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 + ω2 + ω) = a(1 + i )
⇒ |x|2 + |y|2 + |z|2 = 9 |a|2 + 4 |a|2 + 4 |a|2 = 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
is (JEE Adv. 2015)
Ans. 3. (4)
Sol. =
⇒
Similarly
∴
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