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Complex Numbers and Quadratic Equations · Question 1

JEE Advanced 2026 Paper 1MatchingMedium📈 Topic in 33% papers+4−1
1

Match each entry in List-I to the correct entry in List-II and choose the correct option.

  1. (P) → (1), (Q) → (2), (R) → (5), (S) → (4)
  2. (P) → (3), (Q) → (1), (R) → (4), (S) → (5)
  3. (P) → (1), (Q) → (2), (R) → (4), (S) → (5)
  4. (P) → (2), (Q) → (3), (R) → (5), (S) → (4)

Answer

(3) (P) → (1), (Q) → (2), (R) → (4), (S) → (5)

Solution

Roots of x2 + x + 1 = 0 are , 2 and + 1 = - 2.

(P): ( + 1)2026 = 4052 = 2 (4052 = 3 1350 + 2), so 1( + 1)2026 = ; roots , : x2 + x + 1 = 0 → (1).

(Q): ( + 1)2027 = - 4054 = - , so the new roots are - 1 = - and - ; equation x2 - x + 1 = 0 → (2).

(R): = - - 1 = 2, ( - 1)2026 = 4052 = 2, term = ; similarly the other term = 2; sum = -1 → (4).

(S): u = p + 1 satisfies u2 - u - 1 = 0, so v = 1u satisfies v2 + v - 1 = 0: v1 + v2 = -1, v1v2 = -1.

v13 + v23 = (-1)3 - 3(-1)(-1) = -4 → (5)

Answer: (C)

JEE Advanced 2026 Paper 1

Keep practising — Complex Numbers and Quadratic Equations

2

Let a, b, c be positive integers in arithmetic progression such that the equation ax2+bx+c=0 has only integer solutions. Then which of the following statements is (are) TRUE ?

  1. c-b is an integer multiple of a
  2. Both the roots of the equation ax2+bx+c=0 are odd integers
  3. If c=15, then ab=8
  4. If b=8, then x=3 is a root of the equation ax2+bx+c=0
JEE Advanced 2026 Paper 2
3

Let R denote the set of all real numbers. Let ai, bi R for i \1, 2, 3\. Define the functions f: R R, g: R R, and h: R R by f(x) = a1 + 10x + a2x2 + a3x3 + x4, g(x) = b1 + 3x + b2x2 + b3x3 + x4, h(x) = f(x+1) - g(x+2). If f(x) g(x) for every x R, then the coefficient of x3 in h(x) is

  1. 8
  2. 2
  3. -4
  4. -6
JEE Advanced 2025 Paper 1
4

Let R denote the set of all real numbers. Let z1 = 1 + 2i and z2 = 3i be two complex numbers, where i = -1. Let S = \(x, y) R R : |x + iy - z1| = 2|x + iy - z2|\. Then which of the following statements is (are) TRUE?

  1. S is a circle with centre (- 13, 103 )
  2. S is a circle with centre ( 13, 83 )
  3. S is a circle with radius 23
  4. S is a circle with radius 2 23
JEE Advanced 2025 Paper 1
5

For a non-zero complex number z, let (z) denote the principal argument of z, with - < (z) . Let be the cube root of unity for which 0< ( )< . Let = ( n=12025(- )n ). Then the value of 3 is .

JEE Advanced 2025 Paper 2
6

Let f(x)=x4+ax3+bx2+c be a polynomial with real coefficients such that f(1)=-9. Suppose that i 3 is a root of the equation 4x3+3ax2+2bx=0, where i= -1. If 1, 2, 3, and 4 are all the roots of the equation f(x)=0, then | 1|2+| 2|2+| 3|2+| 4|2 is equal to .

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