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1

### IIT-JEE 1983

Subjective
Prove that the complex numbers $${{z_1}}$$, $${{z_2}}$$ and the origin form an equilateral triangle only if $$z_1^2 + z_2^2 - {z_1}\,{z_2} = 0$$.

Solve it.
2

### IIT-JEE 1981

Subjective
Let the complex number $${{z_1}}$$, $${{z_2}}$$ and $${{z_3}}$$ be the vertices of an equilateral triangle. Let $${{z_0}}$$ be the circumcentre of the triangle. Then prove that $$z_1^2 + z_2^2 + z_3^2 = 3z_0^2$$.

Solve it.
3

### IIT-JEE 1980

Subjective
Find the real values of x and y for which the following equation is satisfied $$\,{{(1 + i)x - 2i} \over {3 + i}} + {{(2 + 3i)y + i} \over {3 - i}} = i$$

x = 3, y = - 1
4

### IIT-JEE 1979

Subjective
If x + iy = $$\sqrt {{{a + ib} \over {c + id}}}$$, prove that $${({x^2} + {y^2})^2} = {{{a^2} + {b^2}} \over {{c^2} + {d^2}}}$$.

Solve it.

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Class 12