Show that: `sum_(r=0)^(n-1)|z_1+alpha^r z_2|^2=n(|z_1|^2+|z_2|^2),w h e r ealpha^r ; r=0,1,2, ,(n-1)` , are nth roots of unity and `z_1, z_2` are an tow complex numbers.

Prove that sum_(r=1)^k(-3)^(r-1)^(3n)C_(2r-1)=0,w h e r ek=3n//2 and n is an even integer.

Q. Let z_1 and z_2 be nth roots of unity which subtend a right angle at the origin, then n must be the form .

If z_1, z_2, z_3 are three nonzero complex numbers such that z_3=(1-lambda)z_1+lambdaz_2w h e r elambda in R-{0}, then prove that points corresponding to z_1, z_2a n dz_3 are collinear .

If |z|=2a n d(z_1-z_3)/(z_2-z_3)=(z-2)/(z+2) , then prove that z_1, z_2, z_3 are vertices of a right angled triangle.

Prove that sum_(r=0)^n^n C_r(-1)^r[i^r+i^(2r)+i^(3r)+i^(4r)]=2^n+2^(n/2+1)cos(npi//4),w h e r ei=sqrt(-1)dot

If z_1 + z_2 + z_3 + z_4 = 0 where b_1 in R such that the sum of no two values being zero and b_1 z_1+b_2z_2 +b_3z_3 + b_4z_4 = 0 where z_1,z_2,z_3,z_4 are arbitrary complex numbers such that no three of them are collinear, prove that the four complex numbers would be concyclic if |b_1b_2||z_1-z_2|^2=|b_3b_4||z_3-z_4|^2 .

Let |(( bar z _1)-2( bar z _2))//(2-z_1( bar z _2))|=1 and |z_2|!=1 ,where z_1 and z_2 are complex numbers. Show that |z_1|=2.

z_1a n dz_2 are two distinct points in an Argand plane. If a|z_1|=b|z_2|(w h e r ea ,b in R), then the point (a z_1//b z_2)+(b z_2//a z_1) is a point on the (a)line segment [-2, 2] of the real axis (b)line segment [-2, 2] of the imaginary axis (c)unit circle |z|=1 (d)the line with a rgz=tan^(-1)2

If a m p(z_1z_2)=0a n d|z_1|=|z_2|=1,t h e n z_1+z_2=0 b. z_1z_2=1 c. z_1=z _2 d. none of these

If one root of the equation z^2-a z+a-1=0i s(1+i),w h e r ea is a complex number then find the root.