JEE Advanced Maths: Complex Numbers & Quadratic Equations PYQ (With Solutions)

Q 1 :

Let $θ_{1}, θ_{2}, \dots, θ_{10}$ be positive valued angles (in radian) such that $θ_{1} + θ_{2} + \dots + θ_{10} = 2 π .$ Define the complex numbers $z_{1} = e^{i θ_{1}}, z_{k} = z_{k - 1} e^{i θ_{k}}$ for $k = 2, 3, \dots, 10,$
where $i = \sqrt{- 1}$ . Consider the statements P and Q given below:

$P | z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{10} - z_{9} | + | z_{1} - z_{10} | \leq 2 π$

$Q | z_{2}^{2} - z_{1}^{2} | + | z_{3}^{2} - z_{2}^{2} | + \dots + | z_{10}^{2} - z_{9}^{2} | + | z_{1}^{2} - z_{10}^{2} | \leq 4 π$ [2021]

P is TRUE and Q is FALSE
Q is TRUE and P is FALSE
both P and Q are TRUE
both P and Q are FALSE

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(3)

$Since, | z_{1} | = | z_{2} | = \dots = | z_{10} | = 1$

$θ_{2} = arc (z_{1}, z_{2})$

$| z_{2} - z_{1} | = length of line A B \leq length of arc A B$

$| z_{3} - z_{2} | = length of line B C \leq length of arc B C$

$∴$ $Sum of length of these 10 lines \leq Sum of length of arcs (i.e. 2 π)$

$[∵ θ_{1} + θ_{2} + θ_{3} + \dots + θ_{10} = 2 π]$

$∴$ $P : | z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{1} - z_{10} | \leq 2 π$

$P is true.$

$Now, | z_{2}^{2} - z_{1}^{2} | = | z_{2} - z_{1} | | z_{2} + z_{1} |$

$We know that$

$| z_{2} + z_{1} | \leq | z_{2} | + | z_{1} | \leq 2$

$∴$ $| z_{2}^{2} - z_{1}^{2} | + | z_{3}^{2} - z_{2}^{2} | + \dots + | z_{1}^{2} - z_{10}^{2} | \leq 2 {| z_{2} - z_{1} | + | z_{3} - z_{2} | + \dots + | z_{1} - z_{10} |}$ $\leq 2 (2 π) \Rightarrow Q \leq 4 π$

$Q is also true.$

Q 2 :

Let $S$ be the set of all complex numbers $z$ satisfying $| z - 2 + i |$ $\geq \sqrt{5}$ . If the complex number $z_{0}$ is such that $\frac{1}{| z_{0} - 1 |}$ is the maximum of the set ${\frac{1}{| z - 1 |} : z \in S}$ , then the principal argument of $\frac{4 - z_{0} - \bar{z_{0}}}{z_{0} - \bar{z_{0}} + 2 i}$ is [2019]

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

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(4)

$S :$ $| z - 2 + i |$ $\geq \sqrt{5} represents boundary and outer region of circle with centre (2, - 1) and radius \sqrt{5} units.$

$z_{0} \in S, such that \frac{1}{| z_{0} - 1 |} is the maximum.$

$∴$ $| z_{0} - 1 |$ $is minimum.$

$z_{0} \in S with | z_{0} - 1 | as minimum will be a point on boundary of circle of region S which lies on radius of this circle, which passes through (1, 0) .$

$∴$ $z_{0}, 1, 2 - i are collinear, or (x_{0}, y_{0}), (1, 0), (2, - 1) are collinear.$

$∴$ $Using slopes of parallel lines,$ $x'$

$\frac{y_{0}}{x_{0} - 1} = \frac{- 1}{2 - 1} \Rightarrow y_{0} = 1 - x_{0}$

$Now, \frac{4 - z_{0} - \bar{z_{0}}}{z_{0} - \bar{z_{0}} + 2 i} = \frac{4 - (z_{0} + \bar{z_{0}})}{(z_{0} - \bar{z_{0}}) + 2 i}$

$= \frac{4 - 2 x_{0}}{2 i y_{0} + 2 i} = \frac{4 - 2 x_{0}}{2 i (1 - x_{0}) + 2 i}$

$= \frac{2 (2 - x_{0})}{2 (2 - x_{0}) i} = \frac{1}{i} = - i$

$∴$ $Arg (\frac{4 - z_{0} - \bar{z_{0}}}{z_{0} - \bar{z_{0}} - 2 i}) = Arg (- i) = - \frac{π}{2}$

Q 3 :

Let complex numbers $α$ and $\frac{1}{\bar{α}}$ lie on circles ${(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2}$ and ${(x - x_{0})}^{2} + {(y - y_{0})}^{2} = 4 r^{2}$ respectively. If $z_{0} = x_{0} + i y_{0}$ satisfies the equation $2$ $| z_{0} |^{2}$ $= r^{2} + 2$ , then $| α |$ $=$ [2013]

$\frac{1}{\sqrt{2}}$
$\frac{1}{2}$
$\frac{1}{\sqrt{7}}$
$\frac{1}{3}$

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(3)

$Since, α lies on the circle {(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2}$

$∴$ $| α - z_{0} |^{2}$ $= r^{2}$

$\Rightarrow (α - z_{0}) (\bar{α} - \bar{z_{0}}) = r^{2}$

$\Rightarrow α \bar{α} - α \bar{z_{0}} - \bar{α} z_{0} + z_{0} \bar{z_{0}} = r^{2}$

$\Rightarrow | α |^{2} + | z_{0} |^{2} - α \bar{z_{0}} - \bar{α} z_{0} = r^{2} . . . (i)$

$Also \frac{1}{\bar{α}} lies on the circle {(x - x_{0})}^{2} + {(y - y_{0})}^{2} = 4 r^{2}$

$∴$ $| \frac{1}{\bar{α}} - z_{0} |^{2}$ $= 4 r^{2}$ $\Rightarrow (\frac{1}{\bar{α}} - z_{0}) (\frac{1}{α} - \bar{z_{0}}) = 4 r^{2}$

$\Rightarrow \frac{1}{α \bar{α}} - \frac{z_{0}}{α} - \frac{\bar{z_{0}}}{\bar{α}} + z_{0} \bar{z_{0}} = 4 r^{2}$

$\Rightarrow \frac{1}{| α |^{2}} - \frac{z_{0} \bar{α}}{| α |^{2}} - \frac{\bar{z_{0}} α}{| α |^{2}} + | z_{0} |^{2} = 4 r^{2}$

$\Rightarrow 1 + | α |^{2} | z_{0} |^{2} - z_{0} \bar{α} - \bar{z_{0}} α = 4 r^{2} | α |^{2} . . . (i i)$

$On subtracting equation (i) from (ii), we get$

$1 - | α |^{2} + | z_{0} |^{2} (| α |^{2} - 1) = r^{2} (4 | α |^{2} - 1)$

$or (| α |^{2} - 1) (| z_{0} |^{2} - 1) = r^{2} (4 | α |^{2} - 1)$

$Using | z_{0} |^{2} = \frac{r^{2} + 2}{2}, we get$

$(| α |^{2} - 1) \frac{r^{2}}{2} = r^{2} (4 | α |^{2} - 1)$

$\Rightarrow | α |^{2} - 1 = 8 | α |^{2} - 2$

$\Rightarrow | α |^{2} = \frac{1}{7}$

$\Rightarrow | α | = \frac{1}{\sqrt{7}}$

Q 4 :

Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^{2} + z + 1$ is real. Then a cannot take the value [2012]

$- 1$
$\frac{1}{3}$
$\frac{1}{2}$
$\frac{3}{4}$

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(4)

$∵$ $Im (z) \neq 0$

$\Rightarrow z is non-real$

$and equation z^{2} + z + (1 - a) = 0 will have non-real roots, if D < 0$

$\Rightarrow 1 - 4 (1 - a) < 0 \Rightarrow 4 a < 3 \Rightarrow a < \frac{3}{4}$

$∴$ $a cannot take the value \frac{3}{4} .$

Q 5 :

Let $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation: $z {\bar{z}}^{3} + \bar{z} z^{3} = 350$ is [2009]

48
32
40
80

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(1)

$Given: z = x + i y, where x and y are integers$

$Also, z {\bar{z}}^{3} + \bar{z} z^{3} = 350 \Rightarrow | z |^{2} ({\bar{z}}^{2} + z^{2}) = 350$

$\Rightarrow (x^{2} + y^{2}) (x^{2} - y^{2}) = 175$

$\Rightarrow (x^{2} + y^{2}) (x^{2} - y^{2}) = 25 \times 7 . . . (i)$

$or (x^{2} + y^{2}) (x^{2} - y^{2}) = 35 \times 5 . . . (i i)$

$∵$ $x, y are integers$

$∴$ $x^{2} + y^{2} = 25 and x^{2} - y^{2} = 7 [From eq (i)]$

$\Rightarrow x^{2} = 16, y^{2} = 9$

$\Rightarrow x = \pm 4, y = \pm 3$

$∴$ $Vertices of rectangle are$

$(4, 3), (4, - 3), (- 4, - 3), (- 4, 3)$

$∴$ $Area of rectangle = 8 \times 6 = 48 sq. units$

$Now from eq. (ii),$

$x^{2} + y^{2} = 35 and x^{2} - y^{2} = 5$

$\Rightarrow x^{2} = 20, which is not possible for any integral value of x .$

Q 6 :

Let $z = \cos θ + i \sin θ$ . Then the value of $\sum_{m = 1}^{15} Im (z^{2 m - 1})$ at $θ = 2 °$ is [2009]

$\frac{1}{\sin 2 °}$
$\frac{1}{3 \sin 2 °}$
$\frac{1}{2 \sin 2 °}$
$\frac{1}{4 \sin 2 °}$

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(4)

$z = \cos θ + i \sin θ$

$\Rightarrow z^{2 m - 1} = {(\cos θ + i \sin θ)}^{2 m - 1} = \cos (2 m - 1) θ + i \sin (2 m - 1) θ$

$[By De Moivre's theorem: {(\cos θ + i \sin θ)}^{n} = \cos n θ + i \sin n θ]$

$∴$ $Im (z^{2 m - 1}) = \sin (2 m - 1) θ$

$∴$ $\sum_{m = 1}^{15} Im (z^{2 m - 1}) = \sum_{m = 1}^{15} \sin (2 m - 1) θ$

$= \sin θ + \sin 3 θ + \sin 5 θ + \dots (upto 15 terms)$

$= \frac{\sin [15 (\frac{2 θ}{2})] \cdot \sin [θ + 14 θ]}{\sin θ}$

$[\begin{matrix} ∵ \sin α + \sin (α + β) + \sin (α + 2 β) + \dots (n terms) \\ = \frac{\sin (\frac{n β}{2}) \cdot \sin (α + \frac{(n - 1) β}{2})}{\sin (\frac{β}{2})} \end{matrix}]$

$= \frac{\sin 15 θ \cdot \sin 15 θ}{\sin θ} = \frac{\sin 30 ° \cdot \sin 30 °}{\sin 2 °} = \frac{1}{4 \sin 2 °}$

Q 7 :

A particle $P$ starts from the point $z_{0} = 1 + 2 i$ , where $i = \sqrt{- 1}$ . It moves horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_{1}$ . From $z_{1}$ the particle moves $\sqrt{2}$ units in the direction of the vector $\hat{i} + \hat{j}$ and then it moves through an angle $\frac{π}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point $z_{2}$ . The point $z_{2}$ is given by [2008]

$6 + 7 i$
$- 7 + 6 i$
$7 + 6 i$
$- 6 + 7 i$

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(4)

$The initial position of point is Z_{0} = 1 + 2 i$

$∴$ $Z_{1} = (1 + 5) + (2 + 3) i = 6 + 5 i$

$Now Z_{1} is moved through a distance of \sqrt{2} units in the direction \hat{i} + \hat{j} (i.e. by 1 + i)$

$∴$ $It becomes Z_{1}' = Z_{1} + (1 + i) = 7 + 6 i$

$Now O Z_{1}' is rotated through an angle \frac{π}{2} in anticlockwise direction,$

$therefore Z_{2} = i Z_{1}' = - 6 + 7 i .$

Q 8 :

If $| z |$ $= 1$ and $z \neq \pm 1$ , then all the values of $\frac{z}{1 - z^{2}}$ lie on [2007]

a line not passing through the origin
$| z |$ $= \sqrt{2}$
the x-axis
the y-axis

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(4)

$Given: | z | = 1 and z \neq \pm 1$

$To find the locus of ω = \frac{z}{1 - z^{2}}$

$Now, ω = \frac{z}{1 - z^{2}} = \frac{z}{z \bar{z} - z^{2}}$ $[∵ | z | = 1 \Rightarrow | z |^{2} = z \bar{z} = 1]$

$= \frac{1}{\bar{z} - z} = purely imaginary number$

$∴$ $ω must lie on y-axis.$

Q 9 :

A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Argand plane is [2007]

$3 e^{i π / 4} + 4 i$
$(3 - 4 i) e^{i π / 4}$
$(4 + 3 i) e^{i π / 4}$
$(3 + 4 i) e^{i π / 4}$

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(4)

$\vec{O P} = \vec{O A} + \vec{A P}$

$\Rightarrow \vec{O P} = \vec{O A} + \vec{O B}$

$\Rightarrow \vec{O P} = 3 e^{i π / 4} + 4 e^{i (π / 2 + π / 4)}$

$= 3 e^{i π / 4} + 4 e^{i π / 2} e^{i π / 4}$

$= 3 e^{i π / 4} + 4 i e^{i π / 4}$

$= e^{i π / 4} (3 + 4 i)$

Q 10 :

If $a, b, c$ are integers, not all simultaneously equal and $ω$ is a cube root of unity $(ω \neq 1)$ , then minimum value of $| a + b ω + c ω^{2} |$ is [2005]

$0$
$1$
$\frac{\sqrt{3}}{2}$
$\frac{1}{2}$

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(2)

$Given that a, b, c are integers not all equal and ω is cube root of unity \neq 1,$

$then$ $| a + b ω + c ω^{2} |$

$=$ $| a + b (\frac{- 1 + i \sqrt{3}}{2}) + c (\frac{- 1 - i \sqrt{3}}{2}) |$

$=$ $| (\frac{2 a - b - c}{2}) + i (\frac{b \sqrt{3} - c \sqrt{3}}{2}) |$

$= \frac{1}{2} \sqrt{{(2 a - b - c)}^{2} + 3 {(b - c)}^{2}}$

$= \sqrt{\frac{1}{2} [{(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}]}$

$R.H.S. will be minimum when a = b = c, but according to the question, we cannot take a = b = c .$

$∴$ $The minimum value is obtained when any two are zero and third is a minimum magnitude integer i.e. 1.$

$∴$ $b = c = 0, a = 1 \Rightarrow gives us the minimum value = 1 .$

Q 11 :

The locus of $z$ which lies in shaded region (excluding the boundaries) is best represented by [2005]

$z : | z + 1 | > 2 and | \arg (z + 1) | < \frac{π}{4}$
$z : | z - 1 | > 2 and | \arg (z - 1) | < \frac{π}{4}$
$z : | z + 1 | < 2 and | \arg (z + 1) | < \frac{π}{2}$
$z : | z - 1 | < 2 and | \arg (z + 1) | < \frac{π}{2}$

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(1)

$In the figure, we see that$

$A B = A C = A D = 2$

$∴$ $B C D is an arc of a circle with centre at A and radius 2 . Shaded region is exterior part of this sector A B C D A .$

$∴$ $For any point represented by z on arc B C D we should have$

$| z - (- 1) |$ $= 2$

$and for shaded region, | z + 1 | > 2 . . . (i)$

$For shaded region, we also have$

$- \frac{π}{4} < \arg (z + 1) < \frac{π}{4}$

$or | \arg (z + 1) | < \frac{π}{4} . . . (i i)$

$From (i) and (ii), we get (1) is the correct option.$

Q 12 :

If $ω (\neq 1)$ be a cube root of unity and ${(1 + ω^{2})}^{n} = {(1 + ω^{4})}^{n}$ , then the least positive value of $n$ is [2004]

2
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5
6

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(2)

${(1 + ω^{2})}^{n} = {(1 + ω^{4})}^{n}$

$\Rightarrow {(- ω)}^{n} = {(1 + ω)}^{n} = {(- ω^{2})}^{n}$

$\Rightarrow ω^{n} = 1 \Rightarrow n = 3$

Q 13 :

If $| z | = 1$ and $ω = \frac{z - 1}{z + 1} (where z \neq - 1)$ , then $Re (ω)$ is [2003]

$0$
$- \frac{1}{| z + 1 |^{2}}$
$| \frac{z}{z + 1} |$ $\cdot \frac{1}{| z + 1 |^{2}}$
$\frac{\sqrt{2}}{| z + 1 |^{2}}$

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(1)

$Given that | z | = 1 and ω = \frac{z - 1}{z + 1} (z \neq - 1)$

$Now we know that z \bar{z} = | z |^{2}$

$\Rightarrow z \bar{z} = 1 (for | z | = 1)$

$∴$ $ω = (\frac{z - 1}{z + 1}) \times (\frac{\bar{z} + 1}{\bar{z} + 1})$

$= \frac{z \bar{z} + z - \bar{z} - 1}{z \bar{z} + z + \bar{z} + 1}$ $= \frac{2 i y}{2 + 2 x}$

$[∵ z \bar{z} = 1 and taking z = x + i y so that z + \bar{z} = 2 x and z - \bar{z} = 2 i y]$

$\Rightarrow Re (ω) = 0$

Q 14 :

Let $ω = - \frac{1}{2} + i \frac{\sqrt{3}}{2}$ , then the value of the determinant $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$ is [2002]

$3 ω$
$3 ω (ω - 1)$
$3 ω^{2}$
$3 ω (1 - ω)$

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(2)

$Applying R_{1} \to R_{1} + R_{2} + R_{3}, we get$

$| \begin{matrix} 1 & 1 & 0 \\ 1 & - 1 - ω & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$ $=$ $| \begin{matrix} 3 & 0 & 0 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$

$= 3 [- ω - 1 - ω]$ $= 3 (ω^{2} - ω)$

Q 15 :

The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}$ are the vertices of a triangle which is [2001]

of area zero
right-angled isosceles
equilateral
obtuse-angled isosceles

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(3)

$\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}$

$\Rightarrow \arg (\frac{z_{1} - z_{3}}{z_{2} - z_{3}}) = \arg (\frac{1 - i \sqrt{3}}{2})$

$\Rightarrow \arg (\cos (- \frac{π}{3}) + i \sin (- \frac{π}{3}))$

$\Rightarrow angle between (z_{1} - z_{3}) and (z_{2} - z_{3}) is 60 ° .$

$and$ $| \frac{z_{1} - z_{3}}{z_{2} - z_{3}} |$ $=$ $| \frac{1 - i \sqrt{3}}{2} |$

$\Rightarrow$ $| \frac{z_{1} - z_{3}}{z_{2} - z_{3}} |$ $= 1 \Rightarrow | z_{1} - z_{3} | = | z_{2} - z_{3} | (Imp Step)$

$\Rightarrow The ∆ with vertices z_{1}, z_{2}, z_{3} is isosceles with vertical angle 60 ° .$

$\Rightarrow Hence rest of the two angles should also be 60 ° each.$

$\Rightarrow Required triangle is an equilateral triangle.$

Q 16 :

Let $z_{1}$ and $z_{2}$ be $n^{th}$ roots of unity which subtend a right angle at the origin. Then $n$ must be of the form [2001]

$4 k + 1$
$4 k + 2$
$4 k + 3$
$4 k$

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(4)

$Let z = {(1)}^{1 / n} = {(\cos 2 k π + i \sin 2 k π)}^{1 / n}$

$\Rightarrow z = \cos \frac{2 k π}{n} + i \sin \frac{2 k π}{n}, k = 0, 1, 2, \dots, n - 1 .$

$Let z_{1} = \cos (\frac{2 k_{1} π}{n}) + i \sin (\frac{2 k_{1} π}{n})$

$and z_{2} = \cos (\frac{2 k_{2} π}{n}) + i \sin (\frac{2 k_{2} π}{n})$

$be the two values of z such that they subtend right angle at origin.$

$∴$ $\frac{2 k_{1} π}{n} - \frac{2 k_{2} π}{n} = \pm \frac{π}{2}$ $\Rightarrow 4 (k_{1} - k_{2}) = \pm n$

$As k_{1} and k_{2} are integers and k_{1} \neq k_{2},$

$∴$ $n = 4 k, k \in I .$

Q 17 :

For a complex number $z$ , let $Re (z)$ denote the real part of $z$ . Let $S$ be the set of all complex numbers $z$ satisfying $z^{4} - | z |^{4} = 4 i z^{2},$ where $i = \sqrt{- 1}$ . Then the minimum possible value of $| z_{1} - z_{2} |^{2}$ , where $z_{1}, z_{2} \in S$ with $Re (z_{1}) > 0$ and $Re (z_{2}) < 0$ , is _______ [2020]

(8)

$Let z = x + i y$

$z^{4} - | z |^{4} = 4 i z^{2}$

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

$\Rightarrow z = 0 or z^{2} - {\bar{z}}^{2} = 4 i$

$\Rightarrow 4 i x y = 4 i \Rightarrow x y = 1$

$Locus of z is a rectangular hyperbola x y = 1$

$Given that Re (z_{1}) > 0 and Re (z_{2}) < 0$

$∴$ $| z_{1} - z_{2} |_{\min} = \sqrt{{(1 + 1)}^{2} + {(1 + 1)}^{2}} = \sqrt{8}$

$\Rightarrow | z_{1} - z_{2} |_{\min}^{2} = 8$

Q 18 :

Let $ω \neq 1$ be a cube root of unity. Then the minimum of the set ${| a + b ω + c ω^{2} |^{2} : a, b, c distinct non-zero integers}$ equals ______. [2019]

(3)

$a, b, c are distinct non-zero integers$

$Min. value of | a + b ω + c ω^{2} |^{2} is to be found$ $| a + b ω + c ω^{2} |^{2}$

$= | a + b (\frac{- 1 + i \sqrt{3}}{2}) + c (\frac{- 1 - i \sqrt{3}}{2}) |^{2}$

$= | \frac{1}{2} (2 a - b - c) + \frac{i \sqrt{3}}{2} (b - c) |^{2}$

$= \frac{1}{4} {(2 a - b - c)}^{2} + \frac{3}{4} {(b - c)}^{2}$

$= \frac{1}{4} (4 a^{2} + b^{2} + c^{2} - 4 a b + 2 b c - 4 a c + 3 b^{2} + 3 c^{2} - 6 b c)$

$= a^{2} + b^{2} + c^{2} - a b - b c - c a$

$= \frac{1}{2} [{(a - b)}^{2} + {(b - c)}^{2} + {(c - a)}^{2}]$

$For minimum value, let us consider a = 3, b = 2, c = 1$

$∴$ $minimum value = \frac{1}{2} [1 + 1 + 4] = \frac{6}{2} = 3$

Q 19 :

For a non-zero complex number $z$ , let $\arg (z)$ denote the principal argument of $z$ , with $- π < \arg (z) \leq π$ . Let $ω$ be the cube root of unity for which $0 < \arg (ω) < π$ . Let $α = \arg (\sum_{n = 1}^{2025} {(- ω)}^{n}) .$ Then the value of $\frac{3 α}{π}$ is _________. [2025]

(-2)

$α = \arg (- ω + ω^{2} - ω^{3} + \dots + {(- ω)}^{2025})$

$α = \arg (\frac{- ω ({(- ω)}^{2025} - 1)}{- ω - 1})$ $\Rightarrow α = \arg (2 ω^{2})$

$\Rightarrow α = \frac{- 2 π}{3} \Rightarrow \frac{3 α}{π} = - 2$

Q 20 :

Let $ℝ$ denote the set of all real numbers. Let $z_{1} = 1 + 2 i$ and $z_{2} = 3 i$ be two complex numbers, where $i = \sqrt{- 1}$ .

Let $S = {(x, y) \in ℝ \times ℝ : | x + i y - z_{1} | = 2 | x + i y - z_{2} |} .$ Then which of the following statements is (are) TRUE? [2025]

$S$ is a circle with centre $(- \frac{1}{3}, \frac{10}{3})$
$S$ is a circle with centre $(\frac{1}{3}, \frac{8}{3})$
$S$ is a circle with radius $\frac{\sqrt{2}}{3}$
$S$ is a circle with radius $\frac{2 \sqrt{2}}{3}$

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Select one or more options

(1, 4)

$We have, | x + i y - 1 - 2 i | = 2 | x + i y - 3 i |$

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

$\Rightarrow 3 x^{2} + 3 y^{2} + 2 x - 20 y + 31 = 0$

$\Rightarrow x^{2} + y^{2} + \frac{2 x}{3} - \frac{20 y}{3} + \frac{31}{3} = 0$

$∴$ $S is a circle with centre (- \frac{1}{3}, \frac{10}{3}) and radius$

$= \sqrt{\frac{1}{9} + \frac{100}{9} - \frac{31}{3}} = \sqrt{\frac{8}{9}} = \frac{2 \sqrt{2}}{3}$

Q 21 :

Let $a, b \in ℝ$ and $a^{2} + b^{2} \neq 0$ . Suppose $S = {z \in C : z = \frac{1}{a + i b t}, t \in ℝ^{+}, t \neq 0},$ where $i = \sqrt{- 1}$ . If $z = x + i y$ and $z \in S$ , then $(x, y)$ lies on [2016]

the circle with radius $\frac{1}{2 a}$ and centre $(\frac{1}{2 a}, 0)$ for $a > 0, b \neq 0$
the circle with radius $- \frac{1}{2 a}$ and centre $(- \frac{1}{2 a}, 0)$ for $a < 0, b \neq 0$
the $x$ -axis for $a \neq 0, b = 0$
the $y$ -axis for $a = 0, b \neq 0$

1

2

3

4

Select one or more options

(1, 3, 4)

$z = \frac{1}{a + i b t} = x + i y$

$\Rightarrow x + i y = \frac{a - i b t}{a^{2} + b^{2} t^{2}} \Rightarrow x = \frac{a}{a^{2} + b^{2} t^{2}}, y = \frac{- b t}{a^{2} + b^{2} t^{2}}$

$\Rightarrow x^{2} + y^{2} = \frac{1}{a^{2} + b^{2} t^{2}} = \frac{x}{a} \Rightarrow x^{2} + y^{2} - \frac{x}{a} = 0$

$∴$ $Locus of z is a circle with centre (\frac{1}{2 a}, 0) and radius \frac{1}{2 | a |}$

$irrespective of' a' being + v e or - v e$

$Also for b = 0, a \neq 0, we get y = 0$

$∴$ $locus is x-axis$

$and for a = 0, b \neq 0, we get x = 0$

$∴$ $locus is y-axis.$

$Hence, 1, 3, 4 are the correct options.$

Q 22 :

Let $w = \frac{\sqrt{3} + i}{2}$ and $P = {w^{n} : n = 1, 2, 3, \dots}$ . Further, $H_{1} = {z \in ℂ : Re (z) > \frac{1}{2}} and H_{2} = {z \in ℂ : Re (z) < - \frac{1}{2}},$

where $c$ is the set of all complex numbers. If $z_{1} \in P \cap H_{1}$ , $z_{2} \in P \cap H_{2}$ and $O$ represents the origin, then $∠ z_{1} O z_{2} =$ [2013]

$\frac{π}{2}$
$\frac{π}{6}$
$\frac{2 π}{3}$
$\frac{5 π}{6}$

1

2

3

4

Select one or more options

(3, 4)

$We have w = \frac{\sqrt{3} + i}{2} = \cos \frac{π}{6} + i \sin \frac{π}{6}$

$\Rightarrow w^{n} = \cos \frac{n π}{6} + i \sin \frac{n π}{6}$

$∴$ $P contains all those points which lie on unit circle and have arguments \frac{π}{6}, \frac{2 π}{6}, \frac{3 π}{6} and so on.$

$Since, z_{1} \in P \cap H_{1} and z_{2} \in P \cap H_{2}, therefore z_{1} and z_{2} can have possible positions as shown in the figure.$

$∴$ $∠ z_{1} O z_{2} can be \frac{2 π}{3} or \frac{5 π}{6}$

Q 23 :

Match the statements in Column I with those in Column II. [2010]

[Note: Here $z$ takes values in the complex plane and $Im z$ and $Re z$ denote, respectively, the imaginary part and the real part of $z$ .]

Column I Column II

(A) The set of points $z$ satisfying
$| z - i | z | | = | z + i | z | |$
is contained in or equal to (p) an ellipse with eccentricity $\frac{4}{5}$

(B) The set of points $z$ satisfying
$| z + 4 | + | z - 4 | = 10$
is contained in or equal to (q) the set of points $z$ satisfying $Im z = 0$

(C) If $| w | = 2$ , then the set of points
$z = w - \frac{1}{w}$
is contained in or equal to (r) the set of points $z$ satisfying $| Im z | \leq 1$

(D) If $| w | = 1$ , then the set of points
$z = w + \frac{1}{w}$
is contained in or equal to (s) the set of points $z$ satisfying $| Re z | < 2$

(t) the set of points $z$ satisfying $| z | \leq 3$

$(A) \to (q, r), B \to (p), C \to (p, s, t), D \to (q, r, s, t)$
$(A) \to (q, r, s, t), B \to (p), C \to (p, s, t), D \to (q, r)$
$(A) \to (q, r, s, t), B \to (p, s, t), C \to (p), D \to (q, r)$
$(A) \to (p), B \to (p, s, t), C \to (q, r, s, t), D \to (q, r)$

1

2

3

4

(1)

$(A) \to (q, r)$

$| z - i | z | | = | z + i | z | |$

$\Rightarrow z is equidistant from two points (0, | z |) and (0, - | z |), which lie on imaginary axis.$

$∴$ $z must lie on real axis \Rightarrow Im (z) = 0 . Also | I_{m} (z) | \leq 1$

$(B) \to p$

$Sum of distances of z from two points (- 4, 0) and (4, 0) is 10 which is greater than 8 .$

$∴$ $z traces an ellipse with 2 a = 10 and 2 a e = 8$

$\Rightarrow e = \frac{4}{5}$

$(C) \to (p, s, t)$

$Let ω = 2 (\cos θ + i \sin θ), then$

$z = ω - \frac{1}{ω} = 2 (\cos θ + i \sin θ) - \frac{1}{2} (\cos θ - i \sin θ)$

$\Rightarrow x + i y = \frac{3}{2} \cos θ + i \frac{5}{2} \sin θ$

$Here,$ $| z |$ $= \sqrt{\frac{9 + 25}{4}} = \sqrt{\frac{34}{4}} \leq 3 and$ $| R_{e} (z) |$ $\leq 2$

$Also x = \frac{3}{2} \cos θ, y = \frac{5}{2} \sin θ \Rightarrow \frac{4 x^{2}}{9} + \frac{4 y^{2}}{25} = 1$

$Which is an ellipse with e = \sqrt{1 - \frac{9}{25}} = \frac{4}{5}$

$(D) \to (q, r, s, t)$

$Let ω = \cos θ + i \sin θ then z = 2 \cos θ \Rightarrow Im (z) = 0$

$Also$ $| z | \leq 3 and | Im (z) | \leq 1, | R_{e} (z) | \leq 2$

Topic Question Set

Q 3 :

Let complex numbers $α$ and $\frac{1}{\bar{α}}$ lie on circles ${(x - x_{0})}^{2} + {(y - y_{0})}^{2} = r^{2}$ and ${(x - x_{0})}^{2} + {(y - y_{0})}^{2} = 4 r^{2}$ respectively. If $z_{0} = x_{0} + i y_{0}$ satisfies the equation $2$ $| z_{0} |^{2}$ $= r^{2} + 2$ , then $| α |$ $=$ [2013]

Q 4 :

Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^{2} + z + 1$ is real. Then a cannot take the value [2012]

Q 5 :

Let $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation: $z {\bar{z}}^{3} + \bar{z} z^{3} = 350$ is [2009]

Q 6 :

Let $z = \cos θ + i \sin θ$ . Then the value of $\sum_{m = 1}^{15} Im (z^{2 m - 1})$ at $θ = 2 °$ is [2009]

Q 8 :

If $| z |$ $= 1$ and $z \neq \pm 1$ , then all the values of $\frac{z}{1 - z^{2}}$ lie on [2007]

Q 9 :

A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Argand plane is [2007]

Q 10 :

If $a, b, c$ are integers, not all simultaneously equal and $ω$ is a cube root of unity $(ω \neq 1)$ , then minimum value of $| a + b ω + c ω^{2} |$ is [2005]

Q 11 :

The locus of $z$ which lies in shaded region (excluding the boundaries) is best represented by [2005]

Q 12 :

If $ω (\neq 1)$ be a cube root of unity and ${(1 + ω^{2})}^{n} = {(1 + ω^{4})}^{n}$ , then the least positive value of $n$ is [2004]

Q 13 :

If $| z | = 1$ and $ω = \frac{z - 1}{z + 1} (where z \neq - 1)$ , then $Re (ω)$ is [2003]

Q 14 :

Let $ω = - \frac{1}{2} + i \frac{\sqrt{3}}{2}$ , then the value of the determinant $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$ is [2002]

Q 15 :

The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}$ are the vertices of a triangle which is [2001]

Q 16 :

Let $z_{1}$ and $z_{2}$ be $n^{th}$ roots of unity which subtend a right angle at the origin. Then $n$ must be of the form [2001]

Q 18 :

Let $ω \neq 1$ be a cube root of unity. Then the minimum of the set ${| a + b ω + c ω^{2} |^{2} : a, b, c distinct non-zero integers}$ equals ______. [2019]

Q 20 :

Let $ℝ$ denote the set of all real numbers. Let $z_{1} = 1 + 2 i$ and $z_{2} = 3 i$ be two complex numbers, where $i = \sqrt{- 1}$ .

Let $S = {(x, y) \in ℝ \times ℝ : | x + i y - z_{1} | = 2 | x + i y - z_{2} |} .$ Then which of the following statements is (are) TRUE? [2025]

Q 21 :

Let $a, b \in ℝ$ and $a^{2} + b^{2} \neq 0$ . Suppose $S = {z \in C : z = \frac{1}{a + i b t}, t \in ℝ^{+}, t \neq 0},$ where $i = \sqrt{- 1}$ . If $z = x + i y$ and $z \in S$ , then $(x, y)$ lies on [2016]

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	Column I		Column II
(A)	The set of points $z$ satisfying $\| z - i \| z \| \| = \| z + i \| z \| \|$ is contained in or equal to	(p)	an ellipse with eccentricity $\frac{4}{5}$
(B)	The set of points $z$ satisfying $\| z + 4 \| + \| z - 4 \| = 10$ is contained in or equal to	(q)	the set of points $z$ satisfying $Im z = 0$
(C)	If $\| w \| = 2$ , then the set of points $z = w - \frac{1}{w}$ is contained in or equal to	(r)	the set of points $z$ satisfying $\| Im z \| \leq 1$
(D)	If $\| w \| = 1$ , then the set of points $z = w + \frac{1}{w}$ is contained in or equal to	(s)	the set of points $z$ satisfying $\| Re z \| < 2$
		(t)	the set of points $z$ satisfying $\| z \| \leq 3$

Topic Question Set

Q 2 : Let S be the set of all complex numbers z satisfying |z-2+i|≥5. If the complex number z0 is such that 1|z0-1| is the maximum of the set {1|z-1|:z∈S}, then the principal argument of 4-z0-z0¯z0-z0¯+2i is [2019]

Q 3 : Let complex numbers α and 1α¯ lie on circles (x-x0)2+(y-y0)2=r2 and (x-x0)2+(y-y0)2=4r2 respectively. If z0=x0+iy0 satisfies the equation 2|z0|2=r2+2, then |α|= [2013]

Q 4 : Let z be a complex number such that the imaginary part of z is non-zero and a=z2+z+1 is real. Then a cannot take the value [2012]

Q 5 : Let z=x+iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation: zz¯ 3+z¯z3=350 is [2009]

Q 6 : Let z=cosθ+isinθ. Then the value of ∑m=115Im(z2m-1) at θ=2° is [2009]

Q 8 : If |z|=1 and z≠±1, then all the values of z1-z2 lie on [2007]

Q 9 : A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Argand plane is [2007]

Q 10 : If a,b,c are integers, not all simultaneously equal and ω is a cube root of unity (ω≠1), then minimum value of |a+bω+cω2| is [2005]

Q 11 : The locus of z which lies in shaded region (excluding the boundaries) is best represented by [2005]

Q 12 : If ω(≠1) be a cube root of unity and (1+ω2)n=(1+ω4)n, then the least positive value of n is [2004]

Q 13 : If |z|=1 and ω=z-1z+1 (where z≠-1), then Re(ω) is [2003]

Q 14 : Let ω=-12+i32, then the value of the determinant |1111-1-ω2ω21ω2ω4| is [2002]

Q 15 : The complex numbers z1,z2 and z3 satisfying z1-z3z2-z3=1-i32 are the vertices of a triangle which is [2001]

Q 16 : Let z1 and z2 be nth roots of unity which subtend a right angle at the origin. Then n must be of the form [2001]

Q 17 : For a complex number z, let Re(z) denote the real part of z. Let S be the set of all complex numbers z satisfying z4-|z|4=4iz2, where i=-1. Then the minimum possible value of |z1-z2|2, where z1,z2∈S with Re(z1)>0 and Re(z2)<0, is _______ [2020]

Q 18 : Let ω≠1 be a cube root of unity. Then the minimum of the set {|a+bω+cω2|2:a,b,c distinct non-zero integers} equals ______. [2019]

Q 19 : For a non-zero complex number z, let arg(z) denote the principal argument of z, with -π<arg(z)≤π. Let ω be the cube root of unity for which 0<arg(ω)<π. Let α=arg(∑n=12025(-ω)n). Then the value of 3απ is _________. [2025]

Q 20 : Let ℝ denote the set of all real numbers. Let z1=1+2i and z2=3i be two complex numbers, where i=-1. Let S={(x,y)∈ℝ×ℝ:|x+iy-z1|=2|x+iy-z2|}. Then which of the following statements is (are) TRUE? [2025]

Q 21 : Let a,b∈ℝ and a2+b2≠0. Suppose S={z∈C:z=1a+ibt, t∈ℝ+, t≠0}, where i=-1. If z=x+iy and z∈S, then (x,y) lies on [2016]

Q 22 : Let w=3+i2 and P={wn:n=1,2,3,…}. Further, H1={z∈ℂ:Re(z)>12} and H2={z∈ℂ:Re(z)<-12}, where c is the set of all complex numbers. If z1∈P∩H1, z2∈P∩H2 and O represents the origin, then ∠z1Oz2= [2013]

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Q 4 :

Let $z$ be a complex number such that the imaginary part of $z$ is non-zero and $a = z^{2} + z + 1$ is real. Then a cannot take the value [2012]

Q 5 :

Let $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation: $z {\bar{z}}^{3} + \bar{z} z^{3} = 350$ is [2009]

Q 6 :

Let $z = \cos θ + i \sin θ$ . Then the value of $\sum_{m = 1}^{15} Im (z^{2 m - 1})$ at $θ = 2 °$ is [2009]

Q 8 :

If $| z |$ $= 1$ and $z \neq \pm 1$ , then all the values of $\frac{z}{1 - z^{2}}$ lie on [2007]

Q 9 :

A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Argand plane is [2007]

Q 10 :

If $a, b, c$ are integers, not all simultaneously equal and $ω$ is a cube root of unity $(ω \neq 1)$ , then minimum value of $| a + b ω + c ω^{2} |$ is [2005]

Q 11 :

The locus of $z$ which lies in shaded region (excluding the boundaries) is best represented by [2005]

Q 12 :

If $ω (\neq 1)$ be a cube root of unity and ${(1 + ω^{2})}^{n} = {(1 + ω^{4})}^{n}$ , then the least positive value of $n$ is [2004]

Q 13 :

If $| z | = 1$ and $ω = \frac{z - 1}{z + 1} (where z \neq - 1)$ , then $Re (ω)$ is [2003]

Q 14 :

Let $ω = - \frac{1}{2} + i \frac{\sqrt{3}}{2}$ , then the value of the determinant $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$ is [2002]

Q 15 :

The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z_{1} - z_{3}}{z_{2} - z_{3}} = \frac{1 - i \sqrt{3}}{2}$ are the vertices of a triangle which is [2001]

Q 16 :

Let $z_{1}$ and $z_{2}$ be $n^{th}$ roots of unity which subtend a right angle at the origin. Then $n$ must be of the form [2001]

Q 18 :

Let $ω \neq 1$ be a cube root of unity. Then the minimum of the set ${| a + b ω + c ω^{2} |^{2} : a, b, c distinct non-zero integers}$ equals ______. [2019]

Q 20 :

Let $ℝ$ denote the set of all real numbers. Let $z_{1} = 1 + 2 i$ and $z_{2} = 3 i$ be two complex numbers, where $i = \sqrt{- 1}$ .

Let $S = {(x, y) \in ℝ \times ℝ : | x + i y - z_{1} | = 2 | x + i y - z_{2} |} .$ Then which of the following statements is (are) TRUE? [2025]

Q 21 :

Let $a, b \in ℝ$ and $a^{2} + b^{2} \neq 0$ . Suppose $S = {z \in C : z = \frac{1}{a + i b t}, t \in ℝ^{+}, t \neq 0},$ where $i = \sqrt{- 1}$ . If $z = x + i y$ and $z \in S$ , then $(x, y)$ lies on [2016]