Complex Numbers and Quadratic Equations VBQs Class 11 Mathematics

VBQs Complex Numbers And Quadratic Equations Class 11 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 11 Mathematics with solutions. The following Complex Numbers And Quadratic Equations Class 11 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 11 examinations.

Complex Numbers And Quadratic Equations VBQs Class 11 Mathematics

Question. Let α and β be the roots of the equation x² – x – 1 = 0. If p_k = (α)^k + (β)^k, k ≥ l, then which one of the following statements is not true ?
(a) p₃ = p₅ – p₄
(b) P₅ = 11
(c) (p₁ + p₂ + p₃ + p₄ + p₅) = 26
(d) p₅ = p₂ × p₃

Question. Let α and β be two real roots of the equation (k +1) tan²x – √2 . λ tan x = (1 – k), where k(≠ –1) and λ are real numbers. If tan²(α + β) = 50, then a value of λ is:
(a) 10√2
(b) 10
(c) 5
(d) 5√2

Question. Let z = x + iy be a non- ero complex number such that z2 = i |z|², where i = √-1 , then z lies on the:
(a) line, y = –x
(b) imaginary axis
(c) line, y = x
(d) real axis

Question. If m is chosen in the quadratic equation (m² + 1) x² – 3x + (m² + 1)² = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:
(a) 10√5
(b) 8√3
(c) 8√5
(d) 4√3

Question. The sum of the solutions of the equation | √x – 2 | + √x ( √x – 4) + 2 = 0, (x > 0) is equal to:
(a) 9
(b) 12
(c) 4
(d) 10

Question. If a and b are real numbers such that (2 + α)⁴ = α + bα, where α = (-1 + i√3) / 2 , then a + b is equal to :
(a) 9
(b) 24
(c) 33
(d) 57

Question. Let w (Im w ≠ 0) be a complex number. Then the set of all complex number z satisfying the equation w – w̅z = k (1 – z) , for some real number k, is
(a) { z : |z| = 1}
(b) { z : z = z̅ }
(c) { z : z ≠ 1}
(d) { z : |z| =1, z ≠ 1}

Question. Let α = Im( 1+z² / 2iz ) , where z is any non- ero complex number.
The set A = {α : | z | = 1 and z ≠ ±1 } is equal to:
(a) (– 1, 1)
(b) [– 1, 1]
(c) [0, 1)
(d) (– 1, 0]

Question. Let Z and W be complex numbers such that |Z| = |W|, and arg Z denotes the principal argument of Z.
Statement 1: If arg Z + arg W = p, then Z = -w̅ .
Statement 2: |Z| = |W|, implies arg Z – arg w̅ = π.
(a) Statement 1 is true, Statement 2 is false.
(b) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
(c) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.
(d) Statement 1 is false, Statement 2 is true.

Question. The conjugate of a complex number is 1 / i–1 then that complex number is
(a) –1 / i –1
(b) 1 / i +1
(c) –1 / i +1
(d) 1 / i –1

Question. Let z₀ be a root of the quadratic equation, x² + x + 1 = 0. If z = 3 + 6i z₀⁸¹ – 3i z₀⁹³, then arg z is equal to:
(a) π/4
(b) π/6
(c) π/3
(d) 0

Question. Let p, q and r be real numbers (p ≠ q, r ≠ 0), such that the roots of the equation 1/(x+p) + 1/(x+q) = 1/r are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to.
(a) p² + q² + r²
(b) p² + q²
(c) 2 (p² + q²)
(d) (p² + q²) / 2

Question. If z = x -i y and 1/z³ = p + iq, then (x/p + y/q) / (p²+q²) is equal to
(a) –2
(b) –1
(c) 2
(d) 1

Question. Let and w be complex numbers such that z̅ + iw̅ = 0 and arg zw = π. Then arg equals
(a) 5π/4
(b) π/2
(c) 3π/4
(d) π/4

Question. Let

 If R(z) and I(z) respectively denote the real and imaginary parts of z, then:
(a) I(z) = 0
(b) R(z) > 0 and I(z) > 0
(c) R(z) < 0 and I(z) > 0
(d) R(z) = – (c)

Question. The least positive integer n for which [ (1 + i√3) / (1 – i√3) ]ⁿ = 1, is
(a) 2
(b) 6
(c) 5
(d) 3

Question. If | – 4 | < | – 2 |, its solution is given by
(a) Re(z) > 0
(b) Re(z) < 0
(c) Re(z) > 3
(d) Re(z) > 2

Question. z and w are two non ero complex numbers such that | z | = | w| and Arg z + Arg w = π then equals

Question. Let Z₁ and Z₂ be two roots of the equation Z² + aZ + b = 0 , Z being complex. Further , assume that the origin, Z₁ and Z₂ form an equilateral triangle. Then
(a) a² = 4b
(b) a² = b
(c) a² = 2b
(d) a² = 3b

Question. If p and q are the roots of the equation x² + px + q = 0, then
(a) p = 1, q = –2
(b) p = 0, q = 1
(c) p = –2, q = 0
(d) p = – 2, q = 1

Question. The value of

(a) 1/2 (1-i√3)
(b) 1/2(√3 – i)
(c) – 1/2 (√3 – i)
(d) – 1/2 (1 – i√3)

Question. If z is a non-real complex number, then the minimum value of lmz⁵ / (lmz)⁵ is :
(a) –1
(b) –4
(c) –2
(d) –5

Question. If z is a complex number such that |z| ≥ 2, then the minimum value of | z + 1/2 |
(a) is strictly greater than 5/2
(b) is strictly greater than 3/2 but less than 5/2
(c) is equal to 5/2
(d) lie in the interval (1, 2)

Question. The imaginary part of (3 + 2√-54)^½ – (3 – 2√-54)^½ can be :
(a) – √6
(b) -2√6
(c) 6
(d) √6

Question. Let α = (-1 + i√3) / 2. if

then α and b are the roots of the quadratic equation:
(a) x² + 101x + 100 = 0
(b) x² – 102 x + 101 = 0
(c) x² – 101x + 100 = 0
(d) x² + 102x +101 = 0

Question. If a ∈ R and the equation -3( x – [x] )² + 2( x – [x] ) + α² = 0 (where [x] denotes the greatest integer ≤ x ) has no integral solution, then all possible values of a lie in the interval:
(a) (-2,-1)
(b) (-∞,-2)υ(2,∞)
(c) (-1,0)υ(0,1)
(d) (1, 2)

Question. The values of ‘a’ for which one root of the equation x² – (a + 1 ) x + a² + a – 8 = 0 exceeds 2 and the other is lesser than 2, are given by :
(a) 3 < a < 10
(b) a ≥10
(c) -2 < a < 3
(d) a ≤ -2

Question. If the equations x² + bx – 1 = 0 and x² + x + b = 0 have a common root different from –1, then |b| is equal to :
(a) 2
(b) 3
(c) √3
(d) √2

Question. If λ ∈ R is such that the sum of the cubes of the roots of the equation, x² + (2 – λ) x + (10 – λ) = 0 is minimum, then the magnitude of the difference of the roots of this equation is
(a) 20
(b) 2√5
(c) 2√7
(d) 4√2

Question. If both the roots of the quadratic equation x² – 2kx + k² + k – 5 = 0 are less than 5, then k lies in the interval
(a) (5, 6]
(b) (6, ∞ )
(c) (– ∞ , 4)
(d) [4, 5]

Question. The quadratic equations x² – 6x + a = 0 and x² – cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is 
(a) 1
(b) 4
(c) 3
(d) 2

Question. The least integral value α of x such that (x – 5) / (x² + 5x – 14) > 0, satisfies :
(a) a² + 3a – 4 = 0
(b) a² – 5a + 4 = 0
(c) a² – 7a + 6 = 0
(d) a² + 5a – 6 = 0

Question. If equations ax² + bx + c = 0 (a, b,c ∈ R,a ≠ 0) and 2x² + 3x + 4 = 0 have a common root, then a : b : c equals:
(a) 1 : 2 : 3
(b) 2 : 3 : 4
(c) 4 : 3 : 2
(d) 3 : 2 : 1

Question. The value of k for which the equation (k – 2)x² + 8x + k + 4 = 0 has both roots real, distinct and negative is
(a) 6
(b) 3
(c) 4
(d) 1

Question. Let for a ≠ 1 a ≠ 0, ƒ(x) = ax² + bx + c, g(x) = a₁x² + b₁x + c1 and p(x) = ƒ(x) – g(x).
If p(x) = 0 only for x = -1 and p (– 2) = 2, then the value of p(b) is :
(a) 3
(b) 9
(c) 6
(d) 18

Question. The equation √(3×2 + x 5) = x – 3 , where x is real, has;
(a) no solution
(b) exactly one solution
(c) exactly two solution
(d) exactly four solution

Question. If Re (z-1 / 2z+i) =1, where z = x + iy, then the point (x, y) lies on a:
(a) circle whose centre is at (- 1/2 , 3/2)
(b) straight line whose slope is – 2/3 .
(c) straight line whose slope is 3/2.
(d) circle whose diameter is √5/2.

Question. If Z1 ≠ 0 and Z₂ be two complex numbers such that Z₂ /z₁ is a purely imaginary number, then | 2z₁+3z₂ / 2z₁-3z₂ | is equal to:
(a) 2
(b) 5
(c) 3
(d) 1

Question. | z₁ + z₂ |² + | z₁ – z₂ |² is equal to 
(a) 2 ( | z₁ + z₂ | )
(b) 2 ( | z₁ |² + | z₂ |² )
(c) | z₁ || z₂ |
(d) | z₁ |² + | z₂ |²

Question. Let

where x and y are real numbers then y – x equals :
(a) 91
(b) – 85
(c) 85
(d) – 91

Question. The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there 2√2 units in the south–westwards direction. Then its new position in the Argand plane is at the point represented by :
(a) 1 + i
(b) 2 + 2i
(c) –2 – 2i
(d) –1 – i

Question. If x is a solution of the equation, 

is equal to :
(a) 3/4
(b) 1/2
(c) 2√2
(d) 2

Question. Let α and β be the roots of equation x² – 6x – 2 = 0. If an = αⁿ – βⁿ, for n ≥ 1, then the value of (a₁₀ – 2a₈) / 2a₉ is equal to:
(a) 3
(b) – 3
(c) 6
(d) – 6

Question. A complex z number is said to be unimodular if |z| = 1. Suppose z₁ and z₂ are complex numbers such that (z₁ – 2z₂) / (2 – Z₁z̅₂) is unimodular and z₂ is not unimodular. Then the point z₁ lies on a:
(a) circle of radius 2.
(b) circle of radius √2.
(c) straight line parallel to x-axis
(d) straight line parallel to y-axis.

Question. If z ≠ 1 and z² / (z – 1 ) is real, then the point represented by the complex number z lies :
(a) either on the real axis or on a circle passing through the origin.
(b) on a circle with centre at the origin
(c) either on the real axis or on a circle not passing through the origin.
(d) on the imaginary axis.

Question. If z = √3/2 + i/2 (i = √-1), then (1 + iz + z⁵ + iz⁸)⁹ is equal to:
(a) 0
(b) 1
(c) (– 1 + 2i)⁹
(d) – 1

Question. If ω = z/(z – 1/3 i) and |ω = 1, then z lies on
(a) an ellipse
(b) a circle
(c) a straight line
(d) a parabola

Question. If z₁ and z₂ are two non- ero complex numbers such that | z₁ + z₂ | = | z₁ | + | z₂ | , then arg z₁ – arg z₂ is equal to
(a) π/2
(b) – π
(c) 0
(d) -π/2

Question. If λ be the ratio of the roots of the quadratic equation in x, 3m²x² + m(m – 4)x + 2 = 0, then the least value of m for which λ + 1/λ = 1, is :
(a) 2 – √3
(b) 4 – 3√2
(c) –2 + √2
(d) 4 – 2√3

Question. If one real root of the quadratic equation 81x² + kx + 256 = 0 is cube of the other root, then a value of k is :
(a) –81
(b) 100
(c) 144
(d) – 300

Question. If the cube roots of unity are 1, ω, ω2 then the roots of the equation (x –1)³ + 8 = 0, are
(a) –1, –1 + 2 ω , – 1 – 2 ω²
(b) –1, – 1, – 1
(c) – 1, 1 – 2 ω , 1 – 2 ω²
(d) – 1, 1 + 2 ω , 1 + 2 ω²

Question. The product of the roots of the equation 9x² – 18| x | + 5 = 0, is :
(a) 5/9
(b) 25/81
(c) 5/27
(d) 25/9

Question. If α and β are the roots of the quadratic equation, x² + x sin θ – 2sinθ = 0⁰, θ |∈ (0 , π/2), then

 is equal to :

Question. The number of real roots of the equation 5 + | 2^x – 1 | = 2^x (2^x – 2) is:
(a) 3
(b) 2
(c) 4
(d) 1

Question. If α and β are the roots of the equation, 7x² – 3x – 2 = 0, the the value of α/(1 – α²) + β/(1 – β²) is equal to :
(a) 27/32
(b) 1/24
(c) 3/8
(d) 27/16

Question. If z is a complex number of unit modulus and argument θ, then arg( 1+z / 1+z̅ ) equals:
(a) –θ
(b) π/2 – θ
(c) π
(d) π – θ

Question. Let z satisfy| z | = 1 and = 1– z̅ .
Statement 1 : z is a real number.
Statement 2 : Principal argument of z is π/3
(a) Statement 1 is true Statement 2 is true; Statement 2 is a correct explanation for Statement 1.
(b) Statement 1 is false; Statement 2 is true
(c) Statement 1 is true, Statement 2 is false.
(d) Statement 1 is true; Statement 2 is true; Statement 2 is not a correct explanation for Statement 1.

Question. Let u = (2z + i) / (z – ki), z = x + iy and k > 0. If the curve represented by Re(u) + Im(u) = 1 intersects the y-axis at the points P and Q where PQ = 5, then the value of k is :
(a) 3/2
(b) 1/2
(c) 4
(d) 2

Question. If α and β are roots of the equation, x² – 4√2kx + 2e^4lnk – 1 = 0 for some k, and α² + β² = 66, then α³ + β³ is equal to:
(a) 248√2
(b) 280√2
(c) -32√2
(d) -280√2

Question. If 1/√α and 1/√β are the roots of the equation, ax² + bx + 1 = 0 (a ≠ 0, a, b, ∈ R), then the equation, x( x+b³ ) ( a³ – 3abx ) 0 as roots :

Question. Let λ ≠ 0 be in R. If α and β are roots of the equation, x² – x + 2λ = 0 and α and γ are the roots of the equation, 3x² – 10x + 27λ = 0, then βγ/λ is equal to :
(a) 27
(b) 18
(c) 9
(d) 36

Question. Let α and β be the roots of the equation, 5x² + 6x – 2 = 0. If S_n = αⁿ + βⁿ, n = 1, 2,3, …, then :
(a) 6S₆ + 5S₅ = 2S₄ 
(b) 6S₆ + 5S₅ + 2S₄ = 0
(c) 5S₆ + 6S₅ = 2S₄
(d) 5S₆ + 6S₅ + 2S₄ = 0

Question. If the equation, x² + bx + 45 = 0 (b ∈ R) has con ugate complex roots and they satisfy |z + 1| = 2√10 , then:
(a) b² – b = 30
(b) b² + b = 72
(c) b² – b = 42
(d) b² + b = 12

Question. Let p, q ∈ R. If 2 – √3 is a root of the quadratic equation, x² + px + q = 0, then:
(a) p² – 4q + 12 = 0
(b) q² – 4p – 16 = 0
(c) q² + 4p + 14 = 0
(d) p² – 4q – 12 = 0

Question. If α and β be the roots of the equation x² – 2x + 2 = 0, then the least value of n for which (α/β)ⁿ = 1 is :
(a) 2
(b) 5
(c) 4
(d) 3

Question. If the roots of the quadratic equation x² + px + q = 0 are tan30° and tan15°, respectively, then the value of 2 + q – p is
(a) 2
(b) 3
(c) 0
(d) 1

Question. If z² + z +1 = 0 , where is complex number, then the value of

(a) 18
(b) 54
(c) 6
(d) 12

Question. Consider the quadratic equation (c – 5)x² – 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, 3). Then the number of elements in S is:
(a) 18
(b) 12
(c) 10
(d) 11

Question. The value of λ such that sum of the squares of the roots of the quadratic equation, x² + (3 – λ)x + 2 = l has the least value is:
(a) 15/8
(b) 1
(c) 4/9
(d) 2

Question. Difference between the corresponding roots of x² + ax + b = 0 and x² + bx + a = 0 is same and a ≠ b, then
(a) a + b + 4 = 0
(b) a + b – 4 = 0
(c) a – b – 4 = 0
(d) a – b + 4 = 0

Question. The value of a for which the sum of the squares of the roots of the equation x² – (a – 2) x – a – 1 = 0 assume the least value is
(a) 1
(b) 0
(c) 3
(d) 2

Question. If |z – 4/z| = 2, then the maximum value of |z| is equal to :
(a) √5 +1
(b) 2
(c) 2 + √2
(d) √3 +1

Question. If | z – 3 + 2i | ≤ 4 then the difference between the greatest value and the least value of | z | is
(a) √13
(b) 2√13
(c) 8
(d) 4 + √13

Question. If the equations x² + 2x + 3 = 0 and ax² + bx + c = 0, a,b,c ∈ R, have a common root, then a : b : c is
(a) 1 : 2 : 3
(b) 3 : 2 : 1
(c) 1 : 3 : 2
(d) 3 : 1 : 2

Question. If non- ero real numbers b and c are such that min f(x) > max g(x), where f(x) = x² + 2bx + 2c² and g(x) = – x² – 2cx + b² (x ∈ R); then c/b lies in the interval:
(a) (0 , 1/2)
(b) (1/2 , 1/√2)
(c) (1/√2 , √2)
(d) (√2 , ∞)

Question. If x is real, the maximum value of (3x² + 9x +17) / (3x² + 9x +7)
(a) 1/4
(b) 41
(c) 1
(d) 17/7

Question. If α ≠ β but α2 = 5α – 3 and β2 = 5β – 3 then the equation having α/β and β/α as its roots is
(a) 3x² – 19x + 3 = 0
(b) 3x² + 19x – 3 = 0
(c) 3x² – 19x – 3 = 0
(d) x² – 5x + 3 = 0.

Question. Let α and β be two roots of the equation x² + 2x + 2 = 0, then α¹⁵ + β¹⁵ is equal to:
(a) – 256
(b) 512
(c) – 512
(d) 256

Question. If α and β be two roots of the equation x² – 64x + 256 = 0. Then the value of (α³ / β⁵)^1/8 + (β³ / α⁵)^1/8 is:
(a) 2
(b) 3
(c) 1
(d) 4

Question. Let α, β be real and z be a complex number. If z² + αz + β = 0 has two distinct roots on the line Re z =1, then it is necessary that :
(a) β ∈ (-1,0)
(b) |β| = 1
(c) β ∈ (1,∞)
(d) β ∈ (0,1)

Question. If α and β are the roots of the equation x² – x + 1 = 0, then α²⁰⁰⁹ + β²⁰⁰⁹ =
(a) –1
(b) 1
(c) 2
(d) –2

Question. If α and β are the roots of the equation 2x(2x + 1) = 1, then β is equal to:
(a) 2a(a + 1)
(b) –2a(a + 1)
(c) 2a(a – 1)
(d) 2a²

Question. The number of all possible positive integral values of α for which the roots of the quadratic equation, 6x² – 11x + α = 0 are rational numbers is:
(a) 3
(b) 2
(c) 4
(d) 5

Question. If | z² -1|=| z |² +1, then lies on
(a) an ellipse
(b) the imaginary axis
(c) a circle
(d) the real axis

Question. The locus of the centre of a circle which touches the circle | z – z₁ | = α and | z – z₂ | = b externally (z, z₁ & z₂ are complex numbers) will be
(a) an ellipse
(b) a hyperbola
(c) a circle
(d) none of these

Question. If both the roots of the quadratic equation x² – mx + 4 = 0 are real and distinct and they lie in the interval [1, 5], then m lies in the interval:
(a) (– 5, – 4)
(b) (4, 5)
(c) (5, 6)
(d) (3, 4)

Question. If an angle A of a Δ ABC satisfies 5 cos A + 3 = 0, then the roots of the quadratic equation, 9x² + 27x + 20 = 0 are.
(a) sin A, sec A
(b) sec A, tan A
(c) tan A, cos A
(d) sec A, cot A

Question. The number of real roots of the equation, e^4x + e^3x – 4e^2x + e^x + 1 = 0 is:
(a) 1
(b) 3
(c) 2
(d) 4

Question. The least positive value of ‘α’ for which the equation, 2x² + (α – 10)x + 33/2 = 2α has real roots is _______.

Question. If tan A and tan B are the roots of the quadratic equation, 3x² – 10x – 25 = 0 then the value of 3 sin² (A + B) – 10 sin (A + B). cos (A + B) – 25 cos² (A + B) is
(a) 25
(b) – 25
(c) – 10
(d) 10

Question. If, for a positive integer n, the quadratic equation, 

has two consecutive integral solutions, then n is equal to:
(a) 11
(b) 12
(c) 9
(d) 10

Question. If (1- p) is a root of quadratic equation x² + px + (1- p) = 0 then its root are
(a) –1, 2
(b) –1, 1
(c) 0, –1
(d) 0, 1

Question. The number of real solutions of the equation x² – 3|x| + 2 = 0 is
(a) 3
(b) 2
(c) 4
(d) 1

Question. The sum of all the real values of x satisfying the equation 2^{(x-1)(x2+5x-50)} = 1 is :
(a) 16
(b) 14
(c) – 4
(d) – 5

Question. Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by x–1 and it leaves remainder 6 when divided by x+1; then :
(a) p(b) = 11
(b) p(b) = 19
(c) p(–2) = 19
(d) p(–2) = 11

Question. In a triangle PQR, ∠R = π/2. If tan (p/2) and – tan (Q/2) are the roots of ax² + bx + c = 0, a ≠ 0 then
(a) a = b + c
(b) c = a + b
(c) b = c
(d) b = a + c

Question. If the roots of the equation x² – bx + c = 0 be two consecutive integers, then b² – 4c equals
(a) – 2
(b) 3
(c) 2
(d) 1

Question. The sum of all real values of x satisfying the equation (x2 – 5x + 5)^x2+4x-60 = 1 is :
(a) 6
(b) 5
(c) 3
(d) – 4