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Complex Numbers and Quadratic Equations

JEE Main 2025 (Online) 8th April Evening ShiftMCQMedium📈 Topic in 33% papers+4−1
Ques. 1JEE Main 2025, 8 April (Evening Shift)

The sum of the squares of the roots of |x-2|2 + |x-2| - 2 = 0 and the squares of the roots of x2 - 2|x-3| - 5 = 0, is

  1. 24
  2. 26
  3. 36
  4. 30

Answer

(3) 36

Solution

In the first equation, let t = |x - 2| 0; then

t2 + t - 2 = 0 (t + 2)(t - 1) = 0 t = 1

so x = 1, 3 and the sum of squares = 1 + 9 = 10.

In the second equation, for x 3: x2 - 2x + 1 = 0 x = 1, which is rejected since 1 < 3; for x < 3: x2 + 2x - 11 = 0 x = -1 2 3, both less than 3, so both are accepted.

Their sum of squares = (x1 + x2)2 - 2x1x2 = 4 + 22 = 26

Required sum = 10 + 26 = 36

Answer: (3)

Keep practising — Complex Numbers and Quadratic Equations

Ques. 2JEE Main 2025, 8 April (Evening Shift)

Let A = \ [0, 2 ] : 1 + 10 Re ( 2 + i - 3i ) = 0 \. Then A 2 is equal to

  1. 214 2
  2. 6 2
  3. 274 2
  4. 8 2
Ques. 3JEE Main 2025, 7 April (Morning Shift)

Let the set of all values of p R, for which both the roots of the equation x2 - (p + 2)x + (2p + 9) = 0 are negative real numbers, be the interval ( , ]. Then - 2 is equal to

  1. 5
  2. 0
  3. 20
  4. 9
Ques. 4JEE Main 2025, 7 April (Morning Shift)

Among the statements (S1) : The set \z C - \-i\ : |z| = 1 and z - iz + i is purely real \ contains exactly two elements, and (S2) : The set \z C - \-1\ : |z| = 1 and z - 1z + 1 is purely imaginary \ contains infinitely many elements.

  1. both are incorrect
  2. both are correct
  3. only (S2) is correct
  4. only (S1) is correct
Ques. 5JEE Main 2025, 7 April (Evening Shift)

If the locus of z C, such that Re ( z-12z+i ) + Re ( z-12 z-i ) = 2, is a circle of radius r and center (a, b), then 15abr2 is equal to :

  1. 16
  2. 24
  3. 12
  4. 18
Ques. 6JEE Main 2025, 7 April (Evening Shift)

The number of real roots of the equation x|x-2| + 3|x-3| + 1 = 0 is :

  1. 4
  2. 3
  3. 2
  4. 1
Ques. 7JEE Main 2025, 7 April (Evening Shift)

For t > -1, let t and t be the roots of the equation ((t + 2)1/7 - 1 )x2 + ((t + 2)1/6 - 1 )x + ((t + 2)1/21 - 1 ) = 0 If t -1+ t = a and t -1+ t = b, then 72(a + b)2 is equal to .

Ques. 8JEE Main 2025, 4 April (Morning Shift)

Consider the equation x2 + 4x - n = 0, where n [20, 100] is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to

  1. 6
  2. 5
  3. 8
  4. 7
Ques. 9JEE Main 2025, 4 April (Morning Shift)

Let A = \z C : |z - 2 - i| = 3\, B = \z C : Re(z - iz) = 2\ and S = A B. Then z S |z|2 is equal to .

Ques. 10JEE Main 2025, 4 April (Evening Shift)

Let the product of 1 = (8 + i) + (7 + 4i) and 2 = (1 + 8i) + (4 + 7i) be + i , i = -1. Let p and q be the maximum and the minimum values of + respectively. Then p + q is equal to :

  1. 130
  2. 150
  3. 160
  4. 140
Ques. 11JEE Main 2025, 4 April (Evening Shift)

If is a root of the equation x2+x+1=0 and k=1n ( k+ 1 k )2=20, then n is equal to .

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