Complex Numbers questions
967 published questions with worked solutions.
- For an integer k, let _k= [ k 7 ]+i [ k 7 ], where i= -1 . The value of the expression _ k=1 ^ 12 | _ k+1 - _k| _ k=1 ^3| _ 4k-1...
- Let =e^ i /3 and a,b,c,x,y,z be non-zero complex numbers such that (relations not printed in the source). Then, the value of...
- Let and be the roots of the equation x^2+x+1=0. If the equation whose roots are ^ 58 and ^ 19 is ax^2+bx+c=0, then find the value...
- Let z be a complex number satisfying the equation 2|z-1|+i+z=0. Find the number of such complex numbers.
- A complex number z satisfies ( z z-i )= 3 and |z|=|z-i|, then evaluate [ Re (2z-i) ] where [ ,] represents the greatest integer...
- If is a complex cube root of unity, then evaluate | + ^ ( 15+ 9 50 + 81 500 + 729 5000 + ) |.
- If z=x+iy and x^2+y^2=36 and the range of |x|-|y| is [a,b], then evaluate b^2-a^2.
- Two points A and B in Argand plane represent z and z_1=2 (z+ 3+i ) respectively. If A moves on a circle with centre at origin and...
- Side AB of a square joins z_1=-1 and z_2=3i, and the vertices C(z_3) and z_4 lie in the second quadrant of Argand plane. Evaluate...
- If and are roots of x^2-4x+8=0 and ^ 24 + ^ 24 =2^k( p +i p ), then evaluate k+p.
- If z is a complex number such that |z| 3, then find the minimum value of |z+ 34 |.
- If Re [(1+i)^ -i ]=e^ p (m n), then evaluate pmn.
- If z and w are non-zero complex numbers such that |zw|=1, (z)- (w)= 2 and zw=a+ib, then find a^2+b^2.
- If and are the roots of the equation ix^2-x+12i=0, then find + .
- If |z-25i| 15 and | . (z)- . (z)| is m ^ -1 pq, then find mp-q. (Given that G.C.D. (p,q)=1)
- If is a complex cube root of unity, then find the value of ( a+b +c ^2 c+a +b ^2 + a+b +c ^2 b+c +a ^2 )^2.
- Find the least value of k for which |z-2 3i|=k intersects the curve z= 6.
- If |z+4|^2-|z-4|^2=8, then find Re (z).
- Find the sum of all possible values of Im (z) where | z-12 z-8i |= 53 and | z-4 z-8 |=1.
- Find the digital sum of the value of 1(2- ) (2- ^2 )+2(3- ) (3- ^2 )+ +9(10- ) (10- ^2 ), where is a complex cube root of unity.
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