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

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

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

Also min distance of z from P = Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ Minimum value of |2z – 6 + 5i| = 5

 

Q. 2. Let ω =Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE , 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 ofInteger Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  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 Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE)

Z = a (1 + ω+ ω) = a(1 + i Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE)

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

Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEInteger Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (which is not an integer)

Note : However if ω = Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   then the value of expression can be evaluated as follows

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

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

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

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

= 3           (∵ 1 + ω+ ω2 = 0)

 

Q. 3. For any integer k, let αk = cosInteger Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE+ i sin Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE, where i =Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE,The value of the expression 

Integer Answer Type Questions: Complex Numbers | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is                  (JEE Adv. 2015)

 

Ans. 3. (4)

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

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

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

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

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

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

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