JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank

done Geometry of complex numbers Total Questions - 75

• question_answer1)
  Length of the line segment joining the points \[-1-i\] and \[2+3i\]  is
  A)
  - 5 done clear
  B)
  15 done clear
  C)
  5 done clear
  D)
  25 done clear
  View Solution play_arrow

• question_answer2)
  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)
  \[{{z}_{1}}+{{z}_{4}}={{z}_{2}}+{{z}_{3}}\] done clear
  B)
  \[{{z}_{1}}+{{z}_{3}}={{z}_{2}}+{{z}_{4}}\] done clear
  C)
  \[{{z}_{1}}+{{z}_{2}}={{z}_{3}}+{{z}_{4}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer3)
  The equation \[z\,\overline{z}+a\,\bar{z}+\bar{a}z+b=0,b\in R\] represents a circle if
  A)
  \[|a{{|}^{2}}=b\] done clear
  B)
  \[|a{{|}^{2}}>b\] done clear
  C)
  \[|a{{|}^{2}}<b\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer4)
  Let the complex numbers \[{{z}_{1}},{{z}_{2}}\] and \[{{z}_{3}}\] be the vertices of an equilateral triangle. Let \[{{z}_{0}}\]be the circumcentre of the triangle, then \[z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=\] [IIT 1981]
  A)
  \[z_{0}^{2}\] done clear
  B)
  \[-z_{0}^{2}\] done clear
  C)
  \[3z_{0}^{2}\] done clear
  D)
  \[-3z_{0}^{2}\] done clear
  View Solution play_arrow

• question_answer5)
  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 done clear
  B)
   A straight line done clear
  C)
  A parabola done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer6)
  If three complex numbers are in A.P., then they lie on     [IIT 1985; DCE 2001; Pb. CET 2003]
  A)
  A circle in the complex plane done clear
  B)
  A straight line in the complex plane done clear
  C)
  A parabola in the complex plane done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  If \[a\] and \[b\] are real numbers between 0 and 1 such that the points \[{{z}_{1}}=a+i,{{z}_{2}}=1+bi\] and  \[{{z}_{3}}=0\] form an equilateral triangle, then [IIT 1989]
  A)
  \[a=b=2+\sqrt{3}\] done clear
  B)
  \[a=b=2-\sqrt{3}\] done clear
  C)
  \[a=2-\sqrt{3},b=2+\sqrt{3}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer8)
  If \[|z|=2\], then the points representing the complex numbers \[-1+5z\] will lie on a
  A)
  Circle done clear
  B)
  Straight line done clear
  C)
  Parabola done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer9)
  If the vertices of a quadrilateral be \[A=1+2i,\] \[B=-3+i,\] \[C=-2-3i\] and \[D=2-2i\],  then the quadrilateral is
  A)
  Parallelogram done clear
  B)
  Rectangle done clear
  C)
  Square done clear
  D)
  Rhombus done clear
  View Solution play_arrow

• question_answer10)
  In the Argand plane, the vector \[z=4-3i\] is turned in the clockwise sense through \[{{180}^{o}}\]and stretched three times. The complex number represented by the new vector is [DCE 2005]
  A)
  \[12+9i\] done clear
  B)
  \[12-9i\] done clear
  C)
  \[-12-9i\] done clear
  D)
  \[-12+9i\] done clear
  View Solution play_arrow

• question_answer11)
  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)
  \[2+\sqrt{3}\] done clear
  B)
  \[1+\sqrt{2}\] done clear
  C)
  \[1+\sqrt{3}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  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)
  \[\frac{15}{2}-10i\] done clear
  B)
  \[\frac{-15}{2}+10i\] done clear
  C)
  \[\frac{-15}{2}-10i\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  \[POQ\] 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)
  \[|a+ib|\,=\,|c+id|\] done clear
  B)
  \[a+c=b+d\] done clear
  C)
  \[arg(a+ib)=arg(c+id)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer14)
  Let  \[a\] be a complex number such that \[|a|\,<1\] and \[{{z}_{1}},{{z}_{2}},......\] be vertices of a polygon such that \[{{z}_{k}}=1+a+{{a}^{2}}+.....+{{a}^{k-1}}\]. Then the vertices of the polygon lie within a circle
  A)
  \[|z-a|=a\] done clear
  B)
  \[\left| z-\frac{1}{1-a} \right|=|1-a|\] done clear
  C)
  \[\left| z-\frac{1}{1-a} \right|=\frac{1}{|1-a|}\] done clear
  D)
  \[|z-(1-a)|\,=|\,1-a|\] done clear
  View Solution play_arrow

• question_answer15)
  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)
  \[{{z}_{1}}\left( \cos \frac{2\pi }{n}\pm i\sin \frac{2\pi }{n} \right)\] done clear
  B)
  \[{{z}_{1}}\left( \cos \frac{\pi }{n}\pm i\sin \frac{\pi }{n} \right)\] done clear
  C)
  \[{{z}_{1}}\left( \cos \frac{\pi }{2n}\pm i\sin \frac{\pi }{2n} \right)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer16)
  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)
  \[\frac{5}{2}\] done clear
  B)
  \[3i-\frac{3}{2}\] done clear
  C)
  \[3i-4\] done clear
  D)
  \[3i+4\] done clear
  View Solution play_arrow

• question_answer17)
  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 done clear
  B)
  Vertices of a parallelogram done clear
  C)
  Vertices of a rhombus done clear
  D)
  In a straight line done clear
  View Solution play_arrow

• question_answer18)
  \[ABCD\] is a rhombus. Its diagonals \[AC\] and \[BD\] intersect at the point \[M\] and satisfy \[BD=2AC\].  If the points \[D\] and \[M\] represents the complex numbers \[1+i\] and \[2-i\] respectively, then \[A\] represents the complex number
  A)
  \[3-\frac{1}{2}i\]or \[1-\frac{3}{2}i\] done clear
  B)
  \[\frac{3}{2}-i\]or \[\frac{1}{2}-3i\] done clear
  C)
  \[\frac{1}{2}-i\]or \[1-\frac{1}{2}i\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer19)
  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)
  \[{{z}_{1}}+{{z}_{2}}-{{z}_{3}}\] done clear
  B)
  \[{{z}_{1}}-{{z}_{2}}+{{z}_{3}}\] done clear
  C)
  \[{{z}_{2}}+{{z}_{3}}-{{z}_{1}}\] done clear
  D)
  All the above done clear
  View Solution play_arrow

• question_answer20)
  The equation \[z\overline{z}+(2-3i)z+(2+3i)\overline{z}+4=0\] represents a circle of radius    [Kurukshetra CEE 1996]
  A)
  2 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer21)
  A rectangle is constructed in the complex plane with its sides parallel to the axes and its centre is situated at the origin. If one of the vertices of the rectangle is \[a+ib\sqrt{3}\], then the area of the rectangle is
  A)
  \[ab\sqrt{3}\] done clear
  B)
  \[2ab\sqrt{3}\] done clear
  C)
  \[3ab\sqrt{3}\] done clear
  D)
  \[4ab\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer22)
   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)
  \[{{z}_{1}}+{{z}_{2}}\] done clear
  B)
  \[{{z}_{1}}-{{z}_{2}}\] done clear
  C)
  \[{{z}_{1}}\times {{z}_{2}}\] done clear
  D)
  \[{{z}_{1}}\div {{z}_{2}}\] done clear
  View Solution play_arrow

• question_answer23)
  If \[|z-2|/|z-3|=2\] represents a circle, then its radius is equal to [Kurukshetra CEE 1998]
  A)
  1 done clear
  B)
  \[1/3\] done clear
  C)
  \[3/4\] done clear
  D)
  \[2/3\] done clear
  View Solution play_arrow

• question_answer24)
  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)
  \[{{z}_{1}}^{2}+{{z}_{2}}^{2}+{{z}_{3}}^{2}={{z}_{1}}{{z}_{2}}{{z}_{3}}\] done clear
  B)
  \[{{({{z}_{3}}-{{z}_{1}})}^{2}}={{z}_{3}}-{{z}_{2}}\] done clear
  C)
  \[{{({{z}_{1}}-{{z}_{2}})}^{2}}=({{z}_{1}}-{{z}_{3}})\,({{z}_{3}}-{{z}_{2}})\] done clear
  D)
  \[{{({{z}_{1}}-{{z}_{2}})}^{2}}=2({{z}_{1}}-{{z}_{3}})\,({{z}_{3}}-{{z}_{2}})\] done clear
  View Solution play_arrow

• question_answer25)
  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)
  \[2\sqrt{5}\] done clear
  B)
  \[6\sqrt{2}\] done clear
  C)
  \[4\sqrt{5}\] done clear
  D)
  \[6\sqrt{5}\] done clear
  View Solution play_arrow

• question_answer26)
  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)
  \[\frac{\pi }{4}\] done clear
  B)
  \[\frac{\pi }{3}\] done clear
  C)
  \[\frac{\pi }{2}\] done clear
  D)
  \[\frac{2\pi }{3}\] done clear
  View Solution play_arrow

• question_answer27)
  A circle whose radius is r and centre \[{{z}_{0}}\], then the equation of the circle is [RPET 2000]
  A)
  \[z\bar{z}-z{{\bar{z}}_{0}}-\bar{z}{{z}_{0}}+{{z}_{0}}{{\bar{z}}_{0}}={{r}^{2}}\] done clear
  B)
  \[z\bar{z}+z{{\bar{z}}_{0}}-\bar{z}{{z}_{0}}+{{z}_{0}}{{\bar{z}}_{0}}={{r}^{2}}\] done clear
  C)
  \[z\bar{z}-z{{\bar{z}}_{0}}+\bar{z}{{z}_{0}}-{{z}_{0}}{{\bar{z}}_{0}}={{r}^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer28)
  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)
  \[1+\sqrt{3}\,i\] done clear
  B)
  \[1-\sqrt{3}\,i\] done clear
  C)
  1 done clear
  D)
  - 1\[\] done clear
  View Solution play_arrow

• question_answer29)
  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 done clear
  B)
  2 done clear
  C)
  7 done clear
  D)
  17 done clear
  View Solution play_arrow

• question_answer30)
  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 done clear
  B)
  Square done clear
  C)
  Rhombus done clear
  D)
  Parallelogram done clear
  View Solution play_arrow

• question_answer31)
  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 done clear
  B)
  - 1 done clear
  C)
  1 done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer32)
  If z is a complex number in the Argand plane, then the equation \[|z-2|+|z+2|=8\]represents [Orissa JEE 2004]
  A)
  Parabola done clear
  B)
  Ellipse done clear
  C)
  Hyperbola done clear
  D)
  Circle done clear
  View Solution play_arrow

• question_answer33)
  The points \[1+3i,\,5+i\] and \[3+2i\] in the complex plane are [MP PET 1987]
  A)
  Vertices of a right angled triangle done clear
  B)
  Collinear done clear
  C)
  Vertices of an obtuse angled triangle done clear
  D)
  Vertices of an equilateral triangle done clear
  View Solution play_arrow

• question_answer34)
  If \[{{z}_{1}}\] and \[{{z}_{2}}\] are two complex numbers, then \[|{{z}_{1}}+{{z}_{2}}|\] is                     [RPET 1985; MP PET 1987, 2004;  Kerala (Engg.) 2002]
  A)
  \[\le \,|{{z}_{1}}|+|{{z}_{2}}|\] done clear
  B)
  \[\le \,|{{z}_{1}}|-|{{z}_{2}}|\] done clear
  C)
  \[<\,|{{z}_{1}}|+|{{z}_{2}}|\] done clear
  D)
  \[>\,|{{z}_{1}}|+|{{z}_{2}}|\] done clear
  View Solution play_arrow

• question_answer35)
  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)
  \[2|z{{|}^{2}}\] done clear
  B)
  \[\frac{1}{2}|z{{|}^{2}}\] done clear
  C)
  \[|z{{|}^{2}}\] done clear
  D)
  \[\frac{3}{2}|z{{|}^{2}}\] done clear
  View Solution play_arrow

• question_answer36)
  If  A, B, C are represented by \[3+4i,\] \[5-2i\], \[-1+16i\], then A, B, C are [RPET 1986]
  A)
  Collinear done clear
  B)
  Vertices of equilateral triangle done clear
  C)
  Vertices of isosceles triangle done clear
  D)
  Vertices of right angled triangle done clear
  View Solution play_arrow

• question_answer37)
  If  \[{{z}_{1}},{{z}_{2}}\in C,\] then [MP PET 1995]
  A)
  \[|{{z}_{1}}+{{z}_{2}}|\,\ge \,|{{z}_{1}}|+|{{z}_{2}}|\] done clear
  B)
  \[|{{z}_{1}}-{{z}_{2}}|\,\ge \,|{{z}_{1}}|+|{{z}_{2}}|\] done clear
  C)
  \[|{{z}_{1}}-{{z}_{2}}|\,\le \left| \,|{{z}_{1}}|-|{{z}_{2}}|\, \right|\] done clear
  D)
  \[|{{z}_{1}}+{{z}_{2}}|\,\ge \left| \,|{{z}_{1}}|-|{{z}_{2}}|\, \right|\] done clear
  View Solution play_arrow

• question_answer38)
  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)
  \[{{z}_{1}}+{{z}_{2}}+{{z}_{3}}=0\] done clear
  B)
  \[{{z}_{1}}+4{{z}_{2}}+{{z}_{3}}=0\] done clear
  C)
  \[{{z}_{1}}+{{z}_{2}}+4{{z}_{3}}=0\] done clear
  D)
  \[{{z}_{1}}+{{z}_{2}}+{{z}_{3}}=0\] done clear
  View Solution play_arrow

• question_answer39)
  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 done clear
  B)
  \[{{z}_{1}},{{z}_{2}}\]and the origin form a right angled triangle done clear
  C)
  \[{{z}_{1}},{{z}_{2}}\]and the origin form an equilateral triangle done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer40)
  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 done clear
  B)
  9 done clear
  C)
  \[3\sqrt{2}\] done clear
  D)
  \[2\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer41)
  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 done clear
  B)
  3 done clear
  C)
  4 done clear
  D)
  5 done clear
  View Solution play_arrow

• question_answer42)
  Which one of the following statement is true [RPET 2002]
  A)
  \[|x-y|\,=\,|x|\,-\,|y|\] done clear
  B)
  \[|x+y|\,\le \,|x|\,-\,|y|\] done clear
  C)
  \[|x-y|\,\ge \,|x|\,-\,|y|\] done clear
  D)
  \[|x+y|\,\ge \,|x|\,-\,|y|\] done clear
  View Solution play_arrow

• question_answer43)
  The area of the triangle whose vertices are represented by the complex numbers 0, z, \[z{{e}^{i\alpha }},\] \[(0<\alpha <\pi )\] equals [AMU 2002]
  A)
  \[\frac{1}{2}|z{{|}^{2}}\cos \alpha \] done clear
  B)
  \[\frac{1}{2}|z{{|}^{2}}\sin \alpha \] done clear
  C)
  \[\frac{1}{2}|z{{|}^{2}}\sin \alpha \cos \alpha \] done clear
  D)
  \[\frac{1}{2}|z{{|}^{2}}\] done clear
  View Solution play_arrow

• question_answer44)
  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 done clear
  B)
  Isosceles triangle  done clear
  C)
  Right angled triangle done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer45)
  The complex numbers \[z=x+iy\] which satisfy the equation \[\left| \frac{z-5i}{z+5i} \right|=1\] lie on [IIT 1982]
  A)
  Real axis done clear
  B)
  The line \[y=5\] done clear
  C)
  A circle passing through the origin done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer46)
  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 \[\frac{\sqrt{5}}{2}\] done clear
  B)
  Circle of radius \[\frac{5}{4}\] done clear
  C)
  Straight line done clear
  D)
  Parabola done clear
  View Solution play_arrow

• question_answer47)
  If \[|z+1|\,\,=\sqrt{2}|z-1|,\]then the locus described by the point \[z\] in the Argand diagram is a
  A)
  Straight line done clear
  B)
  Circle done clear
  C)
  Parabola done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer48)
  The region of the complex plane for which \[\left| \frac{z-a}{z+\overline{a}} \right|=1\,\] \[\,[R(a)\ne 0]\] is
  A)
  \[x-\]axis done clear
  B)
  \[y-\]axis done clear
  C)
  The straight line \[x=a\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer49)
  The region of Argand plane defined by  \[|z-1|\,\,+\,\,|z+1|\,\,\le 4\] is
  A)
  Interior of an ellipse done clear
  B)
  Exterior of a circle done clear
  C)
  Interior and boundary of an ellipse done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer50)
  The locus of the points z which satisfy the condition arg \[\left( \frac{z-1}{z+1} \right)\] =\[\frac{\pi }{3}\] is
  A)
  A straight line done clear
  B)
  A circle done clear
  C)
  A parabola done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer51)
  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 done clear
  B)
  A straight line done clear
  C)
  A parabola done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer52)
  If  \[z=(\lambda +3)+i\sqrt{5-{{\lambda }^{2}},}\] then the locus of z is a
  A)
  Circle done clear
  B)
  Straight line done clear
  C)
  Parabola done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer53)
  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 done clear
  B)
  A straight line done clear
  C)
  A circle done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer54)
  If \[z=x+iy\] and \[|z-zi|\,=1,\]then [RPET 1988, 91]
  A)
  \[z\]lies on \[x\]-axis done clear
  B)
  \[z\]lies on \[y\]-axis done clear
  C)
  z lies on a circle done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer55)
  The locus of \[z\] given by \[\left| \frac{z-1}{z-i} \right|=1\],  is [Roorkee 1990]
  A)
  A circle done clear
  B)
  An ellipse done clear
  C)
  A straight line done clear
  D)
  A parabola done clear
  View Solution play_arrow

• question_answer56)
  \[R({{z}^{2}})=1\]is represented by
  A)
  The parabola \[{{x}^{2}}+{{y}^{2}}=1\] done clear
  B)
  The hyperbola \[{{x}^{2}}-{{y}^{2}}=1\] done clear
  C)
  Parabola or a circle done clear
  D)
  All the above done clear
  View Solution play_arrow

• question_answer57)
  The locus represented by \[|z-1|=|z+i|\] is [EAMCET 1991]
  A)
  A circle of radius 1 done clear
  B)
  An ellipse with foci at \[(1,\,0)\] and (0, - 1) done clear
  C)
  A straight line through the origin done clear
  D)
  A circle on the line joining \[(1,\,0),(0,\,1)\] as diameter done clear
  View Solution play_arrow

• question_answer58)
  If  \[{{\log }_{\sqrt{3}}}\left( \frac{|z{{|}^{2}}-|z|+1}{2+|z|} \right)\]\[<2\], then the locus of \[z\] is
  A)
  \[|z|=5\] done clear
  B)
  \[|z|<5\] done clear
  C)
  \[|z|>5\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer59)
  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 done clear
  B)
  Parabola done clear
  C)
  Ellipse done clear
  D)
  Straight line done clear
  View Solution play_arrow

• question_answer60)
  If \[z=x+iy\] and \[|z-2+i|\,=\,|z-3-i|,\] then locus of z is [RPET 1999]
  A)
  \[2x+4y-5=0\] done clear
  B)
  \[2x-4y-5=0\] done clear
  C)
  \[x+2y=0\] done clear
  D)
  \[x-2y+5=0\] done clear
  View Solution play_arrow

• question_answer61)
  Locus of the point z satisfying the equation \[|iz-1|\]+ \[|z-i|=2\] is [Roorkee 1999]
  A)
  A straight line done clear
  B)
  A circle done clear
  C)
  An ellipse done clear
  D)
  A pair of straight lines done clear
  View Solution play_arrow

• question_answer62)
  If \[z=x+iy\] is a complex number satisfying \[{{\left| z+\frac{i}{2} \right|}^{2}}=\] \[\,\,{{\left| z-\frac{i}{2} \right|}^{2}},\] then the locus of z is [EAMCET 2002]
  A)
  \[2y=x\] done clear
  B)
  \[y=x\] done clear
  C)
  y-axis done clear
  D)
  x-axis done clear
  View Solution play_arrow

• question_answer63)
  The locus of the point z satisfying \[arg\left( \frac{z-1}{z+1} \right)=k,\] (where k is non zero) is [Orissa JEE 2002]
  A)
  Circle with centre on y-axis done clear
  B)
  Circle with centre on x-axis done clear
  C)
  A straight line parallel to x-axis done clear
  D)
  A straight line making an angle \[{{60}^{o}}\] with the x-axis done clear
  View Solution play_arrow

• question_answer64)
  If the amplitude of \[z-2-3i\] is \[\pi /4\], then the locus of \[z=x+iy\] is [EAMCET 2003]
  A)
  \[x+y-1=0\] done clear
  B)
  \[x-y-1=0\] done clear
  C)
  \[x+y+1=0\] done clear
  D)
  \[x-y+1=0\] done clear
  View Solution play_arrow

• question_answer65)
  If \[|{{z}^{2}}-1|\,=\,|z{{|}^{2}}+1\], then \[z\]lies on [AIEEE 2004]
  A)
  An ellipse done clear
  B)
  The imaginary axis done clear
  C)
  A circle done clear
  D)
  The real axis done clear
  View Solution play_arrow

• question_answer66)
  If \[\frac{-31}{17}\]and \[\omega =\frac{1-iz}{z-i}\] than \[|\omega |=1\] shows that in complex plane                            [RPET 1985, 97; IIT 1983; DCE 2000, 01; UPSEAT 2003; MP PET 2004]
  A)
  z will be at imaginary axis done clear
  B)
  z will be at real axis done clear
  C)
  z will be at unity circle done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer67)
  The equation \[|z-5i|\div |z+5i|\,=12,\] where \[z=x+iy,\] represents a/an [AMU 1999]
  A)
  Circle done clear
  B)
  Ellipse done clear
  C)
  Parabola done clear
  D)
  No real curve done clear
  View Solution play_arrow

• question_answer68)
  If \[z=x+iy\] and \[arg\,\left( \frac{z-2}{z+2} \right)=\frac{\pi }{6}\], then locus of z  is [RPET 2002]
  A)
  A straight line done clear
  B)
  A circle done clear
  C)
  A parabola done clear
  D)
  An ellipse done clear
  View Solution play_arrow

• question_answer69)
  If \[w=\frac{z}{z-\frac{1}{3}i}\] and \[|w|=1\], then z lies on [AIEEE 2005]
  A)
  A straight line done clear
  B)
  A parabola done clear
  C)
  An ellipse done clear
  D)
  A circle done clear
  View Solution play_arrow

• question_answer70)
  If \[|8+z|+|z-8|=16\] where z is a complex number, then the point z will lie on [J & K 2005]
  A)
  A circle done clear
  B)
  An ellipse done clear
  C)
  A straight line done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer71)
  PQ and PR are two infinite rays. QAR is an arc. Point lying in the shaded region excluding the boundary satisfies  [IIT Screening 2005]
  A)
  \[|z-1|>2;|\arg (z-1)|\,<\frac{\pi }{4}\] done clear
  B)
  \[|z-1|>2;|\arg (z-1)|\,<\frac{\pi }{2}\] done clear
  C)
  \[|z+1|>2;|\arg (z+1)|\,<\frac{\pi }{4}\] done clear
  D)
  \[|z+1|>2;|\arg (z+1)|\,<\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer72)
  . Which of the following equations can represent a triangle [Orissa JEE 2005]
  A)
  \[|z-1|\,=\,|z-2|\] done clear
  B)
  \[|z-1|=|z-2|=|z-i|\] done clear
  C)
  \[|z-1|-|z-2|=2a\] done clear
  D)
  \[|z-1{{|}^{2}}+|z-2{{|}^{2}}=4\] done clear
  View Solution play_arrow

• question_answer73)
  The number of solutions for the equations \[|z-1|=|z-2|=\] \[|z-i|\] is  [Orissa JEE 2005]
  A)
  One solution done clear
  B)
  3 solutions done clear
  C)
  2 solutions done clear
  D)
  No solution done clear
  View Solution play_arrow

• question_answer74)
  If \[|z-2-3i|+|z+2-6i|=4\], where \[i=\sqrt{-1}\], then locus of \[P(z)\] is [DCE 2005]
  A)
  An ellipse done clear
  B)
  \[\varphi \] done clear
  C)
  Line segment joining of point \[2+3i\] and \[-2+6i\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer75)
  If \[z=\sqrt{2}-i\sqrt{2}\] is rotated through an angle \[45{}^\circ \] in the anti-clockwise direction about the origin, then the coordinates of its new position are [Kerala (Engg.) 2005]
  A)
  (2, 0) done clear
  B)
  (\[\sqrt{2},\,\sqrt{2}\]) done clear
  C)
  \[(\sqrt{2},\,-\sqrt{2}\]) done clear
  D)
  \[(\sqrt{2},0)\] done clear
  E)
  (4, 0) done clear
  View Solution play_arrow