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Complex Numbers and Quadratic Equations · Question 53

JEE Main 2025 (Online) 2nd April Morning ShiftMCQEasy📈 Topic in 13% papers+4−1
1

Let z be a complex number such that |z| = 1. If 2 + k2 zk + z = kz, k R, then the maximum distance of k + ik2 from the circle |z - (1 + 2i)| = 1 is :

  1. 5 + 1
  2. 3
  3. 3 + 1
  4. 2

Answer

(1) 5 + 1

Solution

|z| = 1 z = 1z

2 + k2zk + 1z = kz z(2 + k2z)kz + 1 = kz

2z + k2z2 = k2z2 + kz

2z = kz k = 2 ( z 0)

k + ik2 = 2 + 4i, i.e. the point (2, 4)

Circle: centre (1, 2), radius 1

Distance of (2, 4) from the centre = 12 + 22 = 5 > 1 (point outside)

Maximum distance from the circle = 5 + 1

Answer: (1)

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Keep practising — Complex Numbers and Quadratic Equations

2

Let Pn = n + n, n N. If P10 = 123, P9 = 76, P8 = 47 and P1 = 1, then the quadratic equation having roots 1 and 1 is :

  1. x2 + x - 1 = 0
  2. x2 - x + 1 = 0
  3. x2 + x + 1 = 0
  4. x2 - x - 1 = 0
JEE Main 2025 (Online) 2nd April Morning Shift
3

If the set of all a R - \1\, for which the roots of the equation (1 - a)x2 + 2(a - 3)x + 9 = 0 are positive is (- , - ] [ , ), then 2 + + is equal to .

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4

The number of solutions of the equation ( 9x - 9 x + 2 ) ( 2x - 7 x + 3 ) = 0 is :

  1. 3
  2. 2
  3. 1
  4. 4
JEE Main 2025 (Online) 29th January Morning Shift
5

Let |z1 - 8 - 2i| 1 and |z2 - 2 + 6i| 2, z1, z2 C. Then the minimum value of |z1 - z2| is :

  1. 3
  2. 10
  3. 7
  4. 13
JEE Main 2025 (Online) 29th January Morning Shift
6

The sum, of the squares of all the roots of the equation x2 + |2x - 3| - 4 = 0, is

  1. 6(2 - 2)
  2. 3(3 - 2)
  3. 3(2 - 2)
  4. 6(3 - 2)
JEE Main 2025 (Online) 28th January Morning Shift
7

Let O be the origin, the point A be z1 = 3 + 2 2i, the point B(z2) be such that 3|z2| = |z1| and (z2) = (z1) + 6. Then

  1. area of triangle ABO is 114
  2. area of triangle ABO is 11 3
  3. ABO is a scalene triangle
  4. ABO is an obtuse angled isosceles triangle
JEE Main 2025 (Online) 28th January Morning Shift
8

If + i and + i are the roots of x2 - (3 - 2i)x - (2i - 2) = 0, i = -1, then + is equal to:

  1. 2
  2. -6
  3. 6
  4. -2
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