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Complex Numbers and Quadratic Equations

JEE Main 2024 (Online) 1st February Morning ShiftNumericalHard📈 Topic in 40% papers+4
Ques. 1JEE Main 2024, 1 February (Morning Shift)

Let P=\z C: |z+2-3i| 1\ and Q=\z C: z(1+i)+ z(1-i) -8\. Let in P , |z-3+2i| be maximum and minimum at z1 and z2 respectively. If |z1|2+2|z2|2= + 2, where , are integers, then + equals .

Answer

36

Solution

P: disc, centre (-2, 3), radius 1

Q: z=x+iy 2(x-y) -8 x-y+4 0 (half-plane containing C)

∵ distance of C from this line = 1 2<1 ⇒ the line cuts the disc

|z-3+2i| = distance from A(3, -2); line AC (C = centre): x+y=1, perpendicular to x-y+4=0

[figure]

Max: farthest point of the disc on x+y=1: z1= (-2- 1 2, 3+ 1 2 )

Min: foot of perpendicular from A on x-y+4=0: z2= (- 32, 52 )

|z1|2= (2+ 1 2 )2+ (3+ 1 2 )2=14+5 2, 2|z2|2=2 344=17

31+5 2 + =31+5=36

Answer: 36

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE Main 2024, 1 February (Evening Shift)

If z is a complex number such that |z| 1, then the minimum value of |z+ 12(3+4 i) | is :

  1. 2
  2. 52
  3. 32
  4. 3
Ques. 3JEE Main 2024, 1 February (Evening Shift)

Let and be the roots of the equation p x2+q x-r=0, where p 0. If p, q and r be the consecutive terms of a non constant G.P. and 1 + 1 = 34, then the value of ( - )2 is :

  1. 8
  2. 9
  3. 203
  4. 809
Ques. 4JEE Main 2024, 31 January (Morning Shift)

Let S be the set of positive integral values of a for which ax2+2(a+1)x+9a+4x2-8x+32<0, x R. Then, the number of elements in S is :

  1. 0
  3. 3
  4. 1
Ques. 5JEE Main 2024, 31 January (Morning Shift)

For 0<c<b<a, let (a+b-2c)x2+(b+c-2a)x+(c+a-2b)=0 and 1 be one of its roots. Then, among the two statements

  1. only (II) is true
  2. Both (I) and (II) are true
  3. only (I) is true
  4. Neither (I) nor (II) is true
Ques. 6JEE Main 2024, 31 January (Morning Shift)

Let a be the sum of all coefficients in the expansion of (1-2x+2x2 )2023 (3-4x2+2x3 )2024 and b= x 0 ( 0x (1+t)t2024+1dtx2 ). If the equations cx2+dx+e=0 and 2bx2+ax+4=0 have a common root, where c, d, e R, then d : c : e equals

  1. 2 : 1 : 4
  2. 1 : 1 : 4
  3. 1 : 2 : 4
  4. 4 : 1 : 4
Ques. 7JEE Main 2024, 31 January (Morning Shift)

If denotes the number of solutions of |1-i|x=2x and = ( |z| (z) ), where z= 4(1+i)4 [ 1- i +i+ -i1+ i ], i= -1, then the distance of the point ( , ) from the line 4x-3y=7 is .

Ques. 8JEE Main 2024, 31 January (Evening Shift)

Let z1 and z2 be two complex numbers such that z1+z2=5 and z13+z23=20+15i. Then, |z14+z24 | equals -

  1. 15 15
  2. 30 3
  3. 25 3
  4. 75
Ques. 9JEE Main 2024, 31 January (Evening Shift)

Let a, b, c be the lengths of three sides of a triangle satisfying the condition (a2+b2 )x2-2b(a+c)x+ (b2+c2 )=0. If the set of all possible values of x is the interval ( , ), then 12 ( 2+ 2 ) is equal to .

Ques. 10JEE Main 2024, 30 January (Morning Shift)

If z=x+iy, xy 0, satisfies the equation z2+i z=0, then |z2 | is equal to :

  1. 9
  2. 14
  3. 4
  4. 1
Ques. 11JEE Main 2024, 30 January (Morning Shift)

Let , N be roots of the equation x2-70x+ =0, where 2, 3 N. If assumes the minimum possible value, then ( -1+ -1)( +35)| - | is equal to :

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