aligned & (-1+ 2 )^ n = m + 2 n , m , where, n Z \ & (1+ 2 )^ n = m _1+ 2 n _1, where, ~m _1, n _1 Z \& Z T _1 T _2 S aligned but b 2 S for negative b Z . So Z T _1 T _2 S (2) aligned & ( 2 -1)^ n = 1 ( 2 +1)^ n 2024, where, n N \ & T _2 (2024, ) aligned (4) ( (a+b 2 )=0) b=0, where, a Z Options (1), (3), (4) are Correct.

Question 4

Let A 1 , A 2 , A 3 , &#8230; , A 8  be the vertices of a regular octagon that lie on a circle of radius 2 . Let P be a point on the circle and let P A i denote the distance between the points P

Accepted Answer

Given, A 1 , A 2 , A 3 , &#8230; , A 8  vertices of a regular octagon lying on a circle of radius 2 . Now using the concept of  n t h  root of unity, Let any point  P  be,  Z = ( 2 ) ( 1 ) 1 / 8 &#8658; Z 8 = 2 8 &#215; 1 &#8658; Z 8 &#8722; 2 8 = 0 &#8658; Z = 2 , 2 &#945; , 2 &#945; 2 , 2 &#945; 3 , &#8230; , 2 &#945; 7 ; here&#160; &#945; = e i 2 &#960; 8 &#8658; Z 8 &#8722; 2 8 = ( Z &#8722; 2 ) ( Z &#8722; 2 &#945; ) Z &#8722; 2 &#945; 2 Z &#8722; 2 &#945; 3 &#

Question 5

Let A = 1967 + 1686 i &#160; sin &#952; 7 - 3 i &#160; cos &#952; : &#160; &#952; &#8712; R . If A contains exactly one positive integer n , then the value of n is

Accepted Answer

Let  z = 1967 + 1686 i &#160; sin &#952; 7 - 3 i &#160; cos &#952;  is a positive integer. &#8658; z = 1967 + 1686 i &#160; sin &#952; 7 + 3 i &#160; cos &#952; 7 - 3 i &#160; cos &#952; 7 + 3 i &#160; cos &#952; &#8658; z = 1967 &#215; 7 - 1686 &#215; 3 &#160; sin &#952; &#160; cos &#952; + i 1686 &#215; 7 &#160; sin &#952; + 1967 &#215; 3 &#160; cos &#952; 49 + 9 &#160; cos 2 &#952; Now taking imaginary part as zero we get, 1686 &#215; 7 &#160; sin &#952; + 1967 &#215; 3 &#160; cos &

Question 6

Let z be a complex number satisfying z 3 + 2 z 2 + 4 z &#175; - 8 = 0 , where z &#175; denotes the complex conjugate of z . Let the imaginary part of z be nonzero. Match each entry in List-I to the co

Accepted Answer

Given, z 3 + 2 z 2 + 4 z &#175; - 8 = 0 &#160; . . . . . . . 1 Now on taking conjugate both side we get, z 3 + 2 z &#175; 2 + 4 z - 8 = 0 &#160; . . . . . . . . 2 Note &#160; z &#175; = z Now subtracting both equations we get, 2 z 2 - z &#175; 2 + 4 z &#175; - z = 0   &#8658; z - z &#175; 2 z + z &#175; - 4 = 0 &#8757; &#160; z = z &#175; (not possible) or 4 x = 4 &#8658; x = 1   as &#160; z + z &#175; = 2 x So,  z = 1 + &#955; i &#8658; z = 1 + &#955; 2 &#160; &#38; &#160; z &#17

Question 7

Let z &#175; denote the complex conjugate of a complex number z . If z is a non-zero complex number for which both real and imaginary parts of z &#175; 2 + 1 z 2 are integers, then which of the follow

Accepted Answer

Let z = r . e i &#952; So, z &#175; 2 + 1 z 2 = r 2 + 1 r 2 e - 2 i &#952; = a + i b (say), given  a , b &#8712; Z Now taking mod both side and squaring we get, r 2 + 1 r 2 2 = a 2 + b 2 &#8658; r 8 - a 2 + b 2 - 2 r 4 + 1 = 0 &#8658; r 4 = a 2 + b 2 - 2 &#177; a 2 + b 2 - 2 2 - 4 2 Now comparing from option we get, for a 2 + b 2 = 45  (i.e a , b = &#177; 6 , &#177; 3 or &#177; 3 , &#177; 6 We get r = 43 + 3 205 2 1 4

Question 8

Let  z be a complex number with non-zero imaginary part. If  2 + 3 z + 4 z 2 2 - 3 z + 4 z 2  is a real number, then the value of  z 2  is ______.

Accepted Answer

Let  &#945; = 4 z 2 + 3 z + 2 4 z 2 - 3 z + 2 = 1 + 6 z 4 z 2 - 3 z + 2 &#8658; &#160; &#945; = 1 + 6 2 2 z + 1 z - 3 &#8757; &#160; &#945; &#8712; R  then  2 z + 1 z &#8712; R &#8658; 2 z + 1 z = 2 z &#175; + 1 z &#175; &#8658; 2 z - z &#175; - z - z &#175; z 2 = 0 &#8658; z - z &#175; 2 - 1 z 2 = 0 &#8757; &#160; z &#8800; z &#175;   (given) So  z 2 = 1 2

Question 9

Let z &#175;  denote the complex conjugate of a complex number  z and let i = - 1 . In the set of complex numbers, the number of distinct roots of the equation  z &#175; - z 2 = i z

Accepted Answer

Given  z &#175; 1 - i = z 2 1 + i So  z &#175; 1 - i = z 2 1 + i &#8658; z = z 2 &#8658; z = 0  or  z = 1 Also,  z &#175; 1 - i = z 2 1 + i &#8658; z &#175; 1 - i 1 + i = z 2 &#8658; - i z &#175; = z 2 Let  z = x + i y &#8658; x 2 - y 2 - 2 x y i = y - i x i.e.  x 2 - y 2 = y  and  - 2 x y = - x From  - 2 x y = - x , we get x = 0 &#160; or &#160; y = - 1 2 i.e. the points are  0 , 0 ,  0 , 1 ,  3 2 , - 1 2  and  - 3 2 , -

Question 10

Let &#952; 1 , &#952; 2 , &#8230; , &#952; 10 be positive valued angles (in radian) such that &#952; 1 + &#952; 2 + &#8943; + &#952; 10 = 2 &#960; . Define the complex numbers z 1 = e i &#952; 1 , z k

Accepted Answer

Given,  &#952; 1 + &#952; 2 + . . . . . + &#952; 10 = 2 &#960; ,  z 1 = e i &#952; 1 , &#160; z k = z k - 1 e i &#952; k Now,  z k = z k - 1 e i &#952; k &#8658; z 2 = z 1 e i &#952; 1 , &#160; z 3 = z 2 e i &#952; 2 , &#160; &#8230; &#8230; As,  z 1 = e i &#952; 1 &#8658; z 1 = 1 z 2 = z 1 e i &#952; &#8658; z 2 = z 1 = 1 Similarly  z 1 = z 2 &#8230; = z k = 1 &#8658; z 1 , z 2 , &#8230; , z k  lies on a circle of unit radius. We know,  z k - z k - 1 represent

Question 11

For any complex number w = c + i d , let  arg w &#8712; ( - &#960; , &#960; ] , where i = - 1 . Let &#945; and &#946; be real numbers such that for all complex numbers z = x + i y satisfying arg

Accepted Answer

arg z + &#945; z + &#946; = &#960; 4  represents an arc of a circle and  - &#945; , 0 &#160; &#38; &#160; - &#946; , 0  subtends an angle  &#960; 4  on  z Given,  x 2 + y 2 + 5 x - 3 y + 4 = 0 Centre - 5 2 , 3 2 , &#160; r = 25 4 + 9 4 - 4 = 3 2 Angle made by  ( - &#945; , 0 ) &#160; &#38; &#160; ( - &#946; , 0 )  at the centre  C  is  &#960; 2 Using rotation of complex numbers, we get - &#945; - - 5 2 + 3 2 i - &#946; - - 5 2 + 3 2 i =

Question 12

For a complex number z , let  Re z denote the real part of  z . Let S be the set of all complex numbers z satisfying z 4 - | z | 4 = 4 i z 2 ,  where i = - 1 . Then the minimum possible

Accepted Answer

z 4 - | z | 4 = 4 i z 2 &#8658; z 4 - z z &#175; 2 = 4 i z 2 &#8658; &#160;&#160; z 2 z 2 - z &#175; 2 = 4 i z 2 &#160;&#160; &#8658; &#160;&#160; z 2 - z &#175; 2 = 4 i &#8658; &#160;&#160; z + z &#175; z - z &#175; = 4 i &#8658; &#160;&#160; z + z &#175; 2 z - z &#175; 2 i = 1 &#160;&#160; &#8658; &#160;&#160; x y = 1 for  z 1 &#160; &#38; &#160; z 2 &#160;&#160; &#8658; &#160;&#160; x 1 y 1 = 1 and  x 2 y 2 = 1 x 1 &#160; &#38; &#160; x 2 are of opposite sign, similarly  y 1 &n

Question 13

Let S be the set of all complex numbers z satisfying z 2 + z + 1 = 1 . Then which of the following statements is/are TRUE?

Accepted Answer

z 2 + z + 1 = e i &#952; &#8195; &#952; &#8712; ( &#8722; &#960; , &#160; &#960; ] &#8658; z 2 + z + 1 &#8722; e i &#952; = 0 By quadratic formula, we get z = &#8722; 1 &#177; 4 e i &#952; &#8722; 3 2 &#160; . . . . . . . . . . . . . 1 &#8658; z + 1 2 = &#177; 4 cos &#952; &#8722; 3 + i 4 sin &#952; &#8658; z + 1 2 = 4 cos &#952; &#8722; 3 2 + 4 sin &#952; 2 1 4 &#8658; z + 1 2 &#8712; 1 , &#160; 7 Option (C) is correct By equation  1 , 2 z &#8804; 1 + 4 e i &#952; &#8722; 3 &#8658; 2 Z &#8

Question 14

Let S be the set of all complex numbers z satisfying z - 2 + i &#8805; 5 . If the complex number z 0 is such that 1 z 0 - 1 is the maximum of the set 1 z - 1 : z &#8712; S , then the principal argumen

Accepted Answer

The region represented by z - 2 + i ≥ 5 will be region outside and on circle with centre 2 , - 1 and radius 5 . z - 2 + i ≥ 5 Let z = x + i y ⇒ x + i y - 2 + i ≥ 5 ⇒ x - 2 + i y + 1 ≥ 5 ⇒ x - 2 2 + y + 1 2 ≥ 5 ⇒ x - 2 2 + y + 1 2 ≥ 5 Now, for 1 z 0 - 1 to be maximum, z 0 - 1 must be minimum. We need to find the B z 0 which is in given region and nearest to point A 1 , 0 , hence nearest point from A 1 , 0 will be on the line joining A and C . Method 1 Let z 0 = x + i y , then x 2 and y > 0 (from

Question 15

Let &#969; &#8800; 1 be a cube root of unity. Then the minimum of the set a + b &#969; + c &#969; 2 2 : a , &#8201; b , c distinct non &#8211; zero integers equals ________

Accepted Answer

If z is a complex number Then | z | 2 = z z &#175; Now, a + b &#969; + c &#969; 2 2 = a + b &#969; + c &#969; 2 a + b &#969; + c &#969; 2 &#175; a + b &#969; + c &#969; 2 2 = a + b &#969; + c &#969; 2 a + b &#969; - + c &#969; - 2 i f &#8201; a &#8712; R &#8658; a - = a z 1 z 2 &#175; = z - 1 z - 2 &#969; - = &#969; 2 , &#8201;&#8201; &#969; - 2 = &#969; a + b &#969; + c &#969; 2 2 = a + b &#969; + c &#969; 2 a + b &#969; 2 + c &#969; = a 2 + b 2 + c 2 - a b - b c - c a &#8757;&#8201;&#8201; &#9

Complex Number — Previous Year Questions & Solutions