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Complex Numbers and Quadratic Equations

JEE Main 2025 (Online) 28th January Evening ShiftMCQMedium📈 Topic in 32% papers+4−1
Ques. 1JEE Main 2025, 28 January (Evening Shift)

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

Answer

(1) 2

Solution

Discriminant: D = (3 - 2i)2 + 4(2i - 2)

= 9 - 12i - 4 + 8i - 8 = -3 - 4i

(1 - 2i)2 = 1 - 4i - 4 = -3 - 4i

D = (1 - 2i)

x = (3 - 2i) (1 - 2i)2

x = 2 - 2i or x = 1

Taking + i = 2 - 2i and + i = 1 + 0i:

= 2, = -2, = 1, = 0

+ = 2(1) + (-2)(0) = 2

(The value is the same if the two roots are interchanged.)

Answer: (1)

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE Main 2025, 24 January (Morning Shift)

The product of all the rational roots of the equation (x2-9x+11)2-(x-4)(x-5)=3, is equal to

  1. 7
  2. 21
  3. 28
  4. 14
Ques. 3JEE Main 2025, 24 January (Morning Shift)

If and are the roots of the equation 2z2-3z-2i=0, where i= -1, then 16 ( 19+ 19+ 11+ 11 15+ 15 ) ( 19+ 19+ 11+ 11 15+ 15 ) is equal to

  1. 441
  2. 312
  3. 409
  4. 398
Ques. 4JEE Main 2025, 23 January (Morning Shift)

Let | z-i2 z+i |= 13, z C, be the equation of a circle with center at C. If the area of the triangle, whose vertices are at the points (0,0), C and ( ,0) is 11 square units, then 2 equals:

  1. 12125
  2. 100
  3. 8125
  4. 50
Ques. 5JEE Main 2025, 23 January (Morning Shift)

If the equation a(b-c)x2+b(c-a)x+c(a-b)=0 has equal roots, where a+c=15 and b= 365, then a2+c2 is equal to .

Ques. 6JEE Main 2025, 23 January (Evening Shift)

The number of complex numbers z, satisfying |z|=1 and | z z+ zz |=1, is :

  1. 8
  2. 10
  3. 4
  4. 6
Ques. 7JEE Main 2025, 23 January (Evening Shift)

Let , be the roots of the equation x2-ax-b=0 with Im( )<Im( ). Let Pn= n- n. If P3=-5 7 i, P4=-3 7 i, P5=11 7 i and P6=45 7 i, then | 4+ 4 | is equal to .

Ques. 8JEE Main 2025, 22 January (Morning Shift)

Let z1,z2 and z3 be three complex numbers on the circle |z|=1 with (z1)= - 4, (z2)=0 and (z3)= 4. If |z1 z2+z2 z3+z3 z1 |2= + 2, , Z, then the value of 2+ 2 is :

  1. 41
  2. 29
  3. 24
  4. 31
Ques. 9JEE Main 2025, 22 January (Evening Shift)

Let and be the distinct roots of 2x2+( )x-1=0, (0,2 ). If m and M are the minimum and the maximum values of 4+ 4, then 16(M+m) equals :

  1. 27
  2. 17
  3. 25
  4. 24
Ques. 10JEE Main 2025, 22 January (Evening Shift)

Let the curve z(1+i)+ z(1-i)=4, z , divide the region |z-3| 1 into two parts of areas and . Then | - | equals :

  1. 1+ 3
  2. 1+ 6
  3. 1+ 2
  4. 1+ 4

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