1
JEE Advanced 2013 Paper 2 Offline
+4
-1
Let $$S = {S_1} \cap {S_2} \cap {S_3}$$, where $${S_1} = \left\{ {z \in C:\left| z \right| < 4} \right\},{S_2} = \left\{ {z \in C:{\mathop{\rm Im}\nolimits} \left[ {{{z - 1 + \sqrt 3 i} \over {1 - \sqrt 3 i}}} \right] > 0} \right\}$$ and $${S_3} = \left\{ {z \in C:{\mathop{\rm Re}\nolimits} z > 0} \right\}\,$$.

$$\,\mathop {\min }\limits_{z \in S} \left| {1 - 3i - z} \right| =$$

A
$${{2 - \sqrt 3 } \over 2}$$
B
$${{2 + \sqrt 3 } \over 2}$$
C
$${{3 - \sqrt 3 } \over 2}$$
D
$${{3 + \sqrt 3 } \over 2}$$
2
JEE Advanced 2013 Paper 1 Offline
+4
-1
Let complex numbers $$\alpha \,and\,{1 \over {\overline \alpha }}\,$$ lie on circles $${\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2} = {r^2}$$ and $$\,{\left( {x - {x_0}} \right)^2} + \,\,{\left( {y - {y_0}} \right)^2} = 4{r^2}$$ respextively. If $${z_0} = {x_0} + i{y_0}$$ satisfies the equation $$2{\left| {{z_0}} \right|^2}\, = {r^2} + 2,\,then\,\left| a \right| =$$
A
$${1 \over {\sqrt 2 }}$$
B
$${1 \over 2}\,$$
C
$${1 \over {\sqrt 7 }}$$
D
$${1 \over 3}$$
3
IIT-JEE 2012 Paper 1 Offline
+4
-1
Let z be a complex number such that the imaginary part of z is non-zero and $$a\, = \,{z^2} + \,z\, + 1$$ is real. Then a cannot take the value
A
- 1
B
$${1 \over 3}$$
C
$${1 \over 2}$$
D
$${3 \over 4}$$
4
IIT-JEE 2010 Paper 2 Offline
+4
-1
Match the statements in Column I with those in Column II.

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

Column I

(A) The set of points z satisfying $$\left| {z - i} \right|\left. {z\,} \right\|\,\, = \left| {z + i} \right|\left. {\,z} \right\|$$ is contained in or equal to
(B) The set of points z satisfying $$\left| {z + 4} \right| + \,\left| {z - 4} \right| = 10$$ is contained in or equal to
(C) If $$\left| w \right|$$= 2, then the set of points $$z = w - {1 \over w}$$ is contained in or equal to
(D) If $$\left| w \right|$$ = 1, then the set of points $$z = w + {1 \over w}$$ is contained in or equal to.

Column II

(p) an ellipse with eccentricity $${4 \over 5}$$
(q) the set of points z satisfying Im z = 0
(r) the set of points z satisfying $$\left| {{\rm{Im }}\,{\rm{z }}} \right| \le 1$$
(s) the set of points z satisfying $$\,\left| {{\mathop{\rm Re}\nolimits} \,\,z} \right| < 2$$
(t) the set of points z satisfying $$\left| {\,z} \right| \le 3$$
A
(A) - q, s ; (B) - p ; (C) - p, t ; (D) - q, r, s, t
B
(A) - q, r ; (B) - p ; (C) - p, s, t ; (D) - q, r, s, t
C
(A) - p, r ; (B) - p ; (C) - p, t ; (D) -q, r, s, t
D
(A) - p ; (B) - q ; (C) - r, s ; (D) -q, r, s, t
EXAM MAP
Medical
NEET