JEE Advanced Maths – Complex Numbers & Quadratic Equations PYQ with Detailed Solutions

Q 1 :

If $\frac{w - \bar{w} z}{1 - z}$ is purely real where $w = α + i β, β \neq 0$ and $z \neq 1$ , then the set of the values of $z$ is [2006]

${z : | z | = 1}$
${z : z = \bar{z}}$
${z : z \neq 1}$
${z : | z | = 1, z \neq 1}$

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(4)

$∵$ $\frac{w - \bar{w} z}{1 - z} is purely real$

$∴$ $\bar{(\frac{w - \bar{w} z}{1 - z})} = (\frac{w - \bar{w} z}{1 - z}) \Rightarrow \frac{\bar{w} - w \bar{z}}{1 - \bar{z}} = \frac{w - \bar{w} z}{1 - z}$

$\Rightarrow \bar{w} - w \bar{z} - \bar{w} z + w z \bar{z} = w - \bar{w} z - w \bar{z} + \bar{w} z \bar{z}$

$\Rightarrow w - \bar{w} = (w - \bar{w}) | z |^{2}$

$\Rightarrow | z |^{2} = 1$ $(∵ w = α + i β and β \neq 0)$

$\Rightarrow$ $| z |$ $= 1 and also given that z \neq 1$

$∴$ $The required set is {z : | z | = 1, z \neq 1}$

Q 2 :

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that

$| z_{1} |$ $=$ $| z_{2} |$ $=$ $| z_{3} |$ $=$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $= 1,$ then $| z_{1} + z_{2} + z_{3} |$ is [2000]

equal to 1
less than 1
greater than 3
equal to 3

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(1)

$Given :$ $| z_{1} |$ $=$ $| z_{2} |$ $=$ $| z_{3} |$ $= 1$

$Now,$ $| z_{1} |$ $= 1 \Rightarrow$ $| z_{1} |^{2}$ $= 1 \Rightarrow z_{1} \bar{z_{1}} = 1$

$Similarly z_{2} \bar{z_{2}} = 1, z_{3} \bar{z_{3}} = 1$

$Now,$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $= 1 \Rightarrow$ $| \bar{z_{1}} + \bar{z_{2}} + \bar{z_{3}} |$ $= 1$

$\Rightarrow$ $| \bar{z_{1} + z_{2} + z_{3}} |$ $= 1 \Rightarrow$ $| z_{1} + z_{2} + z_{3} |$ $= 1$

Q 3 :

For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} |$ $= 12$ and $| z_{2} - 3 - 4 i |$ $= 5$ , the minimum value of $| z_{1} - z_{2} |$ is [2002]

0
2
7
17

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(2)

$| z_{1} |$ $= 12 \Rightarrow z_{1} lies on a circle with centre (0, 0) and radius 12 units.$

$And$ $| z_{2} - 3 - 4 i |$ $= 5 \Rightarrow z_{2} lies on a circle with centre (3, 4) and radius 5 units.$

[IMAGE 5]

$From figure, it is clear that$ $| z_{1} - z_{2} |$ $i.e., distance between z_{1} and z_{2} will be minimum when they lie at A and B respectively,$

$i.e., O, C, B, A are collinear as shown.$

$Then$ $| z_{1} - z_{2} |$ $= A B = O A - O B = 12 - 2 (5) = 2 .$

$As above is the minimum value, we must have$ $| z_{1} - z_{2} |$ $\geq 2 .$

Q 4 :

If $\arg (z) < 0$ , then $\arg (- z) - \arg (z) =$ [2000]

$π$
$- π$
$- \frac{π}{2}$
$\frac{π}{2}$

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(1)

$Given: \arg (z) < 0 (given) \Rightarrow \arg (z) = - θ$

[IMAGE 6]

$Now, z = r \cos (- θ) + i \sin (- θ) = r [\cos θ - i \sin θ]$

$Again, - z = - r [\cos θ - i \sin θ] = r [\cos (π - θ) + i \sin (π - θ)]$

$∴$ $\arg (- z) = π - θ$

$Thus \arg (- z) - \arg (z) = (π - θ) - (- θ) = π - θ + θ = π$

Q 5 :

Let $A = {\frac{1967 + 1686 i \sin θ}{7 - 3 i \cos θ} : θ \in ℝ} .$ If A contains exactly one positive integer $n$ , then the value of $n$ is [2023]

(281)

$\frac{281 (49 + 18 \sin θ \cos θ + i (21 \cos θ + 42 \sin θ))}{49 + 9 \cos^{2} θ}$

is a positive integer. For positive integer $Im (z) = 0$

$\Rightarrow 21 \cos θ + 42 \sin θ = 0$

$\Rightarrow \tan θ = - \frac{1}{2}, \sin 2 θ = - \frac{4}{5}, \cos^{2} θ = \frac{4}{5}$

$Now R e (2) = \frac{281 (49 - 9 \sin 2 θ)}{49 + 9 \cos^{2} θ}$

$= \frac{281 (49 - 9 \times (- \frac{4}{5}))}{49 + 9 \times \frac{4}{5}}$

$= \frac{281 (49 + \frac{36}{5})}{49 + \frac{36}{5}}$

$= 281$

Q 6 :

For any integer $k$ , let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}),$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||}$ is [2015]

(4)

$Given: α_{k} = \cos \frac{k π}{7} + i \sin \frac{k π}{7} = e^{\frac{i π k}{7}}$

$α_{k + 1} - α_{k} = e^{\frac{i π (k + 1)}{7}} - e^{\frac{i π k}{7}} = e^{\frac{i π k}{7}} (e^{\frac{i π}{7}} - 1)$

$| α_{k + 1} - α_{k} |$ $=$ $| e^{\frac{i π}{7}} - 1 |$

$\Rightarrow \sum_{k = 1}^{12}$ $| α_{k + 1} - α_{k} |$ $= 12$ $| e^{\frac{i π}{7}} - 1 |$

$Similarly, \sum_{k = 1}^{3}$ $| α_{4 k - 1} - α_{4 k - 2} |$ $= 3$ $| e^{\frac{i π}{7}} - 1 |$

$∴$ $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||} = 4$

Q 7 :

If $z$ is any complex number satisfying $| z - 3 - 2 i |$ $\leq 2$ , then the minimum value of $| 2 z - 6 + 5 i |$ is [2011]

(5)

[IMAGE 7]

$Given:$ $| z - 3 - 2 i |$ $< 2,$

which represents a circular region with centre $(3, 2)$ and radius 2.

$Now,$ $| 2 z - 6 + 5 i |$ $= 2$ $| z - (3 - \frac{5}{2} i) |$

$= 2 \times distance of z from P (where z lies in or on the circle)$

$Also min distance of z from P = \frac{5}{2}$

$∴$ $Minimum value of$ $| 2 z - 6 + 5 i |$ $= 5$

Q 8 :

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of $| z |^{2}$ is _______. [2022]

(0.50)

$Let X = \frac{4 z^{2} + 3 z + 2}{4 z^{2} - 3 z + 2}$

$It can be written as = 1 + \frac{6 z}{4 z^{2} - 3 z + 2}; Now X = 1 + \frac{6}{2 (2 z + \frac{1}{z}) - 3}$

$∵$ $X \in ℝ, then 2 z + \frac{1}{z} \in ℝ$

$\Rightarrow 2 z + \frac{1}{z} = 2 \bar{z} + \frac{1}{\bar{z}}$ $\Rightarrow 2 (z - \bar{z}) - \frac{z - \bar{z}}{| z |^{2}} = 0$

$∵$ $(z - \bar{z}) (2 - \frac{1}{| z |^{2}}) = 0$

$∵$ $z \neq \bar{z} (given), So, | z |^{2} = \frac{1}{2}$

Q 9 :

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is _______ . [2022]

(4)

$Given, \bar{z} - z^{2} = i (\bar{z} + z^{2})$

$It can be written as \bar{z} (1 - i) = z^{2} (1 + i)$

$So | \bar{z} | | 1 - i | = | z |^{2} | 1 + i |$

$| z | = | z |^{2} \Rightarrow | z | = 0 or | z | = 1$

$Let \arg (z) = α . So from (i), we get$

$2 n π - α - \frac{π}{4} = 2 α + \frac{π}{4}$

$\Rightarrow α = \frac{1}{3} (\frac{4 n - 1}{2}) π = \frac{(4 n - 1) π}{6}$

So we will get 3 distinct values of $α$ . Hence there will be total 4 possible values of complex number $z$ .

Q 10 :

Let $S = {a + b \sqrt{2} : a, b \in ℤ}, T_{1} = {{(- 1 + \sqrt{2})}^{n} : n \in ℕ}$ and $T_{2} = {{(1 + \sqrt{2})}^{n} : n \in ℕ} .$ Then which of the following statements is (are) TRUE? [2024]

$ℤ \cup T_{1} \cup T_{2} \subset S$
$T_{1} \cap (0, \frac{1}{2024}) = ϕ$ , where $ϕ$ denotes the empty set.
$T_{2} \cap (2024, \infty) \neq ϕ$
For any given $a, b \in ℤ$ , $\cos (π (a + b \sqrt{2})) + i \sin (π (a + b \sqrt{2})) \in ℤ$ if and only if $b = 0$ , where $i = \sqrt{- 1}$ .

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Select one or more options

(1, 3, 4)

$(a) S = {a + b \sqrt{2} : a, b \in ℤ}$

$For b = 0; ℤ \subset S$

$T_{1} = {(- 1 + \sqrt{2})}^{n} = m + \sqrt{2} n, m, n \in ℤ$

$T_{2} = {(1 + \sqrt{2})}^{n} = m_{1} + \sqrt{2} n_{1}, m_{1}, n_{1} \in ℤ$

$For n \in ℕ, elements of T_{1} and T_{2} are of the form a + b \sqrt{2}$

$Hence ℤ \cup T_{1} \cup T_{2} \subset S$

$(b) Now, - 1 + \sqrt{2} < 1 and its higher powers decrease$

$\Rightarrow {(- 1 + \sqrt{2})}^{n} < 1 and can be made in (0, \frac{1}{2024}) for some higher n$

$(c) 1 + \sqrt{2} > 1 and its higher power increases$

$\Rightarrow {(1 + \sqrt{2})}^{n} can be made in (2024, \infty) for some higher n$

$(d) \cos π (a + b \sqrt{2}) + i \sin π (a + b \sqrt{2}) \in ℤ$

$a + b \sqrt{2}$ is an integer $\Rightarrow b = 0$

Q 11 :

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ . If $z$ is a non-zero complex number for which both real and imaginary parts of ${(\bar{z})}^{2} + \frac{1}{z^{2}}$ are integers, then which of the following is/are possible value(s) of $| z |$ ? [2022]

${(\frac{43 + 3 \sqrt{205}}{2})}^{\frac{1}{4}}$
${(\frac{7 + \sqrt{33}}{4})}^{\frac{1}{4}}$
${(\frac{9 + \sqrt{65}}{4})}^{\frac{1}{4}}$
${(\frac{7 + \sqrt{13}}{6})}^{\frac{1}{4}}$

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(1)

$Let z = r . e^{i θ} \Rightarrow \bar{z} = r e^{- i θ}$

$∴$ ${(\bar{z})}^{2} + \frac{1}{z^{2}} = r^{2} e^{- 2 i θ} + \frac{1}{r^{2} e^{2 i θ}} = (r^{2} + \frac{1}{r^{2}}) e^{- 2 i θ} = a + i b (say),$

$where a, b \in ℤ$

$So, {(r^{2} + \frac{1}{r^{2}})}^{2} = a^{2} + b^{2}$ $\Rightarrow r^{8} - (a^{2} + b^{2} - 2) r^{4} + 1 = 0$

$\Rightarrow r^{4} = \frac{(a^{2} + b^{2} - 2) \pm \sqrt{{(a^{2} + b^{2} - 2)}^{2} - 4}}{2}$

$For option (a): | z |^{4} = \frac{43 + 3 \sqrt{205}}{2}$

$\Rightarrow a^{2} + b^{2} = 45 i.e. (a, b) = (\pm 6, \pm 3) or (\pm 3, \pm 6)$

$For option (b): | z |^{4} = \frac{7 + \sqrt{33}}{4} \Rightarrow a^{2} + b^{2} = \frac{11}{2}$

$For option (c): a^{2} + b^{2} = \frac{13}{2}$

$For option (d): a^{2} + b^{2} = \frac{13}{3}$

Q 12 :

Let $S$ be the set of all complex numbers $z$ satisfying $| z^{2} + z + 1 |$ $= 1 .$ Then which of the following statements is/are TRUE? [2020]

$| z + \frac{1}{2} |$ $\leq \frac{1}{2}$ for all $z \in S$
$| z |$ $\leq 2$ for all $z \in S$
$| z + \frac{1}{2} |$ $\geq \frac{1}{2}$ for all $z \in S$
The set $S$ has exactly four elements

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Select one or more options

(2, 3)

$| z^{2} + z + 1 | = 1$

$\Rightarrow$ $| {(z + \frac{1}{2})}^{2} + \frac{3}{4} |$ $\geq 1$ $\Rightarrow$ $| (z + \frac{1}{2}) |^{2}$ $\geq \frac{1}{4}$ $\Rightarrow$ $| z + \frac{1}{2} |$ $\geq \frac{1}{2}$

$also | (z^{2} + z) + 1 | = 1$

$\Rightarrow | z^{2} + z | - 1 \leq 1 \Rightarrow | z^{2} + z | \leq 2$

$\Rightarrow$ $| | z^{2} | - | z | | \leq | z^{2} + z | \leq 2$ $\Rightarrow | r^{2} - r | \leq 2 \Rightarrow r = | z | \leq 2; \forall z \in S$

$Hence, set S is infinite$

Q 13 :

Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y$ $(x, y \in ℝ, i = \sqrt{- 1})$ of the equation

$s z + t \bar{z} + r = 0, where \bar{z} = x - i y .$ Then, which of the following statement(s) is (are) TRUE? [2018]

If $L$ has exactly one element, then $| s |$ $\neq$ $| t |$
If $| s | = | t |$ , then $L$ has infinitely many elements
The number of elements in $L \cap {z : | z - 1 + i | = 5}$ is at most 2
If $L$ has more than one element, then $L$ has infinitely many elements

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Select one or more options

(1, 3, 4)

$We have,$

$s z + t \bar{z} + r = 0 \dots (i)$

$On taking conjugate$

$\bar{s} \bar{z} + \bar{t z} + \bar{r} = 0 \dots (i i)$

$On solving Eqs. (i) and (ii), we get$

$z = \frac{\bar{r} t - r \bar{s}}{| s |^{2} - | t |^{2}}$

$(a) For unique solutions of z$

$| s |^{2} - | t |^{2} \neq 0 \Rightarrow | s | \neq | t |$

$It is true.$

$(b) If | s | = | t |, then \bar{r} t - r \bar{s} may or may not be zero.$

$So, z may have no solution.$

$∴$ $L may be an empty set.$

$It is false.$

$(c)$ If elements of set L represents line, then this line and given circle intersect at maximum two points.

Hence, it is true.

$(d)$ In the case locus of $z$ is a line, so L has infinite elements.

Hence, it is true.

Q 14 :

For a non-zero complex number $z$ , let $\arg (z)$ denote the principal argument with $- π < \arg (z) \leq π$ . Then, which of the following statement(s) is (are) FALSE? [2018]

$\arg (- 1 - i) = \frac{π}{4}$ , where $i = \sqrt{- 1}$
The function $f : ℝ \to (- π, π]$ , defined by $f (t) = \arg (- 1 + i t) for all t \in ℝ,$ is continuous at all points of $ℝ$ , where $i = \sqrt{- 1}$
For any two non-zero complex numbers $z_{1}$ and $z_{2}$ , $\arg (\frac{z_{1}}{z_{2}}) - \arg (z_{1}) + \arg (z_{2})$ is an integer multiple of $2 π$
For any three given distinct complex numbers $z_{1}, z_{2}$ and $z_{3}$ , the locus of the point $z$ satisfying the condition $\arg (\frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})}) = π,$ lies on a straight line

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Select one or more options

(1, 2, 4)

$(a) \arg (- 1 - i) = - \frac{3 π}{4}$

$∴$ $(a) is false$

$(b) f (t) = \arg (- 1 + i t) =$ $[\begin{matrix} π - \tan^{- 1} (t), & t \geq 0 \\ - π + \tan^{- 1} (t), & t < 0 \end{matrix}$

$\lim_{t \to 0^{-}} f (t) = - π and \lim_{t \to 0^{+}} f (t) = π$

$LHL \neq RHL \Rightarrow f is discontinuous at t = 0$

$∴$ $(b) is false$

$(c) \arg (\frac{z_{1}}{z_{2}}) - \arg z_{1} + \arg z_{2}$

$= 2 n π + \arg z_{1} - \arg z_{2} - \arg z_{1} + \arg z_{2}$

$= 2 n π, multiple of 2 π$

$∴$ $(c) is true$

$(d)$ $\arg (\frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})}) = π$

$\Rightarrow \frac{(z - z_{1}) (z_{2} - z_{3})}{(z - z_{3}) (z_{2} - z_{1})} = k, k \in ℝ$

$\Rightarrow (\frac{z - z_{1}}{z - z_{3}}) = k (\frac{z_{2} - z_{1}}{z_{2} - z_{3}})$

$\Rightarrow z, z_{1}, z_{2}, z_{3} are concyclic, i.e. z lies on a circle.$

$∴$ $(d) is false$

Q 15 :

Let $a, b, x$ and $y$ be real numbers such that $a - b = 1$ and $y \neq 0$ . If the complex number $z = x + i y$ satisfies $Im (\frac{a z + b}{z + 1}) = y,$ then which of the following is(are) possible value(s) of $x$ ? [2017]

$- 1 + \sqrt{1 - y^{2}}$
$- 1 - \sqrt{1 - y^{2}}$
$1 + \sqrt{1 + y^{2}}$
$1 - \sqrt{1 + y^{2}}$

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Select one or more options

(1, 2)

$a - b = 1, y \neq 0$

$Im (\frac{a z + b}{z + 1}) = y$

$\Rightarrow Im [(\frac{a (x + i y) + b}{(x + 1) + i y}) \cdot \frac{(x + 1) - i y}{(x + 1) - i y}] = y$

$\Rightarrow \frac{- (a x + b) y + a y (x + 1)}{{(x + 1)}^{2} + y^{2}} = y$

$\Rightarrow \frac{- a x y - b y + a x y + a y}{{(x + 1)}^{2} + y^{2}} = y$

$\Rightarrow a - b = {(x + 1)}^{2} + y^{2}$

$\Rightarrow 1 = {(x + 1)}^{2} + y^{2}$

$∴$ $x = - 1 \pm \sqrt{1 - y^{2}}$

Q 16 :

Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - t) z_{1} + t z_{2}$ for some real number $t$ with $0 < t < 1$ . If $Arg (w)$ denotes the principal argument of a non-zero complex number $w$ , then [2010]

$| z - z_{1} | + | z - z_{2} | = | z_{1} - z_{2} |$
$Arg (z - z_{1}) = Arg (z - z_{2})$
$| \begin{matrix} z - z_{1} & \bar{z} - \bar{z_{1}} \\ z_{2} - z_{1} & \bar{z_{2}} - \bar{z_{1}} \end{matrix} |$
$Arg (z - z_{1}) = Arg (z_{2} - z_{1})$

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Select one or more options

(1, 3, 4)

$Given: z = (1 - t) z_{1} + t z_{2}, where 0 < t < 1$

$\Rightarrow z = \frac{(1 - t) z_{1} + t z_{2}}{(1 - t) + t}$

$\Rightarrow z divides the join of z_{1} and z_{2} internally in the ratio t : (1 - t)$

$∴$ $z_{1}, z, z_{2} are collinear$ [IMAGE 8]

$\Rightarrow | z - z_{1} | + | z - z_{2} | = | z_{1} - z_{2} |$

$Also z = (1 - t) z_{1} + t z_{2}$

$\Rightarrow \frac{z - z_{1}}{z_{2} - z_{1}} = t, which is purely real number$

$∴$ $\arg (\frac{z - z_{1}}{z_{2} - z_{1}}) = 0 \Rightarrow \arg (z - z_{1}) = \arg (z_{2} - z_{1})$

$Also \frac{z - z_{1}}{z_{2} - z_{1}} = t \Rightarrow \frac{\bar{z} - {\bar{z}}_{1}}{{\bar{z}}_{2} - {\bar{z}}_{1}} = t$

$\Rightarrow \frac{z - z_{1}}{z_{2} - z_{1}} = \frac{\bar{z} - {\bar{z}}_{1}}{{\bar{z}}_{2} - {\bar{z}}_{1}}$

$\Rightarrow (z - z_{1}) ({\bar{z}}_{2} - {\bar{z}}_{1}) = (\bar{z} - {\bar{z}}_{1}) (z_{2} - z_{1})$

$\Rightarrow$ $| \begin{matrix} z - z_{1} & \bar{z} - {\bar{z}}_{1} \\ z_{2} - z_{1} & {\bar{z}}_{2} - {\bar{z}}_{1} \end{matrix} |$ $= 0$

Q 17 :

Let $z$ be a complex number satisfying $| z |^{3} + 2 z^{2} + 4 \bar{z} - 8 = 0,$ where $\bar{z}$ denotes the complex conjugate of $z$ . Let the imaginary part of $z$ be non-zero.

Match each entry in List-I to the correct entries in List-II.

List - I List - II

(P) $| z |^{2}$ is equal to (1) 12

(Q) $| z - \bar{z} |^{2}$ is equal to (2) 4

(R) $| z |^{2} + | z + \bar{z} |^{2}$ is equal to (3) 8

(S) $| z + 1 |^{2}$ is equal to (4) 10

(5) 7

The correct option is: [2023]

(P) → (1), (Q) → (3), (R) → (5), (S) → (4)
(P) → (2), (Q) → (1), (R) → (3), (S) → (5)
(P) → (2), (Q) → (4), (R) → (5), (S) → (1)
(P) → (2), (Q) → (3), (R) → (5), (S) → (4)

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(2)

$Given, | z |^{3} + 2 z^{2} + 4 \bar{z} - 8 = 0 \dots (i)$

$| \bar{z} |^{3} + 2 {\bar{z}}^{2} + 4 z - 8 = 0 [Conjugate both sides]$

$\Rightarrow 2 (z^{2} - {\bar{z}}^{2}) + 4 (\bar{z} - z) = 0$

$\Rightarrow 2 (z - \bar{z}) [z + \bar{z} - 2] = 0$

$∵$ $z = \bar{z} (Not possible) or z + \bar{z} = 2$

$∴$ $z = 1 + b i (b \neq 0) \Rightarrow \bar{z} = 1 - b i$

${(1 + b^{2})}^{3 / 2} + 2 (1 - b^{2} + 2 b i) + 4 (1 - b i) - 8 = 0 [from (i)]$

${(1 + b^{2})}^{3 / 2} - 2 (1 + b^{2}) = 0$

$\Rightarrow (1 + b^{2}) (\sqrt{1 + b^{2}} - 2) = 0$

$∵$ $1 + b^{2} \neq 0 \Rightarrow \sqrt{1 + b^{2}} - 2 = 0 \Rightarrow b^{2} = 3$

$(P) | z |^{2} = 1 + b^{2} = 1 + 3 = 4$

$(Q) | z - \bar{z} |^{2} = | 1 + i b - 1 + i b |^{2} = 4 b^{2} = 12$

$(R) | z |^{2} + | z + \bar{z} |^{2} = 4 + | 1 + i b + 1 - i b |^{2} = 4 + 4 = 8$

$(S) | z + 1 |^{2} = | 1 + 1 + i b |^{2} = 4 + b^{2} = 4 + 3 = 7$

Q 18 :

Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where

$S_{1} = {z \in ℂ : | z | < 4}, S_{2} = {z \in ℂ : Im [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0},$ and $S_{3} = {z \in ℂ : Re (z) > 0} .$ [2013]

Q. Area of $S =$

$\frac{10 π}{3}$
$\frac{20 π}{3}$
$\frac{16 π}{3}$
$\frac{32 π}{3}$

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(2)

$S_{1} : x^{2} + y^{2} < 16$

$S_{2} : Im [\frac{(x - 1) + i (y + \sqrt{3})}{1 - i \sqrt{3}}] > 0$

$\Rightarrow \sqrt{3} (x - 1) + (y + \sqrt{3}) > 0 \Rightarrow y + \sqrt{3} x > 0$

$S_{3} : x > 0$

$Then S : S_{1} \cap S_{2} \cap S_{3} is as shown in the figure given below.$

[IMAGE 9]

Area of shaded region

$= \frac{π}{4} \times 4^{2} + \frac{π \times 4^{2} \times 60 °}{360 °}$

$= 4 π + \frac{8 π}{3} = \frac{20 π}{3}$

Q 19 :

Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where

$S_{1} = {z \in ℂ : | z | < 4}, S_{2} = {z \in ℂ : Im [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0},$ and $S_{3} = {z \in ℂ : Re (z) > 0} .$ [2013]

Q. $\underset{z \in S}{m i n}$ $| 1 - 3 i - z |$ $=$

$\frac{2 - \sqrt{3}}{2}$
$\frac{2 + \sqrt{3}}{2}$
$\frac{3 - \sqrt{3}}{2}$
$\frac{3 + \sqrt{3}}{2}$

1

2

3

4

(3)

$S_{1} : x^{2} + y^{2} < 16$

$S_{2} : Im [\frac{(x - 1) + i (y + \sqrt{3})}{1 - i \sqrt{3}}] > 0$

$\Rightarrow \sqrt{3} (x - 1) + (y + \sqrt{3}) > 0 \Rightarrow y + \sqrt{3} x > 0$

$S_{3} : x > 0$

$Then S : S_{1} \cap S_{2} \cap S_{3} is as shown in the figure given below.$

[IMAGE 10]

$\min_{z \in S} | 1 - 3 i - z | = minimum distance between z and (1, - 3)$

$Clearly (from figure) minimum distance between z \in S and (1, - 3)$

$from line y + x \sqrt{3} = 0 i.e.$ $| \frac{\sqrt{3} - 3}{\sqrt{3} + 1} | = \frac{3 - \sqrt{3}}{2}$

Q 20 :

Let $A, B, C$ be three sets of complex numbers as defined below [2008]

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. The number of elements in the set $A \cap B \cap C$ is

0
1
2
$\infty$

1

2

3

4

(2)

$Given : A = {z : Im (z) \geq 1} = {(x, y) : y \geq 1}$

Clearly A is the set of all points lying on or above the line $y = 1$ in cartesian plane.

$B = {z : | z - 2 - i | = 3} = {(x, y) : {(x - 2)}^{2} + {(y - 1)}^{2} = 9}$

$\Rightarrow$ B is the set of all points lying on the boundary of the circle with centre (2,1) and radius 3.

$C = {z : R e [(1 - i) z] = \sqrt{2}} = {(x, y) : x + y = \sqrt{2}}$

$\Rightarrow$ C is the set of all points lying on the straight line represented by $x + y = \sqrt{2}$ .

Graphically, the three sets are represented as shown below :

[IMAGE 11]

From graph $A \cap B \cap C$ consists of only one point P [the common point of the region $y \geq 1, {(x - 2)}^{2} + {(y - 1)}^{2} = 9$ and $x + y = \sqrt{2}$

$∴$ $n (A \cap B \cap C) = 1$

Q 21 :

$Let z_{k} = \cos (\frac{2 k π}{10}) + i \sin (\frac{2 k π}{10}), k = 1, 2, \dots, 9 .$ [2014]

List - I List - II

P. For each $z_{k}$ there exists $z_{j}$ such that $z_{k} . z_{j} = 1$ 1. True

Q. There exists a $k \in {1, 2, \dots, 9}$ such that $z_{1} . z = z_{k}$ has no solution $z$ in the set of complex numbers 2. False

R. $\frac{| 1 - z_{1} | | 1 - z_{2} | \dots | 1 - z_{9} |}{10}$ equals 3. 1

S. $1 - \sum_{k = 1}^{9} \cos (\frac{2 k π}{10})$ equals 4. 2

$P \to 1, Q \to 2, R \to 4, S \to 3$
$P \to 2, Q \to 1, R \to 3, S \to 4$
$P \to 1, Q \to 2, R \to 3, S \to 4$
$P \to 2, Q \to 1, R \to 4, S \to 3$

1

2

3

4

(3)

$(P) \to (1) : z_{k} = \cos \frac{2 k π}{10} + i \sin \frac{2 k π}{10}, k = 1 to 9$

$∴$ $z_{k} = e^{i \frac{2 k π}{10}}$

$Now z_{k} . z_{j} = 1 \Rightarrow z_{j} = \frac{1}{z_{k}} = e^{- i \frac{2 k π}{10}} = \bar{z_{k}}$

$We know if z_{k} is 10th root of unity so will be \bar{z_{k}}$

$∴$ $For every z_{k} there exist z_{i} = \bar{z_{k}}$

$Such that z_{k} . z_{j} = z_{k} . \bar{z_{k}} = 1$

Hence the statement is true.

$(Q) \to (2) : z_{1} = z_{k} \Rightarrow z = \frac{z_{k}}{z_{1}}, z_{1} \neq 0$

$∴$ $We can always find a solution of z_{1} z = z_{k}$

Hence the statement is false.

$(R) \to (3) : We know z^{10} - 1 = (z - 1) (z - z_{1}) \dots (z - z_{9})$

$\Rightarrow (z - z_{1}) (z - z_{2}) \dots (z - z_{9}) = \frac{z^{10} - 1}{z - 1} = 1 + z + z^{2} + \dots + z^{9}$

$For z = 1, we get (1 - z_{1}) (1 - z_{2}) \dots (1 - z_{9}) = 10$

$∴$ $\frac{| 1 - z_{1} | | 1 - z_{2} | \dots | 1 - z_{9} |}{10} = 1$

$(S) \to (4) : 1, z_{1}, z_{2}, \dots, z_{9} are 10th roots of unity.$

$∴$ $z^{10} - 1 = 0$

$From equation 1 + z_{1} + z_{2} + \dots + z_{9} = 0,$

$Re (1) + Re (z_{1}) + Re (z_{2}) + \dots + Re (z_{9}) = 0$

$\Rightarrow Re (z_{1}) + Re (z_{2}) + \dots + Re (z_{9}) = - 1$

$\Rightarrow \sum_{k = 1}^{9} \cos \frac{2 k π}{10} = - 1$ $\Rightarrow 1 - \sum_{k = 1}^{9} \cos \frac{2 k π}{10} = 2$

$Hence (3) is the correct option.$

Q 22 :

Let $A, B, C$ be three sets of complex numbers as defined below

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. $Let z be any point in A \cap B \cap C .$ $Then,$ $| z + 1 - i |^{2}$ $+$ $| z - 5 - i |^{2}$ $lies between$ [2008]

25 and 29
30 and 34
35 and 39
40 and 44

1

2

3

4

(3)

$Given: A = {z : Im (z) \geq 1} = {(x, y) : y \geq 1}$

$Clearly A is the set of all points lying on or above the line y = 1 in Cartesian plane.$

$B = {z : | z - 2 - i | = 3} = {(x, y) : {(x - 2)}^{2} + {(y - 1)}^{2} = 9}$

$\Rightarrow$ B is the set of all points lying on the boundary of the circle with centre (2, 1) and radius 3.

$C = {z : Re [(1 - i) z] = \sqrt{2}} = {(x, y) : x + y = \sqrt{2}}$

$\Rightarrow$ C is the set of all points lying on the straight line represented by $x + y = \sqrt{2} .$

Graphically, the three sets are represented as shown below:

[IMAGE 12]

$Since, z is a point of A \cap B \cap C \Rightarrow z represents the point P$

$∴$ $| z + 1 - i |^{2} + | z - 5 - i |^{2}$

$\Rightarrow$ $| z - (- 1 + i) |^{2} + | z - (5 + i) |^{2}$

$\Rightarrow P Q^{2} + P R^{2} = Q R^{2} = 6^{2} = 36, which lies between 35 and 39$

$∴$ $(3) is correct option.$

Q 23 :

$Let A, B, C be three sets of complex numbers as defined below$

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. $Let z be any point A \cap B \cap C and let w be any point satisfying$ $| w - 2 - i |$ $< 3 .$ $Then, | z | - | w | + 3 lies between$ [2008]

- 6 and 3
- 3 and 6
- 6 and 6
- 3 and 9

1

2

3

4

(4)

$Given: A = {z : Im (z) \geq 1} = {(x, y) : y \geq 1}$

$Clearly A is the set of all points lying on or above the line y = 1 in Cartesian plane.$

$B = {z : | z - 2 - i | = 3} = {(x, y) : {(x - 2)}^{2} + {(y - 1)}^{2} = 9}$

$\Rightarrow$ B is the set of all points lying on the boundary of the circle with centre (2, 1) and radius 3.

$C = {z : Re [(1 - i) z] = \sqrt{2}} = {(x, y) : x + y = \sqrt{2}}$

$\Rightarrow$ C is the set of all points lying on the straight line represented by $x + y = \sqrt{2} .$

Graphically, the three sets are represented as shown below:

[IMAGE 13]

$Given$ $| w - 2 - i |$ $< 3$

$\Rightarrow Distance between w and (2 + i), i.e. S is smaller than 3 .$

$\Rightarrow w is a point lying inside the circle with centre S and radius 3 .$

$\Rightarrow Distance between z (i.e. the point P) and w should be smaller than 6 (the diameter of the circle)$

$i.e. | z - w | < 6$

$But we know that$ $| | z | - | w | |$ $\leq | z - w |$

$\Rightarrow$ $| | z | - | w | |$ $< 6 \Rightarrow - 6 < | z | - | w | < 6$

$\Rightarrow - 3 < | z | - | w | + 3 < 9$

Topic Question Set

Q 1 :

If $\frac{w - \bar{w} z}{1 - z}$ is purely real where $w = α + i β, β \neq 0$ and $z \neq 1$ , then the set of the values of $z$ is [2006]

Q 2 :

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that

$| z_{1} |$ $=$ $| z_{2} |$ $=$ $| z_{3} |$ $=$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $= 1,$ then $| z_{1} + z_{2} + z_{3} |$ is [2000]

Q 3 :

For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} |$ $= 12$ and $| z_{2} - 3 - 4 i |$ $= 5$ , the minimum value of $| z_{1} - z_{2} |$ is [2002]

Q 4 :

If $\arg (z) < 0$ , then $\arg (- z) - \arg (z) =$ [2000]

Q 5 :

Let $A = {\frac{1967 + 1686 i \sin θ}{7 - 3 i \cos θ} : θ \in ℝ} .$ If A contains exactly one positive integer $n$ , then the value of $n$ is [2023]

Q 6 :

For any integer $k$ , let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}),$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||}$ is [2015]

Q 7 :

If $z$ is any complex number satisfying $| z - 3 - 2 i |$ $\leq 2$ , then the minimum value of $| 2 z - 6 + 5 i |$ is [2011]

Q 8 :

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of $| z |^{2}$ is _______. [2022]

Q 9 :

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is _______ . [2022]

Q 10 :

Let $S = {a + b \sqrt{2} : a, b \in ℤ}, T_{1} = {{(- 1 + \sqrt{2})}^{n} : n \in ℕ}$ and $T_{2} = {{(1 + \sqrt{2})}^{n} : n \in ℕ} .$ Then which of the following statements is (are) TRUE? [2024]

Q 11 :

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ . If $z$ is a non-zero complex number for which both real and imaginary parts of ${(\bar{z})}^{2} + \frac{1}{z^{2}}$ are integers, then which of the following is/are possible value(s) of $| z |$ ? [2022]

Q 12 :

Let $S$ be the set of all complex numbers $z$ satisfying $| z^{2} + z + 1 |$ $= 1 .$ Then which of the following statements is/are TRUE? [2020]

Q 13 :

Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y$ $(x, y \in ℝ, i = \sqrt{- 1})$ of the equation

$s z + t \bar{z} + r = 0, where \bar{z} = x - i y .$ Then, which of the following statement(s) is (are) TRUE? [2018]

Q 14 :

For a non-zero complex number $z$ , let $\arg (z)$ denote the principal argument with $- π < \arg (z) \leq π$ . Then, which of the following statement(s) is (are) FALSE? [2018]

Q 15 :

Let $a, b, x$ and $y$ be real numbers such that $a - b = 1$ and $y \neq 0$ . If the complex number $z = x + i y$ satisfies $Im (\frac{a z + b}{z + 1}) = y,$ then which of the following is(are) possible value(s) of $x$ ? [2017]

Q 16 :

Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - t) z_{1} + t z_{2}$ for some real number $t$ with $0 < t < 1$ . If $Arg (w)$ denotes the principal argument of a non-zero complex number $w$ , then [2010]

Q 18 :

Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where

$S_{1} = {z \in ℂ : | z | < 4}, S_{2} = {z \in ℂ : Im [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0},$ and $S_{3} = {z \in ℂ : Re (z) > 0} .$ [2013]

Q. Area of $S =$

Q 19 :

Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where

$S_{1} = {z \in ℂ : | z | < 4}, S_{2} = {z \in ℂ : Im [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0},$ and $S_{3} = {z \in ℂ : Re (z) > 0} .$ [2013]

Q. $\underset{z \in S}{m i n}$ $| 1 - 3 i - z |$ $=$

Q 20 :

Let $A, B, C$ be three sets of complex numbers as defined below [2008]

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. The number of elements in the set $A \cap B \cap C$ is

Q 22 :

Let $A, B, C$ be three sets of complex numbers as defined below

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. $Let z be any point in A \cap B \cap C .$ $Then,$ $| z + 1 - i |^{2}$ $+$ $| z - 5 - i |^{2}$ $lies between$ [2008]

Want Us to Email you About Special Offers & Updates?

	List - I		List - II
(P)	$\| z \|^{2}$ is equal to	(1)	12
(Q)	$\| z - \bar{z} \|^{2}$ is equal to	(2)	4
(R)	$\| z \|^{2} + \| z + \bar{z} \|^{2}$ is equal to	(3)	8
(S)	$\| z + 1 \|^{2}$ is equal to	(4)	10
		(5)	7

	List - I		List - II
P.	For each $z_{k}$ there exists $z_{j}$ such that $z_{k} . z_{j} = 1$	1.	True
Q.	There exists a $k \in {1, 2, \dots, 9}$ such that $z_{1} . z = z_{k}$ has no solution $z$ in the set of complex numbers	2.	False
R.	$\frac{\| 1 - z_{1} \| \| 1 - z_{2} \| \dots \| 1 - z_{9} \|}{10}$ equals	3.	1
S.	$1 - \sum_{k = 1}^{9} \cos (\frac{2 k π}{10})$ equals	4.	2

Topic Question Set

Q 1 : If w-w¯z1-z is purely real where w=α+iβ, β≠0 and z≠1, then the set of the values of z is [2006]

Q 2 : If z1,z2 and z3 are complex numbers such that |z1|=|z2|=|z3|=|1z1+1z2+1z3|=1, then |z1+z2+z3| is [2000]

Q 3 : For all complex numbers z1,z2 satisfying |z1|=12 and |z2-3-4i|=5, the minimum value of |z1-z2| is [2002]

Q 4 : If arg(z)<0, then arg(-z)-arg(z)= [2000]

Q 5 : Let A={1967+1686 isinθ7-3icosθ:θ∈ℝ}. If A contains exactly one positive integer n, then the value of n is [2023]

Q 6 : For any integer k, let αk=cos(kπ7)+isin(kπ7), where i=-1. The value of the expression ∑k=112|αk+1-αk|∑k=13|α4k-1-α4k-2| is [2015]

Q 7 : If z is any complex number satisfying |z-3-2i|≤2, then the minimum value of |2z-6+5i| is [2011]

Q 8 : Let z be a complex number with non-zero imaginary part. If 2+3z+4z22-3z+4z2 is a real number, then the value of |z|2 is _______. [2022]

Q 9 : Let z¯ denote the complex conjugate of a complex number z and let i=-1. In the set of complex numbers, the number of distinct roots of the equation z¯-z2=i(z¯+z2)is _______ . [2022]

Q 10 : Let S={a+b2:a,b∈ℤ}, T1={(-1+2)n:n∈ℕ} and T2={(1+2)n:n∈ℕ}. Then which of the following statements is (are) TRUE? [2024]

Q 11 : Let z¯ denote the complex conjugate of a complex number z. If z is a non-zero complex number for which both real and imaginary parts of (z¯)2+1z2 are integers, then which of the following is/are possible value(s) of |z|? [2022]

Q 12 : Let S be the set of all complex numbers z satisfying |z2+z+1|=1. Then which of the following statements is/are TRUE? [2020]

Q 13 : Let s, t, r be non-zero complex numbers and L be the set of solutions z=x+iy (x,y∈ℝ, i=-1) of the equation sz+tz¯+r=0, where z¯=x-iy. Then, which of the following statement(s) is (are) TRUE? [2018]

Q 14 : For a non-zero complex number z, let arg(z) denote the principal argument with -π<arg(z)≤π. Then, which of the following statement(s) is (are) FALSE? [2018]

Q 15 : Let a, b, x and y be real numbers such that a-b=1 and y≠0. If the complex number z=x+iy satisfies Im(az+bz+1)=y, then which of the following is(are) possible value(s) of x? [2017]

Q 16 : Let z1 and z2 be two distinct complex numbers and let z=(1-t)z1+tz2 for some real number t with 0<t<1. If Arg(w) denotes the principal argument of a non-zero complex number w, then [2010]

Q 18 : Let S=S1∩S2∩S3, where S1={z∈ℂ:|z|<4}, S2={z∈ℂ:Im[z-1+3 i1-3 i]>0}, and S3={z∈ℂ:Re(z)>0}. [2013] Q. Area of S=

Q 19 : Let S=S1∩S2∩S3, where S1={z∈ℂ:|z|<4}, S2={z∈ℂ:Im[z-1+3 i1-3 i]>0}, and S3={z∈ℂ:Re(z)>0}. [2013] Q. minz∈S|1-3i-z|=

Q 20 : Let A,B,C be three sets of complex numbers as defined below [2008] A={z:Im(z)≥1} B={z:|z-2-i|=3} C={z:Re((1-i)z)=2} Q. The number of elements in the set A∩B∩C is

Q 22 : Let A,B,C be three sets of complex numbers as defined below A={z:Im(z)≥1} B={z:|z-2-i|=3} C={z:Re((1-i)z)=2} Q. Let z be any point in A∩B∩C. Then, |z+1-i|2+|z-5-i|2 lies between [2008]

Q 23 : Let A,B,C be three sets of complex numbers as defined below A={z:Im(z)≥1} B={z:|z-2-i|=3} C={z:Re((1-i)z)=2} Q. Let z be any point A∩B∩C and let w be any point satisfying |w-2-i|<3. Then, |z|-|w|+3 lies between [2008]

Want Us to Email you About Special Offers & Updates?

Q 1 :

If $\frac{w - \bar{w} z}{1 - z}$ is purely real where $w = α + i β, β \neq 0$ and $z \neq 1$ , then the set of the values of $z$ is [2006]

Q 2 :

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that

$| z_{1} |$ $=$ $| z_{2} |$ $=$ $| z_{3} |$ $=$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $= 1,$ then $| z_{1} + z_{2} + z_{3} |$ is [2000]

Q 3 :

For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} |$ $= 12$ and $| z_{2} - 3 - 4 i |$ $= 5$ , the minimum value of $| z_{1} - z_{2} |$ is [2002]

Q 4 :

If $\arg (z) < 0$ , then $\arg (- z) - \arg (z) =$ [2000]

Q 5 :

Let $A = {\frac{1967 + 1686 i \sin θ}{7 - 3 i \cos θ} : θ \in ℝ} .$ If A contains exactly one positive integer $n$ , then the value of $n$ is [2023]

Q 6 :

For any integer $k$ , let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}),$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||}$ is [2015]

Q 7 :

If $z$ is any complex number satisfying $| z - 3 - 2 i |$ $\leq 2$ , then the minimum value of $| 2 z - 6 + 5 i |$ is [2011]

Q 8 :

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of $| z |^{2}$ is _______. [2022]

Q 9 :

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is _______ . [2022]

Q 10 :

Let $S = {a + b \sqrt{2} : a, b \in ℤ}, T_{1} = {{(- 1 + \sqrt{2})}^{n} : n \in ℕ}$ and $T_{2} = {{(1 + \sqrt{2})}^{n} : n \in ℕ} .$ Then which of the following statements is (are) TRUE? [2024]

Q 11 :

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ . If $z$ is a non-zero complex number for which both real and imaginary parts of ${(\bar{z})}^{2} + \frac{1}{z^{2}}$ are integers, then which of the following is/are possible value(s) of $| z |$ ? [2022]

Q 12 :

Let $S$ be the set of all complex numbers $z$ satisfying $| z^{2} + z + 1 |$ $= 1 .$ Then which of the following statements is/are TRUE? [2020]

Q 13 :

Let $s, t, r$ be non-zero complex numbers and $L$ be the set of solutions $z = x + i y$ $(x, y \in ℝ, i = \sqrt{- 1})$ of the equation

$s z + t \bar{z} + r = 0, where \bar{z} = x - i y .$ Then, which of the following statement(s) is (are) TRUE? [2018]

Q 14 :

For a non-zero complex number $z$ , let $\arg (z)$ denote the principal argument with $- π < \arg (z) \leq π$ . Then, which of the following statement(s) is (are) FALSE? [2018]

Q 15 :

Let $a, b, x$ and $y$ be real numbers such that $a - b = 1$ and $y \neq 0$ . If the complex number $z = x + i y$ satisfies $Im (\frac{a z + b}{z + 1}) = y,$ then which of the following is(are) possible value(s) of $x$ ? [2017]

Q 16 :

Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - t) z_{1} + t z_{2}$ for some real number $t$ with $0 < t < 1$ . If $Arg (w)$ denotes the principal argument of a non-zero complex number $w$ , then [2010]

Q 18 :

Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where

$S_{1} = {z \in ℂ : | z | < 4}, S_{2} = {z \in ℂ : Im [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0},$ and $S_{3} = {z \in ℂ : Re (z) > 0} .$ [2013]

Q. Area of $S =$

Q 19 :

Let $S = S_{1} \cap S_{2} \cap S_{3}$ , where

$S_{1} = {z \in ℂ : | z | < 4}, S_{2} = {z \in ℂ : Im [\frac{z - 1 + \sqrt{3} i}{1 - \sqrt{3} i}] > 0},$ and $S_{3} = {z \in ℂ : Re (z) > 0} .$ [2013]

Q. $\underset{z \in S}{m i n}$ $| 1 - 3 i - z |$ $=$

Q 20 :

Let $A, B, C$ be three sets of complex numbers as defined below [2008]

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. The number of elements in the set $A \cap B \cap C$ is

Q 22 :

Let $A, B, C$ be three sets of complex numbers as defined below

$A = {z : Im (z) \geq 1}$

$B = {z : | z - 2 - i | = 3}$

$C = {z : Re ((1 - i) z) = \sqrt{2}}$

Q. $Let z be any point in A \cap B \cap C .$ $Then,$ $| z + 1 - i |^{2}$ $+$ $| z - 5 - i |^{2}$ $lies between$ [2008]