Complex Numbers27 questions1 PYQ

Complex Numbers — JEE Maths Practice Questions & Solutions

27 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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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 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 ∣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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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
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
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
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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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 ∣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
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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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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