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Complex Numbers and Quadratic Equations

JEE Main 2025 (Online) 29th January Evening ShiftNumericalHard📈 Topic in 16% papers+4
Ques. 1JEE Main 2025, 29 January (Evening Shift)

Let integers a, b [-3,3] be such that a+b 0. Then the number of all possible ordered pairs (a, b), for which | z-az+b |=1 and vmatrixz+1& & 2\ &z+ 2&1\ 2&1&z+ vmatrix=1, z C, where and 2 are the roots of x2+x+1=0, is equal to .

Answer

10

Solution

C1 C1+C2+C3 and 1+ + 2=0 ⇒ =z3=1 ⇒ z=1, , 2

[figure]

|z-a|=|z+b| ⇒ z on the bisector of a and -b: Re z= a-b2

z=1: a-b=2 ⇒ 5 pairs, reject (1,-1) (a+b=0) ⇒ 4

z= , 2: a-b=-1 ⇒ 6 pairs

∴ 4+6=10

Answer: 10

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE Main 2025, 28 January (Morning Shift)

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)
Ques. 3JEE Main 2025, 28 January (Morning Shift)

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
Ques. 4JEE 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
Ques. 5JEE 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. 6JEE 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. 7JEE 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. 8JEE 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. 9JEE 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. 10JEE 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. 11JEE 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

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