MCQOPTIONS
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MCQOPTIONS
This section includes 8666 Mcqs, each offering curated multiple-choice questions to sharpen your Joint Entrance Exam - Main (JEE Main) knowledge and support exam preparation. Choose a topic below to get started.
| 4501. |
If \[arg\,(z-a)=\frac{\pi }{4}\], where \[a\in R\], then the locus of \[z\in C\] is a [MP PET 1997] |
| A. | Hyperbola |
| B. | Parabola |
| C. | Ellipse |
| D. | Straight line |
| Answer» E. | |
| 4502. |
If \[{{\log }_{\sqrt{3}}}\left( \frac{|z{{|}^{2}}-|z|+1}{2+|z|} \right)\]\[ |
| A. | \[|z|=5\] |
| B. | \[|z|<5\] |
| C. | \[|z|>5\] |
| D. | None of these |
| Answer» C. \[|z|>5\] | |
| 4503. |
The locus represented by \[|z-1|=|z+i|\] is [EAMCET 1991] |
| A. | A circle of radius 1 |
| B. | An ellipse with foci at \[(1,\,0)\] and (0, - 1) |
| C. | A straight line through the origin |
| D. | A circle on the line joining \[(1,\,0),(0,\,1)\] as diameter |
| Answer» D. A circle on the line joining \[(1,\,0),(0,\,1)\] as diameter | |
| 4504. |
\[R({{z}^{2}})=1\]is represented by |
| A. | The parabola \[{{x}^{2}}+{{y}^{2}}=1\] |
| B. | The hyperbola \[{{x}^{2}}-{{y}^{2}}=1\] |
| C. | Parabola or a circle |
| D. | All the above |
| Answer» C. Parabola or a circle | |
| 4505. |
The locus of \[z\] given by \[\left| \frac{z-1}{z-i} \right|=1\], is [Roorkee 1990] |
| A. | A circle |
| B. | An ellipse |
| C. | A straight line |
| D. | A parabola |
| Answer» D. A parabola | |
| 4506. |
A point z moves on Argand diagram in such a way that |z -3i| \[=2,\] then its locus will be [RPET 1992; MP PET 2002] |
| A. | \[y-\]axis |
| B. | A straight line |
| C. | A circle |
| D. | None of these |
| Answer» D. None of these | |
| 4507. |
If \[z=x+iy\] and \[|z-zi|\,=1,\]then [RPET 1988, 91] |
| A. | \[z\]lies on \[x\]-axis |
| B. | \[z\]lies on \[y\]-axis |
| C. | z lies on a circle |
| D. | None of these |
| Answer» D. None of these | |
| 4508. |
If \[z=(\lambda +3)+i\sqrt{5-{{\lambda }^{2}},}\] then the locus of z is a |
| A. | Circle |
| B. | Straight line |
| C. | Parabola |
| D. | None of these |
| Answer» B. Straight line | |
| 4509. |
If the imaginary part of \[\frac{2z+1}{iz+1}\]is -2, then the locus of the point representing \[z\]in the complex plane is [DCE 2001] |
| A. | A circle |
| B. | A straight line |
| C. | A parabola |
| D. | None of these |
| Answer» C. A parabola | |
| 4510. |
The equation \[\overline{b}z+b\overline{z}=c,\]where \[b\] is a non-zero complex constant and c is real, represents |
| A. | A circle |
| B. | A straight line |
| C. | A parabola |
| D. | None of these |
| Answer» C. A parabola | |
| 4511. |
The region of Argand plane defined by \[|z-1|\,\,+\,\,|z+1|\,\,\le 4\] is |
| A. | Interior of an ellipse |
| B. | Exterior of a circle |
| C. | Interior and boundary of an ellipse |
| D. | None of these |
| Answer» D. None of these | |
| 4512. |
If \[|z+1|\,\,=\sqrt{2}|z-1|,\]then the locus described by the point \[z\] in the Argand diagram is a |
| A. | Straight line |
| B. | Circle |
| C. | Parabola |
| D. | None of these |
| Answer» C. Parabola | |
| 4513. |
When \[\frac{z+i}{z+2}\] is purely imaginary, the locus described by the point \[z\] in the Argand diagram is a |
| A. | Circle of radius \[\frac{\sqrt{5}}{2}\] |
| B. | Circle of radius \[\frac{5}{4}\] |
| C. | Straight line |
| D. | Parabola |
| Answer» B. Circle of radius \[\frac{5}{4}\] | |
| 4514. |
If \[{{z}_{1}}=1+2i,{{z}_{2}}=2+3i,{{z}_{3}}=3+4i,\] then \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\] represent the vertices of a/an [Orissa JEE 2004] |
| A. | Equilateral triangle |
| B. | Isosceles triangle |
| C. | Right angled triangle |
| D. | None of these |
| Answer» E. | |
| 4515. |
The area of the triangle whose vertices are represented by the complex numbers 0, z, \[z{{e}^{i\alpha }},\] \[(0 |
| A. | \[\frac{1}{2}|z{{|}^{2}}\cos \alpha \] |
| B. | \[\frac{1}{2}|z{{|}^{2}}\sin \alpha \] |
| C. | \[\frac{1}{2}|z{{|}^{2}}\sin \alpha \cos \alpha \] |
| D. | \[\frac{1}{2}|z{{|}^{2}}\] |
| Answer» C. \[\frac{1}{2}|z{{|}^{2}}\sin \alpha \cos \alpha \] | |
| 4516. |
If \[{{z}_{1}}=1+i,\,{{z}_{2}}=-2+3i\,\,\text{and}\,\,\text{ }{{z}_{3}}=ai/3\], where \[{{i}^{2}}=-1,\] are collinear then the value of a is [AMU 2001] |
| A. | -1 |
| B. | 3 |
| C. | 4 |
| D. | 5 |
| Answer» E. | |
| 4517. |
If the area of the triangle formed by the points \[z,z+iz\] and iz on the complex plane is 18, then the value of \[|z|\] is [MP PET 2001] |
| A. | 6 |
| B. | 9 |
| C. | \[3\sqrt{2}\] |
| D. | \[2\sqrt{3}\] |
| Answer» B. 9 | |
| 4518. |
Let \[{{z}_{1}}\] and \[{{z}_{2}}\] be two complex numbers such that \[\frac{{{z}_{1}}}{{{z}_{2}}}+\frac{{{z}_{2}}}{{{z}_{1}}}=1\]. Then |
| A. | \[{{z}_{1}},{{z}_{2}}\]are collinear |
| B. | \[{{z}_{1}},{{z}_{2}}\]and the origin form a right angled triangle |
| C. | \[{{z}_{1}},{{z}_{2}}\]and the origin form an equilateral triangle |
| D. | None of these |
| Answer» D. None of these | |
| 4519. |
If \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\] are affixes of the vertices \[A,B\] and \[C\] respectively of a triangle \[ABC\] having centroid at \[G\] such that \[z=0\] is the mid point of \[AG,\] then |
| A. | \[{{z}_{1}}+{{z}_{2}}+{{z}_{3}}=0\] |
| B. | \[{{z}_{1}}+4{{z}_{2}}+{{z}_{3}}=0\] |
| C. | \[{{z}_{1}}+{{z}_{2}}+4{{z}_{3}}=0\] |
| D. | \[{{z}_{1}}+{{z}_{2}}+{{z}_{3}}=0\] |
| Answer» E. | |
| 4520. |
If \[{{z}_{1}},{{z}_{2}}\in C,\] then [MP PET 1995] |
| A. | \[|{{z}_{1}}+{{z}_{2}}|\,\ge \,|{{z}_{1}}|+|{{z}_{2}}|\] |
| B. | \[|{{z}_{1}}-{{z}_{2}}|\,\ge \,|{{z}_{1}}|+|{{z}_{2}}|\] |
| C. | \[|{{z}_{1}}-{{z}_{2}}|\,\le \left| \,|{{z}_{1}}|-|{{z}_{2}}|\, \right|\] |
| D. | \[|{{z}_{1}}+{{z}_{2}}|\,\ge \left| \,|{{z}_{1}}|-|{{z}_{2}}|\, \right|\] |
| Answer» E. | |
| 4521. |
If A, B, C are represented by \[3+4i,\] \[5-2i\], \[-1+16i\], then A, B, C are [RPET 1986] |
| A. | Collinear |
| B. | Vertices of equilateral triangle |
| C. | Vertices of isosceles triangle |
| D. | Vertices of right angled triangle |
| Answer» B. Vertices of equilateral triangle | |
| 4522. |
If \[z=x+iy,\] then area of the triangle whose vertices are points \[z,\,iz\] and \[z+iz\] is [MP PET 1997; IIT 1986; AMU 2000; UPSEAT 2002] |
| A. | \[2|z{{|}^{2}}\] |
| B. | \[\frac{1}{2}|z{{|}^{2}}\] |
| C. | \[|z{{|}^{2}}\] |
| D. | \[\frac{3}{2}|z{{|}^{2}}\] |
| Answer» C. \[|z{{|}^{2}}\] | |
| 4523. |
The points \[1+3i,\,5+i\] and \[3+2i\] in the complex plane are [MP PET 1987] |
| A. | Vertices of a right angled triangle |
| B. | Collinear |
| C. | Vertices of an obtuse angled triangle |
| D. | Vertices of an equilateral triangle |
| Answer» C. Vertices of an obtuse angled triangle | |
| 4524. |
If z is a complex number in the Argand plane, then the equation \[|z-2|+|z+2|=8\]represents [Orissa JEE 2004] |
| A. | Parabola |
| B. | Ellipse |
| C. | Hyperbola |
| D. | Circle |
| Answer» C. Hyperbola | |
| 4525. |
If \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\] are three collinear points in argand plane, then \[\left| \,\begin{matrix} {{z}_{1}} & \overline{{{z}_{1}}} & 1 \\ {{z}_{2}} & \overline{{{z}_{2}}} & 1 \\ {{z}_{3}} & \overline{{{z}_{3}}} & 1 \\ \end{matrix}\, \right|=\] [Orissa JEE 2004] |
| A. | 0 |
| B. | -1 |
| C. | 1 |
| D. | 2 |
| Answer» B. -1 | |
| 4526. |
If P, Q, R, S are represented by the complex numbers \[4+i,\,\,1+6i,\,\,-4+3i,\,\,-1-2i\] respectively, then PQRS is a [Orissa JEE 2003] |
| A. | Rectangle |
| B. | Square |
| C. | Rhombus |
| D. | Parallelogram |
| Answer» C. Rhombus | |
| 4527. |
The equation \[z\,\overline{z}+a\,\bar{z}+\bar{a}z+b=0,b\in R\] represents a circle if |
| A. | \[|a{{|}^{2}}=b\] |
| B. | \[|a{{|}^{2}}>b\] |
| C. | \[|a{{|}^{2}}<b\] |
| D. | None of these |
| Answer» C. \[|a{{|}^{2}}<b\] | |
| 4528. |
For all complex numbers \[{{z}_{1}},{{z}_{2}}\] satisfying \[|{{z}_{1}}|\,=12\,\] \[\,\text{and }\,|{{z}_{2}}-3-4i|\,=5,\] the minimum value of \[|{{z}_{1}}-{{z}_{2}}|\] is [IIT Screening 2002] |
| A. | 0 |
| B. | 2 |
| C. | 7 |
| D. | 17 |
| Answer» C. 7 | |
| 4529. |
Let \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\] be three vertices of an equilateral triangle circumscribing the circle \[|z|\]=\[\frac{1}{2}\]. If \[{{z}_{1}}=\frac{1}{2}+\frac{\sqrt{3}\,i}{2}\] and \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\] are in anticlockwise sense then \[{{z}_{2}}\] is [Orissa JEE 2002] |
| A. | \[1+\sqrt{3}\,i\] |
| B. | \[1-\sqrt{3}\,i\] |
| C. | 1 |
| D. | - 1\[\] |
| Answer» E. | |
| 4530. |
A circle whose radius is r and centre \[{{z}_{0}}\], then the equation of the circle is [RPET 2000] |
| A. | \[z\bar{z}-z{{\bar{z}}_{0}}-\bar{z}{{z}_{0}}+{{z}_{0}}{{\bar{z}}_{0}}={{r}^{2}}\] |
| B. | \[z\bar{z}+z{{\bar{z}}_{0}}-\bar{z}{{z}_{0}}+{{z}_{0}}{{\bar{z}}_{0}}={{r}^{2}}\] |
| C. | \[z\bar{z}-z{{\bar{z}}_{0}}+\bar{z}{{z}_{0}}-{{z}_{0}}{{\bar{z}}_{0}}={{r}^{2}}\] |
| D. | None of these |
| Answer» B. \[z\bar{z}+z{{\bar{z}}_{0}}-\bar{z}{{z}_{0}}+{{z}_{0}}{{\bar{z}}_{0}}={{r}^{2}}\] | |
| 4531. |
In the argand diagram, if O, P and Q represents respectively the origin, the complex numbers z and z + iz, then the angle \[\angle OPQ\] is [MP PET 2000] |
| A. | \[\frac{\pi }{4}\] |
| B. | \[\frac{\pi }{3}\] |
| C. | \[\frac{\pi }{2}\] |
| D. | \[\frac{2\pi }{3}\] |
| Answer» D. \[\frac{2\pi }{3}\] | |
| 4532. |
If centre of a regular hexagon is at origin and one of the vertex on argand diagram is 1 + 2i, then its perimeter is [RPET 1999] |
| A. | \[2\sqrt{5}\] |
| B. | \[6\sqrt{2}\] |
| C. | \[4\sqrt{5}\] |
| D. | \[6\sqrt{5}\] |
| Answer» E. | |
| 4533. |
If \[|z-2|/|z-3|=2\] represents a circle, then its radius is equal to [Kurukshetra CEE 1998] |
| A. | 1 |
| B. | \[1/3\] |
| C. | \[3/4\] |
| D. | \[2/3\] |
| Answer» E. | |
| 4534. |
If complex numbers \[{{z}_{1}},{{z}_{2}}\,\text{and }{{z}_{3}}\] represent the vertices A, B and C respectively of an isosceles triangle ABC of which \[\angle C\] is right angle, then correct statement is [RPET 1999] |
| A. | \[{{z}_{1}}^{2}+{{z}_{2}}^{2}+{{z}_{3}}^{2}={{z}_{1}}{{z}_{2}}{{z}_{3}}\] |
| B. | \[{{({{z}_{3}}-{{z}_{1}})}^{2}}={{z}_{3}}-{{z}_{2}}\] |
| C. | \[{{({{z}_{1}}-{{z}_{2}})}^{2}}=({{z}_{1}}-{{z}_{3}})\,({{z}_{3}}-{{z}_{2}})\] |
| D. | \[{{({{z}_{1}}-{{z}_{2}})}^{2}}=2({{z}_{1}}-{{z}_{3}})\,({{z}_{3}}-{{z}_{2}})\] |
| Answer» E. | |
| 4535. |
If the points \[{{P}_{1}}\]and \[{{P}_{2}}\] represent two complex numbers \[{{z}_{1}}\] and \[{{z}_{2}}\], then the point \[{{P}_{3}}\] represents the number |
| A. | \[{{z}_{1}}+{{z}_{2}}\] |
| B. | \[{{z}_{1}}-{{z}_{2}}\] |
| C. | \[{{z}_{1}}\times {{z}_{2}}\] |
| D. | \[{{z}_{1}}\div {{z}_{2}}\] |
| Answer» B. \[{{z}_{1}}-{{z}_{2}}\] | |
| 4536. |
The equation \[z\overline{z}+(2-3i)z+(2+3i)\overline{z}+4=0\] represents a circle of radius [Kurukshetra CEE 1996] |
| A. | 2 |
| B. | 3 |
| C. | 4 |
| D. | 6 |
| Answer» C. 4 | |
| 4537. |
The complex numbers \[{{z}_{1}},{{z}_{2}},{{z}_{3}}\] are the vertices of a triangle. Then the complex numbers \[z\] which make the triangle into a parallelogram is |
| A. | \[{{z}_{1}}+{{z}_{2}}-{{z}_{3}}\] |
| B. | \[{{z}_{1}}-{{z}_{2}}+{{z}_{3}}\] |
| C. | \[{{z}_{2}}+{{z}_{3}}-{{z}_{1}}\] |
| D. | All the above |
| Answer» E. | |
| 4538. |
The points \[{{z}_{1}},\,{{z}_{2}},\,{{z}_{3}},\,{{z}_{4}}\] in the complex plane are the vertices of a parallelogram taken in order, if and only if [IIT 1981, 1983; UPSEAT 2004] |
| A. | \[{{z}_{1}}+{{z}_{4}}={{z}_{2}}+{{z}_{3}}\] |
| B. | \[{{z}_{1}}+{{z}_{3}}={{z}_{2}}+{{z}_{4}}\] |
| C. | \[{{z}_{1}}+{{z}_{2}}={{z}_{3}}+{{z}_{4}}\] |
| D. | None of these |
| Answer» C. \[{{z}_{1}}+{{z}_{2}}={{z}_{3}}+{{z}_{4}}\] | |
| 4539. |
If \[{{z}_{1}},{{z}_{2}},{{z}_{3}},{{z}_{4}}\] are the affixes of four points in the Argand plane and \[z\] is the affix of a point such that \[|z-{{z}_{1}}|\,=\,|z-{{z}_{2}}|\,=\,|z-{{z}_{3}}|\,=|z-{{z}_{4}}|\], then \[{{z}_{1}},{{z}_{2}},{{z}_{3}},{{z}_{4}}\] are |
| A. | Concyclic |
| B. | Vertices of a parallelogram |
| C. | Vertices of a rhombus |
| D. | In a straight line |
| Answer» B. Vertices of a parallelogram | |
| 4540. |
The vertices \[B\] and \[D\] of a parallelogram are \[1-2i\]and \[4+2i\], If the diagonals are at right angles and \[AC=2BD\], the complex number representing \[A\] is |
| A. | \[\frac{5}{2}\] |
| B. | \[3i-\frac{3}{2}\] |
| C. | \[3i-4\] |
| D. | \[3i+4\] |
| Answer» C. \[3i-4\] | |
| 4541. |
Let \[a\] be a complex number such that \[|a|\, |
| A. | \[|z-a|=a\] |
| B. | \[\left| z-\frac{1}{1-a} \right|=|1-a|\] |
| C. | \[\left| z-\frac{1}{1-a} \right|=\frac{1}{|1-a|}\] |
| D. | \[|z-(1-a)|\,=|\,1-a|\] |
| Answer» D. \[|z-(1-a)|\,=|\,1-a|\] | |
| 4542. |
The centre of a regular polygon of \[n\] sides is located at the point \[z=0\] and one of its vertex \[{{z}_{1}}\] is known. If \[{{z}_{2}}\] be the vertex adjacent to \[{{z}_{1}}\], then \[{{z}_{2}}\] is equal to |
| A. | \[{{z}_{1}}\left( \cos \frac{2\pi }{n}\pm i\sin \frac{2\pi }{n} \right)\] |
| B. | \[{{z}_{1}}\left( \cos \frac{\pi }{n}\pm i\sin \frac{\pi }{n} \right)\] |
| C. | \[{{z}_{1}}\left( \cos \frac{\pi }{2n}\pm i\sin \frac{\pi }{2n} \right)\] |
| D. | None of these |
| Answer» B. \[{{z}_{1}}\left( \cos \frac{\pi }{n}\pm i\sin \frac{\pi }{n} \right)\] | |
| 4543. |
The vector \[z=3-4i\] is turned anticlockwise through an angle of \[{{180}^{o}}\] and stretched 2.5 times. The complex number corresponding to the newly obtained vector is |
| A. | \[\frac{15}{2}-10i\] |
| B. | \[\frac{-15}{2}+10i\] |
| C. | \[\frac{-15}{2}-10i\] |
| D. | None of these |
| Answer» C. \[\frac{-15}{2}-10i\] | |
| 4544. |
\[POQ\] is a straight line through the origin \[O,\,\,P\] and \[Q\] represent the complex numbers \[a+ib\] and\[c+id\] respectively and \[OP=OQ\], then |
| A. | \[|a+ib|\,=\,|c+id|\] |
| B. | \[a+c=b+d\] |
| C. | \[arg(a+ib)=arg(c+id)\] |
| D. | None of these |
| Answer» C. \[arg(a+ib)=arg(c+id)\] | |
| 4545. |
If \[\omega \] is a complex number satisfying \[\left| \text{ }\omega +\frac{1}{\omega }\text{ } \right|=2\], then maximum distance of \[\omega \]from origin is |
| A. | \[2+\sqrt{3}\] |
| B. | \[1+\sqrt{2}\] |
| C. | \[1+\sqrt{3}\] |
| D. | None of these |
| Answer» C. \[1+\sqrt{3}\] | |
| 4546. |
The number of triangles that can be formed by choosing the vertices from a set of 12 points, seven of which lie on the same straight line, is [Roorkee 1989; BIT 1989; MP PET 1995;Pb. CET 1997, 98; Roorkee 2000; DCE 2002; AMU 2005] |
| A. | 185 |
| B. | 175 |
| C. | 115 |
| D. | 105 |
| Answer» B. 175 | |
| 4547. |
The number of straight lines joining 8 points on a circle is [MP PET 1984] |
| A. | 8 |
| B. | 16 |
| C. | 24 |
| D. | 28 |
| Answer» E. | |
| 4548. |
The number of diagonals in a polygon of \[m\] sides is [BIT 1992; MP PET 1999; UPSEAT 1999; DCE 1999; Pb. CET 2001] |
| A. | \[\frac{1}{2\ !}m(m-5)\] |
| B. | \[\frac{1}{2\ !}m(m-1)\] |
| C. | \[\frac{1}{2\ !}m(m-3)\] |
| D. | \[\frac{1}{2\ !}m(m-2)\] |
| Answer» D. \[\frac{1}{2\ !}m(m-2)\] | |
| 4549. |
How many triangles can be drawn by means of 9 non-collinear points |
| A. | 84 |
| B. | 72 |
| C. | 144 |
| D. | 126 |
| Answer» B. 72 | |
| 4550. |
If a polygon has 44 diagonals, then the number of its sides are [MP PET 1998; Pb. CET 1996, 2002] |
| A. | 7 |
| B. | 11 |
| C. | 8 |
| D. | None of these |
| Answer» C. 8 | |