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Complex Numbers and Quadratic Equations

JEE Main 2024 (Online) 31st January Morning ShiftMCQMedium📈 Topic in 16% papers+4−1
Ques. 1JEE 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

Answer

(2) 1 : 1 : 4

Solution

Sum of coefficients: put x=1 ⇒ a=(1)2023(1)2024=1

b= x 0 (1+x) (x2024+1 ) 2x= 12 (L'Hôpital's rule)

⇒ second equation: x2+x+4=0, D=1-16<0 ⇒ roots are non-real

c, d, e are real, non-real roots occur in conjugate pairs ⇒ both roots are common

c1= d1= e4 ⇒ d : c : e=1 : 1 : 4

Answer: (2)

Keep practising — Complex Numbers and Quadratic Equations

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

Ques. 11JEE Main 2024, 29 January (Evening Shift)

Let r and respectively be the modulus and amplitude of the complex number z=2-i (2 5 8 ), then (r, ) is equal to

  1. (2 11 8, 11 8 )
  2. (2 3 8, 3 8 )
  3. (2 5 8, 3 8 )
  4. (2 3 8, 5 8 )

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