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Complex Numbers and Quadratic Equations

JEE Main 2024 (Online) 9th April Evening ShiftMCQEasy📈 Topic in 40% papers+4−1
Ques. 1JEE Main 2024, 9 April (Evening Shift)

Let z be a complex number such that the real part of z - 2iz + 2i is zero. Then, the maximum value of |z - (6 + 8i)| is equal to

  1. 8
  2. 12
  3. 10

Answer

(2) 12

Solution

Put z = x + iy. Then z - 2iz + 2i = x + i(y - 2)x + i(y + 2), whose real part is x2 + (y - 2)(y + 2)x2 + (y + 2)2.

Real part = 0 gives x2 + y2 = 4, so z lies on the circle |z| = 2.

[figure]

|z - (6 + 8i)| is the distance of z from Q(6, 8), and OQ = 10.

It is greatest at M, the end of the diameter on the far side of Q, so maximum = 10 + 2 = 12.

Answer: (2)

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE Main 2024, 9 April (Evening Shift)

Let , ; > , be the roots of the equation x2 - 2x - 3 = 0. Let Pn = n - n, n N. Then (11 3 - 10 2)P10 + (11 2 + 10)P11 - 11P12 is equal to

  1. 10 3P9
  2. 11 3P9
  3. 11 2P9
  4. 10 2P9
Ques. 3JEE Main 2024, 8 April (Morning Shift)

Let z be a complex number such that |z + 2| = 1 and Im ( z + 1z + 2 ) = 15. Then the value of |Re ( z + 2 ) | is

  1. 2 65
  2. 245
  3. 65
  4. 1 + 65
Ques. 4JEE Main 2024, 8 April (Morning Shift)

The sum of all the solutions of the equation (8)2x - 16 (8)x + 48 = 0 is :

  1. 1 + 8(6)
  2. 1 + 6(8)
  3. 8(6)
  4. 8(4)
Ques. 5JEE Main 2024, 8 April (Morning Shift)

If the set R = \(a, b) : a + 5b = 42, a, b N\ has m elements and n=1m (1 - in! ) = x + iy, where i = -1, then the value of m + x + y is

  1. 12
  2. 4
  3. 8
  4. 5
Ques. 6JEE Main 2024, 8 April (Evening Shift)

The sum of all possible values of [- , 2 ], for which 1 + i 1 - 2i is purely imaginary, is equal to :

  1. 4
  2. 3
  3. 2
  4. 5
Ques. 7JEE Main 2024, 8 April (Evening Shift)

The number of distinct real roots of the equation |x + 1||x + 3| - 4|x + 2| + 5 = 0, is

Ques. 8JEE Main 2024, 6 April (Morning Shift)

Let , be the distinct roots of the equation x2 - (t2 - 5t + 6 )x + 1 = 0, t R and an = n + n. Then the minimum value of a2023 + a2025a2024 is

  1. -1/2
  2. -1/4
  3. 1/4
  4. 1/2
Ques. 9JEE Main 2024, 6 April (Morning Shift)

Let x1, x2, x3, x4 be the solution of the equation 4x4 + 8x3 - 17x2 - 12x + 9 = 0 and (4 + x12 ) (4 + x22 ) (4 + x32 ) (4 + x42 ) = 12516m. Then the value of m is .

Ques. 10JEE 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.
Ques. 11JEE 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 .

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