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Complex Numbers and Quadratic Equations

JEE Main 2024 (Online) 31st January Morning ShiftMCQMedium📈 Topic in 34% papers+4−1
Ques. 1JEE 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

Answer

(2) Both (I) and (II) are true

Solution

Sum of coefficients =(a+b-2c)+(b+c-2a)+(c+a-2b)=0 ⇒ x=1 is a root

= c+a-2ba+b-2c (product of roots)

Here a+b-2c>0 since a, b>c

(I): <0 b> a+c2 (AM) > ac (GM) ⇒ (I) is true

(II): >0 b< a+c2; <1 3c<3b, i.e. b>c

∵ c< ac< a+c2 too, b may equal ac ⇒ (II) is true

Answer: (2)

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE 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. 3JEE 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. 4JEE 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. 5JEE 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. 6JEE 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. 7JEE 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 :

Ques. 8JEE Main 2024, 30 January (Evening Shift)

If z is a complex number, then the number of common roots of the equations z1985+z100+1=0 and z3+2z2+2z+1=0, is equal to

  1. 0
  2. 2
  3. 1
  4. 3
Ques. 9JEE Main 2024, 30 January (Evening Shift)

The number of real solutions of the equation x (x2+3|x|+5|x-1|+6|x-2| )=0 is .

Ques. 10JEE Main 2024, 29 January (Morning Shift)

If z= 12-2i is such that |z+1|= z+ (1+i), i= -1 and , R, then + is equal to

  1. 2
  2. -4
  3. 3
  4. -1
Ques. 11JEE Main 2024, 29 January (Morning Shift)

Let , be the roots of the equation x2-x+2=0 with Im( )>Im( ). Then 6+ 4+ 4-5 2 is equal to .

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