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1

IIT-JEE 1993

Fill in the Blanks
$$ABCD$$ is a rhombus. Its diagonals $$AC$$ and $$BD$$ intersect at the point $$M$$ and satisfy $$BD$$ = 2$$AC$$. If the points $$D$$ and $$M$$ represent the complex numbers $$1 + i$$ and $$2 - i$$ respectively, then A represents the comp[lex number ..........or..........

Answer

$$3 - {i \over 2}$$ or $$1 - {3 \over 2}i$$
2

IIT-JEE 1989

Fill in the Blanks
If $$a,\,b,\,c,$$ are the numbers between 0 and 1 such that the ponts $${z_1} = a + i,{z_2} = 1 + bi$$ and $${z_3} = 0$$ form an equilateral triangle,
then a= .......and b=..........

Answer

$$2 - \sqrt {3,} $$ $$2 - \sqrt {3} $$
3

IIT-JEE 1988

Fill in the Blanks
For any two complex numbers $${z_1},{z_2}$$ and any real number a and b.

$$\,{\left| {a{z_1} - b{z_2}} \right|^2} + {\left| {b{z_1} + a{z_2}} \right|^2} = .........$$

Answer

$$\left( {{a^2} + {b^2}} \right)\left( {{{\left| {{z_1}} \right|}^2} + {{\left| {{z_2}} \right|}^2}} \right)$$
4

IIT-JEE 1987

Fill in the Blanks
If the expression $$${{\left[ {\sin \left( {{x \over 2}} \right) + \cos {x \over 2} + i\,\tan \left( x \right)} \right]} \over {\left[ {1 + 2\,i\,\sin \left( {{x \over 2}} \right)} \right]}}$$$

is real, then the set of all possible values of $$x$$ is ..............

Answer

$$2n\pi ,\,n\pi + {\pi \over 4}$$

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