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Complex Numbers and Quadratic Equations

JEE Main 2024 (Online) 6th April Evening ShiftMCQMedium📈 Topic in 21% papers+4−1
Ques. 1JEE Main 2024, 6 April (Evening Shift)

If z1, z2 are two distinct complex numbers such that | z1 - 2z2 12 - z1 z2 | = 2, then

  1. either z1 lies on a circle of radius 12 or z2 lies on a circle of radius 1.
  2. z1 lies on a circle of radius 12 and z2 lies on a circle of radius 1.
  3. either z1 lies on a circle of radius 1 or z2 lies on a circle of radius 12.
  4. both z1 and z2 lie on the same circle.

Answer

(3) either z1 lies on a circle of radius 1 or z2 lies on a circle of radius 12.

Solution

|z1 - 2z2| = 2 | 12 - z1 z2 | = |1 - 2z1 z2|

Squaring with |w|2 = w w:

(z1 - 2z2)( z1 - 2 z2) = (1 - 2z1 z2)(1 - 2 z1z2)

∵ -2z1 z2 - 2 z1z2 occurs on both sides and cancels

|z1|2 + 4|z2|2 = 1 + 4|z1|2|z2|2

(|z1|2 - 1)(1 - 4|z2|2) = 0

|z1| = 1 or |z2| = 12

∴ either z1 lies on a circle of radius 1 or z2 lies on a circle of radius 12

Answer: (3)

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE Main 2024, 6 April (Evening Shift)

Let , be roots of x2 + 2x - 8 = 0. If Un = n + n, then U10 + 2 U92U8 is equal to .

Ques. 3JEE Main 2024, 5 April (Morning Shift)

Consider the following two statements : Between the above two statements,

  1. both Statement I and Statement II are incorrect.
  2. Statement I is correct but Statement II is incorrect.
  3. Statement I is incorrect but Statement II is correct.
  4. both Statement I and Statement II are correct.
Ques. 4JEE Main 2024, 5 April (Morning Shift)

The number of distinct real roots of the equation |x||x + 2| - 5|x + 1| - 1 = 0 is .

Ques. 5JEE Main 2024, 5 April (Evening Shift)

Let S1 = \z C : |z| 5\, S2 = \z C : Im ( z + 1 - 3 i1 - 3 i ) 0 \ and S3 = \z C : Re(z) 0\. Then the area of the region S1 S2 S3 is :

  1. 125 24
  2. 125 6
  3. 125 12
  4. 125 4
Ques. 6JEE Main 2024, 5 April (Evening Shift)

The number of real solutions of the equation x|x + 5| + 2|x + 7| - 2 = 0 is .

Ques. 7JEE Main 2024, 4 April (Morning Shift)

If 2 and 6 are the roots of the equation ax2+bx+1=0, then the quadratic equation, whose roots are 12a+b and 16a+b, is :

  1. x2+8x+12=0
  2. 2x2+11x+12=0
  3. 4x2+14x+12=0
  4. x2+10x+16=0
Ques. 8JEE Main 2024, 4 April (Morning Shift)

Let and be the sum and the product of all the non-zero solutions of the equation ( z)2+|z|=0, z . Then 4( 2+ 2) is equal to :

  1. 4
  2. 2
  3. 6
  4. 8
Ques. 9JEE Main 2024, 4 April (Evening Shift)

The area (in sq. units) of the region S = \z C : |z - 1| 2; (z + z) + i(z - z) 2, Im(z) 0\ is

  1. 7 4
  2. 3 2
  3. 7 3
  4. 17 8
Ques. 10JEE Main 2024, 4 April (Evening Shift)

Let S = \ 2 2 : ( 4 + 4 )x2 + ( 2 )x + ( 6 + 6 ) = 0 has real roots\. If and be the smallest and largest elements of the set S, respectively, then 3 (( - 2)2 + ( - 1)2 ) equals .

Ques. 11JEE Main 2024, 1 February (Morning Shift)

Let S= \z :|z-1|=1 and ( 2-1)(z+ z)-i(z- z)=2 2 \. Let z1, z2 be such that |z1|= z S|z| and |z2|= z S|z|. Then | 2z1-z2 |2 equals :

  1. 1
  2. 4
  3. 3
  4. 2

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