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Complex Numbers and Quadratic Equations

JEE Main 2025 (Online) 28th January Morning ShiftMCQMedium📈 Topic in 42% papers+4−1
Ques. 1JEE 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

Answer

(4) ABO is an obtuse angled isosceles triangle

Solution

OA = |z1| = 3 + 8 = 11

OB = |z2| = |z1| 3 = 113

AOB = (z2) - (z1) = 6

Cosine rule: AB2 = OA2 + OB2 - 2(OA)(OB) 6

= 11 + 113 - 2 11 11 3 32

= 11 + 113 - 11 = 113

AB = 113 = OB the triangle is isosceles.

OA2 = 11 > OB2 + AB2 = 223 OBA is obtuse.

Area = 12(OA)(OB) 6 = 12 11 11 3 12 = 114 3

So options (1) and (2) are wrong, and ABO is not scalene.

ABO is an obtuse angled isosceles triangle.

Answer: (4)

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE 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. 3JEE 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. 4JEE 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. 5JEE 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. 6JEE 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. 7JEE 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. 8JEE 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. 9JEE 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. 10JEE 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. 11JEE 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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