Complex Numbers 4 Question 4

4.

The shaded region, where P=(1,0),Q=(1+2,2) R=(1+2,2),S=(1,0) is represented by

(2005,1M)

(a) |z+1|>2,|arg(z+1)|<π4

(b) |z+1|<2,|arg(z+1)|<π2

(c) |z+1|>2, |arg(z+1)|>π4

(d) |z1|<2,|arg(z+1)|>π2

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Answer:

Correct Answer: 4. (a)

Solution:

  1. Since, |PQ|=|PS|=|PR|=2

Shaded part represents the external part of circle having centre (1,0) and radius 2 .

As we know equation of circle having centre z0 and radius r, is |zz0|=r

|z(1+0i)|>2|z+1|>2

Also, argument of z+1 with respect to positive direction of X-axis is π/4.

arg(z+1)π4

and argument of z+1 in anticlockwise direction is π/4.

π/4arg(z+1)

From Eqs. (i) and (ii),

|arg(z+1)|π/4



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