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Complex Numbers and Quadratic Equations

JEE Main 2026 (Online) 28th January Evening ShiftMCQMedium📈 Topic in 16% papers+4−1
Ques. 1JEE Main 2026, 28 January (Evening Shift)

Let the arithmetic mean of 1a and 1b be 516, a > 2. If is such that a, 4, , b are in A.P., then the equation x2 - ax + 2( - 2b) = 0 has :

  1. one root in (1, 4) and another in (-2, 0)
  2. one root in (0, 2) and another in (-4, -2)
  3. both roots in the interval (-2, 0)
  4. complex roots of magnitude less than 2

Answer

(1) one root in (1, 4) and another in (-2, 0)

Solution

Since a, 4, , b are in A.P. with common difference 4 - a, = 8 - a and b = 12 - 2a.

1a + 1b = 2 516 = 58

1a + 112 - 2a = 58 12 - aa(12 - 2a) = 58

96 - 8a = 60a - 10a2 5a2 - 34a + 48 = 0

(5a - 24)(a - 2) = 0 a = 245 (since a > 2)

so = 165 and b = 125. The equation becomes

165x2 - 245x + 2 ( 165 - 245 ) = 0

16x2 - 24x - 16 = 0 2x2 - 3x - 2 = 0

(x - 2)(2x + 1) = 0 x = 2, - 12

∴ One root (2) (1, 4) and the other (- 12 ) (-2, 0).

Answer: (1)

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE Main 2026, 28 January (Evening Shift)

Let A = \z C : |z - 2| 4\ and B = \z C : |z - 2| + |z + 2| = 5\. Then the \|z1 - z2| : z1 A and z2 B\ is :

  1. 172
  2. 8
  3. 9
  4. 152
Ques. 3JEE Main 2026, 24 January (Morning Shift)

The number of the real solutions of the equation: x|x + 3| + |x - 1| - 2 = 0 is

  1. 3
  2. 2
  3. 5
  4. 4
Ques. 4JEE Main 2026, 24 January (Morning Shift)

Let S = \z C : | z-6iz-2i | = 1 and | z-8+2iz+2i | = 35 \. Then z S |z|2 is equal to :

  1. 413
  2. 398
  3. 385
  4. 423
Ques. 5JEE Main 2026, 24 January (Evening Shift)

The smallest positive integral value of a, for which all the roots of x4 - ax2 + 9 = 0 are real and distinct, is equal to

  1. 7
  2. 3
  3. 4
  4. 9
Ques. 6JEE Main 2026, 24 January (Evening Shift)

Let z = (1 + i)(1 + 2i)(1 + 3i) (1 + ni), where i = -1. If |z|2 = 44200, then n is equal to .

Ques. 7JEE Main 2026, 23 January (Morning Shift)

If α and β (α < β) are the roots of the equation (-2 + 3 ) ( | x - 3 | ) + (x - 6 x ) + (9 - 2 3 ) = 0, x 0, then + is equal to :

  1. 8
  2. 10
  3. 9
  4. 11
Ques. 8JEE Main 2026, 23 January (Morning Shift)

Let S = \z : 3 |2z - 3(1 + i)| 7 \ be a set of complex numbers. Then Minz S | (z + 12(5 + 3i) ) | is equal to :

  1. 12
  2. 52
  3. 2
  4. 32
Ques. 9JEE Main 2026, 23 January (Morning Shift)

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

  1. 36 days
  2. 24 days
  3. 30 days
  4. 42 days
Ques. 10JEE Main 2026, 23 January (Evening Shift)

If z = 32 + i2, i = -1, then (z201 - i )8 is equal to

  1. 1
  2. 0
  3. -1
  4. 256
Ques. 11JEE Main 2026, 22 January (Morning Shift)

The number of distinct real solutions of the equation x|x+4| + 3|x+2| + 10 = 0 is

  1. 2
  2. 3
  3. 0
  4. 1

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