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|---|---|---|---|---|---|---|---|---|---|---|---|
WGkJRKvPHKwf6Krn467k9k2k5isco4j | maths | circle | tangent-and-normal | A circle touches the y-axis at the point (0, 4)
and passes through the point (2, 0). Which of
the following lines is not a tangent to this circle? | [{"identifier": "A", "content": "3x \u2013 4y \u2013 24 = 0"}, {"identifier": "B", "content": "4x + 3y \u2013 8 = 0"}, {"identifier": "C", "content": "3x + 4y \u2013 6 = 0"}, {"identifier": "D", "content": "4x \u2013 3y + 17 = 0"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264590/exam_images/u5rwyx3pmckseb9azywf.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2020 (Online) 9th January Morning Slot Mathematics - Circle Question 99 English Explanation">
Equation of family... | mcq | jee-main-2020-online-9th-january-morning-slot | 5,329 |
1zRTbzNUBHDRmyqbal1kmjamamd | maths | circle | tangent-and-normal | The line 2x $$-$$ y + 1 = 0 is a tangent to the circle at the point (2, 5) and the centre of the circle lies on x $$-$$ 2y = 4. Then, the radius of the circle is : | [{"identifier": "A", "content": "5$$\\sqrt 3 $$"}, {"identifier": "B", "content": "4$$\\sqrt 5 $$"}, {"identifier": "C", "content": "3$$\\sqrt 5 $$"}, {"identifier": "D", "content": "5$$\\sqrt 4 $$"}] | ["C"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263455/exam_images/vmasvdkamze2tghom94b.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 17th March Morning Shift Mathematics - Circle Question 85 English Explanation">
<br>$${m_1} \times... | mcq | jee-main-2021-online-17th-march-morning-shift | 5,331 |
dUVWV5PxwRaQBO9JX91kmklu5s8 | maths | circle | tangent-and-normal | Let the tangent to the circle x<sup>2</sup> + y<sup>2</sup> = 25 at the point R(3, 4) meet x-axis and y-axis at points P and Q, respectively. If r is the radius of the circle passing through the origin O and having centre at the incentre of the triangle OPQ, then r<sup>2</sup> is equal to : | [{"identifier": "A", "content": "$${{585} \\over {66}}$$"}, {"identifier": "B", "content": "$${{625} \\over {72}}$$"}, {"identifier": "C", "content": "$${{529} \\over {64}}$$"}, {"identifier": "D", "content": "$${{125} \\over {72}}$$"}] | ["B"] | null | Given equation of circle<br><br>x<sup>2</sup> + y<sup>2</sup> = 25<br><br>$$ \therefore $$ Tangent equation at (3, 4)<br><br>T : 3x + 4y = 25<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267221/exam_images/arrymdsbetsftxxqter6.webp" style="max-width: 100%;height: auto;display: block;margin: 0... | mcq | jee-main-2021-online-17th-march-evening-shift | 5,332 |
APA6e6vHNrsxIPXcgo1kmkme0oa | maths | circle | tangent-and-normal | Two tangents are drawn from a point P to the circle x<sup>2</sup> + y<sup>2</sup> $$-$$ 2x $$-$$ 4y + 4 = 0, such that the angle between these tangents is $${\tan ^{ - 1}}\left( {{{12} \over 5}} \right)$$, where $${\tan ^{ - 1}}\left( {{{12} \over 5}} \right)$$ $$\in$$(0, $$\pi$$). If the centre of the circle is denote... | [{"identifier": "A", "content": "3 : 1"}, {"identifier": "B", "content": "9 : 4"}, {"identifier": "C", "content": "2 : 1"}, {"identifier": "D", "content": "11 : 4"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265875/exam_images/hsjo5h1locnmw5jsj3fw.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 17th March Evening Shift Mathematics - Circle Question 82 English Explanation">
<br>Let $$\theta$$... | mcq | jee-main-2021-online-17th-march-evening-shift | 5,333 |
1ks093wsl | maths | circle | tangent-and-normal | Two tangents are drawn from the point P($$-$$1, 1) to the circle x<sup>2</sup> + y<sup>2</sup> $$-$$ 2x $$-$$ 6y + 6 = 0. If these tangents touch the circle at points A and B, and if D is a point on the circle such that length of the segments AB and AD are equal, then the area of the triangle ABD is equal to : | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "$$(3\\sqrt 2 + 2)$$"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "$$3(\\sqrt 2 - 1)$$"}] | ["C"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265883/exam_images/j4eb4u3olf4aobpzkvfj.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734263397/exam_images/a0quemgl8enbtkiviozm.webp"><img src="https://res.c... | mcq | jee-main-2021-online-27th-july-morning-shift | 5,334 |
1ktkesbfz | maths | circle | tangent-and-normal | Let B be the centre of the circle x<sup>2</sup> + y<sup>2</sup> $$-$$ 2x + 4y + 1 = 0. Let the tangents at two points P and Q on the circle intersect at the point A(3, 1). Then 8.$$\left( {{{area\,\Delta APQ} \over {area\,\Delta BPQ}}} \right)$$ is equal to _____________. | [] | null | 18 | <p> <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1kxvqvz06/35771b20-6f71-497a-bc46-b4444dec857f/6a6f4960-6af6-11ec-b350-33e20cd86462/file-1kxvqvz07.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1kxvqvz06/35771b20-6f71-497a-bc46-b4444dec857f/6a6f4960-6af6-11ec-b350-33e20cd8646... | integer | jee-main-2021-online-31st-august-evening-shift | 5,335 |
1l59kc4xu | maths | circle | tangent-and-normal | <p>A circle touches both the y-axis and the line x + y = 0. Then the locus of its center is :</p> | [{"identifier": "A", "content": "$$y = \\sqrt 2 x$$"}, {"identifier": "B", "content": "$$x = \\sqrt 2 y$$"}, {"identifier": "C", "content": "$${y^2} - {x^2} = 2xy$$"}, {"identifier": "D", "content": "$${x^2} - {y^2} = 2xy$$"}] | ["D"] | null | <p>Let the centre be (h, k)</p>
<p>So, $$\left| h \right| = \left| {{{h + k} \over {\sqrt 2 }}} \right|$$</p>
<p>$$ \Rightarrow 2{h^2} = {h^2} + {k^2} + 2hk$$</p>
<p>Locus will be $${x^2} - {y^2} = 2xy$$</p> | mcq | jee-main-2022-online-25th-june-evening-shift | 5,340 |
1l6kk85jt | maths | circle | tangent-and-normal | <p>A circle $$C_{1}$$ passes through the origin $$\mathrm{O}$$ and has diameter 4 on the positive $$x$$-axis. The line $$y=2 x$$ gives a chord $$\mathrm{OA}$$ of circle $$\mathrm{C}_{1}$$. Let $$\mathrm{C}_{2}$$ be the circle with $$\mathrm{OA}$$ as a diameter. If the tangent to $$\mathrm{C}_{2}$$ at the point $$\mathr... | [{"identifier": "A", "content": "1 : 4"}, {"identifier": "B", "content": "1 : 5"}, {"identifier": "C", "content": "2 : 5"}, {"identifier": "D", "content": "1 : 3"}] | ["A"] | null | <p>Equation of C<sub>1</sub></p>
<p>$${x^2} + {y^2} - 4x = 0$$</p>
<p>Intersection with</p>
<p>$$y = 2x$$</p>
<p>$${x^2} + 4{x^2} - 4x = 0$$</p>
<p>$$5{x^2} - 4x = 0$$</p>
<p>$$ \Rightarrow x = 0,{4 \over 5}$$</p>
<p>$$y = 0,{8 \over 5}$$</p>
<p>$$A:\left( {{4 \over 5},{8 \over 5}} \right)$$</p>
<p><img src="https://ap... | mcq | jee-main-2022-online-27th-july-evening-shift | 5,342 |
1l6nndhie | maths | circle | tangent-and-normal | <p>Let the tangents at two points $$\mathrm{A}$$ and $$\mathrm{B}$$ on the circle $$x^{2}+\mathrm{y}^{2}-4 x+3=0$$ meet at origin $$\mathrm{O}(0,0)$$. Then the area of the triangle $$\mathrm{OAB}$$ is :</p> | [{"identifier": "A", "content": "$$\\frac{3 \\sqrt{3}}{2}$$"}, {"identifier": "B", "content": "$$\\frac{3 \\sqrt{3}}{4}$$"}, {"identifier": "C", "content": "$$\\frac{3}{2 \\sqrt{3}}$$"}, {"identifier": "D", "content": "$$\\frac{3}{4 \\sqrt{3}}$$"}] | ["B"] | null | <p>$${x^2} + {y^2} - 4x + 3 = 0$$</p>
<p>$$ \Rightarrow {(x - 2)^2} + {y^2} = 1$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1l7rrgzao/8d049878-91a0-486a-bf74-ffc8968e3205/b9d7eb00-2ebf-11ed-b92e-01f1dabc9173/file-1l7rrgzap.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/ima... | mcq | jee-main-2022-online-28th-july-evening-shift | 5,343 |
1ldr7omt6 | maths | circle | tangent-and-normal | <p>Let $$y=x+2,4y=3x+6$$ and $$3y=4x+1$$ be three tangent lines to the circle $$(x-h)^2+(y-k)^2=r^2$$. Then $$h+k$$ is equal to :</p> | [{"identifier": "A", "content": "6"}, {"identifier": "B", "content": "5 (1 + $$\\sqrt2$$)"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "5$$\\sqrt2$$"}] | ["C"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1leq10yl8/23860a94-17c3-4a42-9ba1-82deb3b85bcd/09ff05c0-b861-11ed-8195-4f3c56fa1eb5/file-1leq10yl9.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1leq10yl8/23860a94-17c3-4a42-9ba1-82deb3b85bcd/09ff05c0-b861-11ed-8195-4f3c56fa1eb5... | mcq | jee-main-2023-online-30th-january-morning-shift | 5,344 |
1ldu69w0u | maths | circle | tangent-and-normal | <p>Points P($$-$$3, 2), Q(9, 10) and R($$\alpha,4$$) lie on a circle C and PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line $$2x-ky=1$$, then k is equal to ____________.</p> | [] | null | 3 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lef6afmm/c04cf9db-da6e-4771-935e-0ffcde0666c2/fff172f0-b268-11ed-9d4d-b96eca78f2e5/file-1lef6afmn.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lef6afmm/c04cf9db-da6e-4771-935e-0ffcde0666c2/fff172f0-b268-11ed-9d4d-b96eca78f2e5/fi... | integer | jee-main-2023-online-25th-january-evening-shift | 5,346 |
1lgow4qau | maths | circle | tangent-and-normal | <p>Let the centre of a circle C be $$(\alpha, \beta)$$ and its radius $$r < 8$$. Let $$3 x+4 y=24$$ and $$3 x-4 y=32$$ be two tangents and $$4 x+3 y=1$$ be a normal to C. Then $$(\alpha-\beta+r)$$ is equal to :</p> | [{"identifier": "A", "content": "7"}, {"identifier": "B", "content": "9"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "6"}] | ["A"] | null | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lh1kl8kh/309857da-6cb3-42cd-b0c0-e753c4a2dc87/993e3110-e652-11ed-b540-cb85a096fb04/file-1lh1kl8ki.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lh1kl8kh/309857da-6cb3-42cd-b0c0-e753c4a2dc87/993e3110-e652-11ed-b540-cb85a096fb04/fi... | mcq | jee-main-2023-online-13th-april-evening-shift | 5,347 |
1lgykzejb | maths | circle | tangent-and-normal | <p>Let O be the origin and OP and OQ be the tangents to the circle $$x^2+y^2-6x+4y+8=0$$ at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point $$\left( {\alpha ,{1 \over 2}} \right)$$, then a value of $$\alpha$$ is :</p> | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "$$-\\frac{1}{2}$$"}, {"identifier": "C", "content": "$$\\frac{5}{2}$$"}, {"identifier": "D", "content": "$$\\frac{3}{2}$$"}] | ["C"] | null | Centre $(3,-2)$
<br/><br/>Equation of circumcircle is
<br/><br/>$$
\begin{aligned}
& x(x-3)+y(y+2)=0 \\\\
& \Rightarrow x^2-3 x+y^2+2 y=0
\end{aligned}
$$
<br/><br/>Since $\left(\alpha, \frac{1}{2}\right)$ is on the circle
<br/><br/>$$
\begin{aligned}
& \text { So } \alpha^2-3 \alpha+\frac{1}{4}+1=0 \\\\
& \Rightarrow ... | mcq | jee-main-2023-online-8th-april-evening-shift | 5,348 |
1lh2y790j | maths | circle | tangent-and-normal | <p>If the tangents at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the circle $$x^{2}+y^{2}-2 x+y=5$$ meet at the point $$R\left(\frac{9}{4}, 2\right)$$, then the area of the triangle $$\mathrm{PQR}$$ is :</p> | [{"identifier": "A", "content": "$$\\frac{13}{8}$$"}, {"identifier": "B", "content": "$$\\frac{5}{8}$$"}, {"identifier": "C", "content": "$$\\frac{5}{4}$$"}, {"identifier": "D", "content": "$$\\frac{13}{4}$$"}] | ["B"] | null | Equation of circle $x^2+y^2-2 x+y-5=0$
<br><br>On comparing with $x^2+y^2+2 g x+2 f y+c=0$
<br><br>$$
\begin{gathered}
2 g=-2,2 f=1, c=-5 \\\\
g=-1, f=\frac{1}{2}, c=-5
\end{gathered}
$$
<br><br>$\therefore$ Radius of the circle
<br><br>$$
r=\sqrt{(-1)^2+\left(\frac{1}{2}\right)^2+5}=\frac{5}{2}
$$
<br><br><img src="h... | mcq | jee-main-2023-online-6th-april-evening-shift | 5,349 |
jaoe38c1lsconj9i | maths | circle | tangent-and-normal | <p>Consider a circle $$(x-\alpha)^2+(y-\beta)^2=50$$, where $$\alpha, \beta>0$$. If the circle touches the line $$y+x=0$$ at the point $$P$$, whose distance from the origin is $$4 \sqrt{2}$$, then $$(\alpha+\beta)^2$$ is equal to __________.</p> | [] | null | 100 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt1x38le/83e15979-b7dd-4565-bdb1-2ae72cdc0d26/13ec8220-d416-11ee-b9d5-0585032231f0/file-1lt1x38lf.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lt1x38le/83e15979-b7dd-4565-bdb1-2ae72cdc0d26/13ec8220-d416-11ee-b9d5-0585032231f0... | integer | jee-main-2024-online-27th-january-evening-shift | 5,350 |
lv7v3k7d | maths | circle | tangent-and-normal | <p>Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point $$(3,2)$$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $$(5,5)$$ is :</p> | [{"identifier": "A", "content": "4$$\\sqrt2$$"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "5"}, {"identifier": "D", "content": "2$$\\sqrt2$$"}] | ["B"] | null | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwgjwq15/c6544ab2-5387-4f8a-8398-9aeaf56b974c/efedcb90-1786-11ef-9978-292aa9baaa14/file-1lwgjwq16.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwgjwq15/c6544ab2-5387-4f8a-8398-9aeaf56b974c/efedcb90-1786-11ef-9978-292aa9baaa14... | mcq | jee-main-2024-online-5th-april-morning-shift | 5,351 |
xG2EPTYG5Tf9yrpg | maths | complex-numbers | algebra-of-complex-numbers | If $$\left| {z - 4} \right| < \left| {z - 2} \right|$$, its solution is given by : | [{"identifier": "A", "content": "$${\\mathop{\\rm Re}\\nolimits} (z) > 0$$ "}, {"identifier": "B", "content": "$${\\mathop{\\rm Re}\\nolimits} (z) < 0$$"}, {"identifier": "C", "content": "$${\\mathop{\\rm Re}\\nolimits} (z) > 3$$"}, {"identifier": "D", "content": "$${\\mathop{\\rm Re}\\nolimits} (z) > 2$$"}... | ["C"] | null | Given $$\left| {z - 4} \right| < \left| {z - 2} \right|$$
<br><br>Let $$\,\,\,z = x + iy$$
<br><br>$$ \Rightarrow \left| {\left. {\left( {x - 4} \right) + iy} \right)} \right| < \left| {\left( {x - 2} \right) + iy} \right|$$
<br><br>$$ \Rightarrow {\left( {x - 4} \right)^2} + {y^2} < {\left( {x - 2} \right)^2... | mcq | aieee-2002 | 5,352 |
dBRnY1IywoUv56wj | maths | complex-numbers | algebra-of-complex-numbers | If $${z^2} + z + 1 = 0$$, where z is complex number, then value of $${\left( {z + {1 \over z}} \right)^2} + {\left( {{z^2} + {1 \over {{z^2}}}} \right)^2} + {\left( {{z^3} + {1 \over {{z^3}}}} \right)^2} + .......... + {\left( {{z^6} + {1 \over {{z^6}}}} \right)^2}$$ is : | [{"identifier": "A", "content": "18"}, {"identifier": "B", "content": "54"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "12"}] | ["D"] | null | $${z^2} + z + 1 = 0 \Rightarrow z = \omega \,\,\,$$ or $$\,\,\,{\omega ^2}$$
<br><br>So, $$z + {1 \over z} = \omega + {\omega ^2} = - 1$$
<br><br>$${z^2} + {1 \over {{z^2}}} = {\omega ^2} + \omega = - 1,$$
<br><br>$${z^3} + {1 \over {{z^3}}} = {\omega ^3} + {\omega ^3} = 2$$
<br><br>$${z^4} + {1 \over {{z^4}}} =... | mcq | aieee-2006 | 5,353 |
LhUeHRBHKbPCLuNY | maths | complex-numbers | algebra-of-complex-numbers | If $$\,\left| {z - {4 \over z}} \right| = 2,$$ then the maximum value of $$\,\left| z \right|$$ is equal to : | [{"identifier": "A", "content": "$$\\sqrt 5 + 1$$ "}, {"identifier": "B", "content": "2 "}, {"identifier": "C", "content": "$$2 + \\sqrt 2 $$ "}, {"identifier": "D", "content": "$$\\sqrt 3 + 1$$ "}] | ["A"] | null | Given that $$\left| {z - {4 \over z}} \right| = 2$$
<br><br>Now $$\left| z \right| = \left| {z - {4 \over z} + {4 \over { - z}}} \right| \le \left| {z - {4 \over z}} \right| + {4 \over {\left| z \right|}}$$
<br><br>$$ \Rightarrow \left| z \right| \le 2 + {4 \over {\left| z \right|}}$$
<br><br>$$ \Rightarrow {\left| z \... | mcq | aieee-2009 | 5,354 |
cvg6bc045DsjHMd2gYK21 | maths | complex-numbers | algebra-of-complex-numbers | The point represented by 2 + <i>i</i> in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there $$2\sqrt 2 $$ units in the south-westwardsdirection. Then its new position in the Argand plane is at the point represented by : | [{"identifier": "A", "content": "2 + 2i"}, {"identifier": "B", "content": "1 + i"}, {"identifier": "C", "content": "$$-$$1 $$-$$ i"}, {"identifier": "D", "content": "$$-$$2 $$-$$2i"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267287/exam_images/vpkjlli9hudoskpjp7ap.webp" style="max-width: 100%; height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2016 (Online) 9th April Morning Slot Mathematics - Complex Numbers Question 136 English Explanation">
<br><br>... | mcq | jee-main-2016-online-9th-april-morning-slot | 5,355 |
9MmkhQB3ccxn864L | maths | complex-numbers | algebra-of-complex-numbers | Let $$\omega $$ be a complex number such that 2$$\omega $$ + 1 = z where z = $$\sqrt {-3} $$. If
<br/><br/>$$\left| {\matrix{
1 & 1 & 1 \cr
1 & { - {\omega ^2} - 1} & {{\omega ^2}} \cr
1 & {{\omega ^2}} & {{\omega ^7}} \cr
} } \right| = 3k$$,
<br/><br/>then k is equal to : | [{"identifier": "A", "content": "z"}, {"identifier": "B", "content": "-1"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "-z"}] | ["D"] | null | Given 2$$\omega $$ + 1 = z;
<br><br>z = $$\sqrt 3 i$$
<br><br>$$ \Rightarrow $$ $$\omega = {{\sqrt 3 i - 1} \over 2}$$
<br><br>$$ \Rightarrow $$ As $$\omega $$ is complex cube root of unity.
<br><br>$${\omega ^3} = 1$$
<br><br>$$1 + \omega + {\omega ^2} = 0$$
<br><br>$$\left| {\matrix{
1 & 1 & 1 \cr
1... | mcq | jee-main-2017-offline | 5,357 |
90f5UU5PdLZXNwMDULEQU | maths | complex-numbers | algebra-of-complex-numbers | The set of all $$\alpha $$ $$ \in $$ <b>R</b>, for which w = $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$ is purely imaginary number, for all z $$ \in $$ <b>C</b> satisfying |z| = 1 and Re z $$ \ne $$ 1, is : | [{"identifier": "A", "content": "an empty set"}, {"identifier": "B", "content": "{0}"}, {"identifier": "C", "content": "$$\\left\\{ {0,{1 \\over 4}, - {1 \\over 4}} \\right\\}$$"}, {"identifier": "D", "content": "equal to <b>R</b> "}] | ["B"] | null | As w = $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$, w is purely imaginary<br><br>
$$ \therefore w$$ + $$\bar w$$ = 0<br><br>
$$ \Rightarrow $$ $${{1 + \left( {1 - 8\alpha } \right)z} \over {1 - z}}$$ + $${{1 + \left( {1 - 8\alpha } \right)\bar z} \over {1 - \bar z}}$$ = 0<br><br>
$$ \Rightarrow $$ [1 + (1... | mcq | jee-main-2018-online-15th-april-morning-slot | 5,358 |
QOeM4OKvohAunsfuxvotj | maths | complex-numbers | algebra-of-complex-numbers | Let
<br/>A = $$\left\{ {\theta \in \left( { - {\pi \over 2},\pi } \right):{{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}is\,purely\,imaginary} \right\}$$
<br/>. Then the sum of the elements in A is : | [{"identifier": "A", "content": "$${5\\pi \\over 6}$$"}, {"identifier": "B", "content": "$$\\pi $$"}, {"identifier": "C", "content": "$${3\\pi \\over 4}$$"}, {"identifier": "D", "content": "$${{2\\pi } \\over 3}$$"}] | ["D"] | null | Given complex number,
<br><br>$${{3 + 2i\sin \theta } \over {1 - 2i\sin \theta }}$$
<br><br>$$ = {{\left( {3 + 2i\sin \theta } \right)\left( {1 + 2i\sin \theta } \right)} \over {1 + 4{{\sin }^2}\theta }}$$
<br><br>$$ = {{3 + 6i\sin \theta + 2i\sin \theta - 4{{\sin }^2}\theta } \over {1 + 4{{\sin }^2}\theta }}$$
<br><... | mcq | jee-main-2019-online-9th-january-morning-slot | 5,359 |
2kgwaW5saoEQDxFpBDSmf | maths | complex-numbers | algebra-of-complex-numbers | If $${{z - \alpha } \over {z + \alpha }}\left( {\alpha \in R} \right)$$ is a purely imaginary number and | z | = 2, then a value of $$\alpha $$ is : | [{"identifier": "A", "content": "$${1 \\over 2}$$"}, {"identifier": "B", "content": "$$\\sqrt 2 $$"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "1"}] | ["C"] | null | $${{z - \alpha } \over {z + \alpha }} + {{\overline z - \alpha } \over {\overline z + \alpha }} = 0$$
<br><br>$$z\overline z + z\alpha - \alpha \overline z - {\alpha ^2} + z\overline z - z\alpha + \overline z \alpha - {\alpha ^2} = 0$$
<br><br>$${\left| z \right|^2} = {\alpha ^2},$$ $$a = \pm 2$$ | mcq | jee-main-2019-online-12th-january-morning-slot | 5,360 |
zgUc015yKq58iZfHNP18hoxe66ijvwujxie | maths | complex-numbers | algebra-of-complex-numbers | Let z $$ \in $$ C be such that |z| < 1.
<br/><br/> If $$\omega = {{5 + 3z} \over {5(1 - z)}}$$z, then : | [{"identifier": "A", "content": "4Im( $$\\omega$$) > 5"}, {"identifier": "B", "content": "5Im( $$\\omega$$) < 1"}, {"identifier": "C", "content": "5Re( $$\\omega$$) > 4"}, {"identifier": "D", "content": "5Re( $$\\omega$$) > 1"}] | ["D"] | null | $$\omega = {{5 + 3z} \over {5(1 - z)}}$$z
<br><br>$$ \Rightarrow $$ $$5\omega \left( {1 - z} \right) = 5 + 3z$$
<br><br>$$ \Rightarrow $$ $$5\omega - 5\omega z = 5 + 3z$$
<br><br>$$ \Rightarrow $$ $$5\omega - 5 = 5\omega z + 3z$$
<br><br>$$ \Rightarrow $$ $$z\left( {5\omega + 3} \right) = 5\left( {\omega - 1} \rig... | mcq | jee-main-2019-online-9th-april-evening-slot | 5,361 |
3NBa8Gu3laVRzWIBoOjgy2xukfurbxh6 | maths | complex-numbers | algebra-of-complex-numbers | The region represented by<br/> {z = x + iy $$ \in $$ C : |z| – Re(z) $$ \le $$ 1} is also given by the<br/> inequality :
{z = x + iy $$ \in $$ C : |z| – Re(z) $$ \le $$ 1} | [{"identifier": "A", "content": "y<sup>2</sup> $$ \\le $$ $$2\\left( {x + {1 \\over 2}} \\right)$$"}, {"identifier": "B", "content": "y<sup>2</sup> $$ \\le $$ $${x + {1 \\over 2}}$$"}, {"identifier": "C", "content": "y<sup>2</sup> $$ \\ge $$ 2(x + 1)"}, {"identifier": "D", "content": "y<sup>2</sup> $$ \\ge $$ x + 1"}] | ["A"] | null | Given z = x + iy
<br><br> |z| – Re(z) $$ \le $$ 1
<br><br>$$ \Rightarrow $$ $$\sqrt {{x^2} + {y^2}} $$ - x $$ \le $$ 1
<br><br>$$ \Rightarrow $$ $$\sqrt {{x^2} + {y^2}} $$ $$ \le $$ 1 + x
<br><br>$$ \Rightarrow $$ x<sup>2</sup> + y<sup>2</sup> $$ \le $$ 1 + 2x + x<sup>2</sup>
<br><br>$$ \Rightarrow $$ y<sup>2</sup> $$... | mcq | jee-main-2020-online-6th-september-morning-slot | 5,363 |
KD70hRFZdAQ2A6EDwzjgy2xukg3931gx | maths | complex-numbers | algebra-of-complex-numbers | Let z = x + iy be a non-zero complex number
such that $${z^2} = i{\left| z \right|^2}$$, where i = $$\sqrt { - 1} $$ , then z lies
on the : | [{"identifier": "A", "content": "line, y = \u2013x"}, {"identifier": "B", "content": "real axis"}, {"identifier": "C", "content": "line, y = x"}, {"identifier": "D", "content": "imaginary axis"}] | ["C"] | null | Given z = x + iy
<br><br>and $${z^2} = i{\left| z \right|^2}$$
<br><br>$$ \Rightarrow $$ (x + iy)<sup>2</sup>
= i(x<sup>2</sup> + y<sup>2</sup>)
<br><br>$$ \Rightarrow $$ x<sup>2</sup> - y<sup>2</sup> + 2ixy = i(x<sup>2</sup> + y<sup>2</sup>) + 0
<br><br>Comparing both side we get,
<br><br>x<sup>2</sup> - y<sup>2</sup... | mcq | jee-main-2020-online-6th-september-evening-slot | 5,364 |
1krw3pbyy | maths | complex-numbers | algebra-of-complex-numbers | Let $$S = \left\{ {n \in N\left| {{{\left( {\matrix{
0 & i \cr
1 & 0 \cr
} } \right)}^n}\left( {\matrix{
a & b \cr
c & d \cr
} } \right) = \left( {\matrix{
a & b \cr
c & d \cr
} } \right)\forall a,b,c,d \in R} \right.} \right\}$$, where i = $$\sqrt { - 1} $$. ... | [] | null | 11 | Let $$X = \left( {\matrix{
a & b \cr
c & d \cr
} } \right)$$ & $$A = {\left( {\matrix{
0 & i \cr
1 & 0 \cr
} } \right)^n}$$<br><br>$$\Rightarrow$$ AX = IX<br><br>$$\Rightarrow$$ A = I<br><br>$$ \Rightarrow {\left( {\matrix{
0 & i \cr
1 & 0 \cr
} } \right)^... | integer | jee-main-2021-online-25th-july-morning-shift | 5,365 |
1kryfpffu | maths | complex-numbers | algebra-of-complex-numbers | If the real part of the complex number $$z = {{3 + 2i\cos \theta } \over {1 - 3i\cos \theta }},\theta \in \left( {0,{\pi \over 2}} \right)$$ is zero, then the value of sin<sup>2</sup>3$$\theta$$ + cos<sup>2</sup>$$\theta$$ is equal to _______________. | [] | null | 1 | Re $$(z) = {{3 - 6{{\cos }^2}\theta } \over {1 + 9{{\cos }^2}\theta }} = 0$$<br><br>$$\Rightarrow$$ $$\theta$$ = $${{\pi \over 4}}$$<br><br>Hence, sin<sup>2</sup>3$$\theta$$ + cos<sup>2</sup>$$\theta$$ = 1. | integer | jee-main-2021-online-27th-july-evening-shift | 5,366 |
1kteil093 | maths | complex-numbers | algebra-of-complex-numbers | If $$S = \left\{ {z \in C:{{z - i} \over {z + 2i}} \in R} \right\}$$, then : | [{"identifier": "A", "content": "S contains exactly two elements"}, {"identifier": "B", "content": "S contains only one element"}, {"identifier": "C", "content": "S is a circle in the complex plane"}, {"identifier": "D", "content": "S is a straight line in the complex plane"}] | ["D"] | null | Given $${{z - i} \over {z + 2i}} \in R$$<br><br>Then $$\arg \left( {{{z - i} \over {z + 2i}}} \right)$$ is 0 or $$\pi $$<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266523/exam_images/toje9q4ejzuaxxxnmphc.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy... | mcq | jee-main-2021-online-27th-august-morning-shift | 5,367 |
1l544xjnb | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$\alpha$$ and $$\beta$$ be the roots of the equation x<sup>2</sup> + (2i $$-$$ 1) = 0. Then, the value of |$$\alpha$$<sup>8</sup> + $$\beta$$<sup>8</sup>| is equal to :</p> | [{"identifier": "A", "content": "50"}, {"identifier": "B", "content": "250"}, {"identifier": "C", "content": "1250"}, {"identifier": "D", "content": "1500"}] | ["A"] | null | <p>Given equation,</p>
<p>$${x^2} + (2i - 1) = 0$$</p>
<p>$$ \Rightarrow {x^2} = 1 - 2i$$</p>
<p>Let $$\alpha$$ and $$\beta$$ are the two roots of the equation.</p>
<p>As, we know roots of a equation satisfy the equation so</p>
<p>$${\alpha ^2} = 1 - 2i$$</p>
<p>and $${\beta ^2} = 1 - 2i$$</p>
<p>$$\therefore$$ $${\alp... | mcq | jee-main-2022-online-29th-june-morning-shift | 5,368 |
1l545wee0 | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\} $$. Let $$|z - 4i|$$ attains minimum and maximum values, respectively, at z<sub>1</sub> $$\in$$ S and z<sub>2</sub> $$\in$$ S. If $$5(|{z_1}{|^2} + |{z_2}{|^2}) = \alpha + \beta \sqrt 5 $$, where $$\alpha$$ and $$\beta$$ are integers, then ... | [] | null | 26 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lc8b9cup/a3a99286-f2d0-4b94-be08-ef95f78a420f/8454b500-870a-11ed-95c4-a7dd61250e50/file-1lc8b9cuq.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lc8b9cup/a3a99286-f2d0-4b94-be08-ef95f78a420f/8454b500-870a-11ed-95c4-a7dd61250e50/fi... | integer | jee-main-2022-online-29th-june-morning-shift | 5,369 |
1l58gy62s | maths | complex-numbers | algebra-of-complex-numbers | <p>If $${z^2} + z + 1 = 0$$, $$z \in C$$, then <br/><br/>$$\left| {\sum\limits_{n = 1}^{15} {{{\left( {{z^n} + {{( - 1)}^n}{1 \over {{z^n}}}} \right)}^2}} } \right|$$ is equal to _________.</p> | [] | null | 2 | <p>$$\because$$ $${z^2} + z + 1 = 0$$</p>
<p>$$\Rightarrow$$ $$\omega$$ or $$\omega$$<sup>2</sup></p>
<p>$$\because$$ $$\left| {\sum\limits_{n = 1}^{15} {{{\left( {{z^n} + {{( - 1)}^n}{1 \over {{z^n}}}} \right)}^2}} } \right|$$</p>
<p>$$ = \left| {\sum\limits_{n = 1}^{15} {{z^{2n}} + \sum\limits_{n = 1}^{15} {{z^{ - 2n... | integer | jee-main-2022-online-26th-june-evening-shift | 5,370 |
1l6kiejn2 | maths | complex-numbers | algebra-of-complex-numbers | <p>Let S be the set of all $$(\alpha, \beta), \pi<\alpha, \beta<2 \pi$$, for which the complex number $$\frac{1-i \sin \alpha}{1+2 i \sin \alpha}$$ is purely imaginary and $$\frac{1+i \cos \beta}{1-2 i \cos \beta}$$ is purely real. Let $$Z_{\alpha \beta}=\sin 2 \alpha+i \cos 2 \beta,(\alpha, \beta) \in S$$. Then ... | [{"identifier": "A", "content": "3"}, {"identifier": "B", "content": "3 i"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "2 $$-$$ i"}] | ["C"] | null | <p>$$\because$$ $${{1 - i\sin \alpha } \over {1 + 2i\sin \alpha }}$$ is purely imaginary</p>
<p>$$\therefore$$ $${{1 - i\sin \alpha } \over {1 + 2i\sin \alpha }} + {{1 + i\sin \alpha } \over {1 - 2i\sin \alpha }} = 0$$</p>
<p>$$ \Rightarrow 1 - 2{\sin ^2}\alpha = 0$$</p>
<p>$$\therefore$$ $$\alpha = {{5\pi } \over 4}... | mcq | jee-main-2022-online-27th-july-evening-shift | 5,371 |
1l6npfu6z | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$\mathrm{z}=a+i b, b \neq 0$$ be complex numbers satisfying $$z^{2}=\bar{z} \cdot 2^{1-z}$$. Then the least value of $$n \in N$$, such that $$z^{n}=(z+1)^{n}$$, is equal to __________.</p> | [] | null | 6 | <p>$$\because$$ $${z^2} = \overline z \,.\,{2^{1 - |z|}}$$ ...... (1)</p>
<p>$$ \Rightarrow |z{|^2} = |\overline z |\,.\,{2^{1 - |z|}}$$</p>
<p>$$ \Rightarrow |z| = {2^{1 - |z|}}$$,</p>
<p>$$\because$$ $$b \ne 0 \Rightarrow |z| \ne 0$$</p>
<p>$$\therefore$$ $$|z| = 1$$ ...... (2)</p>
<p>$$\because$$ $$z = a + ib$$ then... | integer | jee-main-2022-online-28th-july-evening-shift | 5,372 |
1ldsg85m1 | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$\alpha = 8 - 14i,A = \left\{ {z \in c:{{\alpha z - \overline \alpha \overline z } \over {{z^2} - {{\left( {\overline z } \right)}^2} - 112i}}=1} \right\}$$ and $$B = \left\{ {z \in c:\left| {z + 3i} \right| = 4} \right\}$$. Then $$\sum\limits_{z \in A \cap B} {({\mathop{\rm Re}\nolimits} z - {\mathop{\rm Im}... | [] | null | 14 | <p>Let $$z = x + iy$$</p>
<p>and $$\alpha = 8 - 14i$$</p>
<p>$${{\alpha z - \overline \alpha \,\overline z } \over {{z^2} - {{\overline z }^2} - 112i}} = 1$$</p>
<p>$$\therefore$$ $${{(16y - 28x)} \over {4xy - 112i}} = 1$$</p>
<p>$$(16y - 28x + 112)i = 4xy$$</p>
<p>$$\therefore$$ $$z = - 7i$$ or 4</p>
<p>Now, $$z = ... | integer | jee-main-2023-online-29th-january-evening-shift | 5,373 |
1lgswb5vn | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$\mathrm{S}=\left\{z \in \mathbb{C}-\{i, 2 i\}: \frac{z^{2}+8 i z-15}{z^{2}-3 i z-2} \in \mathbb{R}\right\}$$. If $$\alpha-\frac{13}{11} i \in \mathrm{S}, \alpha \in \mathbb{R}-\{0\}$$, then
$$242 \alpha^{2}$$ is equal to _________.</p> | [] | null | 1680 | Put $z=x+i y$
<br/><br/>$$
\begin{aligned}
& \operatorname{lm}\left(\frac{z^2+8 i z-15}{z^2-3 i z-2}\right)=0 \\\\
& \Rightarrow-\left(x^2-y^2-8 y-15\right)(2 x y-3 x)+(2 x y+8 x)\left(x^2-\right. \left.y^2+3 y-2\right)=0 \\\\
& \Rightarrow\left(x^2-y^2\right)(2 x y+8 x-2 x y+3 x)+(8 y+15)(2 x y- 3 x)+(2 x y+8 x)(3 y-... | integer | jee-main-2023-online-11th-april-evening-shift | 5,375 |
1lgyl12p4 | maths | complex-numbers | algebra-of-complex-numbers | <p> Let $$A=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.$$ is purely imaginary $$\}$$. Then the sum of the elements in $$\mathrm{A}$$ is :</p> | [{"identifier": "A", "content": "$$3 \\pi$$"}, {"identifier": "B", "content": "$$\\pi$$"}, {"identifier": "C", "content": "$$2 \\pi$$"}, {"identifier": "D", "content": "$$4 \\pi$$"}] | ["D"] | null | $$
\begin{aligned}
& \text { Here, } z=\frac{1+2 i \sin \theta}{1-i \sin \theta} \times \frac{1+i \sin \theta}{1+i \sin \theta} \\\\
& \frac{1+i \sin \theta+2 i \sin \theta-2 \sin ^2 \theta}{1-i^2 \sin ^2 \theta} \\\\
& =\frac{\left(1-2 \sin ^2 \theta\right)+i(3 \sin \theta)}{1+\sin ^2 \theta}
\end{aligned}
$$
<br/><br... | mcq | jee-main-2023-online-8th-april-evening-shift | 5,378 |
1lh2xxw75 | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$a \neq b$$ be two non-zero real numbers. Then the number of elements in the set $$X=\left\{z \in \mathbb{C}: \operatorname{Re}\left(a z^{2}+b z\right)=a\right.$$ and $$\left.\operatorname{Re}\left(b z^{2}+a z\right)=b\right\}$$ is equal to :</p> | [{"identifier": "A", "content": "0"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "1"}, {"identifier": "D", "content": "Infinite"}] | ["D"] | null | Let $z=x+i y$
<br/><br/>$$
\begin{array}{ll}
&\Rightarrow z^2=x^2-y^2+2 i x y \\\\
&\therefore a z^2+b z \\\\
& =a\left(x^2-y^2+2 i x y\right)+b(x+i y) \\\\
& =a\left(x^2-y^2\right)+b x+2 a i x y+b i y
\end{array}
$$
<br/><br/>$\begin{array}{ll}\operatorname{Re}\left(a z^2+b z\right)=b \\\\ \Rightarrow a\left(x^2-y^2... | mcq | jee-main-2023-online-6th-april-evening-shift | 5,379 |
lsaognvr | maths | complex-numbers | algebra-of-complex-numbers | Let $\mathrm{S}=|\mathrm{z} \in \mathrm{C}:| z-1 \mid=1$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2} \mid$. Let $z_1, z_2 \in \mathrm{S}$ be such that $\left|z_1\right|=\max\limits_{z \in s}|z|$ and $\left|z_2\right|=\min\limits _{z \in S}|z|$. Then $\left|\sqrt{2} z_1-z_2\right|^2$ equals : | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "2"}] | ["D"] | null | Let $z=x+i y$
<br/><br/>$$
\begin{aligned}
& |z-1|=1 \Rightarrow|x+i y-1|=1 \\\\
& (x-1)^2+y^2=1 .......(1)
\end{aligned}
$$
<br/><br/>$\begin{aligned} & (\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2 \sqrt{2} \text { (Given) } \\\\ & (\sqrt{2}-1)(2 x)-i(2 i y)=2 \sqrt{2} \\\\ & (\sqrt{2}-1) x+y=\sqrt{2} .........(2)\end{align... | mcq | jee-main-2024-online-1st-february-morning-shift | 5,380 |
lsblk2wv | maths | complex-numbers | algebra-of-complex-numbers | If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7=A+B \alpha+C \alpha^2, A, B, C \geqslant 0$, then $5(3 A-2 B-C)$ is equal to ____________. | [] | null | 5 | <p>$$x^2+x+1=0 \Rightarrow x=\omega, \omega^2=\alpha$$</p>
<p>Let $$\alpha=\omega$$</p>
<p>Now $$(1+\alpha)^7=-\omega^{14}=-\omega^2=1+\omega$$</p>
<p>$$\begin{aligned}
& A=1, B=1, C=0 \\
& \therefore 5(3 A-2 B-C)=5(3-2-0)=5
\end{aligned}$$</p> | integer | jee-main-2024-online-27th-january-morning-shift | 5,381 |
jaoe38c1lsd4pjmu | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$z_1+z_2=5$$ and $$z_1^3+z_2^3=20+15 i$$ Then, $$\left|z_1^4+z_2^4\right|$$ equals -</p> | [{"identifier": "A", "content": "$$15 \\sqrt{15}$$\n"}, {"identifier": "B", "content": "$$30 \\sqrt{3}$$\n"}, {"identifier": "C", "content": "$$25 \\sqrt{3}$$\n"}, {"identifier": "D", "content": "75"}] | ["D"] | null | <p>$$\begin{aligned}
& z_1+z_2=5 \\
& z_1^3+z_2^3=20+15 i \\
& z_1^3+z_2^3=\left(z_1+z_2\right)^3-3 z_1 z_2\left(z_1+z_2\right) \\
& z_1^3+z_2^3=125-3 z_1 \cdot z_2(5) \\
& \Rightarrow 20+15 i=125-15 z_1 z_2 \\
& \Rightarrow 3 z_1 z_2=25-4-3 i \\
& \Rightarrow 3 z_1 z_2=21-3 i \\
& \Rightarrow z_1 \cdot z_2=7-i \\
& \R... | mcq | jee-main-2024-online-31st-january-evening-shift | 5,382 |
jaoe38c1lsf0qr88 | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$\alpha, \beta$$ be the roots of the equation $$x^2-x+2=0$$ with $$\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$$. Then $$\alpha^6+\alpha^4+\beta^4-5 \alpha^2$$ is equal to ___________.</p> | [] | null | 13 | <p>$$\begin{aligned}
& \alpha^6+\alpha^4+\beta^4-5 \alpha^2 \\
& =\alpha^4(\alpha-2)+\alpha^4-5 \alpha^2+(\beta-2)^2 \\
& =\alpha^5-\alpha^4-5 \alpha^2+\beta^2-4 \beta+4 \\
& =\alpha^3(\alpha-2)-\alpha^4-5 \alpha^2+\beta-2-4 \beta+4 \\
& =-2 \alpha^3-5 \alpha^2-3 \beta+2 \\
& =-2 \alpha(\alpha-2)-5 \alpha^2-3 \beta+2 \... | integer | jee-main-2024-online-29th-january-morning-shift | 5,384 |
jaoe38c1lsfl3s5i | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$\alpha, \beta$$ be the roots of the equation $$x^2-\sqrt{6} x+3=0$$ such that $$\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$$. Let $$a, b$$ be integers not divisible by 3 and $$n$$ be a natural number such that $$\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+i b), i=\sqrt{-1}$$. Then $$n+a+b$$ is equal... | [] | null | 49 | <p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsr8r3kr/364277a2-5c20-422c-867f-2e93bd294016/10be8ab0-ce37-11ee-9412-cd4f9c6f2c40/file-6y3zli1lsr8r3ks.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsr8r3kr/364277a2-5c20-422c-867f-2e93bd294016/10be8ab0-ce37-11ee... | integer | jee-main-2024-online-29th-january-evening-shift | 5,385 |
1lsgajjz4 | maths | complex-numbers | algebra-of-complex-numbers | <p>If $$z=x+i y, x y \neq 0$$, satisfies the equation $$z^2+i \bar{z}=0$$, then $$\left|z^2\right|$$ is equal to :</p> | [{"identifier": "A", "content": "9"}, {"identifier": "B", "content": "$$\\frac{1}{4}$$"}, {"identifier": "C", "content": "4"}, {"identifier": "D", "content": "1"}] | ["D"] | null | <p>$$\begin{aligned}
& z^2=-i \bar{z} \\
& \left|z^2\right|=|i \bar{z}| \\
& \left|z^2\right|=|z| \\
& |z|^2-|z|=0 \\
& |z|(|z|-1)=0 \\
& |z|=0 \text { (not acceptable) } \\
& \therefore|z|=1 \\
& \therefore|z|^2=1
\end{aligned}$$</p> | mcq | jee-main-2024-online-30th-january-morning-shift | 5,386 |
lv0vxdau | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$\alpha$$ and $$\beta$$ be the sum and the product of all the non-zero solutions of the equation $$(\bar{z})^2+|z|=0, z \in C$$. Then $$4(\alpha^2+\beta^2)$$ is equal to :</p> | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "6"}, {"identifier": "D", "content": "8"}] | ["A"] | null | <p>$$\begin{aligned}
& (\bar{z})^2+|z|=0 \quad \text{... (1)}\\
& z^2+|\bar{z}|=0 \quad \text{... (2)}
\end{aligned}$$</p>
<p>From equation (1) and (2)</p>
<p>$$\begin{aligned}
& \text { as }|z|=|\bar{z}| \\
& \Rightarrow \quad(\bar{z})^2=z^2 \\
& \Rightarrow \quad z=\bar{z} \text { or } z=-\bar{z} \\
& \Rightarrow \op... | mcq | jee-main-2024-online-4th-april-morning-shift | 5,387 |
lv3ve9bf | maths | complex-numbers | algebra-of-complex-numbers | <p>The sum of all possible values of $$\theta \in[-\pi, 2 \pi]$$, for which $$\frac{1+i \cos \theta}{1-2 i \cos \theta}$$ is purely imaginary, is equal to :</p> | [{"identifier": "A", "content": "$$4 \\pi$$\n"}, {"identifier": "B", "content": "$$3 \\pi$$\n"}, {"identifier": "C", "content": "$$2 \\pi$$\n"}, {"identifier": "D", "content": "$$5 \\pi$$"}] | ["B"] | null | <p>$$\frac{1+i \cos \theta}{1-2 i \cos \theta}$$ is purely imaginary</p>
<p>$$n=\frac{1+i \cos \theta}{1-2 i \cos \theta} \times \frac{1+2 i \cos \theta}{1+2 i \cos \theta}=\frac{1+3 i \cos \theta-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}$$</p>
<p>$$n=\frac{1-2 \cos ^2 \theta}{1+4 \cos ^2 \theta}+i\left(\frac{3 \cos \theta... | mcq | jee-main-2024-online-8th-april-evening-shift | 5,388 |
lv5gsxx4 | maths | complex-numbers | algebra-of-complex-numbers | <p>Let $$z$$ be a complex number such that $$|z+2|=1$$ and $$\operatorname{lm}\left(\frac{z+1}{z+2}\right)=\frac{1}{5}$$. Then the value of $$|\operatorname{Re}(\overline{z+2})|$$ is</p> | [{"identifier": "A", "content": "$$\\frac{2 \\sqrt{6}}{5}$$\n"}, {"identifier": "B", "content": "$$\\frac{24}{5}$$\n"}, {"identifier": "C", "content": "$$\\frac{\\sqrt{6}}{5}$$\n"}, {"identifier": "D", "content": "$$\\frac{1+\\sqrt{6}}{5}$$"}] | ["A"] | null | <p>$$\begin{aligned}
& |z+2|=1 \\
& \operatorname{Im}_m\left(\frac{z+1}{z+2}\right)=\frac{1}{5} \\
& |\operatorname{Re}(\overline{z+2})|=?
\end{aligned}$$</p>
<p>Let $$z=x+i y$$</p>
<p>$$\begin{aligned}
& \because|z+2|=1 \Rightarrow(x+2)^2+y^2=1 \quad \ldots(1) \\
& I_m\left(\frac{z+1}{z+2}\right)=\frac{1}{5} \Rightarr... | mcq | jee-main-2024-online-8th-april-morning-shift | 5,389 |
zfiW9PrjVIHHJFRs | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | The locus of the centre of a circle which touches the circle $$\left| {z - {z_1}} \right| = a$$ and$$\left| {z - {z_2}} \right| = b\,$$ externally
<br/><br/>($$z,\,{z_1}\,\& \,{z_2}\,$$ are complex numbers) will be : | [{"identifier": "A", "content": "an ellipse "}, {"identifier": "B", "content": "a hyperbola"}, {"identifier": "C", "content": "a circle "}, {"identifier": "D", "content": "none of these"}] | ["B"] | null | <img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264708/exam_images/uyfgmdfsahcbye8cxeuf.webp" loading="lazy" alt="AIEEE 2002 Mathematics - Complex Numbers Question 166 English Explanation">
Let the circle be $$\left| {z - {z_3}} \right| = r.$$
<br><br>Then according to given con... | mcq | aieee-2002 | 5,390 |
EwK1sFRumluyqFqZ | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let $${Z_1}$$ and $${Z_2}$$ be two roots of the equation $${Z^2} + aZ + b = 0$$, Z being complex. Further , assume that the origin, $${Z_1}$$ and $${Z_2}$$ form an equilateral triangle. Then : | [{"identifier": "A", "content": "$${a^2} = 4b$$ "}, {"identifier": "B", "content": "$${a^2} = b$$ "}, {"identifier": "C", "content": "$${a^2} = 2b$$ "}, {"identifier": "D", "content": "$${a^2} = 3b$$ "}] | ["D"] | null | Given quadratic equation,
<br>$${Z^2} + aZ + b = 0$$
<br>and two roots are $${Z_1}$$ and $${Z_2}$$.
<br><br>$$\therefore$$ $${Z_1}$$ + $${Z_2}$$ = $$-a$$ and $${Z_1}$$$${Z_2}$$ = $$b$$
<br><br>Question says,
<br>There are three complex numbers:
<br>1. Origin (0)
<br>2. $${Z_1}$$
<br>3. $${Z_2}$$
<br>and they form an eq... | mcq | aieee-2003 | 5,391 |
dTNQj6NpNJcqoBXF | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | If $$\,\left| {z + 4} \right|\,\, \le \,\,3\,$$, then the maximum value of $$\left| {z + 1} \right|$$ is : | [{"identifier": "A", "content": "6 "}, {"identifier": "B", "content": "0 "}, {"identifier": "C", "content": "4 "}, {"identifier": "D", "content": "10"}] | ["A"] | null | $$z$$ lies on or inside the circle with center $$(-4,0)$$ and radius $$3$$ units.
<br><br><img class="question-image" src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264288/exam_images/yur8wemdh4kwqpqnxlji.webp" loading="lazy" alt="AIEEE 2007 Mathematics - Complex Numbers Question 154 English Explanation">... | mcq | aieee-2007 | 5,392 |
0Bf9X9z2dXZJuQpg | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let $$\alpha \,,\beta $$ be real and z be a complex number. If $${z^2} + \alpha z + \beta = 0$$ has two distinct roots on the line Re z = 1, then it is necessary that : | [{"identifier": "A", "content": "$$\\beta \\, \\in ( - 1,0)$$ "}, {"identifier": "B", "content": "$$\\left| {\\beta \\,} \\right| = 1$$ "}, {"identifier": "C", "content": "$$\\beta \\, \\in (1,\\infty )$$ "}, {"identifier": "D", "content": "$$\\beta \\, \\in (0,1)$$ "}] | ["C"] | null | As real part of roots is $$1$$
<br><br>Let roots are $$1 + pi,1 + q$$
<br><br>$$\therefore$$ sum of roots $$ = 1 + pi + 1 + qi = - \alpha $$
<br><br>which is real $$ \Rightarrow q = - p\,\,$$
<br><br>or root are $$1+pi$$ and $$1-pi$$
<br><br>product of roots $$ = 1 + {p^2} = \beta \in \left( {1,\infty } \right)$$
<... | mcq | aieee-2011 | 5,393 |
zrx1He6H0PulM6MQ | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | If $$z \ne 1$$ and $$\,{{{z^2}} \over {z - 1}}\,$$ is real, then the point represented by the complex number z lies : | [{"identifier": "A", "content": "either on the real axis or a circle passing through the origin."}, {"identifier": "B", "content": "on a circle with centre at the origin"}, {"identifier": "C", "content": "either on real axis or on a circle not passing through the origin."}, {"identifier": "D", "content": "on the imagin... | ["A"] | null | Let $$z = x + iy$$
<br><br>$$\therefore$$ $$\,\,\,\,{z^2} = {x^2} - {y^2} + 2ixy$$
<br><br>Now $${{{z^2}} \over {z - 1}}$$ is real
<br><br>$$ \Rightarrow {\mathop{\rm Im}\nolimits} \left( {{{{z^2}} \over {z - 1}}} \right) = 0$$
<br><br>$$ \Rightarrow {\mathop{\rm Im}\nolimits} \left( {{{{x^2} - {y^2} + 2ixy} \over {\... | mcq | aieee-2012 | 5,394 |
qbXThOdgWHTXG1fujwjj9 | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let z$$ \in $$C, the set of complex numbers. Then the equation, 2|z + 3i| $$-$$ |z $$-$$ i| = 0 represents : | [{"identifier": "A", "content": "a circle with radius $${8 \\over 3}.$$"}, {"identifier": "B", "content": "a circle with diameter $${{10} \\over 3}.$$"}, {"identifier": "C", "content": "an ellipse with length of major axis $${{16} \\over 3}.$$"}, {"identifier": "D", "content": "an ellipse with length of minor axis $${{... | ["A"] | null | Given,
<br><br>2 $$\,\left| \, \right.$$z + 3i$$\,\left| \, \right.$$ = $$\,\left| \, \right.$$z $$-$$i$$\,\left| \, \right.$$
<br><br>Let z = x + iy
<br><br>$$ \Rightarrow $$$$\,\,\,$$ 2 $$\,\left| \, \right.$$ x + iy + 3i $$\,\left| \, \right.$$ = $$\,\left| \, \right.$$ x + iy $$-$$ i $$\,\left| \, \right.$$
<br><b... | mcq | jee-main-2017-online-8th-april-morning-slot | 5,395 |
r2VEe8qSKWNdEek63f6ds | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | The equation
<br/>Im $$\left( {{{iz - 2} \over {z - i}}} \right)$$ + 1 = 0, z $$ \in $$ <b>C</b>, z $$ \ne $$ i
<br/>represents a part of a circle having radius
equal to :
| [{"identifier": "A", "content": "2 "}, {"identifier": "B", "content": "1"}, {"identifier": "C", "content": "$${3 \\over 4}$$"}, {"identifier": "D", "content": "$${1 \\over 2}$$"}] | ["C"] | null | Let z = x + iy
<br><br>Then,
<br><br>Im $$\left( {{{iz - 2} \over {z - i}}} \right)$$ + 1 = 0
<br><br>$$ \Rightarrow $$ $${\mathop{\rm Im}\nolimits} \left[ {\left( {{{i\left( {x + iy} \right) - 2} \over {x + iy - i}}} \right)} \right] + 1 = 0$$
<br><br>$$ \Rightarrow $$$${\mathop{\rm Im}\nolimits} \left[ {\left( {{{ix... | mcq | jee-main-2017-online-9th-april-morning-slot | 5,396 |
BSbow338TPfoEJmZ5O7Hf | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | All the points in the set<br/>
$$S = \left\{ {{{\alpha + i} \over {\alpha - i}}:\alpha \in R} \right\}(i = \sqrt { - 1} )$$ lie on a : | [{"identifier": "A", "content": "straight line whose slope is \u20131"}, {"identifier": "B", "content": "straight line whose slope is 1."}, {"identifier": "C", "content": "circle whose radius is 1."}, {"identifier": "D", "content": "circle whose radius is $$\\sqrt 2$$ ."}] | ["C"] | null | Let h + ik = $${{\alpha + i} \over {\alpha - i}}$$
<br><br>= $${{\left( {\alpha + i} \right)\left( {\alpha + i} \right)} \over {\left( {\alpha - i} \right)\left( {\alpha + i} \right)}}$$
<br><br>= $${{\left( {{\alpha ^2} - 1} \right) + 2i\alpha } \over {{\alpha ^2} + 1}}$$
<br><br>$$ \therefore $$ h = $${{{\alpha... | mcq | jee-main-2019-online-9th-april-morning-slot | 5,397 |
sDfdRhll5oVpjbpMGu7k9k2k5e2q3wg | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | If $${\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {2z + i}}} \right) = 1$$, where z = x + iy, then the point (x, y) lies on a :
| [{"identifier": "A", "content": "straight line whose slope is $${3 \\over 2}$$"}, {"identifier": "B", "content": "straight line whose slope is $$-{2 \\over 3}$$"}, {"identifier": "C", "content": "circle whose diameter is $${{\\sqrt 5 } \\over 2}$$"}, {"identifier": "D", "content": "circle whose centre is at $$\\left( {... | ["C"] | null | $${\mathop{\rm Re}\nolimits} \left( {{{z - 1} \over {2z + i}}} \right) = 1$$
<br><br>Put z = x + iy
<br><br>$$ \therefore $$ $${\mathop{\rm Re}\nolimits} \left( {{{\left( {x + iy} \right) - 1} \over {2\left( {x + iy} \right) + i}}} \right) = 1$$
<br><br>$$ \Rightarrow $$ $${\mathop{\rm Re}\nolimits} \left( {\left( {{{\... | mcq | jee-main-2020-online-7th-january-morning-slot | 5,399 |
TBLslHD22EPp6lLBJjjgy2xukf8z8vmo | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let $$u = {{2z + i} \over {z - ki}}$$, z = x + iy and k > 0. If the curve represented<br/> by Re(u) + Im(u) = 1 intersects the y-axis at the points P and Q where PQ = 5, then the value of k is : | [{"identifier": "A", "content": "2"}, {"identifier": "B", "content": "4"}, {"identifier": "C", "content": "1/2"}, {"identifier": "D", "content": "3/2"}] | ["A"] | null | Given, z = x + iy<br><br>and $$u = {{2z + i} \over {z - ki}}$$<br><br>$$ = {{2(x + iy) + i} \over {(x + iy) - ki}}$$<br><br>$$ = {{2x + i(2y + 1)} \over {x + i(y - k)}} \times {{x - i(y - k)} \over {x - i(y - k)}}$$<br><br>$$ = {{2{x^2} + (2y + 1)(y - k) + i(2xy + x - 2xy + 2kx)} \over {{x^2} + {{(y - k)}^2}}}$$<br><br... | mcq | jee-main-2020-online-4th-september-morning-slot | 5,400 |
mhVOjrFE4VXwW6yfRE1kls4xtsd | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let the lines (2 $$-$$ i)z = (2 + i)$$\overline z $$ and (2 $$+$$ i)z + (i $$-$$ 2)$$\overline z $$ $$-$$ 4i = 0, (here i<sup>2</sup> = $$-$$1) be normal to a circle C. If the line iz + $$\overline z $$ + 1 + i = 0 is tangent to this circle C, then its radius is : | [{"identifier": "A", "content": "$${3 \\over {2\\sqrt 2 }}$$"}, {"identifier": "B", "content": "$$3\\sqrt 2 $$"}, {"identifier": "C", "content": "$${1 \\over {2\\sqrt 2 }}$$"}, {"identifier": "D", "content": "$${3 \\over {\\sqrt 2 }}$$"}] | ["A"] | null | $$(2 - i)z = (2 + i)\overline z $$<br><br>$$ \Rightarrow (2 - i)(x + iy) = (2 + i)(x - iy)$$<br><br>$$ \Rightarrow 2x - ix + 2iy + y = 2x + ix - 2 - iy + y$$<br><br>$$ \Rightarrow 2ix - 4iy = 0$$<br><br>$${L_1}:x - 2y = 0$$<br><br>$$ \Rightarrow (2 + i)z + (i - 2)\overline z - 4i = 0$$<br><br>$$ \Rightarrow (2 + i)(x ... | mcq | jee-main-2021-online-25th-february-morning-slot | 5,402 |
t3QY3WP7FQprViAD1n1kluybz0d | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let z be those complex numbers which satisfy<br/><br/>| z + 5 | $$ \le $$ 4 and z(1 + i) + $$\overline z $$(1 $$-$$ i) $$ \ge $$ $$-$$10, i = $$\sqrt { - 1} $$.<br/><br/>If the maximum value of | z + 1 |<sup>2</sup> is $$\alpha$$ + $$\beta$$$$\sqrt 2 $$, then the value of ($$\alpha$$ + $$\beta$$) is ____________. | [] | null | 48 | Let, z = x + iy<br><br>Given, z(1 + i) + $$\overline z $$ (1 $$-$$ i) $$ \ge $$ $$-$$ 10<br><br>$$ \Rightarrow $$ z + $$\overline z $$ + i (z $$-$$ $$\overline z $$) $$ \ge $$ $$-$$ 10<br><br>$$ \Rightarrow $$ 2x + i (2iy) $$ \ge $$ $$-$$ 10<br><br>$$ \Rightarrow $$ x + i<sup>2</sup> y $$ \ge $$ $$-$$ 5<br><br>$$ \Righ... | integer | jee-main-2021-online-26th-february-evening-slot | 5,403 |
Y5JJ3LyDE1P4FfE39O1kmjbgw76 | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | The area of the triangle with vertices A(z), B(iz) and C(z + iz) is : | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "$${1 \\over 2}$$| z |<sup>2</sup>"}, {"identifier": "C", "content": "$${1 \\over 2}$$| z + iz |<sup>2</sup>"}, {"identifier": "D", "content": "$${1 \\over 2}$$"}] | ["B"] | null | <img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734267682/exam_images/cpesvpbbsxejbpma67r4.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 17th March Morning Shift Mathematics - Complex Numbers Question 95 English Explanation">
<br>Each ... | mcq | jee-main-2021-online-17th-march-morning-shift | 5,404 |
tYCQViOYWriDal2fUG1kmklebtf | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let S<sub>1</sub>, S<sub>2</sub> and S<sub>3</sub> be three sets defined as<br/><br/>S<sub>1</sub> = {z$$\in$$C : |z $$-$$ 1| $$ \le $$ $$\sqrt 2 $$}<br/><br/>S<sub>2</sub> = {z$$\in$$C : Re((1 $$-$$ i)z) $$ \ge $$ 1}<br/><br/>S<sub>3</sub> = {z$$\in$$C : Im(z) $$ \le $$ 1}<br/><br/>Then the set S<sub>1</sub> $$\cap$$ ... | [{"identifier": "A", "content": "has exactly three elements"}, {"identifier": "B", "content": "is a singleton"}, {"identifier": "C", "content": "has infinitely many elements"}, {"identifier": "D", "content": "has exactly two elements"}] | ["C"] | null | Let, z = x + iy<br><br><img src="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266724/exam_images/cnzpzty76azz6qp0mhza.webp" style="max-width: 100%;height: auto;display: block;margin: 0 auto;" loading="lazy" alt="JEE Main 2021 (Online) 17th March Evening Shift Mathematics - Complex Numbers Question 94 English ... | mcq | jee-main-2021-online-17th-march-evening-shift | 5,405 |
krFSobiivVYaEzRzw11kmlinglj | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | If the equation $$a|z{|^2} + \overline {\overline \alpha z + \alpha \overline z } + d = 0$$ represents a circle where a, d are real constants then which of the following condition is correct? | [{"identifier": "A", "content": "|$$\\alpha$$|<sup>2</sup> $$-$$ ad $$\\ne$$ 0"}, {"identifier": "B", "content": "|$$\\alpha$$|<sup>2</sup> $$-$$ ad > 0 and a$$\\in$$R $$-$$ {0}"}, {"identifier": "C", "content": "|$$\\alpha$$|<sup>2</sup> $$-$$ ad $$ \\ge $$ 0 and a$$\\in$$R"}, {"identifier": "D", "content": "$$\\al... | ["B"] | null | $$a|z{|^2} + \alpha \overline z + \overline \alpha z + d = 0$$<br><br>$$ \Rightarrow $$ $$z\overline z + \left( {{\alpha \over a}} \right)\overline z + \left( {{{\overline \alpha } \over a}} \right)z + {d \over a} = 0$$<br><br>$$ \therefore $$ Centre $$ = - {\alpha \over a}$$<br><br>$$r = \sqrt {{{\left| {{\alp... | mcq | jee-main-2021-online-18th-march-morning-shift | 5,406 |
Hc7j5aGTrUNOVMU3TI1kmllzn5y | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let z<sub>1</sub>, z<sub>2</sub> be the roots of the equation z<sup>2</sup> + az + 12 = 0 and z<sub>1</sub>, z<sub>2</sub> form an equilateral triangle with origin. Then, the value of |a| is : | [] | null | 6 | For equilateral triangle with vertices z<sub>1</sub>, z<sub>2</sub> and z<sub>3</sub>,<br><br>$$z_1^2 + z_2^2 + z_3^3 = {z_1}{z_2} + {z_2}{z_3} + {z_3}{z_1}$$<br><br>Here one vertex z<sub>3</sub> is 0<br><br>$$ \therefore $$ $$z_1^2 + z_2^2 = {z_1}{z_2} + 0 + 0$$<br><br>Given, z<sub>1</sub>, z<sub>2</sub> are roots of ... | integer | jee-main-2021-online-18th-march-morning-shift | 5,407 |
ACZzZlXbS3L9a3R9ns1kmm2z0b8 | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let a complex number be w = 1 $$-$$ $${\sqrt 3 }$$i. Let another complex number z be such that |zw| = 1 and arg(z) $$-$$ arg(w) = $${\pi \over 2}$$. Then the area of the triangle with vertices origin, z and w is equal to : | [{"identifier": "A", "content": "4"}, {"identifier": "B", "content": "$${1 \\over 4}$$"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "$${1 \\over 2}$$"}] | ["D"] | null | <picture><source media="(max-width: 1227px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265980/exam_images/quuj4fvtbuln79e3uwpk.webp"><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265539/exam_images/ttoow4xxjqahazhug2ch.webp"><source media="(max-wi... | mcq | jee-main-2021-online-18th-march-evening-shift | 5,408 |
1krxi95ji | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let C be the set of all complex numbers. Let<br/><br/>S<sub>1</sub> = {z$$\in$$C : |z $$-$$ 2| $$\le$$ 1} and <br/><br/>S<sub>2</sub> = {z$$\in$$C : z(1 + i) + $$\overline z $$(1 $$-$$ i) $$\ge$$ 4}.<br/><br/>Then, the maximum value of $${\left| {z - {5 \over 2}} \right|^2}$$ for z$$\in$$S<sub>1</sub> $$\cap$$ S<sub>2<... | [{"identifier": "A", "content": "$${{3 + 2\\sqrt 2 } \\over 4}$$"}, {"identifier": "B", "content": "$${{5 + 2\\sqrt 2 } \\over 2}$$"}, {"identifier": "C", "content": "$${{3 + 2\\sqrt 2 } \\over 2}$$"}, {"identifier": "D", "content": "$${{5 + 2\\sqrt 2 } \\over 4}$$"}] | ["D"] | null | |t $$-$$ 2| $$\le$$ 1<br><br>Put t = x + iy<br><br><picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264870/exam_images/xng9d9n757qbnimzbcss.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734264992/exam_images... | mcq | jee-main-2021-online-27th-july-evening-shift | 5,409 |
1ks061tqe | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let C be the set of all complex numbers. Let<br/><br/>$${S_1} = \{ z \in C||z - 3 - 2i{|^2} = 8\} $$<br/><br/>$${S_2} = \{ z \in C|{\mathop{\rm Re}\nolimits} (z) \ge 5\} $$ and <br/><br/>$${S_3} = \{ z \in C||z - \overline z | \ge 8\} $$.<br/><br/>Then the number of elements in $${S_1} \cap {S_2} \cap {S_3}$$ is equal ... | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "0"}, {"identifier": "C", "content": "2"}, {"identifier": "D", "content": "Infinite"}] | ["A"] | null | $${S_1}:|z - 3 - 2i{|^2} = 8$$<br><br>$$|z - 3 - 2i| = 2\sqrt 2 $$<br><br>$${(x - 3)^2} + {(y - 2)^2} = {(2\sqrt 2 )^2}$$<br><br>$${S_2}:x \ge 5$$<br><br>$${S_3}:|z - \overline z | \ge 8$$<br><br>$$|2iy| \ge 8$$<br><br>$$2|y| \ge 8$$<br><br>$$\therefore$$ $$y \ge 4$$, $$y \le - 4$$<br><br><img src="https://res.cloudin... | mcq | jee-main-2021-online-27th-july-morning-shift | 5,411 |
1ktbe0f2q | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | The equation $$\arg \left( {{{z - 1} \over {z + 1}}} \right) = {\pi \over 4}$$ represents a circle with : | [{"identifier": "A", "content": "centre at (0, $$-$$1) and radius $$\\sqrt 2 $$"}, {"identifier": "B", "content": "centre at (0, 1) and radius $$\\sqrt 2 $$"}, {"identifier": "C", "content": "centre (0, 0) and radius $$\\sqrt 2 $$"}, {"identifier": "D", "content": "centre at (0, 1) and radius 2"}] | ["B"] | null | <picture><source media="(max-width: 320px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734266796/exam_images/f29lrw36fjmxmetgizum.webp"><source media="(max-width: 500px)" srcset="https://res.cloudinary.com/dckxllbjy/image/upload/v1734265873/exam_images/uw851jxzxx4ohkmsafk9.webp"><img src="https://res.c... | mcq | jee-main-2021-online-26th-august-morning-shift | 5,412 |
1l54akox4 | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <p>Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z $$-$$ 1) $$-$$ arg(z + 1) = $${\pi \over 4}$$ intersect :</p> | [{"identifier": "A", "content": "exactly at one point."}, {"identifier": "B", "content": "exactly at two points."}, {"identifier": "C", "content": "nowhere."}, {"identifier": "D", "content": "at infinitely many points."}] | ["C"] | null | <p>Let $$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}$$</p>
<p>This represent a circle with center at (0, 0) and radius = 3</p>
<p>Now, given</p>
<p>$$\arg (z - 1) - \arg... | mcq | jee-main-2022-online-29th-june-evening-shift | 5,414 |
1l567v516 | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <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> | [] | null | 40 | <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-92bb-e57e64e1a06d/fi... | integer | jee-main-2022-online-28th-june-morning-shift | 5,415 |
1l6ggfmy6 | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <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> | [{"identifier": "A", "content": "$$\\arg {z_2} = \\pi - {\\tan ^{ - 1}}3$$"}, {"identifier": "B", "content": "$$\\arg ({z_1} - 2{z_2}) = - {\\tan ^{ - 1}}{4 \\over 3}$$"}, {"identifier": "C", "content": "$$|{z_2}| = \\sqrt {10} $$"}, {"identifier": "D", "content": "$$|2{z_1} - {z_2}| = 5$$"}] | ["D"] | null | <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-11ed-9dc0-a1792fcc650... | mcq | jee-main-2022-online-26th-july-morning-shift | 5,417 |
1l6rez1b9 | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <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> | [{"identifier": "A", "content": "$$\\left(-\\sqrt{2}, \\frac{1}{2 \\sqrt{2}}\\right]$$"}, {"identifier": "B", "content": "$$\\left(-\\frac{1}{\\sqrt{2}}, \\frac{1}{4}\\right]$$"}, {"identifier": "C", "content": "$$\\left(-\\sqrt{2}, \\frac{1}{2}\\right]$$"}, {"identifier": "D", "content": "$$\\left(-\\frac{1}{\\sqrt{2}... | ["B"] | null | $S:\{z=x+i y:|z-1+i| \geq|z|,|z|<2,|z-i|=|z-1|\}$
<br><br>
$$
|z-1+i| \geq|z|
$$<br><br>
<img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lc8c3waq/4ba6cc0d-ceab-4ef1-baed-c4899ebff5b2/d5a07220-870d-11ed-ae18-7336d1cc7e9d/file-1lc8c3war.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/... | mcq | jee-main-2022-online-29th-july-evening-shift | 5,418 |
1ldpswr3h | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <p>For all $$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}$$. Then :</p> | [{"identifier": "A", "content": "the curves $$C_{1}$$ and $$C_{2}$$ intersect at 4 points"}, {"identifier": "B", "content": "the curve $$C_{2}$$ lies inside $$C_{1}$$"}, {"identifier": "C", "content": "the curve $$C_{1}$$ lies inside $$C_{2}$$"}, {"identifier": "D", "content": "the curves $$C_{1}$$ and $$C_{2}$$ inters... | ["A"] | null | Let $\mathrm{w}=\mathrm{z}+\frac{1}{\mathrm{z}}=4 \mathrm{e}^{\mathrm{i} \theta}+\frac{1}{4} \mathrm{e}^{-\mathrm{i} \theta}$
<br/><br/>$\Rightarrow \mathrm{w}=\frac{17}{4} \cos \theta+\mathrm{i} \frac{15}{4} \sin \theta$
<br/><br/>So locus of $w$ is ellipse $\frac{x^{2}}{\left(\frac{17}{4}\right)^{2}}+\frac{y^{2}}{\... | mcq | jee-main-2023-online-31st-january-morning-shift | 5,421 |
1ldv236hn | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <p>Let $$\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\}}$$ represents a</p> | [{"identifier": "A", "content": "hyperbola with the length of the transverse axis 7"}, {"identifier": "B", "content": "hyperbola with eccentricity 2"}, {"identifier": "C", "content": "straight line with the sum of its intercepts on the coordinate axes equals $$-18$$"}, {"identifier": "D", "content": "straight line with... | ["D"] | null | $\left|z-z_{1}\right|^{2}-\left|z-z_{2}\right|^{2}=\left|z_{1}-z_{2}\right|^{2}$
<br/><br/>
$\Rightarrow(x-2)^{2}+(y-3)^{2}-(x-3)^{2}-(y-4)^{2}=1+1$
<br/><br/>
$\Rightarrow-4 x+4+9-6 y-9+6 x-16+8 y=2$
<br/><br/>
$\Rightarrow 2 x+2 y=14$
<br/><br/>
$\Rightarrow x+y=7$ | mcq | jee-main-2023-online-25th-january-morning-shift | 5,422 |
1lh2z0axk | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <p>For $$\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|$$ is equal to __________.</p> | [] | null | 2 | Given equation of circle,
<br/><br/>$$
\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
$$
<br/><br/>$\begin{array}{lll}\text { Radius,... | integer | jee-main-2023-online-6th-april-evening-shift | 5,425 |
lsaq2wp0 | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | Let $\mathrm{P}=\{\mathrm{z} \in \mathbb{C}:|z+2-3 i| \leq 1\}$ and $\mathrm{Q}=\{\mathrm{z} \in \mathbb{C}: z(1+i)+\bar{z}(1-i) \leq-8\}$. Let in $\mathrm{P} \cap \mathrm{Q}$, $|z-3+2 i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $\left|z_1\right|^2+2\left|z_2\right|^2=\alpha+\beta \sqrt{2}$, where $\... | [] | null | 36 | <img src="https://app-content.cdn.examgoal.net/fly/@width/image/6y3zli1lsz1bfzj/da67c7ec-984a-45c9-af11-5958669a974a/42ad13f0-d280-11ee-9b77-fbceb54c8042/file-6y3zli1lsz1bfzk.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/6y3zli1lsz1bfzj/da67c7ec-984a-45c9-af11-5958669a974a/42ad13f0-d280-11ee-9b... | integer | jee-main-2024-online-1st-february-morning-shift | 5,426 |
lsbkg21j | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | If $S=\{z \in C:|z-i|=|z+i|=|z-1|\}$, then, $n(S)$ is : | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "2"}, {"identifier": "C", "content": "3"}, {"identifier": "D", "content": "0"}] | ["A"] | null | <p>$$|z-i|=|z+i|=|z-1|$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lt1dbkuq/8fa39b20-7b31-407e-a4e3-48710c870daf/c548b320-d3c8-11ee-a50b-bb659a2e1d74/file-1lt1dbkur.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lt1dbkuq/8fa39b20-7b31-407e-a4e3-48710c870daf/c548b32... | mcq | jee-main-2024-online-27th-january-morning-shift | 5,427 |
jaoe38c1lscorvog | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <p>Let the complex numbers $$\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$$ is __________.</p> | [] | null | 20 | <p>$$\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)} \\
& \left|z-z_0... | integer | jee-main-2024-online-27th-january-evening-shift | 5,428 |
luxwe3b8 | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <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> | [{"identifier": "A", "content": "8"}, {"identifier": "B", "content": "12"}, {"identifier": "C", "content": "10"}, {"identifier": "D", "content": "$$\\infty$$"}] | ["B"] | null | <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>$$\begin{aligned}
& \Rightarrow x^2+y^2-4=0 \\... | mcq | jee-main-2024-online-9th-april-evening-shift | 5,429 |
lv2er3qp | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <p>The area (in sq. units) of the region $$S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{lm}(z) \geq 0\}$$ is</p> | [{"identifier": "A", "content": "$$\\frac{7 \\pi}{4}$$\n"}, {"identifier": "B", "content": "$$\\frac{3 \\pi}{2}$$\n"}, {"identifier": "C", "content": "$$\\frac{7 \\pi}{3}$$\n"}, {"identifier": "D", "content": "$$\\frac{17 \\pi}{8}$$"}] | ["B"] | null | <p>$$|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}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwh... | mcq | jee-main-2024-online-4th-april-evening-shift | 5,430 |
lv9s1zxt | maths | complex-numbers | applications-of-complex-numbers-in-coordinate-geometry | <p>Let $$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$$ is :</p> | [{"identifier": "A", "content": "$$\\frac{125 \\pi}{24}$$\n"}, {"identifier": "B", "content": "$$\\frac{125 \\pi}{6}$$\n"}, {"identifier": "C", "content": "$$\\frac{125 \\pi}{12}$$\n"}, {"identifier": "D", "content": "$$\\frac{125 \\pi}{4}$$"}] | ["C"] | null | <p>$$S_1=\{z \in C:|z| \leq 5\}$$</p>
<p><img src="https://app-content.cdn.examgoal.net/fly/@width/image/1lwekhn5c/6d7750d9-cc34-4e40-9c38-cefbaf7666fe/a4650700-166f-11ef-8416-25c08f86a011/file-1lwekhn5d.png?format=png" data-orsrc="https://app-content.cdn.examgoal.net/image/1lwekhn5c/6d7750d9-cc34-4e40-9c38-cefbaf7666f... | mcq | jee-main-2024-online-5th-april-evening-shift | 5,431 |
6qtmLybxMm3QFvTz | maths | complex-numbers | argument-of-complex-numbers | z and w are two nonzero complex numbers such that $$\,\left| z \right| = \left| w \right|$$ and Arg z + Arg w =$$\pi $$ then z equals | [{"identifier": "A", "content": "$$\\overline \\omega $$ "}, {"identifier": "B", "content": "$$ - \\overline \\omega $$ "}, {"identifier": "C", "content": "$$\\omega $$ "}, {"identifier": "D", "content": "$$ - \\omega $$ "}] | ["B"] | null | Let $$\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 }} = - \overline \omega .... | mcq | aieee-2002 | 5,432 |
9qdSyFDNne1uVHsv | maths | complex-numbers | argument-of-complex-numbers | Let z and w be complex numbers such that $$\overline z + i\overline w = 0$$ and arg zw = $$\pi $$. Then arg z equals : | [{"identifier": "A", "content": "$${{5\\pi } \\over 4}$$ "}, {"identifier": "B", "content": "$${{\\pi } \\over 2}$$ "}, {"identifier": "C", "content": "$${{3\\pi } \\over 4}$$"}, {"identifier": "D", "content": "$${{\\pi } \\over 4}$$ "}] | ["C"] | null | 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>$$ \Rightarrow z = - \overline i w$$
<b... | mcq | aieee-2004 | 5,434 |
HB8aSQvg51aWaI5u | maths | complex-numbers | argument-of-complex-numbers | 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 : | [{"identifier": "A", "content": "$${\\pi \\over 2}\\,$$ "}, {"identifier": "B", "content": "$$ - \\pi $$ "}, {"identifier": "C", "content": "0 "}, {"identifier": "D", "content": "$${{ - \\pi } \\over 2}$$ "}] | ["C"] | null | 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 $$<$$ $$\left| {{z_1}} \right| + \left| {{z_2}} \right|$$ but here they are e... | mcq | aieee-2005 | 5,435 |
TErZStb9eMi3WgFR | maths | complex-numbers | argument-of-complex-numbers | If z is a complex number of unit modulus and argument $$\theta $$, then arg $$\left( {{{1 + z} \over {1 + \overline z }}} \right)$$ equals : | [{"identifier": "A", "content": "$$ - \\theta \\,\\,$$ "}, {"identifier": "B", "content": "$${\\pi \\over 2} - \\theta \\,$$ "}, {"identifier": "C", "content": "$$\\theta \\,$$ "}, {"identifier": "D", "content": "$$\\,\\pi - \\theta \\,\\,$$ "}] | ["C"] | null | 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>$$ = \arg \left( z \right) = ... | mcq | jee-main-2013-offline | 5,436 |
8tnVLW8PvIETENfbmt3rsa0w2w9jx23bxo1 | maths | complex-numbers | argument-of-complex-numbers | If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = $${\pi \over 2}$$
, then : | [{"identifier": "A", "content": "$$z\\overline w = {{1 - i} \\over {\\sqrt 2 }}$$"}, {"identifier": "B", "content": "$$\\overline z w = i$$"}, {"identifier": "C", "content": "$$z\\overline w = {{ - 1 + i} \\over {\\sqrt 2 }}$$"}, {"identifier": "D", "content": "$$\\overline z w = -i$$"}] | ["D"] | null | $$\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\theta }} = {e^{ - i(\pi ... | mcq | jee-main-2019-online-10th-april-evening-slot | 5,437 |
n1yDVpjIDl2kKgsiz97k9k2k5fo1l04 | maths | complex-numbers | argument-of-complex-numbers | 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 : | [{"identifier": "A", "content": "$$\\pi - {\\tan ^{ - 1}}\\left( {{3 \\over 4}} \\right)$$"}, {"identifier": "B", "content": "$$ - {\\tan ^{ - 1}}\\left( {{3 \\over 4}} \\right)$$"}, {"identifier": "C", "content": "$${\\tan ^{ - 1}}\\left( {{4 \\over 3}} \\right)$$"}, {"identifier": "D", "content": "$$\\pi - {\\tan ^... | ["D"] | null | Let z = $${{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\cos \theta } \right... | mcq | jee-main-2020-online-7th-january-evening-slot | 5,438 |
1krpubj40 | maths | complex-numbers | argument-of-complex-numbers | 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) | [{"identifier": "A", "content": "$${\\pi \\over 4}$$"}, {"identifier": "B", "content": "$$ - {{3\\pi } \\over 4}$$"}, {"identifier": "C", "content": "$$ - {\\pi \\over 4}$$"}, {"identifier": "D", "content": "$${{3\\pi } \\over 4}$$"}] | ["B"] | null | 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>$$ \Rightarrow \overl... | mcq | jee-main-2021-online-20th-july-morning-shift | 5,440 |
1ktgosazo | maths | complex-numbers | argument-of-complex-numbers | Let z<sub>1</sub> and z<sub>2</sub> be two complex numbers such that $$\arg ({z_1} - {z_2}) = {\pi \over 4}$$ and z<sub>1</sub>, z<sub>2</sub> satisfy the equation | z $$-$$ 3 | = Re(z). Then the imaginary part of z<sub>1</sub> + z<sub>2</sub> is equal to ___________. | [] | null | 6 | Let z<sub>1</sub> = x<sub>1</sub> + iy ; z<sub>2</sub> = x<sub>2</sub> + iy<sub>2</sub><br><br>z<sub>1</sub> $$-$$ z<sub>2</sub> = (x<sub>1</sub> $$-$$ x<sub>2</sub>) + i(y<sub>1</sub> $$-$$ y<sub>2</sub>)<br><br>$$\therefore$$ $$\arg ({z_1} - {z_2}) = {\pi \over 4}$$ $$\Rightarrow$$ $${\tan ^{ - 1}}\left( {{{{y_1} - ... | integer | jee-main-2021-online-27th-august-evening-shift | 5,441 |
1l587df22 | maths | complex-numbers | argument-of-complex-numbers | <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> | [{"identifier": "A", "content": "a portion of a circle centred at $$\\left( {0, - {1 \\over {\\sqrt 3 }}} \\right)$$ that lies in the second and third quadrants only"}, {"identifier": "B", "content": "a portion of a circle centred at $$\\left( {0, - {1 \\over {\\sqrt 3 }}} \\right)$$ that lies in the second quadrant on... | ["B"] | null | $$
\left|\frac{z+1}{z-1}\right|<1 \Rightarrow|z+1|<|z-1| \Rightarrow \operatorname{Re}(z)<0
$$<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-7111-4568-85c5-b0eec7b3bc91/5b4... | mcq | jee-main-2022-online-26th-june-morning-shift | 5,442 |
1l59jqnkg | maths | complex-numbers | argument-of-complex-numbers | <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> | [{"identifier": "A", "content": "$$\\arg {z_2} = {\\pi \\over 4}$$"}, {"identifier": "B", "content": "$$\\arg {z_2} = - {{3\\pi } \\over 4}$$"}, {"identifier": "C", "content": "$$\\arg {z_1} = {\\pi \\over 4}$$"}, {"identifier": "D", "content": "$$\\arg {z_1} = - {{3\\pi } \\over 4}$$"}] | ["C"] | null | <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>From (i) and (ii),... | mcq | jee-main-2022-online-25th-june-evening-shift | 5,443 |
1l5aj6yrj | maths | complex-numbers | argument-of-complex-numbers | <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> | [{"identifier": "A", "content": "$${\\tan ^{ - 1}}\\left( {{2 \\over {\\sqrt 5 }}} \\right) - \\pi $$"}, {"identifier": "B", "content": "$${\\tan ^{ - 1}}\\left( {{{24} \\over 7}} \\right) - \\pi $$"}, {"identifier": "C", "content": "$${\\tan ^{ - 1}}\\left( 3 \\right) - \\pi $$"}, {"identifier": "D", "content": "$${\\... | ["B"] | null | <p>$${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$$ z is intersection ... | mcq | jee-main-2022-online-25th-june-morning-shift | 5,444 |
1ldr7wgx2 | maths | complex-numbers | argument-of-complex-numbers | <p>Let $$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)$$ is equal to __________.</p> | [] | null | 9 | <p>$$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( {{{ - 1} \over 1}} \r... | integer | jee-main-2023-online-30th-january-morning-shift | 5,445 |
1lgutz71b | maths | complex-numbers | argument-of-complex-numbers | <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 $$w_{1... | [{"identifier": "A", "content": "$$-\\pi+\\tan ^{-1} \\frac{8}{9}$$"}, {"identifier": "B", "content": "$$-\\pi+\\tan ^{-1} \\frac{33}{5}$$"}, {"identifier": "C", "content": "$$\\pi-\\tan ^{-1} \\frac{8}{9}$$"}, {"identifier": "D", "content": "$$\\pi-\\tan ^{-1} \\frac{33}{5}$$"}] | ["C"] | null | <p>To solve the problem, let's break it down step by step.</p>
<p><strong>Step 1 :</strong> Find $w_{1}$ </p>
<p>Given $z_{1} = 5 + 4i$. </p>
<p>When you rotate $z_{1}$ by $90^{\circ}$ anticlockwise about the origin, the real part becomes negative of the imaginary part of $z_{1}$, and the imaginary part becomes the... | mcq | jee-main-2023-online-11th-april-morning-shift | 5,446 |
jaoe38c1lsfkvcni | maths | complex-numbers | argument-of-complex-numbers | <p>Let $$\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)$$ is equal to</p> | [{"identifier": "A", "content": "$$\\left(2 \\sec \\frac{11 \\pi}{8}, \\frac{11 \\pi}{8}\\right)$$\n"}, {"identifier": "B", "content": "$$\\left(2 \\sec \\frac{3 \\pi}{8}, \\frac{3 \\pi}{8}\\right)$$\n"}, {"identifier": "C", "content": "$$\\left(2 \\sec \\frac{5 \\pi}{8}, \\frac{3 \\pi}{8}\\right)$$\n"}, {"identifier":... | ["B"] | null | <p>$$\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| \\
& =2 \sec \frac{3 \pi}... | mcq | jee-main-2024-online-29th-january-evening-shift | 5,448 |
PM4qMlStSHAcQdgu | maths | complex-numbers | conjugate-of-complex-numbers | The conjugate of a complex number is $${1 \over {i - 1}}$$ then that complex number is : | [{"identifier": "A", "content": "$${{ - 1} \\over {i - 1}}$$ "}, {"identifier": "B", "content": "$${1 \\over {i + 1}}\\,$$ "}, {"identifier": "C", "content": "$${{ - 1} \\over {i + 1}}$$ "}, {"identifier": "D", "content": "$${1 \\over {i - 1}}$$ "}] | ["C"] | null | $$\left( {{1 \over {i - 1}}} \right) = {1 \over { - i - 1}} = {{ - 1} \over {i + 1}}$$ | mcq | aieee-2008 | 5,449 |
I5pK1RCzpUnmvnIyw53rsa0w2w9jwxjodf8 | maths | complex-numbers | conjugate-of-complex-numbers | 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 : | [{"identifier": "A", "content": "$$ - {1 \\over 5} + {3 \\over 5}i$$"}, {"identifier": "B", "content": "$$ - {1 \\over 5} - {3 \\over 5}i$$"}, {"identifier": "C", "content": "$${1 \\over 5} - {3 \\over 5}i$$"}, {"identifier": "D", "content": "$$ - {3 \\over 5} - {1 \\over 5}i$$"}] | ["B"] | null | $$z = {{{{\left( {1 + i} \right)}^2}} \over {a - i}} \times {{a + i} \over {a + i}}$$<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>
$$ \Rightarrow \left| z \right| = \sqrt {{{\left( {{{ - 2} \over {{a^2} + 1}}} \right)}^2} ... | mcq | jee-main-2019-online-10th-april-morning-slot | 5,450 |
LNvcjihjOACPdCzciI7k9k2k5grsymq | maths | complex-numbers | conjugate-of-complex-numbers | If the equation, x<sup>2</sup> + bx + 45 = 0 (b $$ \in $$ R) has
conjugate complex roots and they satisfy
|z +1| = 2$$\sqrt {10} $$ , then : | [{"identifier": "A", "content": "b<sup>2</sup> \u2013 b = 42"}, {"identifier": "B", "content": "b<sup>2</sup> + b = 12"}, {"identifier": "C", "content": "b<sup>2</sup> + b = 72"}, {"identifier": "D", "content": "b<sup>2</sup> \u2013 b = 30"}] | ["D"] | null | x<sup>2</sup>
+ bx = 45 = 0 (b $$ \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 {10} $$
<br><br>$${\... | mcq | jee-main-2020-online-8th-january-morning-slot | 5,451 |
LbJQB0EUlx4IaWnLwY1kmhzri06 | maths | complex-numbers | conjugate-of-complex-numbers | 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 ______________. | [] | null | 4 | 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 $$-$$ 1)<sup>2</s... | integer | jee-main-2021-online-16th-march-morning-shift | 5,453 |
1kru4nljk | maths | complex-numbers | conjugate-of-complex-numbers | 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 : | [{"identifier": "A", "content": "1"}, {"identifier": "B", "content": "$${4 \\over 3}$$"}, {"identifier": "C", "content": "$${3 \\over 2}$$"}, {"identifier": "D", "content": "2"}] | ["B"] | null | 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 ..... (1)<br><br>2xy... | mcq | jee-main-2021-online-22th-july-evening-shift | 5,454 |
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