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Complex Numbers and Quadratic Equations · Question 12

JEE Main 2026 (Online) 2nd April Morning ShiftMCQHard📈 Topic in 26% papers+4−1
1

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

Answer

(3) 2

Solution

LHS = r=0n-1(x+r)(x+r+2)

=nx2+x r=0n-1(2r+2)+ r=0n-1r(r+2)

=nx2+n(n+1)x+ n(n-1)(2n+5)6

∴ Dividing the equation by n:

x2+(n+1)x+ (n-1)(2n+5)6-4=0, whose roots are and +2.

Sum of roots: 2 +2=-(n+1)

=- n+32 and +2=- n-12

Product of roots: ( +2)= (n+3)(n-1)4= (n-1)(2n+5)6-4

3(n+3)(n-1)=2(n-1)(2n+5)-48

3n2+6n-9=4n2+6n-58

n2=49 n=7 (∵ n )

∴ =- 7+32=-5 (roots -5 and -3)

n+ =7-5=2

Answer: (3)

JEE Main 2026 (Online) 2nd April Morning Shift

Keep practising — Complex Numbers and Quadratic Equations

2

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
3

Let , be the roots of the equation x2-3x+r=0, and 2, 2 be the roots of the equation x2+3x+r=0. If the roots of the equation x2+6x=m are 2 + +2r and -2 - r2, then m is equal to :

  1. -135
  2. -567
  3. 135
  4. 567
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4

Let the circles C1 : |z|=r and C2 : |z-3-4i|=5, z C, be such that C2 lies within C1. If z1 moves on C1, z2 moves on C2 and |z1-z2|=2, then |z1-z2| is equal to :

  1. 12
  2. 17
  3. 22
  4. 24
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5

If α, β, where α < β, are the roots of the equation x2 - ( + 3)x + 3 = 0 such that 1 - 1 = 13, then the sum of all possible values of λ is

  1. 2
  2. 6
  3. 8
  4. 4
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6

Let S = \x3 + ax2 + bx + c : a, b, c N and a, b, c 20\ be a set of polynomials. Then the number of polynomials in S, which are divisible by x2 + 2, is

  1. 6
  2. 120
  3. 20
  4. 10
JEE Main 2026 (Online) 28th January Morning Shift
7

Let z be a complex number such that |z − 6| = 5 and |z + 2 − 6i| = 5. Then the value of z3 + 3z2 - 15z + 141 is equal to :

  1. 61
  2. 37
  3. 42
  4. 50
JEE Main 2026 (Online) 28th January Morning Shift
8

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
JEE Main 2026 (Online) 28th January Evening Shift
9

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
JEE Main 2026 (Online) 28th January Evening Shift
10

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
JEE Main 2026 (Online) 24th January Morning Shift
11

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
JEE Main 2026 (Online) 24th January Morning Shift

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