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The equation $$\arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4}$$ represents a circle with :
Options:
[{"identifier": "A", "content": "centre at (0, $$-$$1) and radius $$\\sqrt 2 $$"}, {"identifier": "B", "content": "centre at (0, 1) and radius $$\\sqrt 2 $$"}, {"identifier": "C", "content": "centre (0, ... | ["B"]
Explanation:
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266796/exam_images/f29lrw36fjmxmetgizum.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265873/exam_images/uw851jxzxx4ohkmsafk9.webp"><im... |
A point z moves in the complex plane such that $$\arg \left( {{{z - 2} \over {z + 2}}} \right) = {\pi \over 4}$$, then the minimum value of $${\left| {z - 9\sqrt 2 - 2i} \right|^2}$$ is equal to _______________.
Options:
[] | 98
Explanation:
Let $$z = x + iy$$<br><br>$$\arg \left( {{{x - 2 + iy} \over {x + 2 + iy}}} \right) = {\pi \over 4}$$<br><br>$$\arg (x - 2 + iy) - \arg (x + 2 + iy) = {\pi \over 4}$$<br><br>$${\tan ^{ - 1}}\left( {{y \over {x - 2}}} \right) - {\tan ^{ - 1}}\left( {{y \over {x + 2}}} \right) = {\pi \over 4}$$<br><br... |
<p>Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z $$-$$ 1) $$-$$ arg(z + 1) = $${\pi \over 4}$$ intersect :</p>
Options:
[{"identifier": "A", "content": "exactly at one point."}, {"identifier": "B", "content": "exactly at two points."}, {"identifier": "C", "content": "now... | ["C"]
Explanation:
<p>Let $$z = x + iy$$</p>
<p>$$\therefore$$ $$|z| = \sqrt {{x^2} + {y^2}} $$</p>
<p>Given, $$|z| = 3$$</p>
<p>$$\therefore$$ $$\sqrt {{x^2} + {y^2}} = 3$$</p>
<p>$$ \Rightarrow {x^2} + {y^2} = 9 = {3^2}$$</p>
<p>This represent a circle with center at (0, 0) and radius = 3</p>
<p>Now, given</p>
<p>$... |
<p>The number of elements in the set {z = a + ib $$\in$$ C : a, b $$\in$$ Z and 1 < | z $$-$$ 3 + 2i | < 4} is __________.</p>
Options:
[] | 40
Explanation:
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lc8e8f52/d1800b00-4e42-4501-bb81-7122062be06f/25ad5460-8716-11ed-92bb-e57e64e1a06d/file-1lc8e8f53.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lc8e8f52/d1800b00-4e42-4501-bb81-7122062be06f/25ad5460-8716-11ed-92b... |
<p>The number of points of intersection of <br/><br/>$$|z - (4 + 3i)| = 2$$ and $$|z| + |z - 4| = 6$$, z $$\in$$ C, is :</p>
Options:
[{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "3"}] | ["C"]
Explanation:
<p>$${C_1}:|z - (4 + 3i)| = 2$$ and $${C_2}:|z| + |z - 4| = 6$$, $$z \in C$$</p>
<p>C<sub>1</sub> represents a circle with centre (4, 3) and radius 2 and C<sub>2</sub> represents a ellipse with focii at (0, 0) and (4, 0) and length of major axis = 6, and length of semi-major axis 2$$\sqrt5$$ and (4,... |
<p>Let O be the origin and A be the point $${z_1} = 1 + 2i$$. If B is the point $${z_2}$$, $${\mathop{\rm Re}\nolimits} ({z_2}) < 0$$, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?</p>
Options:
[{"identifier": "A", "content": "$$\\arg {z_2} = \\pi... | ["D"]
Explanation:
<p> <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7nbq29i/e583a584-7e22-4939-89d9-b1f38618bbd4/01c10960-2c4f-11ed-9dc0-a1792fcc650d/file-1l7nbq29j.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7nbq29i/e583a584-7e22-4939-89d9-b1f38618bbd4/01c10960-2c4f-1... |
<p>Let $$\mathrm{S}=\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z+i|=|z-1|\}$$. Then the set of all values of $$x$$, for which $$w=2 x+i y \in \mathrm{S}$$ for some $$y \in \mathbb{R}$$, is :</p>
Options:
[{"identifier": "A", "content": "$$\\left(-\\sqrt{2}, \\frac{1}{2 \\sqrt{2}}\\right]$$"}, {"identifier": "B", "content": "... | ["B"]
Explanation:
$S:\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z-i|=|z-1|\}$
<br><br>
$$
|z-1+i| \geq|z|
$$<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lc8c3waq/4ba6cc0d-ceab-4ef1-baed-c4899ebff5b2/d5a07220-870d-11ed-ae18-7336d1cc7e9d/file-1lc8c3war.png?format=png" data-orsrc="https://app-conte... |
<p>Let $$a,b$$ be two real numbers such that $$ab < 0$$. IF the complex number $$\frac{1+ai}{b+i}$$ is of unit modulus and $$a+ib$$ lies on the circle $$|z-1|=|2z|$$, then a possible value of $$\frac{1+[a]}{4b}$$, where $$[t]$$ is greatest integer function, is :</p>
Options:
[{"identifier": "A", "content": "$\\left... | ["C"]
Explanation:
$\begin{aligned} & \left|\frac{1+a i}{b+i}\right|=1 \\\\ & |1+i a|=|b+i| \\\\ & a^2+1=b^2+1 \Rightarrow \mathrm{a}=\pm \mathrm{b} \Rightarrow \mathrm{b}=-\mathrm{a} \quad \text { as } \mathrm{ab}<0 \\\\ & (\mathrm{a} +\mathrm{ib}) \text { lies on }|z-1|=|2 z| \\\\ & |a+i b-1|=2|a+i b| \\\\ & (a-1)^2... |
<p>If the center and radius of the circle $$\left| {{{z - 2} \over {z - 3}}} \right| = 2$$ are respectively $$(\alpha,\beta)$$ and $$\gamma$$, then $$3(\alpha+\beta+\gamma)$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "12"}, {"identifier": "B", "content": "10"}, {"identifier": "C", "content": "11"}, {"... | ["A"]
Explanation:
$$\left| {{{z - 2} \over {z - 3}}} \right| = 2$$
<br/><br/>$$
\begin{aligned}
& \sqrt{(x-2)^2+y^2}=2 \sqrt{(x-3)^2+y^2} \\\\
& \Rightarrow x^2+y^2-4 x+4=4 x^2+4 y^2-24 x+36 \\\\
& \Rightarrow 3 x^2+3 y^2-20 x+32=0 \\\\
& \Rightarrow x^2+y^2-\frac{20}{3} \mathrm{x}+\frac{32}{3}=0 \\\\
& \Rightarrow (... |
<p>For all $$z \in C$$ on the curve $$C_{1}:|z|=4$$, let the locus of the point $$z+\frac{1}{z}$$ be the curve $$\mathrm{C}_{2}$$. Then :</p>
Options:
[{"identifier": "A", "content": "the curves $$C_{1}$$ and $$C_{2}$$ intersect at 4 points"}, {"identifier": "B", "content": "the curve $$C_{2}$$ lies inside $$C_{1}$$"}... | ["A"]
Explanation:
Let $\mathrm{w}=\mathrm{z}+\frac{1}{\mathrm{z}}=4 \mathrm{e}^{\mathrm{i} \theta}+\frac{1}{4} \mathrm{e}^{-\mathrm{i} \theta}$
<br/><br/>$\Rightarrow \mathrm{w}=\frac{17}{4} \cos \theta+\mathrm{i} \frac{15}{4} \sin \theta$
<br/><br/>So locus of $w$ is ellipse $\frac{x^{2}}{\left(\frac{17}{4}\right)... |
<p>Let $$\mathrm{z_1=2+3i}$$ and $$\mathrm{z_2=3+4i}$$. The set $$\mathrm{S = \left\{ {z \in \mathbb{C}:{{\left| {z - {z_1}} \right|}^2} - {{\left| {z - {z_2}} \right|}^2} = {{\left| {{z_1} - {z_2}} \right|}^2}} \right\}}$$ represents a</p>
Options:
[{"identifier": "A", "content": "hyperbola with the length of the tra... | ["D"]
Explanation:
$\left|z-z_{1}\right|^{2}-\left|z-z_{2}\right|^{2}=\left|z_{1}-z_{2}\right|^{2}$
<br/><br/>
$\Rightarrow(x-2)^{2}+(y-3)^{2}-(x-3)^{2}-(y-4)^{2}=1+1$
<br/><br/>
$\Rightarrow-4 x+4+9-6 y-9+6 x-16+8 y=2$
<br/><br/>
$\Rightarrow 2 x+2 y=14$
<br/><br/>
$\Rightarrow x+y=7$ |
<p>Let $$w=z \bar{z}+k_{1} z+k_{2} i z+\lambda(1+i), k_{1}, k_{2} \in \mathbb{R}$$. Let $$\operatorname{Re}(w)=0$$ be the circle $$\mathrm{C}$$ of radius 1 in the first quadrant touching the line $$y=1$$ and the $$y$$-axis. If the curve $$\operatorname{Im}(w)=0$$ intersects $$\mathrm{C}$$ at $$\mathrm{A}$$ and $$\mathr... | 24
Explanation:
Given the expression for $w$ as :
<br/><br/>$$w = z\bar{z} + k_1z + k_2iz + \lambda(1+i), \quad \text{where } k_1, k_2 \in \mathbb{R}.$$
<br/><br/>1. If $w = x+iy$, we can separate this into the real and imaginary parts :
<br/><br/> The real part is: $$\text{Re}(w) = x^2 + y^2 + k_1x - k_2y + \lamb... |
<p>Let $$\mathrm{C}$$ be the circle in the complex plane with centre $$\mathrm{z}_{0}=\frac{1}{2}(1+3 i)$$ and radius $$r=1$$. Let $$\mathrm{z}_{1}=1+\mathrm{i}$$ and the complex number $$z_{2}$$ be outside the circle $$C$$ such that $$\left|z_{1}-z_{0}\right|\left|z_{2}-z_{0}\right|=1$$. If $$z_{0}, z_{1}$$ and $$z_{2... | ["B"]
Explanation:
Given, $$
z_0=\frac{1+3 i}{2}, z_1=(1+i)
$$
<br/><br/>$$
\left|z_1-z_0\right|=\left|\frac{1-i}{2}\right|=\frac{1}{\sqrt{2}}
$$
<br/><br/>$$
\begin{aligned}
& \left|z_1-z_0\right|\left|z_2-z_0\right|=1 \\\\
& \Rightarrow \frac{1}{\sqrt{2}}\left|z_2-z_0\right|=1 \\\\
& \Rightarrow\left|z_2-z_0\right|=... |
<p>For $$\alpha, \beta, z \in \mathbb{C}$$ and $$\lambda > 1$$, if $$\sqrt{\lambda-1}$$ is the radius of the circle $$|z-\alpha|^{2}+|z-\beta|^{2}=2 \lambda$$, then $$|\alpha-\beta|$$ is equal to __________.</p>
Options:
[] | 2
Explanation:
Given equation of circle,
<br/><br/>$$
\begin{aligned}
& \quad|z-\alpha|^2+|z-\beta|^2=2 \lambda \\\\
& \therefore 2 \lambda=|\alpha-\beta|^2 .........(i)
\end{aligned}
$$
<br/><br/>For circle,
<br/><br/>$$
\left|z-z_1\right|^2+\left|z-z_2\right|^2=\left|z_1-z_2\right|^2
$$
<br/><br/>$\begin{array}{lll... |
Let $\mathrm{P}=\{\mathrm{z} \in \mathbb{C}:|z+2-3 i| \leq 1\}$ and $\mathrm{Q}=\{\mathrm{z} \in \mathbb{C}: z(1+i)+\bar{z}(1-i) \leq-8\}$. Let in $\mathrm{P} \cap \mathrm{Q}$, $|z-3+2 i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $\left|z_1\right|^2+2\left|z_2\right|^2=\alpha+\beta \sqrt{2}$, where $\... | 36
Explanation:
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsz1bfzj/da67c7ec-984a-45c9-af11-5958669a974a/42ad13f0-d280-11ee-9b77-fbceb54c8042/file-6y3zli1lsz1bfzk.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsz1bfzj/da67c7ec-984a-45c9-af11-5958669a974a/42ad... |
If $S=\{z \in C:|z-i|=|z+i|=|z-1|\}$, then, $n(S)$ is :
Options:
[{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "0"}] | ["A"]
Explanation:
<p>$$|z-i|=|z+i|=|z-1|$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt1dbkuq/8fa39b20-7b31-407e-a4e3-48710c870daf/c548b320-d3c8-11ee-a50b-bb659a2e1d74/file-1lt1dbkur.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lt1dbkuq/8fa39b20-7b31-407e-a4e3-... |
<p>Let the complex numbers $$\alpha$$ and $$\frac{1}{\bar{\alpha}}$$ lie on the circles $$\left|z-z_0\right|^2=4$$ and $$\left|z-z_0\right|^2=16$$ respectively, where $$z_0=1+i$$. Then, the value of $$100|\alpha|^2$$ is __________.</p>
Options:
[] | 20
Explanation:
<p>$$\begin{aligned}
& \left|z-z_0\right|^2=4 \\
& \Rightarrow\left(\alpha-z_0\right)\left(\bar{\alpha}-\bar{z}_0\right)=4 \\
& \Rightarrow \alpha \bar{\alpha}-\alpha \bar{z}_0-z_0 \bar{\alpha}+\left|z_0\right|^2=4 \\
& \Rightarrow|\alpha|^2-\alpha \bar{z}_0-z_0 \bar{\alpha}=2 \quad\text{......... (1)}... |
<p>Let $$z$$ be a complex number such that the real part of $$\frac{z-2 i}{z+2 i}$$ is zero. Then, the maximum value of $$|z-(6+8 i)|$$ is equal to</p>
Options:
[{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "12"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "$$\\infty$$"... | ["B"]
Explanation:
<p>$$\begin{aligned}
& n=\frac{z-2 i}{z+2 i} \\
& \text { Let } z=x+i y \\
& n=\frac{x+(y-2) i}{x+(y+2) i} \times\left(\frac{x-(y+2) i}{x-(y+2) i}\right) \\
& \operatorname{Re}(n)=\frac{x^2+(y-2)(y+2)}{x^2+(y+2)^2}=0 \\
& \Rightarrow x^2+(y-2)(y+2)=0
\end{aligned}$$</p>
<p>$$\begin{aligned}
& \Right... |
<p>The area (in sq. units) of the region $$S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{lm}(z) \geq 0\}$$ is</p>
Options:
[{"identifier": "A", "content": "$$\\frac{7 \\pi}{4}$$\n"}, {"identifier": "B", "content": "$$\\frac{3 \\pi}{2}$$\n"}, {"identifier": "C", "content": "$$\\frac{... | ["B"]
Explanation:
<p>$$|z-1| \leq 2 \quad \Rightarrow \quad(x-1)^2+y^2=4$$</p>
<p>$$\begin{aligned}
& z+\bar{z}+i(z-\bar{z}) \leq 2 \\
\Rightarrow \quad & x-y \leq 1 \\
& \operatorname{Im}(z) \geq 0 \\
\Rightarrow \quad & y \geq 0
\end{aligned}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/f... |
<p>Let $$S_1=\{z \in \mathbf{C}:|z| \leq 5\}, S_2=\left\{z \in \mathbf{C}: \operatorname{Im}\left(\frac{z+1-\sqrt{3} i}{1-\sqrt{3} i}\right) \geq 0\right\}$$ and $$S_3=\{z \in \mathbf{C}: \operatorname{Re}(z) \geq 0\}$$. Then the area of the region $$S_1 \cap S_2 \cap S_3$$ is :</p>
Options:
[{"identifier": "A", "cont... | ["C"]
Explanation:
<p>$$S_1=\{z \in C:|z| \leq 5\}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwekhn5c/6d7750d9-cc34-4e40-9c38-cefbaf7666fe/a4650700-166f-11ef-8416-25c08f86a011/file-1lwekhn5d.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwekhn5c/6d7750d9-cc34-4... |
z and w are two nonzero complex numbers such that $$\,\left| z \right| = \left| w \right|$$ and Arg z + Arg w =$$\pi $$ then z equals
Options:
[{"identifier": "A", "content": "$$\\overline \\omega $$ "}, {"identifier": "B", "content": "$$ - \\overline \\omega $$ "}, {"identifier": "C", "content": "$$\\omega $$ "}, {... | ["B"]
Explanation:
Let $$\left| z \right| = \left| \omega \right| = r$$
<br><br>$$\therefore$$ $$z = r{e^{i\theta }},\omega = r{e^{i\phi }}$$
<br><br>where $$\,\,\theta + \phi = \pi .$$
<br><br>$$\therefore$$ $$z = r{e^{i\left( {\pi - \phi } \right)}} = r{e^{i\pi }}.$$
$${e^{ - i\phi }} = - r{e^{ - i\phi }} = -... |
If $$z$$ and $$\omega $$ are two non-zero complex numbers such that $$\left| {z\omega } \right| = 1$$ and $$Arg(z) - Arg(\omega ) = {\pi \over 2},$$ then $$\,\overline {z\,} \omega $$ is equal to
Options:
[{"identifier": "A", "content": "$$- i$$"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "... | ["A"]
Explanation:
Given that,
<br>$$\left| {z\omega } \right| = 1$$
<br>$$ \Rightarrow $$ $$\left| z \right|\left| \omega \right|$$ = 1
<br>$$ \Rightarrow $$ $$\left| z \right|$$ = $${1 \over {\left| \omega \right|}}$$
<br><br>and $$Arg(z) - Arg(\omega ) = {\pi \over 2}$$
<br>$$ \Rightarrow $$ $$Arg\left( {{z \ove... |
Let z and w be complex numbers such that $$\overline z + i\overline w = 0$$ and arg zw = $$\pi $$. Then arg z equals :
Options:
[{"identifier": "A", "content": "$${{5\\pi } \\over 4}$$ "}, {"identifier": "B", "content": "$${{\\pi } \\over 2}$$ "}, {"identifier": "C", "content": "$${{3\\pi } \\over 4}$$"}, {"identifi... | ["C"]
Explanation:
Given $$\overline z + i\overline w = 0$$
<br><br>$$ \Rightarrow \overline z = - i\overline w $$
<br><br>$$ \Rightarrow \overline{\overline z} = - \overline {i\overline w } $$
<br><br>$$ \Rightarrow \overline{\overline z} = - \overline i \overline{\overline w} $$
<br><br>$$ \Rightarrow z = ... |
If $${z_1}$$ and $${z_2}$$ are two non-zero complex numbers such that $$\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$$, then arg $${z_1}$$ - arg $${z_2}$$ is equal to :
Options:
[{"identifier": "A", "content": "$${\\pi \\over 2}\\,$$ "}, {"identifier": "B", "content": "$$ - \\pi ... | ["C"]
Explanation:
Given that, $$\,\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$$
<br><br>$$\,\left| {{z_1} + {z_2}} \right|$$ is the vector sum of $${z_1}$$ and $${z_2}$$. So $$\,\left| {{z_1} + {z_2}} \right|$$ should be $$<$$ $$\left| {{z_1}} \right| + \left| {{z_2}} \right|$$... |
If z is a complex number of unit modulus and argument $$\theta $$, then arg $$\left( {{{1 + z} \over {1 + \overline z }}} \right)$$ equals :
Options:
[{"identifier": "A", "content": "$$ - \\theta \\,\\,$$ "}, {"identifier": "B", "content": "$${\\pi \\over 2} - \\theta \\,$$ "}, {"identifier": "C", "content": "$$\\the... | ["C"]
Explanation:
Given $$\,\,\,\,\left| z \right| = 1,\,\,\arg \,z = \theta $$
<br><br>As we know, $$\,\,\,\,\overrightarrow z = {1 \over z}$$
<br><br>$$\therefore$$ $$\,\,\,\,\arg \left( {{{1 + z} \over {1 + \overrightarrow z }}} \right) = \arg \left( {{{1 + z} \over {1 + {1 \over z}}}} \right)$$
<br><br>$$ = \arg... |
If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = $${\pi \over 2}$$
, then :
Options:
[{"identifier": "A", "content": "$$z\\overline w = {{1 - i} \\over {\\sqrt 2 }}$$"}, {"identifier": "B", "content": "$$\\overline z w = i$$"}, {"identifier": "C", "content": "$$z\\overline w = {{ - 1 + i}... | ["D"]
Explanation:
$$\left| {zw} \right| = 1$$<br><br>
$$ \Rightarrow $$ $$\left| z \right|\left| w \right| = 1$$<br><br>
Let $$w = {1 \over r}{e^{i\theta }}$$<br><br>
then z = $$r{e^{i\left( {\theta + {\pi \over 2}} \right)}}$$<br><br>
$$\overline z w = {e^{ - i\left( {\theta + {\pi \over 2}} \right)}}.{e^{i\thet... |
If $${{3 + i\sin \theta } \over {4 - i\cos \theta }}$$, $$\theta $$ $$ \in $$ [0, 2$$\theta $$], is a real number, then an argument of <br/>sin$$\theta $$ + icos$$\theta $$ is :
Options:
[{"identifier": "A", "content": "$$\\pi - {\\tan ^{ - 1}}\\left( {{3 \\over 4}} \\right)$$"}, {"identifier": "B", "content": "$$ - ... | ["D"]
Explanation:
Let z = $${{3 + i\sin \theta } \over {4 - i\cos \theta }}$$
<br><br>= $${{3 + i\sin \theta } \over {4 - i\cos \theta }} \times {{\left( {4 + i\cos \theta } \right)} \over {\left( {4 + i\cos \theta } \right)}}$$
<br><br>= $${{\left( {12 - \sin \theta \cos \theta } \right) + i\left( {4\sin \theta + 3... |
If z<sub>1</sub>
, z<sub>2</sub>
are complex numbers such that
<br/>Re(z<sub>1</sub>) = |z<sub>1</sub> – 1|, Re(z<sub>2</sub>) = |z<sub>2</sub> – 1| , and
<br/>arg(z<sub>1</sub> - z<sub>2</sub>) = $${\pi \over 6}$$, then Im(z<sub>1</sub>
+ z<sub>2</sub>
) is equal to :
Options:
[{"identifier": "A", "content": "$${... | ["D"]
Explanation:
Let $${z_1} = {x_1} + i{y_1},\,{z_2} = {x_2} + i{y_2}$$
<br><br>Given Re(z<sub>1</sub>) = |z<sub>1</sub> – 1|
<br><br>$$ \therefore $$ x<sub>1</sub> = |(x<sub>1</sub> - 1) + iy<sub>1</sub>|
<br><br>$$ \Rightarrow $$ x<sub>1</sub> = $$\sqrt {{{\left( {{x_1} - 1} \right)}^2} + y_1^2} $$
<br><br>$$ \Ri... |
If z and $$\omega$$ are two complex numbers such that $$\left| {z\omega } \right| = 1$$ and $$\arg (z) - \arg (\omega ) = {{3\pi } \over 2}$$, then $$\arg \left( {{{1 - 2\overline z \omega } \over {1 + 3\overline z \omega }}} \right)$$ is :<br/><br/>(Here arg(z) denotes the principal argument of complex number z)
Opti... | ["B"]
Explanation:
As $$\left| {z\omega } \right| = 1$$<br><br>$$\Rightarrow$$ If $$\left| z \right| = r$$, then $$\left| \omega \right| = {1 \over r}$$<br><br>Let $$\arg (z) = \theta $$<br><br>$$\therefore$$ $$\arg (\omega ) = \left( {\theta - {{3\pi } \over 2}} \right)$$<br><br>So, $$z = r{e^{i\theta }}$$<br><br>$... |
Let z<sub>1</sub> and z<sub>2</sub> be two complex numbers such that $$\arg ({z_1} - {z_2}) = {\pi \over 4}$$ and z<sub>1</sub>, z<sub>2</sub> satisfy the equation | z $$-$$ 3 | = Re(z). Then the imaginary part of z<sub>1</sub> + z<sub>2</sub> is equal to ___________.
Options:
[] | 6
Explanation:
Let z<sub>1</sub> = x<sub>1</sub> + iy ; z<sub>2</sub> = x<sub>2</sub> + iy<sub>2</sub><br><br>z<sub>1</sub> $$-$$ z<sub>2</sub> = (x<sub>1</sub> $$-$$ x<sub>2</sub>) + i(y<sub>1</sub> $$-$$ y<sub>2</sub>)<br><br>$$\therefore$$ $$\arg ({z_1} - {z_2}) = {\pi \over 4}$$ $$\Rightarrow$$ $${\tan ^{ - 1}}\l... |
<p>Let $$A = \left\{ {z \in C:\left| {{{z + 1} \over {z - 1}}} \right| < 1} \right\}$$ and $$B = \left\{ {z \in C:\arg \left( {{{z - 1} \over {z + 1}}} \right) = {{2\pi } \over 3}} \right\}$$. Then A $$\cap$$ B is :</p>
Options:
[{"identifier": "A", "content": "a portion of a circle centred at $$\\left( {0, - {1 \\... | ["B"]
Explanation:
$$
\left|\frac{z+1}{z-1}\right|<1 \Rightarrow|z+1|<|z-1| \Rightarrow \operatorname{Re}(z)<0
$$<br><br>
and $\arg \left(\frac{z-1}{z+1}\right)=\frac{2 \pi}{3}$ is a part of circle as shown.<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lc6ndrpb/ccdf08bc-7111-4568-8... |
<p>Let z<sub>1</sub> and z<sub>2</sub> be two complex numbers such that $${\overline z _1} = i{\overline z _2}$$ and $$\arg \left( {{{{z_1}} \over {{{\overline z }_2}}}} \right) = \pi $$. Then :</p>
Options:
[{"identifier": "A", "content": "$$\\arg {z_2} = {\\pi \\over 4}$$"}, {"identifier": "B", "content": "$$\\arg ... | ["C"]
Explanation:
<p>$$\because$$ $${{{z_1}} \over {{z_2}}} = - i \Rightarrow {z_1} = - i{z_2}$$</p>
<p>$$ \Rightarrow \arg ({z_1}) = - {\pi \over 2} + \arg ({z_2})$$ ..... (i)</p>
<p>Also $$\arg ({z_1}) - \arg ({\overline z _2}) = \pi $$</p>
<p>$$ \Rightarrow \arg ({z_1}) + \arg ({z_2}) = \pi $$ ..... (ii)</p>
<... |
<p>Let a circle C in complex plane pass through the points $${z_1} = 3 + 4i$$, $${z_2} = 4 + 3i$$ and $${z_3} = 5i$$. If $$z( \ne {z_1})$$ is a point on C such that the line through z and z<sub>1</sub> is perpendicular to the line through z<sub>2</sub> and z<sub>3</sub>, then $$arg(z)$$ is equal to :</p>
Options:
[{"i... | ["B"]
Explanation:
<p>$${z_1} = 3 + 4i$$, $${z_2} = 4 + 3i$$ and $${z_3} = 5i$$</p>
<p>Clearly, $$C \equiv {x^2} + {y^2} = 25$$</p>
<p>Let $$z(x,y)$$</p>
<p>$$ \Rightarrow \left( {{{y - 4} \over {x - 3}}} \right)\left( {{2 \over { - 4}}} \right) = - 1$$</p>
<p>$$ \Rightarrow y = 2x - 2 \equiv L$$</p>
<p>$$\therefore$... |
<p>Let $$z=1+i$$ and $$z_{1}=\frac{1+i \bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$$. Then $$\frac{12}{\pi} \arg \left(z_{1}\right)$$ is equal to __________.</p>
Options:
[] | 9
Explanation:
<p>$$z = 1 + i$$</p>
<p>$${z_1} = {{1 + i\overline z } \over {\overline z (1 - z) + {1 \over z}}}$$</p>
<p>$$ = {{z(1 + i\overline z )} \over {|z{|^2}(1 - z) + 1}}$$</p>
<p>$$ = {{(1 + i)(1 + i(1 - i))} \over {2(1 - 1 - i) + 1}}$$</p>
<p>$${z_1} = 1 - i$$</p>
<p>$$\arg {z_1} = {\tan ^{ - 1}}\left( {{{ -... |
<p>Let $$w_{1}$$ be the point obtained by the rotation of $$z_{1}=5+4 i$$ about the origin through a right angle in the anticlockwise direction, and $$w_{2}$$ be the point obtained by the rotation of $$z_{2}=3+5 i$$ about the origin through a right angle in the clockwise direction. Then the principal argument of $$w_{1... | ["C"]
Explanation:
<p>To solve the problem, let's break it down step by step.</p>
<p><strong>Step 1 :</strong> Find $w_{1}$ </p>
<p>Given $z_{1} = 5 + 4i$. </p>
<p>When you rotate $z_{1}$ by $90^{\circ}$ anticlockwise about the origin, the real part becomes negative of the imaginary part of $z_{1}$, and the imagin... |
<p>If $$\alpha$$ denotes the number of solutions of $$|1-i|^x=2^x$$ and $$\beta=\left(\frac{|z|}{\arg (z)}\right)$$, where $$z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right], i=\sqrt{-1}$$, then the distance of the point $$(\alpha, \beta)$$ from the line $$4 x-3... | 3
Explanation:
<p>$$\begin{aligned}
& (\sqrt{2})^x=2^x \Rightarrow x=0 \Rightarrow \alpha=1 \\
& z=\frac{\pi}{4}(1+i)^4\left[\frac{\sqrt{\pi}-\pi i-i-\sqrt{\pi}}{\pi+1}+\frac{\sqrt{\pi}-i-\pi i-\sqrt{\pi}}{1+\pi}\right] \\
& =-\frac{\pi i}{2}\left(1+4 i+6 i^2+4 i^3+1\right) \\
& =2 \pi i \\
& \beta=\frac{2 \pi}{\frac{... |
<p>Let $$\mathrm{r}$$ and $$\theta$$ respectively be the modulus and amplitude of the complex number $$z=2-i\left(2 \tan \frac{5 \pi}{8}\right)$$, then $$(\mathrm{r}, \theta)$$ is equal to</p>
Options:
[{"identifier": "A", "content": "$$\\left(2 \\sec \\frac{11 \\pi}{8}, \\frac{11 \\pi}{8}\\right)$$\n"}, {"identifier"... | ["B"]
Explanation:
<p>$$\begin{aligned}
& z=2-i\left(2 \tan \frac{5 \pi}{8}\right)=x+i y(\text { let }) \\
& r=\sqrt{x^2+y^2} ~\& ~\theta=\tan ^{-1} \frac{y}{x} \\
& r=\sqrt{(2)^2+\left(2 \tan \frac{5 \pi}{8}\right)^2} \\
& =\left|2 \sec \frac{5 \pi}{8}\right|=\left|2 \sec \left(\pi-\frac{3 \pi}{8}\right)\right| \\
& ... |
The conjugate of a complex number is $${1 \over {i - 1}}$$ then that complex number is :
Options:
[{"identifier": "A", "content": "$${{ - 1} \\over {i - 1}}$$ "}, {"identifier": "B", "content": "$${1 \\over {i + 1}}\\,$$ "}, {"identifier": "C", "content": "$${{ - 1} \\over {i + 1}}$$ "}, {"identifier": "D", "content":... | ["C"]
Explanation:
$$\left( {{1 \over {i - 1}}} \right) = {1 \over { - i - 1}} = {{ - 1} \over {i + 1}}$$ |
If a > 0 and z = $${{{{\left( {1 + i} \right)}^2}} \over {a - i}}$$, has magnitude $$\sqrt {{2 \over 5}} $$, then $$\overline z $$ is equal to :
Options:
[{"identifier": "A", "content": "$$ - {1 \\over 5} + {3 \\over 5}i$$"}, {"identifier": "B", "content": "$$ - {1 \\over 5} - {3 \\over 5}i$$"}, {"identifier": "C"... | ["B"]
Explanation:
$$z = {{{{\left( {1 + i} \right)}^2}} \over {a - i}} \times {{a + i} \over {a + i}}$$<br><br>
$$ \Rightarrow z = {{\left( {1 - 1 + 2i} \right)\left( {a + i} \right)} \over {{a^2} + 1}} = {{2ai - 2} \over {{a^2} + 1}}$$ <br><br>
$$ \Rightarrow \left| z \right| = \sqrt {{{\left( {{{ - 2} \over {{a^2}... |
If the equation, x<sup>2</sup> + bx + 45 = 0 (b $$ \in $$ R) has
conjugate complex roots and they satisfy
|z +1| = 2$$\sqrt {10} $$ , then :
Options:
[{"identifier": "A", "content": "b<sup>2</sup> \u2013 b = 42"}, {"identifier": "B", "content": "b<sup>2</sup> + b = 12"}, {"identifier": "C", "content": "b<sup>2</sup> +... | ["D"]
Explanation:
x<sup>2</sup>
+ bx = 45 = 0 (b $$ \in $$ R)
<br>has roots $$\alpha $$ + i$$\beta $$, $$\alpha $$ – i$$\beta $$
<br>sum of roots = – b = 2$$\alpha $$
<br>product of roots = 45 =
$$\alpha $$<sup>2</sup>
+ $$\beta $$<sup>2</sup>
<br><br>Let z = x + iy
<br><br>$$ \therefore $$ |x + iy +1| = 2$$\sqrt ... |
If $$\alpha$$, $$\beta$$ $$\in$$ R are such that 1 $$-$$ 2i (here i<sup>2</sup> = $$-$$1) is a root of z<sup>2</sup> + $$\alpha$$z + $$\beta$$ = 0, then ($$\alpha$$ $$-$$ $$\beta$$) is equal to :
Options:
[{"identifier": "A", "content": "$$-$$7"}, {"identifier": "B", "content": "7"}, {"identifier": "C", "content": "3"... | ["A"]
Explanation:
1 $$-$$ 2i is the root of the equation. So other root is 1 $$+$$ 2i
<br><br>$$ \therefore $$ Sum of roots = 1 $$-$$ 2i + 1 $$+$$ 2i = 2 = -$$\alpha $$
<br><br>Product of roots = (1 $$-$$ 2i)(1 $$+$$ 2i) = 1 - 4i<sup>2</sup> = 5 = $$\beta $$
<br><br>$$ \therefore $$ $$\alpha $$ - $$\beta $$ = -2 - 5 ... |
Let z and $$\omega$$ be two complex numbers such that $$\omega = z\overline z - 2z + 2,\left| {{{z + i} \over {z - 3i}}} \right| = 1$$ and Re($$\omega$$) has minimum value. Then, the minimum value of n $$\in$$ N for which $$\omega$$<sup>n</sup> is real, is equal to ______________.
Options:
[] | 4
Explanation:
Let z = x + iy<br><br>| z + i | = | z $$-$$ 3i |<br><br>$$ \Rightarrow $$ y = 1<br><br>Now <br><br>$$\omega$$ = x<sup>2</sup> + y<sup>2</sup> $$-$$ 2x $$-$$ 2iy + 2<br><br>$$\omega$$ = x<sup>2</sup> + 1 $$-$$ 2x $$-$$ 2i + 2<br><br>Re($$\omega$$) = x<sup>2</sup> $$-$$ 2x + 3<br><br>Re($$\omega$$) = (x $... |
Let n denote the number of solutions of the equation z<sup>2</sup> + 3$$\overline z $$ = 0, where z is a complex number. Then the value of $$\sum\limits_{k = 0}^\infty {{1 \over {{n^k}}}} $$ is equal to :
Options:
[{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "$${4 \\over 3}$$"}, {"identifier":... | ["B"]
Explanation:
z<sup>2</sup> + 3$$\overline z $$ = 0<br><br>Put z = x + iy<br><br>$$\Rightarrow$$ x<sup>2</sup> $$-$$ y<sup>2</sup> + 2ixy + 3(x $$-$$ iy) = 0<br><br>$$\Rightarrow$$ (x<sup>2</sup> $$-$$ y<sup>2</sup> + 3x) + i(2xy $$-$$ 3y) = 0 + i0<br><br>$$\therefore$$ x<sup>2</sup> $$-$$ y<sup>2</sup> + 3x = 0 ... |
<p>Sum of squares of modulus of all the complex numbers z satisfying $$\overline z = i{z^2} + {z^2} - z$$ is equal to ___________.</p>
Options:
[] | 2
Explanation:
Let $z=x+i y$
<br/><br/>
So $2 x=(1+i)\left(x^{2}-y^{2}+2 x y i\right)$
<br/><br/>
$\Rightarrow 2 x=x^{2}-y^{2}-2 x y\quad$ ...(i) and
<br/><br/>
$$
x^{2}-y^{2}+2 x y=0\quad\dots(ii)
$$
<br/><br/>
From (i) and (ii) we get
<br/><br/>
$$
x=0 \text { or } y=-\frac{1}{2}
$$
<br/><br/>
When $x=0$ we get $y=0... |
<p>The area of the polygon, whose vertices are the non-real roots of the equation $$\overline z = i{z^2}$$ is :</p>
Options:
[{"identifier": "A", "content": "$${{3\\sqrt 3 } \\over 4}$$"}, {"identifier": "B", "content": "$${{3\\sqrt 3 } \\over 2}$$"}, {"identifier": "C", "content": "$${3 \\over 2}$$"}, {"identifier":... | ["A"]
Explanation:
<p>$$\overline z = i{z^2}$$</p>
<p>Let $$z = x + iy$$</p>
<p>$$x - iy = i({x^2} - {y^2} + 2xiy)$$</p>
<p>$$x - iy = i({x^2} - {y^2}) - 2xy$$</p>
<p>$$\therefore$$ $$x = - 2yx$$ or $${x^2} - {y^2} = - y$$</p>
<p>$$x = 0$$ or $$y = - {1 \over 2}$$</p>
<p>Case - I</p>
<p>$$x = 0$$</p>
<p>$$ - {y^2}... |
<p>The real part of the complex number $${{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "$${{500} \\over {13}}$$"}, {"identifier": "B", "content": "$${{110} \\over {13}}$$"}, {"identifier": "C", "content": "$${{55} \\over {6... | ["D"]
Explanation:
<p>Given,</p>
<p>$${{{{(1 + 2i)}^8}\,.\,{{(1 - 2i)}^2}} \over {(3 + 2i)\,.\,\overline {(4 - 6i)} }}$$</p>
<p>$$ = {{{{(1 + 2i)}^2}{{(1 - 2i)}^2}{{(1 + 2i)}^6}} \over {(3 + 2i)(4 + 6i)}}$$</p>
<p>$$ = {{{{(1 - 4{i^2})}^2}{{(1 + 2i)}^6}} \over {12 + 18i + 8i + 12{i^2}}}$$</p>
<p>$$ = {{{{(1 + 5)}^2}{{... |
<p>Let $$S=\left\{z \in \mathbb{C}: z^{2}+\bar{z}=0\right\}$$. Then $$\sum\limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))$$ is equal to ______________.</p>
Options:
[] | 0
Explanation:
<p>$$\because$$ $${z^2} + \overline z = 0$$</p>
<p>Let $$z = x + iy$$</p>
<p>$$\therefore$$ $${x^2} - {y^2} + 2ixy + x - iy = 0$$</p>
<p>$$({x^2} - {y^2} + x) + i(2xy - y) = 0$$</p>
<p>$$\therefore$$ $${x^2} + {y^2} = 0$$ and $$(2x - 1)y = 0$$</p>
<p>if $$x = \, + \,{1 \over 2}$$ then $$y = \, \pm \,{{... |
<p>If $$z=2+3 i$$, then $$z^{5}+(\bar{z})^{5}$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "244"}, {"identifier": "B", "content": "224"}, {"identifier": "C", "content": "245"}, {"identifier": "D", "content": "265"}] | ["A"]
Explanation:
<p>$$z = (2 + 3i)$$</p>
<p>$$ \Rightarrow {z^5} = (2 + 3i){\left( {{{(2 + 3i)}^2}} \right)^2}$$</p>
<p>$$ = (2 + 3i){( - 5 + 12i)^2}$$</p>
<p>$$ = (2 + 3i)( - 119 - 120i)$$</p>
<p>$$ = - 238 - 240i - 357i + 360$$</p>
<p>$$ = 122 - 597i$$</p>
<p>$${\overline z ^5} = 122 + 597i$$</p>
<p>$${z^5} + {\o... |
If the set $\left\{\operatorname{Re}\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right): z \in \mathbb{C}, \operatorname{Re}(z)=3\right\}$ is equal to<br/><br/> the interval $(\alpha, \beta]$, then $24(\beta-\alpha)$ is equal to :
Options:
[{"identifier": "A", "content": "36"}, {"identifier": "B", "content": "27"... | ["D"]
Explanation:
Let $z_1=\left(\frac{z-\bar{z}+z \bar{z}}{2-3 z+5 \bar{z}}\right)$<br/><br/>
Let $\mathrm{z}=3+\mathrm{iy}$<br/><br/> $\bar{z}=3-i y$<br/><br/>
$$
\begin{aligned}
& z_1=\frac{2 i y+\left(9+y^2\right)}{2-3(3+i y)+5(3-i y)} \\\\
& =\frac{9+y^2+i(2 y)}{8-8 i y} \\\\
& =\frac{\left(9+y^2\right)+i(2 y)}{... |
<p>Let $$S=\left\{z \in \mathbb{C}: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}$$. Then $$\sum_\limits{z \in \mathrm{S}}|z|^{2}$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "$$\\frac{7}{2}$$"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "3"}, {"identifier": "... | ["B"]
Explanation:
Let $z=x+i y$
<br/><br/>$$
\begin{aligned}
&\bar{z}=i\left(z^2+\operatorname{Re}(\bar{z})\right) \\\\
&\Rightarrow x-i y=i\left(x^2-y^2+2 i x y+x\right) \\\\
& x-i y=i\left(x^2-y^2+x\right)-2 x y \\\\
&x=-2 x y \Rightarrow x(2 y+1)=0 \\\\
&\Rightarrow x=0, y=\frac{-1}{2} ........(i)\\\\
&-y=x^2-y^... |
<p>For $$a \in \mathbb{C}$$, let $$\mathrm{A}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z}) > \operatorname{Im}(\bar{a}+z)\}$$ and $$\mathrm{B}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z})<\operatorname{Im}(\bar{a}+z)\}$$. Then among the two statements :</p>
<p>(S1): If $$\operatorname{Re}(a), \operatornam... | ["A"]
Explanation:
We are given that $a \in \mathbb{C}$ and $z \in \mathbb{C}$.
<br/><br/>Let $a = x_1 + iy_1$ and $z = x_2 + iy_2$ where $x_1, y_1, x_2, y_2 \in \mathbb{R}$
<br/><br/>We are also given two sets A and B defined as follows :
<br/><br/>- A is the set of all complex numbers $z$ for which the real part o... |
If $$z = x - iy$$ and $${z^{{1 \over 3}}} = p + iq$$, then
<br/><br/>$${{\left( {{x \over p} + {y \over q}} \right)} \over {\left( {{p^2} + {q^2}} \right)}}$$ is equal to :
Options:
[{"identifier": "A", "content": "- 2"}, {"identifier": "B", "content": "- 1"}, {"identifier": "C", "content": "2 "}, {"identifier": "D",... | ["A"]
Explanation:
Given $${z^{{1 \over 3}}} = p + iq$$
<br><br>$$ \Rightarrow $$ z = (p + iq)<sup>3</sup>
<br><br>= p<sup>3</sup> + (iq)<sup>3</sup> +3p(iq)(p + iq)
<br><br>= p<sup>3</sup> - iq<sup>3</sup> +3ip<sup>2</sup>q - 3pq<sup>2</sup>
<br><br>= p(p<sup>2</sup> - 3q<sup>2</sup>) - iq(q<sup>2</sup> - 3p<sup>2</s... |
If the cube roots of unity are 1, $$\omega \,,\,{\omega ^2}$$ then the roots of the equation $${(x - 1)^3}$$ + 8 = 0, are :
Options:
[{"identifier": "A", "content": "$$ - 1, - 1 + 2\\,\\,\\omega , - 1 - 2\\,\\,{\\omega ^2}$$ "}, {"identifier": "B", "content": "$$ - 1, - 1, - 1$$ "}, {"identifier": "C", "content": "$$ ... | ["C"]
Explanation:
$${\left( {x - 1} \right)^3} + 8 = 0$$
<br><br>$$ \Rightarrow \left( {x - 1} \right) = \left( { - 2} \right){\left( 1 \right)^{1/3}}$$
<br><br>$$ \Rightarrow x - 1 = - 2\,\,\,$$ or $$\,\,\, - 2\omega \,\,\,\,$$ or $$\,\,\,\, - 2{\omega ^2}$$
<br><br>or $$\,\,\,x = - 1\,\,\,$$ or $$\,\,\,1 - 2\omeg... |
If $$\omega ( \ne 1)$$ is a cube root of unity, and $${(1 + \omega )^7} = A + B\omega \,$$. Then $$(A,B)$$ equals :
Options:
[{"identifier": "A", "content": "(1 ,1)"}, {"identifier": "B", "content": "(1, 0)"}, {"identifier": "C", "content": "(- 1 ,1)"}, {"identifier": "D", "content": "(0 ,1)"}] | ["A"]
Explanation:
$${\left( {1 + \omega } \right)^7} = A + B\omega ;\,\,\,\,{\left( { - {\omega ^2}} \right)^7} = A + B\omega $$
<br><br>$$ - {\omega ^2} = A + B\omega ;\,\,\,\,\,\,\,\,\,\,1 + \omega = A + B\omega $$
<br><br>$$ \Rightarrow A = 1,B = 1.$$ |
If $$\alpha ,\beta \in C$$ are the distinct roots of the equation
<br/>x<sup>2</sup> - x + 1 = 0, then $${\alpha ^{101}} + {\beta ^{107}}$$ is equal to :
Options:
[{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "-1"}, {"identifier": "C", "content": "0"}, {"identifier": "D", "content": "1"}] | ["D"]
Explanation:
Given equation,
<br><br>x<sup>2</sup> $$-$$ x + 1 = 0
<br><br>Roots of this equation
<br><br>x = $${{1 \pm \sqrt 3 i} \over 2}$$
<br><br>$$\therefore\,\,\,$$ $$ \propto \, = \,{{1 + \sqrt 3 \,i} \over 2}$$
<br><br>and $$\beta = \,{{1 - \sqrt 3 \,i} \over 2}$$
<br><br>We know;
<br><br>$$\omega = ... |
If a and b are real numbers such that<br/> $${\left( {2 + \alpha } \right)^4} = a + b\alpha $$ <br/>where $$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$ then a + b is
equal to :
Options:
[{"identifier": "A", "content": "33"}, {"identifier": "B", "content": "9"}, {"identifier": "C", "content": "24"}, {"identifier": "D", "... | ["B"]
Explanation:
$$\alpha = \omega $$ as given $$\alpha = {{ - 1 + i\sqrt 3 } \over 2}$$
<br><br>$$ \Rightarrow {(2 + \omega )^4} = a + b\omega \,({\omega ^3} = 1)$$<br><br>$$ \Rightarrow {2^4} + {4.2^3}\omega + {6.2^2}{\omega ^3} + 4.2.\,{\omega ^3} + {\omega ^4} = a + b\omega $$<br><br>$$ \Rightarrow 16 + 32\om... |
If $${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$$ then :
Options:
[{"identifier": "A", "content": "x = 2n + 1, where n is any positive integer"}, {"identifier": "B", "content": "x = 4n , where n is any positive integer"}, {"identifier": "C", "content": "x = 2n, where n is any positive integer\n"}, {"identifier":... | ["B"]
Explanation:
$${\left( {{{1 + i} \over {1 - i}}} \right)^x} = 1$$
<br><br>$$ \Rightarrow $$ $${\left[ {{{\left( {1 + i} \right)\left( {1 + i} \right)} \over {\left( {1 - i} \right)\left( {1 + i} \right)}}} \right]^x} = 1$$
<br><br>$$ \Rightarrow $$ $${\left[ {{{{{\left( {1 + i} \right)}^2}} \over {1 - {i^2}}}} \... |
The least positive integer n for which $${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1,$$ is :
Options:
[{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "3"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "6"}] | ["B"]
Explanation:
$${\left( {{{1 + i\sqrt 3 } \over {1 - i\sqrt 3 }}} \right)^n} = 1$$
<br><br>$$ \Rightarrow \,\,\,\,\,{\left( {{{{{1 + i\sqrt 3 } \over 2}} \over {{{1 - i\sqrt 3 } \over 2}}}} \right)^n} = 1$$
<br><br>We know, $$\omega $$ = $$-$$ $${{1 - i\sqrt 3 } \over 2}$$
<br><br>and $$\omega $$<sup>2</sup> = $... |
Let z<sub>0</sub> be a root of the quadratic equation, x<sup>2</sup> + x + 1 = 0, If z = 3 + 6iz$$_0^{81}$$ $$-$$ 3iz$$_0^{93}$$, then arg z is equal to :
Options:
[{"identifier": "A", "content": "$${\\pi \\over 4}$$"}, {"identifier": "B", "content": "$${\\pi \\over 6}$$"}, {"identifier": "C", "content": "$${\\pi ... | ["A"]
Explanation:
1 + x + x<sup>2</sup> = 0
<br><br>x = $${{ - 1 \pm \sqrt {1 - 4} } \over 2} = {{ - 1 \pm i\sqrt 3 } \over 2}$$
<br><br>z<sub>0</sub> = w, w<sup>2</sup>
<br><br>Now
<br><br>z = 3 + 6iz$$_0^{81}$$ $$-$$ 3iz$$_0^{93}$$
<br><br>z = 3 + 6iw<sup>81</sup> $$-$$ 3iw<sup>93</sup>  ... |
Let $$z = {\left( {{{\sqrt 3 } \over 2} + {i \over 2}} \right)^5} + {\left( {{{\sqrt 3 } \over 2} - {i \over 2}} \right)^5}.$$ If R(z) and 1(z) respectively denote the real and imaginary parts of z, then :
Options:
[{"identifier": "A", "content": "R(z) = $$-$$ \n3"}, {"identifier": "B", "content": "R(z) < 0 and I(z... | ["C"]
Explanation:
$$z = {\left( {{{\sqrt 3 + i} \over 2}} \right)^5} + {\left( {{{\sqrt 3 - i} \over 2}} \right)^5}$$
<br><br>$$z = {\left( {{e^{i\pi /6}}} \right)^5} + {\left( {{e^{ - i\pi /6}}} \right)^5}$$
<br><br>$$ = {e^{i5\pi /6}} + {e^{ - i5\pi /6}}$$
<br><br>$$ = \cos {{5\pi } \over 6} + i{{\sin 5\pi } \ove... |
Let $${\left( { - 2 - {1 \over 3}i} \right)^3} = {{x + iy} \over {27}}\left( {i = \sqrt { - 1} } \right),\,\,$$ where x and y are real numbers, then y $$-$$ x equals :
Options:
[{"identifier": "A", "content": "$$-$$ 85 "}, {"identifier": "B", "content": "85"}, {"identifier": "C", "content": "$$-$$ 91 "}, {"identifie... | ["D"]
Explanation:
$${\left( { - 2 - {i \over 3}} \right)^3} = - {{{{\left( {6 + i} \right)}^3}} \over {27}}$$
<br><br>$$ = {{ - 198 - 107i} \over {27}} = {{x + iy} \over {27}}$$
<br><br>Hence, $$y - x = 198 - 107 = 91$$ |
If $$z = {{\sqrt 3 } \over 2} + {i \over 2}\left( {i = \sqrt { - 1} } \right)$$,
<br/><br/>then (1 + iz + z<sup>5</sup> + iz<sup>8</sup>)<sup>9</sup> is equal to :
Options:
[{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "\u20131"}, {"identifier": "C", "content": "0"}, {"identifier": "D", "conten... | ["B"]
Explanation:
$$z = {{\sqrt 3 } \over 2} + {i \over 2}$$
<br><br>$$ \Rightarrow $$ z = $$\cos {\pi \over 6}$$ + i $$\sin {\pi \over 6}$$
<br><br>$$ \Rightarrow $$ z = $${e^{i{\pi \over 6}}}$$
<picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267414/exam_... |
The value of <br/><br/>$${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}$$ is :
Options:
[{"identifier": "A", "content": "$${1 \\over 2}\\left( {\\sqrt 3 - i} \\right)$$"}, {"identifier": "B", "content": "-$${1 \\over 2}\\left(... | ["B"]
Explanation:
$${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}$$
<br><br>= $${\left( {{{1 + \sin \left( {{\pi \over 2} - {{5\pi } \over {18}}} \right) + i\cos \left( {{\pi \over 2} - {{5\pi } \over {18}}} \right)} \over ... |
The value of $${\left( {{{ - 1 + i\sqrt 3 } \over {1 - i}}} \right)^{30}}$$ is :
Options:
[{"identifier": "A", "content": "\u20132<sup>15</sup>i"}, {"identifier": "B", "content": "\u20132<sup>15</sup>"}, {"identifier": "C", "content": "2<sup>15</sup>i"}, {"identifier": "D", "content": "6<sup>5</sup>"}] | ["A"]
Explanation:
$${\left( {{{ - 1 + i\sqrt 3 } \over {1 - i}}} \right)^{30}}$$
<br><br>= $${\left( {{{2\omega } \over {1 - i}}} \right)^{30}}$$
<br><br>= $${{{2^{30}}.{\omega ^{30}}} \over {{{\left( {{{\left( {1 - i} \right)}^2}} \right)}^{15}}}}$$
<br><br>= $${{{2^{30}}.1} \over {{{\left( {1 + {i^{^2}} - 2i} \righ... |
Let $$i = \sqrt { - 1} $$. If $${{{{\left( { - 1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 - i)}^{24}}}} + {{{{\left( {1 + i\sqrt 3 } \right)}^{21}}} \over {{{(1 + i)}^{24}}}} = k$$, and $$n = [|k|]$$ be the greatest integral part of | k |. Then $$\sum\limits_{j = 0}^{n + 5} {{{(j + 5)}^2} - \sum\limits_{j = 0}^{n + 5} ... | 310
Explanation:
$${(1 + i)^2} = 1 + {i^2} + 2i = 1 - 1 + 2i = 2i$$<br><br>$${(1 - i)^2} = 1 + {i^2} - 2i = 1 - 1 - 2i = - 2i$$<br><br>We know,<br><br>$$ - {1 \over 2} + {{i\sqrt 3 } \over 2} = \omega $$<br><br>$$ \Rightarrow - 1 + i\sqrt 3 = 2\omega $$<br><br>and $$ - {1 \over 2} - {{i\sqrt 3 } \over 2} = {\omega ... |
If $${\left( {\sqrt 3 + i} \right)^{100}} = {2^{99}}(p + iq)$$, then p and q are roots of the equation :
Options:
[{"identifier": "A", "content": "$${x^2} - \\left( {\\sqrt 3 - 1} \\right)x - \\sqrt 3 = 0$$"}, {"identifier": "B", "content": "$${x^2} + \\left( {\\sqrt 3 + 1} \\right)x + \\sqrt 3 = 0$$"}, {"identif... | ["A"]
Explanation:
$${\left( {2{e^{i\pi /6}}} \right)^{100}} = {2^{99}}(p + iq)$$<br><br>$${2^{100}}\left( {\cos {{50\pi } \over 3} + i\sin {{50\pi } \over 3}} \right) = {2^{99}}(p + iq)$$<br><br>$$p + iq = 2\left( {\cos {{2\pi } \over 3} + i\sin {{2\pi } \over 3}} \right)$$<br><br>p = $$-$$1, q = $$\sqrt 3 $$<br><br>... |
The least positive integer n such that $${{{{(2i)}^n}} \over {{{(1 - i)}^{n - 2}}}},i = \sqrt { - 1} $$ is a positive integer, is ___________.
Options:
[] | 6
Explanation:
$${{{{(2i)}^n}} \over {{{(1 - i)}^{n - 2}}}} = {{{{(2i)}^n}} \over {{{( - 2i)}^{{{n - 2} \over 2}}}}}$$<br><br>$$ = {{{{(2i)}^{{{n + 2} \over 2}}}} \over {{{( - 1)}^{{{n - 2} \over 2}}}}} = {{{2^{{{n + 2} \over 2}}};{i^{{{n + 2} \over 2}}}} \over {{{( - 1)}^{{{n - 2} \over 2}}}}}$$<br><br>This is positi... |
The complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ is equal to :
Options:
[{"identifier": "A", "content": "$\\cos \\frac{\\pi}{12}-i \\sin \\frac{\\pi}{12}$"}, {"identifier": "B", "content": "$\\sqrt{2}\\left(\\cos \\frac{\\pi}{12}+i \\sin \\frac{\\pi}{12}\\right)$"}, {"identifier": "C", "conte... | ["D"]
Explanation:
$\mathrm{Z}=\frac{\mathrm{i}-1}{\cos \frac{\pi}{3}+\mathrm{i} \sin \frac{\pi}{3}}=\frac{\mathrm{i}-1}{\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}}$
<br/><br/>$=\frac{i-1}{\frac{1}{2}+\frac{\sqrt{3}}{2} \mathrm{i}} \times \frac{\frac{1}{2}-\sqrt{\frac{3}{2}} \mathrm{i}}{\frac{1}{2}-\sqrt{3 / 2} \mathr... |
<p>The value of $${\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}$$ is</p>
Options:
[{"identifier": "A", "content": "$$ - {1 \\over 2}\\left( {1 - i\\sqrt 3 } \\right)$$"}, {"identifier": "B", "content": "$$ - {1 \\over 2}\\left... | ["B"]
Explanation:
$z=\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^{3}$
<br/><br/>
$$
\begin{aligned}
& 1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}=1+\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18} \\\\
& =1+2 \cos ^{2} \frac{5 \pi}{36}-1+2 i \sin \fr... |
<p>Let $$\mathrm{p,q\in\mathbb{R}}$$ and $${\left( {1 - \sqrt 3 i} \right)^{200}} = {2^{199}}(p + iq),i = \sqrt { - 1} $$ then $$\mathrm{p+q+q^2}$$ and $$\mathrm{p-q+q^2}$$ are roots of the equation.</p>
Options:
[{"identifier": "A", "content": "$${x^2} + 4x - 1 = 0$$"}, {"identifier": "B", "content": "$${x^2} - 4x + ... | ["B"]
Explanation:
<p>$${\left( {1 - \sqrt 3 i} \right)^{200}}$$</p>
<p>$$ = {\left[ {2\left( {{1 \over 2} - {{\sqrt 3 } \over 2}i} \right)} \right]^{200}}$$</p>
<p>$$ = {2^{200}}{\left( {\cos {\pi \over 3} - i\sin {\pi \over 3}} \right)^{200}}$$</p>
<p>$$ = {2^{200}}\left( {\cos {{200\pi } \over 3} - i\sin {{200\pi... |
If $$\,\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1$$, then z lies on :
Options:
[{"identifier": "A", "content": "an ellipse"}, {"identifier": "B", "content": "the imaginary axis"}, {"identifier": "C", "content": "a circle"}, {"identifier": "D", "content": "the real axis"}] | ["B"]
Explanation:
Given $$\,\left| {{z^2} - 1} \right| = {\left| z \right|^2} + 1$$,
<br><br>By squaring both sides we get,
<br><br>$${\left| {{z^2} - 1} \right|^2}$$ = $${\left( {{{\left| z \right|}^2} + 1} \right)^2}$$
<br><br>$$ \Rightarrow $$ $$\left( {{z^2} - 1} \right)$$$$\overline {\left( {{z^2} - 1} \right)} ... |
If $$\,\omega = {z \over {z - {1 \over 3}i}}\,$$ and $$\left| \omega \right| = 1$$, then $$z$$ lies on :
Options:
[{"identifier": "A", "content": "an ellipse"}, {"identifier": "B", "content": "a circle"}, {"identifier": "C", "content": "a straight line"}, {"identifier": "D", "content": "a parabola"}] | ["C"]
Explanation:
Given $$\,\omega = {z \over {z - {1 \over 3}i}}\,$$ and $$\left| \omega \right| = 1$$
<br><br>$$\therefore$$ $${{\left| z \right|} \over {\left| {z - {1 \over {\sqrt 3 }}i} \right|}} = \left| \omega \right|$$
<br><br>$$ \Rightarrow $$ $${{\left| z \right|} \over {\left| {z - {1 \over {\sqrt 3 }}i... |
The number of complex numbers z such that $$\left| {z - 1} \right| = \left| {z + 1} \right| = \left| {z - i} \right|$$ equals :
Options:
[{"identifier": "A", "content": "1 "}, {"identifier": "B", "content": "2 "}, {"identifier": "C", "content": "$$\\infty $$ "}, {"identifier": "D", "content": "0 "}] | ["A"]
Explanation:
Let $$z=x+iy$$
<br><br>$$\left| {z - 1} \right| = \left| {z + 1} \right|{\left( {x - 1} \right)^2} + {y^2}$$
<br><br>$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = {\left( {x + 1} \right)^2} + {y^2}$$
<br><br>$$ \Rightarrow {\mathop{\rm Re}\nolimits} \,z = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow ... |
If z is a complex number such that $$\,\left| z \right| \ge 2\,$$, then the minimum value of $$\,\,\left| {z + {1 \over 2}} \right|$$ :
Options:
[{"identifier": "A", "content": "is strictly greater that $${{5 \\over 2}}$$"}, {"identifier": "B", "content": "is strictly greater that $${{3 \\over 2}}$$ but less than $${{... | ["B"]
Explanation:
We know minimum value of
<br><br>$$\,\,\,\left| {{Z_1} + {Z_2}} \right|\,\,\,$$ is $$\,\,\,\left| {\left| {{Z_1}} \right| - \left| {{Z_2}} \right|} \right|$$
<br><br>Thus minimum value of
<br><br>$$\,\,\,\left| {Z + {1 \over 2}} \right|\,\,\,$$ is $$\,\,\,\left| {\left| Z \right| - {1 \over 2}} \... |
A complex number z is said to be unimodular if $$\,\left| z \right| = 1$$. Suppose $${z_1}$$ and $${z_2}$$ are complex numbers such that $${{{z_1} - 2{z_2}} \over {2 - {z_1}\overline {{z_2}} }}$$ is unimodular and $${z_2}$$ is not unimodular. Then the point $${z_1}$$ lies on a :
Options:
[{"identifier": "A", "content"... | ["A"]
Explanation:
$$\left| {{{{z_1} - 2{z_2}} \over {2 - {z_1}{{\overline z }_2}}}} \right| = 1 \Rightarrow {\left| {{z_1} - 2{z_2}} \right|^2} = {\left| {2 - {z_1}{{\overline z }_2}} \right|^2}$$
<br><br>$$ \Rightarrow \left( {{z_1} - 2{z_2}} \right)\left( {\overline {{z_1} - 2{z_2}} } \right)$$
<br><br>$$\,\,\,\,\,... |
If |z $$-$$ 3 + 2i| $$ \le $$ 4 then the difference between the greatest value and the least value of |z| is :
Options:
[{"identifier": "A", "content": "$$2\\sqrt {13} $$"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "content": "4 + $$\\sqrt {13} $$"}, {"identifier": "D", "content": "$$\\sqrt {13} $$"}] | ["A"]
Explanation:
$$\left| {z - \left( {3 - 2i} \right)} \right| \le 4$$ represents a circle whose center is (3, $$-$$2) and radius = 4.
<br><br>$$\left| z \right|$$ = $$\left| z -0\right|$$ represents the distance of point 'z' from origin (0, 0)
<br><br><img src="https://res.cloudinary.com/dckxllbjy/image... |
Let z<sub>1</sub> and z<sub>2</sub> be any two non-zero complex numbers such that $$3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|.$$ If $$z = {{3{z_1}} \over {2{z_2}}} + {{2{z_2}} \over {3{z_1}}}$$ then :
Options:
[{"identifier": "A", "content": "$${\\rm I}m\\left( z \\right) = 0$$"}, {"identifier": "B", "con... | ["C"]
Explanation:
Given, $$3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|$$
<br><br>$$ \Rightarrow $$ $${{\left| {{z_1}} \right|} \over {\left| {{z_2}} \right|}} = {4 \over 3}$$
<br><br>$$ \Rightarrow $$ $${{\left| {3{z_1}} \right|} \over {\left| {2{z_2}} \right|}} = {4 \over 3} \times {3 \over 2} = 2$$
<br><br>As... |
Let z be a complex number such that |z| + z = 3 + i (where i = $$\sqrt { - 1} $$). Then |z| is equal to :
Options:
[{"identifier": "A", "content": "$${{\\sqrt {34} } \\over 3}$$"}, {"identifier": "B", "content": "$${5 \\over 3}$$"}, {"identifier": "C", "content": "$${5 \\over 4}$$"}, {"identifier": "D", "content": "$... | ["B"]
Explanation:
$$\left| z \right| + z = 3 + i$$
<br><br>$$z = 3 - \left| z \right| + i$$
<br><br>Let $$3 - \left| z \right| = a \Rightarrow \left| z \right| = \left( {3 - a} \right)$$
<br><br>$$ \Rightarrow z = a + i \Rightarrow \left| z \right| = \sqrt {{a^2} + 1} $$
<br><br>$$ \Rightarrow 9 + {a^2} - ... |
Let z<sub>1</sub> and z<sub>2</sub> be two complex numbers satisfying | z<sub>1</sub> | = 9 and | z<sub>2</sub> – 3 – 4i | = 4. Then the minimum value of
| z<sub>1</sub> – z<sub>2</sub> | is :
Options:
[{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "2"}, {"ide... | ["A"]
Explanation:
$$\left| {{z_1}} \right| = 9,\,\,\left| {{z_2} - \left( {3 + 4i} \right)} \right| = 4$$
<br><br>$${C_1},(0,0)$$ radius r<sub>1</sub> = 9
<br><br>C<sub>2</sub> (3, 4), radius r<sub>2</sub> = 4
<br><br>C<sub>1</sub>C<sub>2</sub> = $$\left| {{r_1} - {r_2}} \right| = 5$$
<br><br>$$ \therefore $$ &n... |
Let z be complex number such that
<br/> $$\left| {{{z - i} \over {z + 2i}}} \right| = 1$$ and |z| = $${5 \over 2}$$.<br/> Then the value of |z + 3i| is :
Options:
[{"identifier": "A", "content": "$$2\\sqrt 3 $$"}, {"identifier": "B", "content": "$$\\sqrt {10} $$"}, {"identifier": "C", "content": "$${{15} \\over 4}$$"}... | ["D"]
Explanation:
Given $$\left| {{{z - i} \over {z + 2i}}} \right| = 1$$
<br><br>|z – i| = |z + 2i|
<br><br>(let z = x + iy)
<br><br>$$ \Rightarrow $$ x<sup>2</sup>
+ (y – 1)<sup>2</sup>
= x<sup>2</sup>
+ (y + 2)<sup>2</sup>
<br><br>$$ \Rightarrow $$ y = $$ - {1 \over 2}$$
<br><br>Also given |z| = $${5 \over 2}$... |
If z be a complex number satisfying
|Re(z)| + |Im(z)| = 4, then |z| cannot be :
Options:
[{"identifier": "A", "content": "$$\\sqrt {10} $$"}, {"identifier": "B", "content": "$$\\sqrt {7} $$"}, {"identifier": "C", "content": "$$\\sqrt {{{17} \\over 2}} $$"}, {"identifier": "D", "content": "$$\\sqrt {8} $$"}] | ["B"]
Explanation:
Let z = x + iy
<br><br>given that |Re(z)| + |Im(z)| = 4
<br><br>$$ \therefore $$ |x| + |y| = 4
<img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264539/exam_images/tkloovrsmfpi6fjklioa.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Ma... |
If the least and the largest real values of a, for which the <br/>equation z + $$\alpha $$|z – 1| + 2i = 0
(z $$ \in $$ C and i = $$\sqrt { - 1} $$) has a solution, are p and q respectively; then 4(p<sup>2</sup> + q<sup>2</sup>) is equal to __________.
Options:
[] | 10
Explanation:
$$x + iy + \alpha \sqrt {{{(x - 1)}^2} + {y^2}} + 2i = 0$$<br><br>$$ \therefore $$ y + 2 = 0 and $$x + \alpha \sqrt {{{(x - 1)}^2} + {y^2}} = 0$$<br><br>y = $$-$$2 & $${x^2} = {\alpha ^2}({x^2} - 2x + 1 + 4)$$<br><br>$${\alpha ^2} = {{{x^2}} \over {{x^2} - 2x + 5}} \Rightarrow {x^2}({\alpha ^2} -... |
Let a complex number z, |z| $$\ne$$ 1, <br/><br/>satisfy $${\log _{{1 \over {\sqrt 2 }}}}\left( {{{|z| + 11} \over {{{(|z| - 1)}^2}}}} \right) \le 2$$. Then, the largest value of |z| is equal to ____________.
Options:
[{"identifier": "A", "content": "5"}, {"identifier": "B", "content": "8"}, {"identifier": "C", "conte... | ["D"]
Explanation:
$${{|z| + 11} \over {{{(|z| - 1)}^2}}} \ge {1 \over 2}$$<br><br>$$2|z| + 22 \ge {(|z| - 1)^2}$$<br><br>$$2|z| + 22 \ge \,|z{|^2} - 2|z| + 1$$<br><br>$$|z{|^2} - 4|z| - 21 \le 0$$<br><br>$$(|z| - 7)(|z| + 3) \le 0$$<br><br>$$ \Rightarrow \,|z| \le 7$$<br><br>$$ \therefore $$ $$|z{|_{\max }} = 7$$ |
The least value of |z| where z is complex number which satisfies the inequality $$\exp \left( {{{(|z| + 3)(|z| - 1)} \over {||z| + 1|}}{{\log }_e}2} \right) \ge {\log _{\sqrt 2 }}|5\sqrt 7 + 9i|,i = \sqrt { - 1} $$, is equal to :
Options:
[{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "3"}, {"id... | ["B"]
Explanation:
Let | z | = t, t $$ \ge $$ 0<br><br>$${e^{{{(t + 3)(t - 1)} \over {t + 1}}{{\log }_e}2}} \ge {\log _{\sqrt 2 }}16 = 8$$ ($$ \because $$ t + 1 > 0)<br><br>$${2^{{{(t + 3)(t - 1)} \over {t + 1}}}} \ge {2^3}$$<br><br>$${{(t + 3)(t - 1)} \over {t + 1}} \ge 3$$<br><br>$${t^2} + 2t - 3 \ge 3t + 3$$<br>... |
If z is a complex number such that $${{z - i} \over {z - 1}}$$ is purely imaginary, then the minimum value of | z $$-$$ (3 + 3i) | is :
Options:
[{"identifier": "A", "content": "$$2\\sqrt 2 - 1$$"}, {"identifier": "B", "content": "$$3\\sqrt 2 $$"}, {"identifier": "C", "content": "$$6\\sqrt 2 $$"}, {"identifier": "D",... | ["D"]
Explanation:
$${{z - i} \over {z - 1}}$$ is purely imaginary number<br><br>Let $$z = x + iy$$<br><br>$$\therefore$$ $${{x + i(y - 1)} \over {(x - 1) + i(y)}} \times {{(x - 1) - iy} \over {(x - 1) - iy}}$$<br><br>$$ \Rightarrow {{x(x - 1) + y(y - 1) + i( - y - x + 1)} \over {{{(x - 1)}^2} + {y^2}}}$$ is purely im... |
If for the complex numbers z satisfying | z $$-$$ 2 $$-$$ 2i | $$\le$$ 1, the maximum value of | 3iz + 6 | is attained at a + ib, then a + b is equal to ______________.
Options:
[] | 5
Explanation:
| z $$-$$ 2 $$-$$ 2i | $$\le$$ 1<br><br>| x + iy $$-$$ 2 $$-$$ 2i | $$\le$$ 1<br><br>|(x $$-$$ 2) + i(y $$-$$ 2)| $$\le$$ 1<br><br>(x $$-$$ 2)<sup>2</sup> + (y $$-$$ 2)<sup>2</sup> $$\le$$ 1<br><br>| 3iz + 6 |<sub>max</sub> at a + ib<br><br>$$\left| {3i} \right|\left| {z + {6 \over {3i}}} \right|$$<br><... |
<p>Let S = {z $$\in$$ C : |z $$-$$ 3| $$\le$$ 1 and z(4 + 3i) + $$\overline z $$(4 $$-$$ 3i) $$\le$$ 24}. If $$\alpha$$ + i$$\beta$$ is the point in S which is closest to 4i, then 25($$\alpha$$ + $$\beta$$) is equal to ___________.</p>
Options:
[] | 80
Explanation:
<p>Here $$|z - 3| < 1$$</p>
<p>$$ \Rightarrow {(x - 3)^2} + {y^2} < 1$$</p>
<p>and $$z = (4 + 3i) + \overline z (4 - 3i) \le 24$$</p>
<p>$$ \Rightarrow 4x - 3y \le 12$$</p>
<p>$$\tan \theta = {4 \over 3}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l5v605na/b9bf1f33-... |
<p>Let $$A = \{ z \in C:1 \le |z - (1 + i)| \le 2\} $$</p>
<p>and $$B = \{ z \in A:|z - (1 - i)| = 1\} $$. Then, B :</p>
Options:
[{"identifier": "A", "content": "is an empty set"}, {"identifier": "B", "content": "contains exactly two elements"}, {"identifier": "C", "content": "contains exactly three elements"}, {"ide... | ["D"]
Explanation:
<p>Let, $$z = x + iy$$</p>
<p>Given, $$1 \le \left| {z - (1 + i)} \right| \le 2$$</p>
<p>$$ \Rightarrow 1 \le \left| {x + iy - 1 - i} \right| \le 2$$</p>
<p>$$ \Rightarrow 1 \le \left| {(x - 1) + i(y - 1)} \right| \le 2$$</p>
<p>$$ \Rightarrow 1 \le \sqrt {{{(x - 1)}^2} + {{(y - 1)}^2}} \le 2$$</p>... |
<p>For $$\mathrm{n} \in \mathbf{N}$$, let $$\mathrm{S}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-3+2 i|=\frac{\mathrm{n}}{4}\right\}$$ and $$\mathrm{T}_{\mathrm{n}}=\left\{z \in \mathbf{C}:|z-2+3 i|=\frac{1}{\mathrm{n}}\right\}$$. Then the number of elements in the set $$\left\{n \in \mathbf{N}: S_{n} \cap T_{n}=\phi\rig... | ["D"]
Explanation:
$S_{n}=\left\{z \in C:|z-3+2 i|=\frac{n}{4}\right\}$ represents a circle with centre $C_{1}(3,-2)$ and radius $r_{1}=\frac{n}{4}$
<br/><br/>
Similarly $T_{n}$ represents circle with centre $C_{2}(2,-3)$ and radius $r_{2}=\frac{1}{n}$
<br/><br/>
$\text { As } S_{n} \cap T_{n}=\phi$<br/><br/>
$$
\begi... |
<p>For $$z \in \mathbb{C}$$ if the minimum value of $$(|z-3 \sqrt{2}|+|z-p \sqrt{2} i|)$$ is $$5 \sqrt{2}$$, then a value Question: of $$p$$ is _____________.</p>
Options:
[{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "$$\\frac{7}{2}$$"}, {"identifier": "C", "content": "4"}, {"identifier": "D", ... | ["C"]
Explanation:
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7bug80c/85c5d49d-7e9c-43b0-b928-429394532f1d/d5c2fcc0-25fe-11ed-b97b-f3de74335fe6/file-1l7bug80d.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1l7bug80c/85c5d49d-7e9c-43b0-b928-429394532f1d/d5c2fcc0-25fe-11... |
<p>If $$z=x+i y$$ satisfies $$|z|-2=0$$ and $$|z-i|-|z+5 i|=0$$, then :</p>
Options:
[{"identifier": "A", "content": "$$x+2 y-4=0$$"}, {"identifier": "B", "content": "$$x^{2}+y-4=0$$"}, {"identifier": "C", "content": "$$x+2 y+4=0$$"}, {"identifier": "D", "content": "$$x^{2}-y+3=0$$"}] | ["C"]
Explanation:
<p>$$|z - i| = |z + 5i|$$</p>
<p>So, $$\mathrm{z}$$ lies on $${ \bot ^r}$$ bisector of $$(0,1)$$ and $$(0, - 5)$$</p>
<p>i.e., line $$y = - 2$$</p>
<p>as $$|z| = 2$$</p>
<p>$$ \Rightarrow z = - 2i$$</p>
<p>$$x = 0$$ and $$y = - 2$$</p>
<p>so, $$x + 2y + 4 = 0$$</p> |
<p>Let the minimum value $$v_{0}$$ of $$v=|z|^{2}+|z-3|^{2}+|z-6 i|^{2}, z \in \mathbb{C}$$ is attained at $${ }{z}=z_{0}$$. Then $$\left|2 z_{0}^{2}-\bar{z}_{0}^{3}+3\right|^{2}+v_{0}^{2}$$ is equal to :</p>
Options:
[{"identifier": "A", "content": "1000"}, {"identifier": "B", "content": "1024"}, {"identifier": "C",... | ["A"]
Explanation:
<p>Let $$z = x + iy$$</p>
<p>$$v = {x^2} + {y^2} + {(x - 3)^2} + {y^2} + {x^2} + {(y - 6)^2}$$</p>
<p>$$ = (3{x^2} - 6x + 9) + (3{y^2} - 12y + 36)$$</p>
<p>$$ = 3({x^2} + {y^2} - 2x - 4y + 15)$$</p>
<p>$$ = 3[{(x - 1)^2} + {(y - 2)^2} + 10]$$</p>
<p>$${v_{\min }}$$ at $$z = 1 + 2i = {z_0}$$ and $${v... |
<p>Let $$S_{1}=\left\{z_{1} \in \mathbf{C}:\left|z_{1}-3\right|=\frac{1}{2}\right\}$$ and $$S_{2}=\left\{z_{2} \in \mathbf{C}:\left|z_{2}-\right| z_{2}+1||=\left|z_{2}+\right| z_{2}-1||\right\}$$. Then, for $$z_{1} \in S_{1}$$ and $$z_{2} \in S_{2}$$, the least value of $$\left|z_{2}-z_{1}\right|$$ is :</p>
Options:
[... | ["C"]
Explanation:
<p>$$\because$$ $${\left| {{Z_2} + |{Z_2} - 1|} \right|^2} = {\left| {{Z_2} - |{Z_2} + 1|} \right|^2}$$</p>
<p>$$ \Rightarrow \left( {{Z_2} + |{Z_2} - 1|} \right)\left( {{{\overline Z }_2} + |{Z_2} - 1|} \right) = \left( {{Z_2} - |{Z_2} + 1|} \right)\left( {{{\overline Z }_2} - |{Z_2} + 1|} \right)$... |
<p>If $$z \neq 0$$ be a complex number such that $$\left|z-\frac{1}{z}\right|=2$$, then the maximum value of $$|z|$$ is :</p>
Options:
[{"identifier": "A", "content": "$$\\sqrt{2}$$"}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "$$\\sqrt{2}-1$$"}, {"identifier": "D", "content": "$$\\sqrt{2}+1$... | ["D"]
Explanation:
<p>We know,</p>
<p>$$\left| {|{z_1}| - |{z_2}|} \right| \le \left| {{z_1} + {z_2}} \right| \le |{z_1}| + |{z_2}|$$</p>
<p>$$\therefore$$ $$\left| {|z| - {1 \over {|z|}}} \right| \le \left| {z - {1 \over z}} \right|$$</p>
<p>$$ \Rightarrow \left| {|z| - {1 \over {|z|}}} \right| \le 2$$ [Given $$\left... |
<p>Let $$z$$ be a complex number such that $$\left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i$$. Then $$z$$ lies on the circle of radius 2 and centre :</p>
Options:
[{"identifier": "A", "content": "(0, $$-$$2)"}, {"identifier": "B", "content": "(0, 0)"}, {"identifier": "C", "content": "(0, 2)"}, {"identifier":... | ["A"]
Explanation:
$\left|\frac{z-2 i}{z+i}\right|=2$
<br/><br/>
$\Rightarrow (z-2 i)(\bar{z}+2 i)=4(z+i)(\bar{z}-i)$
<br/><br/>
$\Rightarrow z \bar{z}+2 i z-2 i \bar{z}+4=4(z \bar{z}-z i+\overline{z i}+1)$
<br/><br/>
$\Rightarrow 3 z \bar{z}-6 i z+6 i \bar{z}=0$
<br/><br/>
$\Rightarrow z \bar{z}-2 i z+2 i \bar{z}=0$
... |
<p>If for $$z=\alpha+i \beta,|z+2|=z+4(1+i)$$, then $$\alpha+\beta$$ and $$\alpha \beta$$ are the roots of the equation :</p>
Options:
[{"identifier": "A", "content": "$$x^{2}+2 x-3=0$$"}, {"identifier": "B", "content": "$$x^{2}+3 x-4=0$$"}, {"identifier": "C", "content": "$$x^{2}+x-12=0$$"}, {"identifier": "D", "cont... | ["D"]
Explanation:
Given : $|z+2|=z+4(1+i)$
<br/><br/>Also, $z=\alpha+i \beta$
<br/><br/>$$
\begin{aligned}
& \therefore|z+2|=|\alpha+i \beta+2|=(\alpha+i \beta)+4+4 i \\\\
& \Rightarrow|(\alpha+2)+i \beta|=(\alpha+4)+i(\beta+4) \\\\
& \Rightarrow \sqrt{(\alpha+2)^2+\beta^2}=(\alpha+4)+i(\beta+4) \\\\
& \Rightarrow \b... |
If $z$ is a complex number such that $|z| \leqslant 1$, then the minimum value of $\left|z+\frac{1}{2}(3+4 i)\right|$ is :
Options:
[{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$\\frac{5}{2}$"}, {"identifier": "C", "content": "$\\frac{3}{2}$"}, {"identifier": "D", "content": "3"}] | ["C"]
Explanation:
<p>To find the minimum value of $\left|z+\frac{1}{2}(3+4i)\right|$, where $z$ is a complex number with $|z| \leqslant 1$, we can think of this geometrically as the distance from any point inside or on the boundary of the unit circle in the complex plane to the fixed point $\frac{1}{2}(3+4i)$.</p>
<... |
<p>The sum of the square of the modulus of the elements in the set $$\{z=\mathrm{a}+\mathrm{ib}: \mathrm{a}, \mathrm{b} \in \mathbf{Z}, z \in \mathbf{C},|z-1| \leq 1,|z-5| \leq|z-5 \mathrm{i}|\}$$ is __________.</p>
Options:
[] | 9
Explanation:
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lw3in2th/8eab885e-7362-426e-8f81-a5a9157aded7/ca24c350-105b-11ef-9330-f7b57aaea8e9/file-1lw3in2ti.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lw3in2th/8eab885e-7362-426e-8f81-a5a9157aded7/ca24c350-105b-11ef-9... |
<p>Consider the following two statements :</p>
<p>Statement I: For any two non-zero complex numbers $$z_1, z_2,(|z_1|+|z_2|)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right) \text {, and }$$</p>
<p>Statement II : If $$x, y, z$$ are three distinct... | ["B"]
Explanation:
<p>$$\begin{aligned}
& \frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}=\lambda \\
& \Rightarrow a^2=\lambda^2|(y-z)|^2 \\
& b^2=\lambda^2|(z-x)|^2 \\
& c^2=\lambda^2|(x-y)|^2 \\
& \frac{a^2(\overline{y-z})}{(y-z)(y-z)}=\frac{a^2(\bar{y}-\bar{z})}{|y-z|^2}=\frac{a^2(\bar{y}-\bar{z})}{\frac{a^2}{\lamb... |
<p>If $$z_1, z_2$$ are two distinct complex number such that $$\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2$$, then</p>
Options:
[{"identifier": "A", "content": "either $$z_1$$ lies on a circle of radius $$\\frac{1}{2}$$ or $$z_2$$ lies on a circle of radius 1.\n"}, {"identifier": "B", "content": "$$z_1$... | ["C"]
Explanation:
<p>$$\begin{aligned}
& \left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2 \\
& \left|z_1-2 z_2\right|=\left|1-2 z_1 \bar{z}_2\right| \\
& \Rightarrow\left(z_1-2 z_2\right)\left(\bar{z}_1-2 \bar{z}_2\right)=\left(1-2 z_1 \bar{z}_2\right)\left(1-2 \bar{z}_1 z_2\right) \\
& \Rightarrow\left|z_1... |
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