Complex Numbers and Quadratic Equations — practice questions

129 curated practice questions from Complex Numbers and Quadratic Equations (Mathematics JEE Advanced, NTA Exams) — 117 hard, 12 medium. Every question includes the final answer free; step-by-step worked solutions with a free account.

• If the roots of the equation, x 3 + Px 2 + Qx – 19 = 0 are each one more than the roots of the equation, x 3 –Ax 2…
• x 2 + x + 1 is a factor of ax 3 + bx 2 + cx + d = 0, then the real root of above equation is (a, b, c, d ∈ R)
• The value of ( 1+i 3 1-i 3 )^ 6 + ( 1-i 3 1+i 3 )^ 6 is
• If z ∈ C, then m ∈ R represents a straight line if 10 m =
• If a < b < c < d, then for any positive λ, the quadratic equation (x – a) (x – c) + λ(x – b) (x – d) = 0 has
• 3+2 i 1-2 i will be purely imaginary, if θ is equal to
• The value of a for which the equations x 3 + ax + 1 = 0 and x 4 + ax 2 + 1 = 0 have a common root is
• If | z_ 1 z-z_ 2 z_ 1 z+z_ 2 |=K, K > 0 (z 1 , z 2 ≠ 0), then
• Locus of z, if
• If z is a complex number of unit modulus and argument θ, then arg ( 1+z 1+ z ) is equal to
• Let P denotes a complex number z on the Argand’s plane, and Q denotes a complex number 2|z|^ 2 CiS ( 4 + ) where θ =…
• If |z 2 – 1| = |z| 2 + 1, then z lies on
• Match the following The correct matching is
• If z 2 + z + 1 = 0, where z is a complex number, then the value of (z+ 1 z )^ 2 + (z^ 2 + 1 z^ 2 )^ 2 + (z^ 3 + 1 z^ 3…
• The area of the triangle whose vertices are i, α, β, where i= -1 and α, β are the non-real cube roots of unity, is
• The system of equations . array r |z+1+ i |= 2 |z|=3 array , (where i = -1 ) has
• If a < 0, then root of the equation x 2 – 2a |x – a| – 3a 2 = 0 is
• If α, β, γ are the cube roots of a negative number p, then for any three real numbers x, y, z the value of x +y +z x +y…
• If z = re i θ , then | i iz | is equal to
• If z_ n = n(n+1)(n+2) +i n(n+1)(n+2) for n=1, 2, 3, ....., k, then the value of k Lim (z 1 z 2 .....z k ) is
• Let w = cos and α = w + w 2 + w 4 and β = w 3 + w 5 + w 6 . _ k=0 ^ 6 w^ k^ 2 equals :
• If a, b, p, q are non-zero real numbers, the two equations, 2 a 2 x 2 – 2 abx + b 2 = 0 and p 2 x 2 + 2 pqx + q 2 = 0…
• The value of the expression 2 (1+ 1 ) (1+ 1 ^ 2 )+3 (2+ 1 ) (2+ 1 ^ 2 ) +4 (3+ 1 ) (3+ 1 ^ 2 )+ +(n+1) (n+ 1 ) (n+ 1 ^…
• If z lies on the circle centered at origin and if area of the triangle, whose vertices are z, ωz and z + ωz, (ω being…
• Match the equation on the left with the curve they represent on the right The correct matching is
• If z 1 lies in |z–3| 2 on |z–1| + |z +1| = 3 and A = |z 1 –z 2 |, then
• If a, b, c ∈ R and α is a real root of the equation ax 2 + bx + c = 0, and β is the real root of the equation –ax 2 +…
• If (a + ib) (c + id) (e + if) (g + ih) = A + i B, then (a 2 + b 2 ) (c 2 + d 2 ) (e 2 + f 2 ) (g 2 + h 2 ) is equal to
• Let w = cos and α = w + w 2 + w 4 and β = w 3 + w 5 + w 6 . 2α equals :
• Which of the following is a fourth root of 1 2 + i 3 2 ?
• If both roots of the quadratic equation (2 – x) (x + 1) = p are distinct and positive then p must lie in the interval
• The value of k for which the equation x 3 – 3x + k = 0 has The correct matching is :
• If z 0 , z 1 represents points P, Q on the locus |z–1|=1 and the line segment PQ subtends an angle 2 at the points z =…
• 5^ x +(2 3 )^ 2 x -169 0 is true in the interval.
• b > 0 and θ ∈ (–π, π], then the value of θ is
• The roots of the equation, (x 2 + 1) 2 = x(3x 2 + 4x + 3), are given by
• The value of ‘a’ for which the sum of the squares of the roots of the equation x 2 – (a – 2) x – a – 1 = 0 assume the…
• In the given figure vertices of ∆ ABC lie on y = f (x) = ax 2 + bx + c. The ∆ ABC is right angled isosceles triangle…
• Assertion : If |z| ≥ 2, then the least value of |z+ 1 2 | is 3 2 . Reason : |z 1 + z 2 | ≤ |z 1 | + |z 2 |
• If α, β are the roots of x 2 – p (x + 1) – c = 0 then ^ 2 +2 +1 ^ 2 +2 +c + ^ 2 +2 +1 ^ 2 +2 +c is equal to
• Let z 1 , z 2 and z 3 be three points on |z| = 1. If θ 1 , θ 2 and θ 3 be the arguments of z 1 , z 2 , z 3…
• For a R , | a | > 1, let Then the possible value(s) of a is/are
• When x 100 is divided by x 2 – 3x + 2, the remainder is (2 k+1 –1) x – 2(2 k – 1) where k is a numerical quantity, then…
• Let z 1 = a 1 + ib 1 (a 1 , b 1 ) and z 2 = a 2 + ib 2 (a 2 , b 2 ); where i = -1 , be two complex numbers. If ∠POQ =…
• Let w = cos and α = w + w 2 + w 4 and β = w 3 + w 5 + w 6 . αβ equals :
• The value of m for which one of the roots of x 2 – 3x + 2m = 0 is double of one of the roots of x 2 – x + m = 0 is
• If |z- 4 z |=2, then the maximum value of | z | is equal to
• Column–I Column–II (A) Number of real solution of is (P) (B) The number of non-negative real roots of equal to (Q) (C)…
• If both roots of the quadratic equation x 2 + x + p = 0 exceed p where p ∈ R then p must lie in the interval
• If a , b , c , d are positive real numbers such that a + b + c + d = 2, then M = ( a + b ) ( c + d ) satisfies the…
• If roots of the equation x 4 – 12x 3 + bx 2 + cx + 81 = 0 are positive then
• Number of complex numbers z such that |z| = 1 and | z z + z z |=1 is
• If z 1 , z 2 , z 3 , z 4 are roots of the equation a 0 z 4 + a 1 z 3 + a 2 z 2 + a 3 z + a 4 = 0, where a 0 , a 1 , a 2…
• Let =- 1 2 +i 3 2 . Then the value of | array ccc 1 & 1 & 1 1 & -1- ^ 2 & ^ 2 1 & ^ 2 & ^ 4 array |
• The equations x 3 + 5x 2 + px + q = 0 and x 3 + 7x 2 + px + r = 0 have two roots in common. If the third root of each…
• If the equation (k – 2) x 2 – (k – 4) x – 2 = 0 has difference of roots as 3 then the sum of all the values of k is :
• If 6 x^ 2 -5 x-3 x^ 2 -2 x+6 4, then the sum of the least and the highest values of 4 x 2 is
• If 1, x 1 , x 2 , x 3 are the roots of x 4 - 1 = 0 and ω is an imaginary cube root of unity, then the value of ( ^ 2…
• For a complex number z the minimum value of | z | + | z – 2 | is
• cos α is a root of the equation 25x 2 + 5x – 12 = 0, –1 < x < 0, then the value of sin 2α is
• The maximum integral part of positive value of a for which, the least value of 4x 2 – 4 a x + a 2 – 2 a + 2 on [0, 2]…
• If ω ≠ 1 is a complex cube root of unity, then sum of the series S = 1 + 2ω + 3ω 2 + ... + 3nω 3n-1 (n ∈ N) is
• If α, β are the roots of ax 2 + bx + c = 0 and α + β, α 2 + β 2 , α 3 + β 3 are in G.P. and ∆ = b 2 – 4 ac , then
• If 0 ax 2 + bx + c = 0 are non real complex roots, then
• Assertion: If one roots is 5 - 2 is then the equation of lowest degree with rational coefficient is x 4 – 14x 2 + 9 =…
• The least value of a ∈ R for which 4 a x^ 2 + 1 x 1, for all x > 0, is
• Let a, b, c ∈ R. If ax 2 + bx + c = 0 has two real roots A and B where A 1, then
• If p,q,r,s,∈ R and α,β are roots of the equation x 2 + px + q = 0 and α 4 and β 4 are roots of x 2 – rx + s = 0, then…
• If α and β are different complex numbers with |β| = l, then | - 1- | is
• Let a , b and c be three real numbers satisfying [ array lll a & b & c array ] [ array lll 1 & 9 & 7 8 & 2 & 7 7 & 3 &…
• If roots x 1 and x 2 of x 2 + 1 = x a satisfy | x _ 1 ^ 2 - x _ 2 ^ 2 |> 1 a , then a (- 1 k , 0 ) (0, 1 k ) the…
• If α, β are the roots of the equation ax 2 + bx + c = 0 and s n = α n + β n , then a s n + 1 + c S n – 1 =
• If the quadratic equations, 3x 2 + ax + 1 = 0 and 2x 2 + bx + 1 = 0 have a common root, then the value of the…
• If a, b, c ∈ R, a > 0 and c ≠ 0 Let α and β be the real and distinct roots of the equation ax 2 + bx + c = |c| and p, q…
• If θ ∈ (0, π), the principal value of the arg (z) and |z| of the complex number z= (1+ +i )^ 5 ( +i )^ 3 is
• If p(x) = ax 2 + bx and q(x) = l x 2 + mx + n with p(1) = q(1); p(2) – q(2) = 1 and p(3) – q(3) = 4, then p(4) – q(4) is
• If | array ccc 6 i & -3 i & 1 4 & 3 i & -1 20 & 3 & i array |=x+i y, then
• The harmonic mean of the roots of the equation (5 + ) x 2 – (4 + ) x + 8 + = 0 is
• If a(p + q) 2 + 2bpq + c = 0 and a(p + r) 2 + 2bpr + c = 0, (a ≠ 0) then
• The complex number z = 1 + i is rotated through an angle 3 2 in anticlockwise direction about the origin and stretched…
• If p & q are roots of the equation x 2 – 2x + A = 0 and r & s be roots of the equation x 2 – 18 x + B = 0 if p < q < r…
• Let S= a+b 2 : a, b Z , T_ 1 = (-1+ 2 )^ n : n N , and T_ 2 = (1+ 2 )^ n : n N . Then which of the following statements…
• Assertion: If a > b > c and a 3 + b 3 + c 3 = 3abc, then the equation ax 2 + bx + c = 0 has one positive and one…
• If z is a complex number satisfying |z 2 – 1|=4 |z|, then the minimum value of |z| is
• If z 2 – z + 1 = 0, then z n – z –n , where n is a multiple of 3, is
• 8 x^ 2 +16 x-51 (2 x-3)(x+4) >3 if x is such that
• If α, β are the roots of the equation, x 2 – 2mx + m 2 – 1 = 0 then the range of values of m for which α, β ∈ (–2, 4) is
• If a point z 1 is the reflection of a point z 2 through the line b z + b z = c b ≠ 0, in the Argand plane, then b z _ 2…
• Let f ( x ) = (1+ b 2 ) x 2 + 2 bx + 1 and let m ( b ) be the minimum value of f ( x ). As b varies, the range of m ( b…
• Two real numbers α and β are such that α + β = 3 and |α – β| = 4, then α and β are the roots of the quadratic equation
• If roots of the equation x 4 – 12x 3 + bx 2 + cx + 81 = 0 are positive then Value of b is
• If a complex number z has modulus 1 and argument π/3, then z 2 + z
• If z = x + iy and w = 1- iz z - i , then |w| = 1 implies that in the complex plane
• If roots of the equation x 4 – 12x 3 + bx 2 + cx + 81 = 0 are positive then Value of c is
• If α is an imaginary constant such that αz 2 + z + = 0 has a real root, then
• Suppose A(z 1 ), B(z 2 ) and C(z 3 ) are vertices of a triangle lying on the unit circle |z| = 1. AD is altitude of the…
• If the complex numbers z 1 , z 2 , z 3 satisfying z_ 1 -z_ 3 z_ 2 -z_ 3 = 1-i 3 2 then triangle is
• If (x + 1) 2 is greater then 5x – 1 and less than 7x – 3 then the integral value of x is equal to
• In the given figure vertices of ∆ ABC lie on y = f (x) = ax 2 + bx + c. The ∆ ABC is right angled isosceles triangle…
• Equation has
• If A (z 1 ), B (z 2 ) and C (z 3 ) be the vertices of a triangle ABC in which ABC = 4 and AB BC = 2 , then the value of…
• The area of the triangle on the complex plane formed by the complex numbers z, iz and z + iz is
• Number of common roots of the equations and z^ 1985 +z^ 100 +1=0 , z being a complex number, is
• If arg ( z _ 1 ) = arg (z 2 ), then
• z and ω are two non zero complex number such that |z| = |ω| and Arg z + Arg ω = π, then z equals
• The roots of (z-1)^ n =2 (z+1)^ n (where n ≥ 3 and ω is complex cube root of unity) lie on a
• If α, β are the roots of the quadratic equation, x 2 – 2p (x – 4) – 15 = 0 then the set of values of p for which one…
• If A and B be two complex numbers satisfying . Then the two points represented by A and B and the origin form the…
• Assertion : If z is a complex number (z ≠ 1), then | z | z | -1 | | z | Reason : In a unit radius circle chord (AP) ≤…
• If ‘x’ is real and satisfying the inequality, | x |< a x ( a R ), then
• Let z 1 = a 1 + ib 1 (a 1 , b 1 ) and z 2 = a 2 + ib 2 (a 2 , b 2 ); where i = -1 , be two complex numbers. If ∠POQ =…
• Let z 1 and z 2 be two roots of the equation z 2 + az + b = 0, z being complex. Further, assume that the origin, z 1…
• If the quadratic equations ax 2 + 2cx + b = 0 and ax 2 + 2bx + c = 0 (b ≠ c) have a common root, then a + 4b + 4c is…
• In the given figure vertices of ∆ ABC lie on y = f (x) = ax 2 + bx + c. The ∆ ABC is right angled isosceles triangle…
• Let z 1 = a 1 + ib 1 (a 1 , b 1 ) and z 2 = a 2 + ib 2 (a 2 , b 2 ); where i = -1 , be two complex numbers. If ∠POQ =…
• If a, b ∈ R, a b . The roots of the quadratic equation, x 2 – 2 (a + b) x + 2 (a 2 + b 2 ) = 0 are
• If z 1 = a + ib and z 2 = c + id are complex numbers such that |z 1 | = |z 2 | = 1 and Re ( z _ 1 z _ 2 ) = 0, then the…
• Let z 1 , z 2 be two complex numbers represented by points on the circle |z| = 1 and |z| = 2, respectively, then
• If b 1 and x 2 of the equation 2x 2 + 6x + b = 0, satisfy the condition ( x_ 1 x_ 2 )+ ( x_ 2 x_ 1 ) < k where k is…
• If 0 ≤ x ≤ π, then the solution of the equation 16^ ^ 2 x + 16^ ^ 2 x = 10 is given by x equal to
• Let z 1 and z 2 be complex numbers such that Z _ 1 Z _ 2 and | z 1 | = | z 2 |. If z 1 has positive real part and z 2…
• If the expression x 2 – 11x + a and x 2 – 14x + 2a must have a common factor and a ≠ 0, then, the common factor is
• If α, β, γ, δ are the roots of the equation, x 4 – Kx 3 + Kx 2 + Lx + M = 0 where K, L and M are real numbers then the…
• Let α( a ) and β( a ) be the roots of the equation where a > –1 . Then _ a 0^ + (a) ext and _ a 0^ + (a) are
• Let w = cos and α = w + w 2 + w 4 and β = w 3 + w 5 + w 6 . α and β are roots of the equations :
• If a,b,c are three distinct real numbers and ω ≠ 1 is a complex cube root of unity, then the value of | a+b +c ^ 2 a ^…
• If z 1 and z 2 are two complex numbers satisfying the equation | z_ 1 +i z_ 2 z_ 1 -i z_ 2 |=1 , then z_ 1 z_ 2 is
• For a > 0, the roots of the equation l og ax a + l og x a 2 + l og a 2 x a 3 = 0, are given by
• If α, β are roots of the equation ax 2 + 3x + 2 = 0 (a < 0), then α 2 / β + β 2 / α is greater than

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