Complex Numbers jee mains questions | JEE Main Maths Complex Numbers Previous Year Question

Q 11 :
Let $A = {z \in C : | z - 2 - i | = 3}$ , $B = {z \in C : R e (z - i z) = 2}$ and $S = A \cap B$ . Then $\sum_{z \in S} | z |^{2}$ is equal to __________. [2025]

0%

(22)

Let z = x + iy.

We have, A : |z – 2 – i| = 3

$\Rightarrow$ $| (x - 2) + i (y - 1) |$ $= 3 \Rightarrow {(x - 2)}^{2} + {(y - 1)}^{2} = 9$ ... (i)

Also, B : Re(z – iz) = 2

$R e ((x + y) + i (y - x)) = 2 \Rightarrow x + y = 2$ ... (ii)

From (i) and (ii), we get $x = \frac{3 \pm \sqrt{17}}{2}, y = \frac{1 \mp \sqrt{17}}{2}$

$∴ \sum_{z \in S} | z |^{2} = \frac{1}{4} [2 \times 26 + 2 \times 18] = 22$ .

Q 12 :
$If for z = α + i β, | z + 2 | = z + 4 (1 + i), then α + β and α β are the roots of the equation$ [2023]

$x^{2} + 7 x + 12 = 0$
$x^{2} + x - 12 = 0$
$x^{2} + 3 x - 4 = 0$
$x^{2} + 2 x - 3 = 0$

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(1)

We have, $| z + 2 |$ $= z + 4 (1 + i)$

$\Rightarrow$ $| α + i β + 2 |$ $= α + i β + 4 + 4 i$

$\Rightarrow$ $\sqrt{{(α + 2)}^{2} + β^{2}} = (α + 4) + i (β + 4)$

Comparing real and imaginary parts, we get

$β + 4 = 0 \Rightarrow β = - 4$

Also, ${(α + 2)}^{2} + 4^{2} = {(α + 4)}^{2}$

$\Rightarrow α^{2} + 4 + 4 α + 16 = α^{2} + 16 + 8 α \Rightarrow 4 α = 4 \Rightarrow α = 1$

Now, $α + β = - 3, α β = - 4$

Equation having roots - 3 and - 4 is $(x + 3) (x + 4) = 0$

i.e. $x^{2} + 7 x + 12 = 0$

Q 13 :
Let the complex number $z = x + i y$ be such that $\frac{2 z - 3 i}{2 z + i}$ is purely imaginary. If $x + y^{2} = 0$ , then $y^{4} + y^{2} - y$ is equal to [2023]

$\frac{3}{2}$
$\frac{2}{3}$
$\frac{3}{4}$
$\frac{4}{3}$

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(3)

$Given, \frac{2 z - 3 i}{2 z + i} is purely imaginary.$

$∴$ $\frac{2 z - 3 i}{2 z + i} + \frac{2 \bar{z} + 3 i}{2 \bar{z} - i} = 0$

$\Rightarrow (2 z - 3 i) (2 \bar{z} - i) + (2 \bar{z} + 3 i) (2 z + i) = 0$

$\Rightarrow 4 z \bar{z} - 2 i z - 6 i \bar{z} - 3 + 4 z \bar{z} + 2 i \bar{z} + 6 i z - 3 = 0$

$\Rightarrow 8 z \bar{z} + 4 i z - 4 i \bar{z} - 6 = 0$ $[∵ z = x + i y]$

$\Rightarrow 4 (x^{2} + y^{2}) + 2 i (x + i y) - 2 i (x - i y) - 3 = 0$

$\Rightarrow 4 x^{2} + 4 y^{2} - 4 y - 3 = 0$

$\Rightarrow 4 y^{4} + 4 y^{2} - 4 y - 3 = 0$ $[∵ x + y^{2} = 0 (Given)]$

$\Rightarrow y^{4} + y^{2} - y = \frac{3}{4}$

Q 14 :
Let $S = {z = x + i y : \frac{2 z - 3 i}{4 z + 2 i} is a real number} .$ Then which of the following is NOT correct? [2023]

$(x, y) = (0, - \frac{1}{2})$
$y + x^{2} + y^{2} \neq - \frac{1}{4}$
$y \in (- \infty, - \frac{1}{2}) \cup (- \frac{1}{2}, \infty)$
$x = 0$

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(1)

$As, \frac{2 z - 3 i}{4 z + 2 i} is a real number.$

$∴$ $\frac{2 z - 3 i}{4 z + 2 i} = \frac{\bar{2 z - 3 i}}{4 z + 2 i} \Rightarrow \frac{2 z - 3 i}{4 z + 2 i} = \frac{2 \bar{z} + 3 i}{4 \bar{z} - 2 i}$

$\Rightarrow (2 z - 3 i) (4 \bar{z} - 2 i) = (2 \bar{z} + 3 i) (4 z + 2 i)$

$\Rightarrow 8 z \bar{z} - 4 i z - 12 i \bar{z} - 6 = 8 z \bar{z} + 4 i \bar{z} + 12 i z - 6$

$\Rightarrow - 4 i (z + \bar{z}) - 12 i (z + \bar{z}) = 0$

$\Rightarrow 16 i (z + \bar{z}) = 0 \Rightarrow z + \bar{z} = 0 \Rightarrow 2 x = 0 \Rightarrow x = 0$

$As, y can not be \frac{- 1}{2} .$

$∴$ $Option (1) is not possible.$

Q 15 :
For $a \in ℂ$ , let $A = {z \in ℂ : R e (a + \bar{z}) > I m (\bar{a} + z)} and B = {z \in ℂ : R e (a + \bar{z}) < I m (\bar{a} + z)} .$ Then among the two statements:

(S1) : If $R e (a), I m (a) > 0$ , then the set A contains all the real numbers.

(S2) : If $R e (a), I m (a) < 0$ , then the set B contains all the real numbers. [2023]

only (S2) is true
both are true
only (S1) is true
both are false

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(4)

$Let a = x_{1} + i y_{1} and z = x + i y$

$Now Re (a + \bar{z}) > Im (\bar{a} + z)$

$\Rightarrow Re (x_{1} + i y_{1} + x - i y) > Im (x_{1} - i y_{1} + x + i y)$

$\Rightarrow Re (x_{1} + x + i (y_{1} - y)) > Im (x_{1} + x + i (y - y_{1}))$

$∴ x_{1} + x > - y_{1} + y$

$Let x_{1} = 2, y_{1} = 10, x = - 12, y = 0$

Given inequality is not valid for these values.

$S 1 is false.$

$Now Re (a + \bar{z}) < Im (\bar{a} + z)$

$\Rightarrow Re (x_{1} + x + i (y_{1} - y)) < Im (x_{1} + x + i (y - y_{1}))$

$\Rightarrow x_{1} + x < - y_{1} + y$

$Let x_{1} = - 2, y_{1} = - 10, x = 12, y = 0$

Given inequality is not valid for these values.

$S 2 is false.$

Q 16 :
Let $S = {z \in ℂ : \bar{z} = i (z^{2} + R e (\bar{z}))} .$ Then $\sum_{z \in S} | z |^{2}$ is equal to [2023]

4
$\frac{5}{2}$
$\frac{7}{2}$
3

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(1)

$\bar{z} = i (z^{2} + Re (\bar{z}))$

$x - i y = i (x^{2} - y^{2} + 2 i x y + x); x - i y = i (x^{2} - y^{2} + x) - 2 x y$

$\Rightarrow x = - 2 x y and y = y^{2} - x^{2} - x$

$x = 0 \Rightarrow y = y^{2} \Rightarrow y = 0 or 1$

$y = \frac{- 1}{2} \Rightarrow \frac{- 1}{2} = \frac{1}{4} - x^{2} \Rightarrow x = \frac{1}{2} or \frac{- 3}{2}$

$Possible places of z = 0 + i 0, 0 + i, \frac{1}{2} - \frac{1}{2} i, \frac{- 3}{2} - \frac{i}{2}$

$\Rightarrow \sum | z |^{2} = 0 + 1 + (\frac{1}{4} + \frac{1}{4}) + (\frac{9}{4} + \frac{1}{4}) = 1 + 3 = 4$

Q 17 :
If the set ${R e (\frac{z - \bar{z} + z \bar{z}}{2 - 3 z + 5 \bar{z}}) : z \in C, R e (z) = 3}$ is equal to the interval $(α, β]$ , then 24 $(β - α)$ is equal to [2023]

27
36
42
30

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(4)

Let $z_{1} = (\frac{z - \bar{z} + z \bar{z}}{2 - 3 z + 5 \bar{z}})$

Let $z = 3 + i y$ $[∵ Re (z) = 3]$

$\Rightarrow \bar{z} = 3 - i y$

$z_{1} = \frac{2 i y + (9 + y^{2})}{2 - 3 (3 + i y) + 5 (3 - i y)}$

$\frac{9 + y^{2} + i (2 y)}{8 - 8 i y} = \frac{(9 + y^{2}) + i (2 y)}{8 (1 - i y)} \times \frac{(1 + i y)}{(1 + i y)}$

$= \frac{9 + 9 i y + y^{2} - 2 y^{2} + i y^{3} + 2 i y}{8 (1 + y^{2})}$

$So, Re (z_{1}) = \frac{(9 + y^{2}) - 2 y^{2}}{8 (1 + y^{2})} = \frac{9 - y^{2}}{8 (1 + y^{2})}$

$= \frac{1}{8} [\frac{10 - (1 + y^{2})}{1 + y^{2}}] = \frac{1}{8} [\frac{10}{1 + y^{2}} - 1]$

Thus, $1 + y^{2} \in [1, \infty) \Rightarrow \frac{1}{1 + y^{2}} \in (0, 1]$

$\Rightarrow \frac{10}{1 + y^{2}} \in (0, 10] \Rightarrow \frac{10}{1 + y^{2}} - 1 \in (- 1, 9]$

$∴ Re (z_{1}) \in (\frac{- 1}{8}, \frac{9}{8}]$ ; $α = \frac{- 1}{8}, β = \frac{9}{8}$

$∴$ $24 (β - α) = 24 (\frac{9}{8} + \frac{1}{8}) = 30$

Q 18 :
Let $z_{1} = 2 + 3 i$ and $z_{2} = 3 + 4 i$ . The set $S = {Z \in ℂ : | z - z_{1} |^{2} - | z - z_{2} |^{2} = | z_{1} - z_{2} |^{2}}$ represents a [2023]

straight line with the sum of its intercepts on the coordinate axes equals −18
hyperbola with eccentricity 2
hyperbola with the length of the transverse axis 7
straight line with the sum of its intercepts on the coordinate axes equals 14

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(4)

Let $z = x + i y$ ; $z_{1} = 2 + 3 i$ ; $z_{2} = 3 + 4 i$

$| (x + i y) - (2 + 3 i) |^{2} - | (x + i y) - (3 + 4 i) |^{2} = | (2 + 3 i) - (3 + 4 i) |^{2}$

${(x - 2)}^{2} + {(y - 3)}^{2} - {(x - 3)}^{2} - {(y - 4)}^{2} = 1^{2} + 1^{2}$

$2 x - 5 + 2 y - 7 = 2 \Rightarrow 2 x + 2 y = 14 \Rightarrow x + y = 7;$

$x intercept = y intercept = 7; sum of intercepts = 14$

Q 19 :
Let $z$ be a complex number such that $| \frac{z - 2 i}{z + i} |$ $= 2, z \neq - i .$ Then $z$ lies on the circle of radius 2 and centre [2023]

(2, 0)
(0, 2)
(0, 0)
(0, -2)

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(4)

Q 20 :
The complex number $z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}}$ is equal to [2023]

$\sqrt{2} i (\cos \frac{5 π}{12} - i \sin \frac{5 π}{12})$
$\cos \frac{π}{12} - i \sin \frac{π}{12}$
$\sqrt{2} (\cos \frac{π}{12} + i \sin \frac{π}{12})$
$\sqrt{2} (\cos \frac{5 π}{12} + i \sin \frac{5 π}{12})$

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(4)

$z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}} = \frac{\sqrt{2} e^{i \frac{3 π}{4}}}{e^{i \frac{π}{3}}}$

$= \sqrt{2} e^{i (\frac{3 π}{4} - \frac{π}{3})} = \sqrt{2} e^{i (\frac{5 π}{12})} = \sqrt{2} (\cos \frac{5 π}{12} + i \sin \frac{5 π}{12})$

Topic Question Set

Q 11 :
Let $A = {z \in C : | z - 2 - i | = 3}$ , $B = {z \in C : R e (z - i z) = 2}$ and $S = A \cap B$ . Then $\sum_{z \in S} | z |^{2}$ is equal to __________. [2025]

0%

Q 12 :
$If for z = α + i β, | z + 2 | = z + 4 (1 + i), then α + β and α β are the roots of the equation$ [2023]

Q 13 :
Let the complex number $z = x + i y$ be such that $\frac{2 z - 3 i}{2 z + i}$ is purely imaginary. If $x + y^{2} = 0$ , then $y^{4} + y^{2} - y$ is equal to [2023]

Q 14 :
Let $S = {z = x + i y : \frac{2 z - 3 i}{4 z + 2 i} is a real number} .$ Then which of the following is NOT correct? [2023]

Q 16 :
Let $S = {z \in ℂ : \bar{z} = i (z^{2} + R e (\bar{z}))} .$ Then $\sum_{z \in S} | z |^{2}$ is equal to [2023]

Q 17 :
If the set ${R e (\frac{z - \bar{z} + z \bar{z}}{2 - 3 z + 5 \bar{z}}) : z \in C, R e (z) = 3}$ is equal to the interval $(α, β]$ , then 24 $(β - α)$ is equal to [2023]

Q 18 :
Let $z_{1} = 2 + 3 i$ and $z_{2} = 3 + 4 i$ . The set $S = {Z \in ℂ : | z - z_{1} |^{2} - | z - z_{2} |^{2} = | z_{1} - z_{2} |^{2}}$ represents a [2023]

Q 19 :
Let $z$ be a complex number such that $| \frac{z - 2 i}{z + i} |$ $= 2, z \neq - i .$ Then $z$ lies on the circle of radius 2 and centre [2023]

(4)

Q 20 :
The complex number $z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}}$ is equal to [2023]

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Topic Question Set

Q 11 : Let A={z∈C:|z–2–i|=3}, B={z∈C:Re(z–iz)=2} and S=A∩B. Then ∑z∈S|z|2 is equal to __________. [2025]

0%

Q 12 : If for z=α+iβ, |z+2|=z+4(1+i), then α+β and αβ are the roots of the equation [2023]

Q 13 : Let the complex number z=x+iy be such that 2z-3i2z+i is purely imaginary. If x+y2=0, then y4+y2-y is equal to [2023]

Q 14 : Let S={z=x+iy:2z-3i4z+2i is a real number}. Then which of the following is NOT correct? [2023]

Q 15 : For a∈ℂ, let A={z∈ℂ:Re(a+z¯)>Im(a¯+z)} and B={z∈ℂ:Re(a+z¯)<Im(a¯+z)}. Then among the two statements: (S1) : If Re(a),Im(a)>0, then the set A contains all the real numbers. (S2) : If Re(a),Im(a)<0, then the set B contains all the real numbers. [2023]

Q 16 : Let S={z∈ℂ:z¯=i(z2+Re(z¯))}. Then ∑z∈S|z|2 is equal to [2023]

Q 17 : If the set {Re(z-z¯+zz¯2-3z+5z¯) :z∈C,Re(z)=3} is equal to the interval (α,β], then 24 (β-α) is equal to [2023]

Q 18 : Let z1=2+3i and z2=3+4i. The set S={Z∈ℂ:|z-z1|2-|z-z2|2=|z1-z2|2} represents a [2023]

Q 19 : Let z be a complex number such that |z-2iz+i|=2, z≠-i. Then z lies on the circle of radius 2 and centre [2023]

(4)

Q 20 : The complex number z=i-1cosπ3+isinπ3 is equal to [2023]

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Q 11 :
Let $A = {z \in C : | z - 2 - i | = 3}$ , $B = {z \in C : R e (z - i z) = 2}$ and $S = A \cap B$ . Then $\sum_{z \in S} | z |^{2}$ is equal to __________. [2025]

Q 12 :
$If for z = α + i β, | z + 2 | = z + 4 (1 + i), then α + β and α β are the roots of the equation$ [2023]

Q 13 :
Let the complex number $z = x + i y$ be such that $\frac{2 z - 3 i}{2 z + i}$ is purely imaginary. If $x + y^{2} = 0$ , then $y^{4} + y^{2} - y$ is equal to [2023]

Q 14 :
Let $S = {z = x + i y : \frac{2 z - 3 i}{4 z + 2 i} is a real number} .$ Then which of the following is NOT correct? [2023]

Q 16 :
Let $S = {z \in ℂ : \bar{z} = i (z^{2} + R e (\bar{z}))} .$ Then $\sum_{z \in S} | z |^{2}$ is equal to [2023]

Q 17 :
If the set ${R e (\frac{z - \bar{z} + z \bar{z}}{2 - 3 z + 5 \bar{z}}) : z \in C, R e (z) = 3}$ is equal to the interval $(α, β]$ , then 24 $(β - α)$ is equal to [2023]

Q 18 :
Let $z_{1} = 2 + 3 i$ and $z_{2} = 3 + 4 i$ . The set $S = {Z \in ℂ : | z - z_{1} |^{2} - | z - z_{2} |^{2} = | z_{1} - z_{2} |^{2}}$ represents a [2023]

Q 19 :
Let $z$ be a complex number such that $| \frac{z - 2 i}{z + i} |$ $= 2, z \neq - i .$ Then $z$ lies on the circle of radius 2 and centre [2023]

Q 20 :
The complex number $z = \frac{i - 1}{\cos \frac{π}{3} + i \sin \frac{π}{3}}$ is equal to [2023]