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JEE Main Complex Numbers and Quadratic Equations questions

108 Mathematics questions on Complex Numbers and Quadratic Equations as asked in JEE Main, newest first. 58 come with a worked solution.

  1. The number of real solutions of the equation 3 (x^2 + 1/(x^2) ) - 2 (x + 1/x ) + 5 = 0, is

    2023 · with solution

  2. The value of ((1+sin(2π)/9+icos(2π)/9)/(1+sin(2π)/9-icos(2π)/9) )^3 is

    2023 · with solution

  3. Let z be a complex number such that |(z-2i)/(z+i) | = 2, z ≠ -i. Then z lies on the circle of radius 2 and centre

    2023 · with solution

  4. Let λ ≠ 0 be a real number. Let α, β be the roots of the equation 14x^2 - 31x + 3λ = 0 and α, γ be the roots of the equation 35x^2 - 53x + 4λ = 0. Then (3α)/(β) and (4α)/(γ) are the roots of the equation

    2023

  5. For two non-zero complex numbers z_1 and z_2, if Re(z_1 z_2) = 0 and Re(z_1 + z_2) = 0, then which of the following are possible? A. Im(z_1) > 0 and Im(z_2) > 0 B. Im(z_1) 0 C. Im(z_1) > 0 and Im(z_2) < 0 D. Im(z_1) < 0…

    2023 · with solution

  6. Let p, q ∈ R and (1-√(3)i )^200 = 2^199(p+iq), i = √(-1). Then p + q + q^2 and p - q + q^2 are roots of the equation.

    2023 · with solution

  7. Let a ∈ R, and let α, β be the roots of the equation x^2 + 60^1/4x + a = 0. If α^4 + β^4 = -30, then the product of all possible values of a is _______

    2023 · with solution

  8. Let λ ∈ R and let the equation E be |x|^2 - 2|x| + |λ - 3| = 0. Then the largest element in the set S = \x + λ : x is an integer solution of E\ is

    2023

  9. If z and w are two complex numbers such that |zw| = 1 and arg(z) - arg(w) = (π)/2, then

    2019 · with solution

  10. Let z = (dfrac√(3)2 + i/2 )^5 + (dfrac√(3)2 - i/2 )^5. If R(z) and I(z) respectively denote the real and imaginary parts of z, then

    2019

  11. The equation |z - i| = |z - 1|, i = √(-1), represents

    2019 · with solution

  12. Let z ∈ C with Im(z) = 10 and it satisfies (2z - n)/(2z + n) = 2i - 1 for some natural number n, then

    2019

  13. The number of real roots of the equation 5 + |2^x - 1| = 2^x(2^x - 2) is

    2019 · with solution

  14. If α, β and γ are three consecutive terms of a non-constant GP such that the equations α x^2 + 2β x + γ = 0 and x^2 + x - 1 = 0 have a common root, then, α(β + γ) is equal to

    2019

  15. All the points in the set S = \(α + i)/(α - i) : α ∈ R \ (i = √(-1)) lie on a

    2019 · with solution

  16. If z = dfrac√(3)2 + i/2 (i = √(-1) ), then (1 + iz + z^5 + iz^8)^9 is equal to

    2019

  17. Consider the quadratic equation, (c-5)x^2 - 2cx + (c-4) = 0, c ≠ 5. Let S be the set of all integral values of c for which one root of the equation lies in the interval (0, 2) and its other root lies in the interval (2,…

    2019 · with solution

  18. If a > 0 and z = ((1+i)^2)/(a-i), has magnitude √(2/5), then z is equal to

    2019

  19. Let z_1 and z_2 be two complex numbers satisfying |z_1| = 9 and |z_2 - 3 - 4i| = 4. Then, the minimum value of |z_1 - z_2| is

    2019 · with solution

  20. The sum of the solutions of the equation |√(x) - 2| + √(x)(√(x) - 4) + 2 = 0 (x > 0) is equal to

    2019 · with solution

  21. If m is chosen in the quadratic equation (m^2+1)x^2-3x+(m^2+1)^2=0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is

    2019

  22. Let z ∈ C be such that |z| < 1. If ω = (5 + 3z)/(5(1-z)), then

    2019 · with solution

  23. Let z_0 be a root of the quadratic equation, x^2 + x + 1 = 0. If z = 3 + 6iz_0^81 - 3iz_0^93, then arg z is equal to

    2019

  24. If both the roots of the quadratic equation x^2 - mx + 4 = 0 are real and distinct and they lie in the interval [1, 5] then m lies in the interval

    2019 · with solution

  25. If (z-α)/(z+α) (α ∈ R) is a purely imaginary number and |z| = 2, then a value of α is

    2019 · with solution

  26. Let (-2 - 1/3i )^3 = (x+iy)/27 (i = √(-1)), where x and y are real numbers, then y - x equals

    2019

  27. The number of integral values of m for which the equation (1 + m^2)x^2 - 2(1 + 3m)x + (1 + 8m) = 0, has no real root is

    2019

  28. Let z be a complex number such that |z| + z = 3 + i (where i = √(-1)). Then, |z| is equal to

    2019 · with solution

  29. Let α and β be the roots of the quadratic equation x^2 sinθ - x(sinθcosθ + 1) + cosθ = 0 (0 < θ < 45°) and α < β. Then sum_n=0^∞ (α^n + ((-1)^n)/(β^n) ) is equal to

    2019

  30. Let A = \θ ∈ (-(π)/2, π ) : (3 + 2isinθ)/(1 - 2isinθ) is purely imaginary \ Then, the sum of the elements in A is

    2019

  31. The number of integral values of m for which the quadratic expression, (1 + 2m)x^2 - 2(1 + 3m)x + 4(1 + m), x ∈ R, is always positive, is

    2019 · with solution

  32. If λ be the ratio of the roots of the quadratic equation in x, 3m^2x^2 + m(m-4)x + 2 = 0, then the least value of m for which λ + 1/(λ) = 1, is

    2019

  33. If one real root of the quadratic equation 81x^2 + kx + 256 = 0 is cube of the other root, then a value of k is

    2019 · with solution

  34. The value of λ such that sum of the squares of the roots of the quadratic equation, x^2 + (3 - λ)x + 2 = λ has the least value is

    2019

  35. The number of all possible positive integral values of α for which the roots of the quadratic equation 6x^2 - 11x + α = 0 are rational numbers is

    2019

  36. Let α and β be two roots of the equation x^2 + 2x + 2 = 0, then α^15 + β^15 is equal to

    2019 · with solution

  37. Let S = \x ∈ R : x ≥ 0 and 2|√(x) - 3| + √(x)(√(x) - 6) + 6 = 0\. Then, S

    2018 · with solution

  38. If α, β ∈ C are the distinct roots of the equation x^2 - x + 1 = 0, then α^101 + β^107 is equal to

    2018

  39. A value of θ for which (2 + 3isinθ)/(1 - 2isinθ) is purely imaginary, is

    2016 · with solution

  40. The sum of all real values of x satisfying the equation (x^2 - 5x + 5)^x^2 + 4x - 60 = 1 is

    2016 · with solution

  41. Let -(π)/6 β_1 and α_2 > β_2, then α_1 + β_2 equals

    2016

  42. A complex number z is said to be unimodular, if |z| ≠ 1. If z_1 and z_2 are complex numbers such that dfracz_1 - 2z_22 - z_1z_2 is unimodular and z_0 is not unimodular. Then, the point z_1 lies on a

    2015

  43. Let α and β be the roots of equation x^2 - 6x - 2 = 0. If a_n = α^n - β^n, for n ≥ 1, then the value of dfraca_10 - 2a_82a_9 is

    2015 · with solution

  44. If a ∈ R and the equation -3(x - [x])^2 + 2(x - [x]) + a^2 = 0 (where, [x] denotes the greatest integer ≤ x) has no integral solution, then all possible values of a lie in the interval

    2014 · with solution

  45. If z is a complex number such that |z| ≥ 2, then the minimum value of |z + 1/2 |

    2014

  46. In the quadratic equation p(x) = 0 with real coefficients has purely imaginary roots. Then, the equation p[p(x)] = 0 has

    2014

  47. Let α and β be the roots of equation px^2 + qx + r = 0, p ≠ 0. If p, q and r are in AP and 1/(α) + 1/(β) = 4, then the value of |α - β| is

    2014

  48. If z is a complex number of unit modulus and argument θ, then arg (dfrac1+z1+z ) is equal to

    2013 · with solution

  49. If the equations x^2 + 2x + 3 = 0 and ax^2 + bx + c = 0, a, b, c ∈ R have a common root, then a : b : c is

    2013 · with solution

  50. The real number k for which the equation, 2x^3 + 3x + k = 0 has two distinct real roots in [0, 1]

    2013

  51. Let z be a complex number such that the imaginary part of z is non-zero and a = z^2 + z + 1 is real. Then, a cannot take the value

    2012 · with solution

  52. A value of b for which the equations x^2 + bx - 1 = 0, x^2 + x + b = 0 have one root in common is

    2011

  53. Let z = cosθ + isinθ. Then, the value of sum_m=1^15 Im(z^2m-1) at θ = 2° is

    2009 · with solution

  54. Let z = x + iy be a complex number where, x and y are integers. Then, the area of the rectangle whose vertices are the root of the equation zz^3 + zz^3 = 350, is

    2009

  55. A particle P starts from the point z_0 = 1 + 2i, where i = √(-1). It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z_1. From z_1 the particle moves…

    2008

  56. A man walks a distance of 3 units from the origin towards the North-East (N 45° E) direction. From there, he walks a distance of 4 units towards the North-West (N 45° W) direction to reach a point P. Then, the position…

    2007

  57. If |z| = 1 and z ≠ ± 1, then all the values of z/(1-z^2) lie on

    2007 · with solution

  58. Let α, β be the roots of the equation x^2 - px + r = 0 and (α)/2, 2β be the roots of the equation x^2 - qx + r = 0. Then, the value of r is

    2007

  59. If w = α + iβ, where β ≠ 0 and z ≠ 1, satisfies the condition that (dfracw - wz1-z ) is purely real, then the set of values of z is

    2006 · with solution

  60. If a, b, c are the sides of a triangle ABC such that x^2 - 2(a+b+c)x + 3λ(ab+bc+ca) = 0 has real roots, then

    2006 · with solution

  61. The minimum value of |a + bω + cω^2|, where a, b and c are all not equal integers and ω (≠ 1) is a cube root of unity, is

    2005

  62. For all 'x', x^2 + 2ax + (10 - 3a) > 0, then the interval in which 'a' lies is

    2004 · with solution

  63. If ω (≠ 1) be a cube root of unity and (1 + ω^2)^n = (1 + ω^4)^n, then the least positive value of n is

    2004 · with solution

  64. If one root is square of the other root of the equation x^2 + px + q = 0, then the relation between p and q is

    2004

  65. If |z| = 1 and w = (z-1)/(z+1) (where, z ≠ -1), then Re(w) is

    2003

  66. If f(x) = x^2 + 2bx + 2c^2 and g(x) = -x^2 - 2cx + b^2, such that min f(x) > max g(x), then the relation between b and c is

    2003 · with solution

  67. If 0 < α < (π)/2 is a fixed angle. If P = (cosθ, sinθ) and Q = (cos(α - θ), sin(α - θ)), then Q is obtained from P by

    2002

  68. For all complex numbers z_1, z_2 satisfying |z_1| = 12 and |z_2 - 3 - 4i| = 5, the minimum value of |z_1 - z_2| is

    2002 · with solution

  69. The set of all real numbers x for which x^2 - |x + 2| + x > 0 is

    2002 · with solution

  70. The complex numbers z_1, z_2 and z_3 satisfying (z_1 - z_3)/(z_2 - z_3) = frac1 - i√(3)2 are the vertices of a triangle which is

    2001 · with solution

  71. Let z_1 and z_2 be nth roots of unity which subtend a right angled at the origin, then n must be of the form (where, k is an integer)

    2001

  72. The number of solutions of log_4(x-1) = log_2(x-3)

    2001 · with solution

  73. If arg(z) < 0, then arg(-z) - arg(z) equals

    2000 · with solution

  74. If b > a, then the equation (x - a)(x - b) - 1 = 0 has

    2000

  75. If z_1, z_2 and z_3 are complex numbers such that |z_1| = |z_2| = |z_3| = |1/z_1 + 1/z_2 + 1/z_3 | = 1, then |z_1 + z_2 + z_3| is

    2000

  76. For the equation 3x^2 + px + 3 = 0, p > 0, if one of the root is square of the other, then p is equal to

    2000 · with solution

  77. If α and β (α < β) are the roots of the equation x^2 + bx + c = 0, where c < 0 < b, then

    2000

  78. If the roots of the equation x^2 - 2ax + a^2 + a - 3 = 0 are real and less than 3, then

    1999 · with solution

  79. If i = √(-1), then 4 + 5 (-1/2 + dfraci√(3)2 )^334 + 3 (-1/2 + dfraci√(3)2 )^365 is equal to

    1999

  80. If ω is an imaginary cube root of unity, then (1 + ω - ω^2)^7 is equal to

    1998 · with solution

  81. The value of sum_k=1^6 (sin(2π k)/7 - icos(2π k)/7 ) is

    1998 · with solution

  82. If A_0, A_1, A_2, A_3, A_4 and A_5 be a regular hexagon inscribed in a circle of unit radius. Then, the product of the lengths of the line segments A_0A_1, A_0A_2 and A_0A_4 is

    1998 · with solution

  83. The equation √(x+1) - √(x-1) = √(4x-1) has

    1997 · with solution

  84. For positive integers n_1, n_2 the value of expression (1+i)^n_1 + (1+i^3)^n_1 + (1+i^5)^n_2 + (1+i^7)^n_2, here i = √(-1) is a real number, if and only if

    1996 · with solution

  85. Let z and w be two complex numbers such that |z| ≤ 1, |w| ≤ 1 and |z + iw| = |z - iw| = 2, then z equals

    1995

  86. Let z and w be two non-zero complex numbers such that |z| = |w| and arg(z) + arg(w) = π, then z equals

    1995

  87. If ω (≠ 1) is a cube root of unity and (1 + ω)^7 = A + Bω, then A and B are respectively

    1995 · with solution

  88. Let α, β be the roots of the equation, (x-a)(x-b) = c, c ≠ 0. Then the roots of the equation (x-α)(x-β) + c = 0 are

    1992

  89. The equation (cos p - 1)x^2 + (cos p)x + sin p = 0 in the variable x, has real roots. Then, p can take any value in the interval

    1990

  90. Let f(x) be a quadratic expression which is positive for all real values of x. If g(x) = f(x) + f'(x) + f''(x), then for any real x

    1990

  91. Let a, b, c be real numbers, a ≠ 0. If α is a root of a^2x^2 + bx + c = 0, β is the root of a^2x^2 - bx - c = 0 and 0 < α < β, then the equation a^2x^2 + 2bx + 2c = 0 has a root γ that always satisfies

    1989

  92. The equation x^3/4(log_2 x)^2 + log_2 x - 5/4 = √(2) has

    1989 · with solution

  93. If α and β are the roots of x^2 + px + q = 0 and α^4, β^4 are the roots of x^2 - rx + s = 0, then the equation x^2 - 4qx + 2q^2 - r = 0 has always

    1989

  94. The complex numbers sin x + i cos 2x and cos x - i sin 2x are conjugate to each other, for

    1988

  95. If z_1 and z_2 are two non-zero complex numbers such that |z_1 + z_2| = |z_1| + |z_2|, then arg(z_1) - arg(z_2) is equal to

    1987 · with solution

  96. If a, b, c and u, v, w are the complex numbers representing the vertices of two triangles such that c = (1-r)a + rb and w = (1-r)u + rv, where r is a complex number, then the two triangles

    1985

  97. The equation x - 2/(x-1) = 1 - 2/(x-1) has

    1984 · with solution

  98. For real x, the function ((x-a)(x-b))/((x-c)) will assume all real values provided

    1984 · with solution

  99. The points z_1, z_2, z_3 and z_4 in the complex plane are the vertices of a parallelogram taken in order, if and only if

    1983 · with solution

  100. If z = x + iy and w = (1 - iz)/(z - i), then |w| = 1 implies that, in the complex plane

    1983 · with solution

  101. The inequality |z-4| < |z-2| represents the region given by

    1982

  102. If z = (dfrac√(3)2 + i/2 )^5 + (dfrac√(3)2 - i/2 )^5, then

    1982

  103. The number of real solutions of the equation |x|^2 - 3|x| + 2 = 0 is

    1982 · with solution

  104. The complex numbers z = x + iy which satisfy the equation |(z - 5i)/(z + 5i) | = 1, lie on

    1981 · with solution

  105. Both the roots of the equation (x-b)(x-c) + (x-a)(x-c) + (x-a)(x-b) = 0 are always

    1980 · with solution

  106. Let a > 0, b > 0 and c > 0. Then, both the roots of the equation ax^2 + bx + c = 0

    1979

  107. If a + b + c = 0, then the quadratic equation 3ax^2 + 2bx + c = 0 has

  108. The largest interval for which x^12 - x^9 + x^4 - x + 1 > 0 is

    with solution

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