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Complex Numbers and Quadratic Equations · Question 2

JEE Advanced 2026 Paper 2Multiple CorrectHard📈 Topic in 17% papers+4−1
1

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

Answer

(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

Solution

Integer roots r,s: r+s=- ba, rs= ca are integers, so b=ka, c=la with k,l N; hence c-b=(l-k)a. (A) true.

AP: 2b=a+c l=2k-1. Equation becomes x2+kx+(2k-1)=0.

Discriminant k2-8k+4=(k-4)2-12 must be a perfect square m2: (k-4-m)(k-4+m)=12 with factors of equal parity \2,6\ or \-6,-2\ k=8 (since k>0).

So l=15: x2+8x+15=0 x=-3,-5, both odd. (B) true.

c=15 15a=15 a=1, b=8 ab=8. (C) true.

b=8 a=1, roots -3,-5; x=3 is not a root. (D) false.

Answer: (A), (B), (C)

JEE Advanced 2026 Paper 2

Keep practising — Complex Numbers and Quadratic Equations

2

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
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3

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
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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 .

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5

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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