# Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q. 1. If z is any complex number satisfying |z – 3 – 2i| ≤ 2, then the minimum value of |2z – 6 + 5i| is     (2011)

Ans. (5)
Sol. Given |z – 3 – 2i| < 2 which represents a circular region with centre (3, 2) and radius 2.

Now |2z – 6 + 5i| =

= 2 × distance of z from P (where Z lies in or on the circle)

Also min distance of z from P =

∴ Minimum value of |2z – 6 + 5i| = 5

Q. 2. Let ω = , and a, b, c, x, y, z be non-zero complex numbers such that     (2011)
a + b + c = x
a + bω + cω2 = y
a + bω2 + cω = z

Then the value of  is

Ans.  (3)
Sol. The expression may not attain integral value for all a, b, c.
If we consider a = b = c then

x = 3a, y = a (1 + ω + ω2) = a (1 + i )

Z = a (1 + ω+ ω) = a(1 + i )

⇒ |x|2 + |y|2 + |z|2 = 9 |a|2 + 4 |a|2 + 4 |a|= 17 |a|2

(which is not an integer)

Note : However if ω =    then the value of expression can be evaluated as follows

= 3           (∵ 1 + ω+ ω2 = 0)

Q. 3. For any integer k, let αk = cos+ i sin , where i =,The value of the expression

Ans. 3. (4)

Sol.

⇒

Similarly

∴

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