1
IIT-JEE 2003
Subjective
+2
-0
Prove that there exists no complex number z such that $$\left| z \right| < {1 \over 3}\,and\,\sum\limits_{r = 1}^n {{a_r}{z^r}} = 1$$ where $$\left| {{a_r}} \right| < 2$$.
2
IIT-JEE 2002
Subjective
+5
-0
Let a complex number $$\alpha ,\,\alpha \ne 1$$, be a root of the equation $${z^{p + q}} - {z^p} - {z^q} + 1 = 0$$, where p, q are distinct primes. Show that either $$1 + \alpha + {\alpha ^2} + .... + {\alpha ^{p - 1}} = 0\,or\,1 + \alpha + {\alpha ^2} + .... + {\alpha ^{q - 1}} = 0$$, but not both together.
3
IIT-JEE 1999
Subjective
+10
-0
For complex numbers z and w, prove that $${\left| z \right|^2}w - {\left| w \right|^2}z = z - w$$ if and only if $$z = w\,or\,z\overline {\,w} = 1$$.
4
IIT-JEE 1997
Subjective
+5
-0
Let $${z_1}$$ and $${z_2}$$ be roots of the equation $${z^2} + pz + q = 0\,$$ , where the coefficients p and q may be complex numbers. Let A and B represent $${z_1}$$ and $${z_2}$$ in the complex plane. If $$\angle AOB = \alpha \ne 0\,$$ and OA = OB, where O is the origin, prove that $${p^2} = 4q\,{\cos ^2}\left( {{\alpha \over 2}} \right)$$.
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