If x = 2 + 5 i then the value of x 3 − 5 x 2 + 33 x − 19 is equal to
Let z 1 , z 2 be two complex numbers such that z 1 ≠ 0 and z 2 / z 1 1 is purely real, then 2 i z 1 + 5 z 2 2 i z 1 − 5 z 2 is equal to
If Z be a complex number satisfying Z 4 + Z 3 + 2 Z 2 + Z + 1 = 0 , then set of all possible value of Z is
Let A , B be two sets, A = { z / | z + 6 | + | z − 2 | = 10 } , B = { z / | z − 6 | = 3 } , then the number of elements common to both A and B is
Let i = − 1 , define a sequence of a complex number by z 1 = 0 , z n + 1 = z n 2 + i for n ≥ 1 .In the complex plane, how far from the origin z 111 ?
Let ω = − 1 2 + 3 2 i Then the value of the determinant Δ = 1 1 1 1 − 1 − ω 2 ω 2 1 ω 2 ω 4 is
Let z 1 , z 2 , z 3 be three complex numbers such that z 1 = z 2 = z 3 = 1 and z = z 1 + z 2 + z 3 1 z 1 + 1 z 2 + 1 z 3 , than | z | cannot exceed
If z 1 , z 2 , z 3 be three complex number such than z 1 + z 2 + z 3 = 0 and z 1 = z 2 = z 3 = 1 than z 1 2 + 2 z 2 2 + z 3 2 equals
If z ( 1 + a ) = b + ic and a 2 + b 2 + c 2 = 1 then ( 1 + iz ) ( 1 − iz ) =
If the equation, x 2 + b x + 45 = 0 , ( b ∈ R ) has conjugate complex roots and they satisfy | z + 1 | = 2 10 then:
Let z ∈ C , if A = z : arg ( z ) = π 4 and B = z : arg ( z – 3 – 3 i ) = 2 π 3 then n ( A ∩ B ) is equal to
If x=a+ib is a complex number such that x 2 = 3 + 4 i a n d x 3 = 2 + 11 i w h e r e i = – 1 , t h e n ( a + b ) =
It is given that complex numbers z 1 and z 2 satisfy z 1 = 2 and z 2 = 3 .If the included angle of their corresponding vectors is 60 ∘ then z 1 + z 2 z 1 − z 2 can be expressed as N 7 , where N is a natural number, then N =
I f x = 3 + i t h e n t h e v a l u e o f x 3 – 3 x 2 – 8 x + 15
If 1 + 2 z 2 = z 2 + 1 2 + 2 z 2 + 1 2 ,then the calue of z z + 1 is
If z − 4 z = 2 then the maximum value of | z | is equal to
If | z | ≤ 4 , then the maximum value of | i z + 3 − 4 i | is equal to
The sequence S = i + 2 i 2 + 3 i 3 + 4 i 4 + ⋯ up to 100 terms simplifies to, where i = − 1
If 1 + x + x 2 n = a 0 + a 1 x + a 2 x 2 + … + a 2 n x 2 n , then value of a 0 + a 3 + a 6 + … is
Let z 1 and z 2 be two non-zero complex numbers such that z 1 z 2 + z 2 z 1 = 1 , then the origin and points represented by z 1 and z 2
If ( x + i y ) 1 / 3 = a + i b , then x a + y b equals
Let z = 1 1 − 2 i 3 + 5 i 1 + 2 i − 5 10 i 3 − 5 i − 10 i 11 , then
If 1 + i 1 − i n = − 1 , n ∈ N then least value of n is
If x 2 + y 2 = 1 and x ≠ − 1 then 1 + y + i x 1 + y − i x equals
If | z | = z + 3 − 2 i , then z equals
Principal argument of z = i − 1 i 1 cos 2 π 7 + sin 2 π 7 is
If x + i y = a + i b c + i d , then x 2 + y 2 2 c 2 + d 2 equal
All the roots of ( z + 1 ) 4 = z 4 lie on
Suppose a , b , c ∈ C , and | a | = | b | = | c | = 1 and a b c = a + b + c then b c + c a + a b is equal to
Let z , w be two complex numbers such that z ¯ + i w ¯ = 0 and a r g ( z w ) = π , then a r g z equal
The number of complex numbers satisfying z ¯ = i z 2 is
If z = 1 , z ≠ 1 , then value of a r g 1 1 – z cannot exceed
If 3 49 ( x + i y ) = 3 2 + i 2 3 100 , y ∈ N and x = k y , then value of k is
If z ≠ 1 , z 2 z – 1 is real, then point represented by the complex number z lies
The system of equations | z + 1 − i | = 2 and | z | = 3 has
If ( 4 + i ) ( z + z ¯ ) − ( 3 + i ) ( z − z ¯ ) + 26 i = 0 , then the value of z 2 is
If ω = 2 , then the set of points x + i y = ω – 1 ω lie on
Suppose a ∈ R and the equation z + a | z | + 2 i = 0 has no solution in C , then α satisfies the relation.
Let z ∈ C be such that R e ( z 2 ) = 0 , then
If a > 0 and z | z | + a z + 3 i = 0 ,
If x + i y = 3 cos θ + i sin θ + 2 than 4 x − x 2 − y 2
Area of a triangle with vertices given by z , i z z + i z , where z is a complex number, is
Let z be a complex number such that Z = 2 then maximum possible value of z + 2 z is
If a , b , x , y ∈ R , ω ≠ 1 , is a cube root of unity and ( a + b ω ) 7 = x + y ω , then ( b + a ω ) 7 equals
5 + i sin θ 5 − 3 i sin θ is a real number when
The number of solutions of z 2 + | z | = 0 is
If the complex numbers z 1 , z 2 and z 3 represent the vertices of an equilateral triangle such that z 1 = z 2 = z 3 , then
suppose z 1 , z 2 , z 3 represently the vertices A , B , and C respectively of a ∆ A B C with centroid at G . If the mid point of AG is the origin, then,
If z = i ( i + 2 ) ,then value of z 4 + 4 z 3 + 6 z 2 + 4 z is
If α , β , γ are the cube roots of p , p < 0 , then for any x, y and z which does not make denominator zero, the expression x α + y β + z γ x β + y γ + z α
If z 1 and z 2 are two complex numbers and a , b are two real numbers, then a z 1 − b z 2 2 + b z 1 + a z 2 2 equals
Number of complex numbers such that | z | = 1 and | z = 1 − 2 z is
If the imaginary part of 2 z + 1 i z + 1 is – 4 , then the locus of the point representing z in the complex plane is
Suppose that three points z 1 , z 2 , z 3 are connected by the relation a z 1 + b z 2 + c z 3 = 0 , where a + b + c = 0 then the points are
If ( a + b i ) 11 = x + i y , where a , b , x , y ∈ R , then ( b + a i ) 11 equals
If Δ = 6 i − 3 i 1 4 3 i − 1 20 3 i = x + i y , then
If | z | = 1 and w = z − 1 z + 1 ( where z ≠ − 1 ) , then R e ( w ) equals
If w ≠ 1 is a cube root of unity, then 1 , ω , ω 2
If a and b are real numbers between 0 and 1 such that the points z 1 = a + i , z 2 = 1 + b i and z 3 = 0 form an equilateral triangle, then
If z 1 = z 3 = z 3 = 1 then value of z 2 − z 3 2 + z 3 − z 1 2 + z 1 − z 2 2 cannot exceed
Let z and w be two complex numbers such that | z | = | w | = 1 and | z + i w | = | z − i w ¯ | = 2 . Then z equals
If z 1 and z 2 are two complex numbers such that z 1 − z 2 z 1 + z 2 = 1 , then
If z 1 , z 2 are two complex numbers such that z 1 = z 2 = 2 and z 1 + z 2 = 3 than z 1 − z 2 equals;
The real part of z = 1 1 − cos θ + i sin θ is
If z 2 + z + 1 = 0 , where z is a complex number, then values of S = z + 1 z 2 + z 2 + 1 z 2 2 + z 3 + 1 z 3 2 + ⋯ + z 6 + 1 z 6 2 is
The number of values of θ ∈ ( 0 , π ] , such that ( cos θ + i sin θ ) ( cos 3 θ + i sin 3 θ ) ( cos 5 θ + i sin 5 θ ) ( cos 7 θ + i sin 7 θ ) ( cos 9 θ + i sin 9 θ ) = − 1 is
If n ∈ N , then ( 1 + i ) n ( 1 − i ) n − 2 is equal to-
Let ω ≠ 1 , be a cube root of unity, and f : I C be defined by f ( n ) = 1 + ω n + ω 2 n then range of f is
If z + 1 z = 2 cos θ , z ∈ C then z 2 n − 2 z n cos ( n θ ) is equal to
Im 2 z + 1 i z + 1 = 5 represents
If z = x + i y and x 2 + y 2 = 16 then the range of | | x | − | y | | is
Number of solutions of the equation z 3 + 3 ( z ¯ ) 2 | z | = 0 where z is a complex number is
If for complex numbers z 1 and z 2 , arg z 1 − arg z 2 = 0 then z 1 − z 2 is equal to
2 cos A = x + 1 x , 2 cos B = y + 1 y . Then find the value of k if k cos A − B = x y + y x
If Re z − 1 2 z + i = 1 , where z = x + iy, then the point (x, y) lies on a
If 3 + i sin θ 4 − i cos θ , θ ∈ 0 , 2 π , is a real number, then an argument of sin θ + i cos θ is:
If the equation, x 2 + b x + 45 = 0 , ( b ∈ R ) has conjugate complex roots and they satisfy | z + 1 | = 2 10 then:
Let z be a complex number such that z – i z + 2 i = 1 and | z | = 5 2 , Then the value of z + 3 i is:
If m and x are two real numbers and i = – 1 , then e 2 m i cot – 1 x x i + 1 x i – 1 m =
If log 1 / 3 | z | 2 – | z | + 1 2 + | z | > – 2 , then
Let z 1 , z 2 be the roots of the equation z 2 + a z + 12 = 0 and z 1 , z 2 form an equilateral triangle with origin. Then, the value of | a | is
If Z = 3 + i 2 , then Z 101 + i 103 105 is equal to
If | z − 2 − i | = | z | sin π 4 − arg z , then locus of z is
If ‘ z ‘lies on the circle | z − 3 i | = 3 2 , then the value of arg z − 3 z + 3 is equal to
If | 2 z − 4 − 2 i | = | z | sin π 4 − arg z , then locus of z is/an
If a complex number z satisfies | z | 2 + 4 | z | 2 − 2 z z ¯ + z ¯ z − 16 = 0 , then the maximum value of Z is
If complex number z lies on the curve | z − ( − 1 + i ) | = 1 , then the locus of the complex number ω = z + i 1 − i , i = − 1 is a circle having
Let α & β are roots of x 2 + ω x + ω 2 = 0 , where ω is an imaginary cube root of unity and z = α 9 + i β 9 , then value of | z | is
if x=a+ib is a complx number such that x 2 = 3 + 4 i and x 3 = 2 + 11 i where i = – 1 , then a + b equals to
If z 1 , z 2 and z 3 are three points lying on the circle z = 2 t nimum value of z 1 + z 2 2 + z 2 + z 3 2 + z 3 + z 1 2 is
Let x + 1 x = 1 and α , β , γ are distinct positive integers such that x a + 1 x α + x β + 1 x β + x γ + 1 x γ = 0 . Then minimum value of α + β + γ =
The real value of θ for which the expression 1 + i cos θ 1 − 2 i cos θ is a real number, where i = − 1 , is
Least positive argument of the 4 th root of the complex number 2 − i 12 is:
The magnitude of 3 + 4 i 5
The magnitude of 3 + 4 i 5
If α ( ≠ 1 ) is a fifth root of unity and β ≠ 1 is a fourth root of unity then z = ( 1 + α ) ( 1 + β ) 1 + α 2 1 + β 2 1 + α 3 1 + β 3 equals
If z lies on the circle z – 1 = 1 then z – 2 z equals
For any complex number z, the minimum value of | z | + | z − 2 i | is
If a + i b = ∑ k = 1 101 i k , then ( a , b ) equals
If 1 , ω , . . . . . ω n – 1 are the nth roots of unity, then value of 1 2 – ω + 1 2 – ω 2 + . . . + 1 2 – ω n – 1 equals
The conjugate of a complex number z is 2 1 − i then Re ( z ) equals
If ω = cos π n + i sin π n , then value of 1 + ω + ω 2 + . . . . + ω n – 1 is
The number of complex numbers z such that | z − i | = | z + i | = | z + 1 | is
If z 2 – 1 = z 2 + 1 , then z is lies on
If z = 1 and z ≠ ± 1 , then all the values of z 1 – z 2 lie on:
If ω ( ≠ 1 ) is a root of unity and 1 + ω 2 11 = a + b ω + c ω 2 , then ( a , b , c ) equals
The number of complex numbers z such that ( 1 + i ) z = i z
If z is a non-zero complex number, then arg ( z ) + a r g ( z ) equals
If z ∈ C and 2 z = | z | + i , then z equals
If z = 1 3 + 1 2 i 7 + 1 3 − 1 2 i 7 then
If roots of the equation z 2 + a z + b = 0 are purely imaginary then
If ω ( ≠ 1 ) is a complex cube root of unity and 1 + ω 4 n = 1 + ω 8 n then the least positive integral value of n is
If z 1 + z 2 + z 3 = 0 and z 1 = z 2 = z 3 = 1 , then value of z 1 2 + z 2 2 + z 3 2 equals
If z = 1 + cos θ + i sin θ sin θ + i ( 1 + cos θ ) ( 0 < θ < π / 2 ) then | z | equals
The equation ( 1 + i ) z − 2 ( 1 + i ) z + 4 = k does not represent a circle when k is
If |z| ≥ 5, then least value of z − 1 z is
If α , β are distinct complex numbers with β = 1 , then value of β – α 1 – α ¯ β equals
If α ( ≠ 1 ) is a fifth root of unity and β ( ≠ 1 ) is a fourth root of unity then z = ( 1 + α ) ( 1 + β ) 1 + α 2 1 + β 2 1 + α 3 1 + β 3 equals
Suppose z 1 , z 2 , z 3 are vertices of an equi-lateral triangle whose circumcentre − 3 + 4 i , then ∣ z 1 + z 2 + z 3 ∣ is equal to
If z − 4 z = 2 , then the maximum value of z is equal to
Suppose z 1 , z 2 , z 3 are three complex numbers, and Δ = 1 4 i 1 z 1 z ¯ 1 1 z 2 z ¯ 2 1 z 2 z ¯ 3 then
If z ≠ 0 lies on the circle | z − 1 | = 1 and ω = 5 / z , then ω lies on
If z ¯ = 3 i + 25 z + 3 i then | z | cannot exceed
If ω ( ≠ 1 ) is a cube root of unity, then the value of tan ω 2017 + ω 2225 π − π / 3
The number of complex numbers z which satisfy z 2 + 2 | z | 2 = 2 is
The inequality a + i b > c + i d is true when
If 8 i z 3 + 12 z 2 − 18 z + 27 i = 0
Let z = a cos π 5 + i sin π 5 , a ∈ R , | a | < 1 , then S = z 2015 + z 2016 + z 2017 + … equals
If z 1 , z 2 and z 3 , z 4 are two pairs of conjugate complex numbers, then arg z 1 z 4 + arg z 2 z 3 equal
If z = 20 i – 21 + 20 i + 21 , than one of the possible value of | a r g ( z ) | equals
If z ≠ 0 is a complex number such that R e ( z ) = 0 , then
If z k = cos k π 10 + i sin k π 10 then z 1 z 2 z 3 z 4 is equal
The point 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
If i = − 1 , then 4 + 3 − 1 2 + i 3 2 127 + 5 − 1 2 + i 3 2 124 is equal to
If z 1 = z 2 = 1 , z 1 z 2 ≠ − 1 and z = z 1 + z 2 1 + z 1 z 2
The value of S = ∑ k = 1 6 sin 2 π k 7 − i cos 2 π k 7
The complex numbers sin x + i cos 2 x and cos x − i sin 2 x are conjugate to each other for
Suppose z 1 , z 2 are two distinct complex numbers and a , b are real numbers. Statement – 1 : If z 1 + z 2 = a , z 1 z 2 = b , then a r g ( z 1 z 2 ) = 0 Statement – 2 : If z 1 + z 2 = a , z 1 z 2 = b , then z 1 ¯ = z 2
If the complex number z − 1 z + 1 is purely imaginary, then
Let α a n d β be the roots of the equation x 2 + x + 1 = 0 . The equation whose roots are α 19 , β 7 is
If z is a complex number such that − π / 2 ≤ arg z ≤ π / 2 then which of the following inequality is true.
If ω is an imaginary cube root of unity, then the value of sin ω 10 + ω 23 π − π 4 is
If z ≠ 0 is s a complex number such that arg ( z ) = π / 4 then
If z 1 = z 2 = z 3 = 1 and z 1 + z 2 + z 3 = 2 + i then the number z 1 z ¯ 2 + z 2 z ¯ 3 + z 3 z ¯ 1 is:
If ω ≠ 1 is a cube root of unity and ( 1 + ω ) 2017 = A + B ( ω ) Then A and B are respectively the numbers
Let z and w be two non-zero complex numbers such that | z | = | w | and arg ( z ) + arg ( w ) = π . Then z equal
Let ω ≠ 1 , be a cube root of unity, and a , b ∈ R . Statement – 1 : a 3 + b 3 = ( a + b ) a ω + b ω 2 a ω 2 + b ω Statement – 2 : x 3 − 1 = ( x − 1 ) x ω 2 − ω x ω − ω 2 for each x ∈ R .
If | ω | = 1 , then the set of points z = ω + 1 ω is satisfies
If ω ≠ 1 is a cube root of unity, then 1 1 + i + ω 2 ω 2 1 − i − 1 ω 2 − 1 − i − i + ω − 1 − 1 equal
If ω ≠ 1 is a cube root of unity, then value of 1 + ω − ω 2 7 equal
If z 1 , ,z 2 , z 3 are complex numbers such that z 1 = z 2 = z 3 = 1 z 1 + 1 z 2 + 1 z 3 = 1 then z 1 + z 2 + z 3 is
Let z 1 and z 2 be nth roots of unity which subtend a right angle at the origin. Then n must be of the form
The complex numbers z 1 , z 2 a n d z 3 satisfying z 1 − z 3 z 2 − z 3 = 1 2 ( 1 − 3 i ) are vertices of a triangle which is
For any complex number z , the minimum value of | z | + | z − 2 i | is
The inequality | z − i | < | z + i | represents the region
If i z 3 + z 2 − z + i = 0 , then
If x + i y = 1 1 − cos θ + 2 i sin θ , θ ≠ 2 n π , n ∈ I then maximum value of x is
The equation z 3 = z ¯ has
If z = x + i y and w = 1 − i z z − i , then | w | = 1 implies, that, in the complex plane
If z = 5 + t + i 25 − t 2 , ( − 5 ≤ t ≤ 5 ) then locus of z is a curve which passes through
The area of the triangle whose vertices are the points represented by the complex number z , i z and z i + z is
If ω ≠ 1 is a cube root of unity and satisfies 1 a + ω + 1 b + ω + 1 c + ω = 2 ω 2 and 1 a + ω 2 + 1 b + ω 2 + 1 c + ω 2 = 2 ω , then the value of 1 a + 1 + 1 b + 1 + 1 c + 1 then
Suppose z is a complex number such that z ≠ − 1 , | z | = 1 , and arg ( z ) = θ Let ω = z ( 1 − z ¯ ) z ¯ ( 1 + z ) ,than Re ( ω ) ) is equal to
If z = i ( 1 + 3 ) then z 4 + 2 z 3 + 4 z 2 + 5 is equal to-
Suppose a < 0 and z 1 , z 2 , z 3 , z 4 4 be the fourth roots of a then z 1 2 + z 2 2 + z 3 2 + z 4 2 is equal
Suppose arg ( z ) = − 5 π / 13 then, arg z + z 1 + z z is
If z ∈ C − { 0 , − 2 } is such that log ( 1 / 7 ) | z − 2 | > log ( 1 / 7 ) | z | then
. Let z 1 , z 2 be two complex numbers such that I m ( z 1 + z 2 ) = 0 and Im z 1 z 2 = 0 then
If x = 9 1 / 3 9 1 / 9 9 1 / 27 ⋯ ∞ , y = 4 1 / 3 4 − 1 / 9 4 1 / 27 ⋯ ∞ , and z = ∑ r = 1 ∞ ( 1 + i ) − r then arg(x + yz) is equal to
The expression 1 + sin π 8 + icos π 8 1 + sin π 8 − icos π 8 8 =
If cos α + 2 cos β + 3 cos γ = sin α + 2 sin β + 3 sin γ = 0 then the value of sin 3 α + 8 sin 3 β + 27 sin 3 γ is
If z = x + iy ( x , y ∈ R , x ≠ − 1 / 2 ) the number of values of z satisfying | z | n = z 2 | z | n − 2 + z | z | n − 2 + 1 ⋅ ( n ∈ N , n > 1 ) is
