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Complex Numbers and Quadratic Equations · Question 3

JEE Main 2026 (Online) 6th April Morning ShiftMCQMedium📈 Topic in 13% papers+4−1
1

Let the set of all values of k R such that the equation z ( z+2+i )+k(2+3i)=0, z C, has at least one solution, be the interval [ , ]. Then 9( + ) is equal to :

  1. -10
  2. -8
  3. 10 13
  4. 8 13

Answer

(1) -10

Solution

Put z=x+iy : z z=x2+y2, z(2+i)=(2x-y)+i(x+2y)

(x2+y2+2x-y+2k )+i(x+2y+3k)=0

Real part : x2+y2+2x-y+2k=0 … (i)

circle, centre (-1, 12 ), r2=1+ 14-2k= 54-2k

Imaginary part : x+2y+3k=0 … (ii), a straight line

A solution exists line (ii) meets circle (i)

distance of centre from the line r

|-1+1+3k| 5 54-2k

9k25 54-2k 36k2+40k-25 0

k [ , ], where , are the roots of 36k2+40k-25=0

+ =- 4036=- 109

9( + )=-10

Answer: (1)

JEE Main 2026 (Online) 6th April Morning Shift

Keep practising — Complex Numbers and Quadratic Equations

2

Let S= \z C : z2+ 6 iz-3=0 \. Then z S z8 is equal to :

  1. 162
  2. 184
  3. 262
  4. 324
JEE Main 2026 (Online) 6th April Evening Shift
3

Let a, b C. Let , be the roots of the equation x2+ax+b=0. If - = 11 and 2- 2=3i 11, then ( 3- 3 )2 is equal to:

  1. 160
  2. 176
  3. 194
  4. 187
JEE Main 2026 (Online) 5th April Morning Shift
4

Let , be the roots of the equation x2-x+p=0 and , be the roots of the equation x2-4x+q=0; p, q Z. If , , , are in G.P., then |p+q| equals :

  1. 16
  2. 32
  3. 34
  4. 38
JEE Main 2026 (Online) 5th April Evening Shift
5

Let z1, z2 C be the distinct solutions of the equation z2+4z-(1+12i)=0. Then |z1|2+|z2|2 is equal to :

  1. 18
  2. 22
  3. 29
  4. 34
JEE Main 2026 (Online) 5th April Evening Shift
6

If the set of all solutions of |x2+x-9|=|x|+|x2-9| is [ , ] [ , ), then ( 2+ 2+ 2 ) is equal to:

  1. 9
  2. 18
  3. 36
  4. 72
JEE Main 2026 (Online) 4th April Morning Shift
7

Let z be a complex number such that |z+2|=|z-2| and ( z+3z-i )= 4. Then |z|2 is equal to:

  1. 9
  2. 4
  3. 5
  4. 1
JEE Main 2026 (Online) 4th April Morning Shift
8

Let S=\z C : z2+4z+16=0\. Then z S |z+ 3 i |2 is equal to :

  1. 42
  2. 23
  3. 27
  4. 38
JEE Main 2026 (Online) 4th April Evening Shift
9

If the quadratic equation ( +2)x2-3 x+4 =0, -2, has two positive roots, then the number of possible integral values of is :

  1. 1
  2. 2
  3. 3
  4. 4
JEE Main 2026 (Online) 4th April Evening Shift
10

Let , +2, Z, be the roots of the quadratic equation x(x+2)+(x+1)(x+3)+(x+2)(x+4)+ +(x+n-1)(x+n+1)=4n for some n N. Then n+ is equal to :

  1. 0
  2. 1
  3. 2
  4. 3
JEE Main 2026 (Online) 2nd April Morning Shift
11

Let x and y be real numbers such that 50 ( 2x1+3i- y1-2i )=31+17i, i= -1. Then the value of 10(x-3y) is :

  1. 20
  2. 31
  3. 35
  4. 75
JEE Main 2026 (Online) 2nd April Morning Shift

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