Complex Numbers47 questions8 PYQ

Complex Numbers — JEE Maths Practice Questions & Solutions

47 questions on Complex Numbers with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

hardPYQ · JEE Advanced 2014
Let zk=cos⁡(2kπ10)+isin⁡(2kπ10)z_k = \cos\left(\dfrac{2k\pi}{10}\right) + i\sin\left(\dfrac{2k\pi}{10}\right); k=1,2,…,9k = 1, 2, \ldots, 9. Match each entry in List-I with the correct entry in List-II.
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hardPYQ · JEE Main 2026
Let the set of all values of k∈Rk\in\mathbb{R} such that the equation z(zˉ+2+i)+k(2+3i)=0z(\bar z+2+i)+k(2+3i)=0, z∈Cz\in\mathbb{C}, has at least one solution, be the interval [α,β][\alpha,\beta]. Then 9(α+β)9(\alpha+\beta) is equal to
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mediumPYQ · JEE Main 2026
Let z1,z2∈Cz_1, z_2 \in \mathbb{C} be the distinct solutions of z2+4z−(1+12i)=0z^2 + 4z - (1+12i) = 0. Then ∣z1∣2+∣z2∣2|z_1|^2 + |z_2|^2 is equal to:
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mediumPYQ · JEE Main 2026
Let xx and yy be real numbers such that 50(2x1+3i−y1−2i)=31+17i50\left(\dfrac{2x}{1+3i}-\dfrac{y}{1-2i}\right)=31+17i, i=−1i=\sqrt{-1}. Then the value of 10(x−3y)10(x-3y) is
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mediumPYQ · JEE Main 2026
Let a,b∈Ca,b\in\mathbb{C}. Let α,β\alpha,\beta be the roots of x2+ax+b=0x^2+ax+b=0. If β−α=11\beta-\alpha=\sqrt{11} and β2−α2=3i11\beta^2-\alpha^2=3i\sqrt{11}, then (β3−α3)2(\beta^3-\alpha^3)^2 is equal to
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mediumPYQ · JEE Main 2026
Let the circle C1:∣z∣=rC_1:|z|=r and C2:∣z−3−4i∣=5, z∈CC_2:|z-3-4i|=5,\ z\in\mathbb{C}, be such that C2C_2 lies within C1C_1. If z1z_1 moves on C1C_1, z2z_2 moves on C2C_2 and min⁡∣z1−z2∣=2\min|z_1-z_2|=2, then max⁡∣z1−z2∣\max|z_1-z_2| is equal to
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easyPYQ · JEE Main 2026
Let zz be a complex number such that ∣z+2∣=∣z−2∣|z+2|=|z-2| and arg⁡(z+3z−i)=π4\arg\left(\dfrac{z+3}{z-i}\right)=\dfrac{\pi}{4}. Then ∣z∣2|z|^2 is equal to
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easyPYQ · JEE Main 2026
Let S={z∈C:z2+4z+16=0}S=\{z\in\mathbb{C}:z^2+4z+16=0\}. Then ∑z∈S∣z+3 i∣2\displaystyle\sum_{z\in S}\left|z+\sqrt3\,i\right|^2 is equal to
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hard
If z1,z2,z3z_1, z_2, z_3 are the non-zero complex numbers representing the points A,B,CA, B, C such that
2z1=1z2+1z3,\frac{2}{z_1} = \frac{1}{z_2} + \frac{1}{z_3},
 then which of the following is true?
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hard
Let C=cos⁡2π7+cos⁡4π7+cos⁡8π7C = \cos\dfrac{2\pi}{7} + \cos\dfrac{4\pi}{7} + \cos\dfrac{8\pi}{7} and S=sin⁡2π7+sin⁡4π7+sin⁡8π7S = \sin\dfrac{2\pi}{7} + \sin\dfrac{4\pi}{7} + \sin\dfrac{8\pi}{7}, then
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hard
If ∣z1∣=2|z_1| = 2, ∣z2∣=3|z_2| = 3, ∣z3∣=4|z_3| = 4 and ∣z1+z2+z3∣=5|z_1 + z_2 + z_3| = 5, then ∣4z2z3+9z3z1+16z1z2∣|4z_2 z_3 + 9z_3 z_1 + 16z_1 z_2| equals
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hard
If log⁡tan⁡30° ⁣(2∣z∣2+2∣z∣−3∣z∣+1)<−2\log_{\tan 30°}\!\left(\dfrac{2|z|^2 + 2|z| - 3}{|z| + 1}\right) < -2, then
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hard
If the roots of z3+iz2+2i=0z^3 + iz^2 + 2i = 0 represent the vertices of △ABC\triangle ABC in the Argand plane, then the area of the triangle is (in square units)
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hard
If z=x+iyz = x + iy, then the equation ∣2z−iz+1∣=m\left|\dfrac{2z - i}{z + 1}\right| = m does not represent a circle when
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hard
If P=cos⁡2π7+isin⁡2π7P = \cos\dfrac{2\pi}{7} + i\sin\dfrac{2\pi}{7}, a=P+P2+P4a = P + P^{2} + P^{4} and b=P3+P5+P6b = P^{3} + P^{5} + P^{6}, then the value of ∣(a+b)+abi∣\left|(a+b) + abi\right| (where i=−1i = \sqrt{-1}) is
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hard
If z1z_1, z2z_2, z3z_3 are the roots of the equation
z3−z2(1+3i)+z(3i−2)+2=0,z^{3} - z^{2}(1+3i) + z(3i-2) + 2 = 0 ,
 then Im⁡(z1)+Im⁡(z2)+Im⁡(z3)\operatorname{Im}(z_1) + \operatorname{Im}(z_2) + \operatorname{Im}(z_3) is
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medium
The number of values of zz satisfying ∣z−2i∣=2|z - 2i| = 2 and z(1−i)−zˉ(1+i)=4iz(1-i) - \bar{z}(1+i) = 4i simultaneously is
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medium
If z1,z2,z3z_1, z_2, z_3 and z4z_4 are the consecutive vertices of a square, then z12+z22+z32+z42z_1^2 + z_2^2 + z_3^2 + z_4^2 equals
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medium
If z1,z2,z3z_1, z_2, z_3 are the vertices of an isosceles right-angled triangle, right-angled at the vertex z2z_2, then (z1−z2)2+(z3−z2)2(z_1 - z_2)^2 + (z_3 - z_2)^2 equals
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medium
If cos⁡α+cos⁡β+cos⁡γ=0=sin⁡α+sin⁡β+sin⁡γ\cos\alpha + \cos\beta + \cos\gamma = 0 = \sin\alpha + \sin\beta + \sin\gamma, then sin⁡3α+sin⁡3β+sin⁡3γsin⁡(α+β+γ)\dfrac{\sin 3\alpha + \sin 3\beta + \sin 3\gamma}{\sin(\alpha + \beta + \gamma)} is equal to
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medium
If zz is a complex number satisfying z4+z3+2z2+z+1=0z^4 + z^3 + 2z^2 + z + 1 = 0, then the set of possible values of ∣z∣|z| is
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medium
The complex number 3+4i3 + 4i is rotated about the origin by an angle π4\dfrac{\pi}{4} in the anticlockwise direction and then stretched twice. The complex number corresponding to the new position is
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medium
If (a+ib)5=α+iβ(a + ib)^5 = \alpha + i\beta, then (b+ia)5(b + ia)^5 is equal to
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medium
The complex numbers a+ia + i, a−ia - i, 1+ai1 + ai and 1−ai1 - ai, where a∈Ra \in \mathbb{R}, taken in that order on the Argand plane, represent the vertices of a parallelogram if
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medium
If (1+i)(1+2i)(1+3i)⋯(1+ni)=α+iβ(1+i)(1+2i)(1+3i)\cdots(1+ni) = \alpha + i\beta, then 2⋅5⋅10⋯(1+n2)2 \cdot 5 \cdot 10 \cdots (1+n^2) (where α,β,n∈R\alpha, \beta, n \in \mathbb{R}) is equal to
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medium
If z=(1+3 i)10+(1−3 i)10z = (1 + \sqrt{3}\,i)^{10} + (1 - \sqrt{3}\,i)^{10}, then Arg(z)\mathrm{Arg}(z) may be
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medium
If the complex numbers z1,z2,z3z_1, z_2, z_3 satisfy 3z1=5z2−2z33z_1 = 5z_2 - 2z_3, then z1,z2,z3z_1, z_2, z_3 lie in a/on
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medium
The value of sin⁡ ⁣[log⁡e ⁣{(cos⁡π2+isin⁡π2) ⁣z}]\sin\!\left[\log_e\!\left\{\left(\cos\dfrac{\pi}{2} + i\sin\dfrac{\pi}{2}\right)^{\!z}\right\}\right], where zz satisfies ∣z−2i∣=1|z - 2i| = 1 and has least modulus.
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medium
If ω≠1\omega \neq 1 is a cube root of unity and (1+ω)7=l+mω(1 + \omega)^7 = l + m\omega, then the value of l+ml + m is
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medium
The number of common roots of the equations x3+2x2+2x+1=0x^3 + 2x^2 + 2x + 1 = 0 and x2012+x2014+1=0x^{2012} + x^{2014} + 1 = 0 is
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medium
If x=a+ibx = a + ib is a complex number such that x2=3+4ix^2 = 3 + 4i and x3=2+11ix^3 = 2 + 11i, where i=−1i = \sqrt{-1}, then a+ba + b is
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medium
If the complex number zz satisfies z+∣z∣=2+8iz + |z| = 2 + 8i, then the value of ∣z∣|z| will be
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medium
If f(x)=2x3+2x2−7x+72f(x) = 2x^3 + 2x^2 - 7x + 72, then f ⁣(3−5i2)f\!\left(\dfrac{3 - 5i}{2}\right) is
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medium
The value of
∑k=1100(i k!+ω k!),\sum_{k=1}^{100}\left(i^{\,k!} + \omega^{\,k!}\right),
 where i=−1i = \sqrt{-1} and ω\omega is a complex cube root of unity, is
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medium
If z1z_1 and z2z_2 are two complex numbers satisfying ∣z1+z2z1−z2∣=1\left|\dfrac{z_1 + z_2}{z_1 - z_2}\right| = 1, then z1z2\dfrac{z_1}{z_2} is a number which is
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medium
If ∣z+2−i∣=5|z + 2 - i| = 5, then the maximum value of ∣3z+9−7i∣|3z + 9 - 7i| is
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medium
All roots of the equation (1+z)6+z6=0(1+z)^{6} + z^{6} = 0
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medium
If z2z_2 and z4z_4 are conjugates of z1z_1 and z3z_3 respectively, then
arg⁡(z1z4)+arg⁡(z2z3)\arg\left(\frac{z_1}{z_4}\right) + \arg\left(\frac{z_2}{z_3}\right)
 is equal to
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medium
If one of the vertices of the regular hexagon circumscribing the circle ∣z−1−i∣=3|z - 1 - i| = \sqrt3 is 2+(1+3)i2 + \left(1+\sqrt3\right)i, then the complex number which is NOT representing any vertex of the hexagon is
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medium
Given z1,z2,z3,z4z_1, z_2, z_3, z_4 are four points in the complex plane such that ∣z1∣<1\left|z_1\right| < 1, ∣z2∣=1\left|z_2\right| = 1 and ∣z3∣≤1\left|z_3\right| \le 1, and
z3=z2(z1−z4)z1ˉz4−1,z_3 = \frac{z_2\left(z_1 - z_4\right)}{\bar{z_1}z_4 - 1} ,
 then ∣z4∣\left|z_4\right| can be (z1≠z4)\left(z_1 \ne z_4\right)
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medium
Consider the curves
C1:∣z−2∣=2+Re⁡(z)andC2:∣z∣=3,C_1 : |z-2| = 2 + \operatorname{Re}(z) \qquad \text{and} \qquad C_2 : |z| = 3 ,
 where z=x+iyz = x + iy, x,y∈Rx, y \in \mathbb{R} and i=−1i = \sqrt{-1}. They intersect at PP and QQ in the first and fourth quadrants respectively. Tangents to C1C_1 at PP and QQ intersect the xx-axis at RR, and tangents to C2C_2 at PP and QQ intersect the xx-axis at SS. If the area of △PRS\triangle PRS is λ2\lambda\sqrt2 sq. units, then find (λ2)\left(\lambda^{2}\right).
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medium
If ∣z1∣=1\left|z_1\right| = 1, ∣z2−2∣=3\left|z_2 - 2\right| = 3 and ∣z3−5∣=6\left|z_3 - 5\right| = 6, then the maximum value of ∣2z1−3z2−4z3∣\left|2z_1 - 3z_2 - 4z_3\right| is not less than (where z1,z2,z3z_1, z_2, z_3 are complex numbers)
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medium
If ∣z−3−2i∣=∣z+2i∣|z - 3 - 2i| = |z + 2i|, where zz is a complex number, then the minimum value of ∣z∣|z| will be
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medium
Consider complex numbers zz and ww satisfying the equations
z+w=zˉwand1w+z=wzˉ,z + w = \frac{\bar z}{w} \qquad \text{and} \qquad \frac{1}{w} + z = w\bar z ,
 then which of the following is/are correct?
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medium
One vertex of an equilateral triangle is at the origin and the other two vertices are the roots of 2z2+2z+k=02z^2 + 2z + k = 0. Then the value of kk is
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easy
If ∣z∣=min⁡{∣z−1∣, ∣z+1∣}|z| = \min\{|z - 1|,\, |z + 1|\}, then the value of ∣z+zˉ∣|z + \bar{z}| is
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easy
If α\alpha is a root of the equation x2−x+1=0x^{2} - x + 1 = 0, then find the value of
∣α333+α666+α999∣.\left|\alpha^{333} + \alpha^{666} + \alpha^{999}\right| .
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