MCQ (Single Correct Answer)

1

The value of $${(1 - \omega + {\omega ^2})^5} + {(1 + \omega - {\omega ^2})^5}$$, where $$\omega$$ and $$\omega$$2 are the complex cube roots of unity is

WB JEE 2008
2

Let $$\alpha$$, $$\beta$$ be the roots of $${x^2} - 2x\cos \phi + 1 = 0$$, then the equation whose roots are $${\alpha ^n},{\beta ^n}$$ is

WB JEE 2008
3

The principal amplitude of $${(\sin 40^\circ + i\cos 40^\circ )^5}$$ is

WB JEE 2008
4

A and B are two points on the Argand plane such that the segment AB is bisected at the point (0, 0). If the point A, which is in the third quadrant has principal amplitude $$\theta$$, then the principal amplitude of the point B is

WB JEE 2008
5

For two complex numbers z1, z2 the relation $$\left| {{z_1} + {z_2}} \right| = \left| {{z_1}} \right| + \left| {{z_2}} \right|$$ holds if

WB JEE 2008
6

If 1, $$\omega$$, $$\omega$$2 are cube roots of unity, then $$\left| {\matrix{ 1 & {{\omega ^n}} & {{\omega ^{2n}}} \cr {{\omega ^{2n}}} & 1 & {{\omega ^n}} \cr {{\omega ^n}} & {{\omega ^{2n}}} & 1 \cr } } \right|$$ has value

WB JEE 2008
7

If $$i = \sqrt { - 1} $$ and n is positive integer, then $${i^n} + {i^{n + 1}} + {i^{n + 2}} + {i^{n + 3}}$$ is equal to

WB JEE 2009
8

The modulus of $${{1 - i} \over {3 + i}} + {{4i} \over 5}$$ is

WB JEE 2009
9

For any complex number z, the minimum value of $$|z| + |z - 1|$$ is

WB JEE 2009
10

If $$z = {4 \over {1 - i}}$$, then $$\overline z $$ is (where $$\overline z $$ is complex conjugate of z)

WB JEE 2010
11

If $$ - \pi < \arg (z) < - {\pi \over 2}$$, then $$\arg \overline z - \arg ( - \overline z )$$ is

WB JEE 2010
12

For the real parameter t, the locus of the complex number $$z = (1 - {t^2}) + i\sqrt {1 + {t^2}} $$ in the complex plane is

WB JEE 2011
13

If $$x + {1 \over x} = 2\cos \theta $$, then for any integer n, $${x^n} + {1 \over {{x^n}}} = $$

WB JEE 2011
14

If $$\omega$$ $$\ne$$ 1 is a cube root of unity, then the sum of the series $$S = 1 + 2\omega + 3{\omega ^2} + \,\,.....\,\, + 3n{\omega ^{3n - 1}}$$ is

WB JEE 2011
15

If $z_1, z_2$ are complex numbers such that $\frac{2 z_1}{3 z_2}$ is a purely imaginary number, then the value of $\left|\frac{z_1-z_2}{z_1+z_2}\right|$ is

WB JEE 2025
16

Let $\omega(\neq 1)$ be a cubic root of unity. Then the minimum value of the set $\left\{\mid a+b \omega+c \omega^2\right\}^2 ; a, b, c$ are distinct non-zero integers} equals

WB JEE 2025
17

If $\left|Z_1\right|=\left|Z_2\right|=\left|Z_3\right|=1$ and $Z_1+Z_2+Z_3=0$, then the area of the triangle whose vertices are $Z_1, Z_2, Z_3$ is

WB JEE 2025
18

If $$z_1$$ and $$z_2$$ be two roots of the equation $$z^2+a z+b=0, a^2<4 b$$, then the origin, $$\mathrm{z}_1$$ and $$\mathrm{z}_2$$ form an equilateral triangle if

WB JEE 2024
19

If $$\cos \theta+i \sin \theta, \theta \in \mathbb{R}$$, is a root of the equation

$$a_0 x^n+a_1 x^{n-1}+\ldots .+a_{n-1} x+a_n=0, a_0, a_1, \ldots . a_n \in \mathbb{R}, a_0 \neq 0,$$

then the value of $$a_1 \sin \theta+a_2 \sin 2 \theta+\ldots .+a_n \sin n \theta$$ is

WB JEE 2024
20

If the vertices of a square are $${z_1},{z_2},{z_3}$$ and $${z_4}$$ taken in the anti-clockwise order, then $${z_3} = $$

WB JEE 2023
21

Reflection of the line $$\overline a z + a\overline z = 0$$ in the real axis is given by :

WB JEE 2023
22

If $$|z - 25i| \le 15$$, then Maximum arg(z) $$-$$ Minimum arg(z) is equal to

(arg z is the principal value of argument of z)

WB JEE 2022
23

If z = x $$-$$ iy and $${z^{{1 \over 3}}} = p + iq(x,y,p,q \in R)$$, then $${{\left( {{x \over p} + {y \over q}} \right)} \over {({p^2} + {q^2})}}$$ is equal to

WB JEE 2022
24
If |z| = 1 and z $$\ne$$ $$\pm$$ 1, then all the points representing $${z \over {1 - {z^2}}}$$ lie on
WB JEE 2021
25
Let C denote the set of all complex numbers. Define A = {(z, w) | z, w$$\in$$C and |z| = |w|}, B = {z, w} | z, w$$\in$$C and z2 = w2}. Then
WB JEE 2021
26
The number of complex numbers p such that $$\left| p \right| = 1$$ and imaginary part of p4 is 0, is
WB JEE 2020
27
The equation $$z\bar z + (2 - 3i)z + (2 + 3i)\bar z + 4 = 0$$ represents a circle of radius
WB JEE 2020
28
Let z be a complex number such that the principal value of argument, arg z > 0. Then, arg z $$-$$ arg($$-$$ z) is
WB JEE 2019
29
The general value of the real angle $$\theta$$, which satisfies the equation, $$(\cos \theta + i\sin \theta )(\cos 2\theta + i\sin 2\theta )...(\cos n\theta + i\sin n\theta ) = 1$$ is given by, (assuming k is an integer)
WB JEE 2019
30
For any non-zero complex number z, the minimum value of | z | + | z $$-$$ 1 | is
WB JEE 2019
31
The polar coordinate of a point P is $$\left( {2, - {\pi \over 4}} \right)$$. The polar coordinate of the point Q which is such that line joining PQ is bisected perpendicularly by the initial line, is
WB JEE 2019
32
If $${Z_r} = \sin {{2\pi r} \over {11}} - i\cos {{2\pi r} \over {11}}$$, then $$\sum\limits_{r = 0}^{10} {{Z_r}} $$ is equal to
WB JEE 2018
33
If z1 and z2 be two non-zero complex numbers such that $${{{z_1}} \over {{z_2}}} + {{{z_2}} \over {{z_1}}} = 1$$, then the origin and the points represented by z1 and z2
WB JEE 2018
34
If $${a_r} = {(\cos 2r\pi + i\sin 2r\pi )^{1/9}}$$,

then the value of $$\left| {\matrix{ {{a_1}} & {{a_2}} & {{a_3}} \cr {{a_4}} & {{a_5}} & {{a_6}} \cr {{a_7}} & {{a_8}} & {{a_9}} \cr } } \right|$$ is equal to
WB JEE 2018
35
Let z1 and z2 be complex numbers such that z1 $$ \ne $$ z2 and |z1| = |z2|. If Re(z1) > 0 and Im(z2) < 0, then $${{{z_1} + {z_2}} \over {{z_1} - {z_2}}}$$ is
WB JEE 2018
36
The expression $${{{{(1 + i)}^n}} \over {{{(1 - i)}^{n - 2}}}}$$ equals
WB JEE 2017
37
Let z = x + iy, where x and y are real. The points (x, y) in the X-Y plane for which $${{{z + i} \over {z - i}}}$$ is purely imaginary, lie on
WB JEE 2017
38
The value of $$\sum\limits_{n = 1}^{13} {({i^n} + {i^{n + 1}})} $$, $$i = \sqrt { - 1} $$ is
WB JEE 2016
39
If $$\omega$$ is an imaginary cube root of unity, then the value of (2 $$-$$ $$\omega$$) (2 $$-$$ $$\omega$$2) + 2(3 $$-$$ $$\omega$$)(3 $$-$$ $$\omega$$2) + ... + (n $$-$$ 1) (n $$-$$ $$\omega$$)(n $$-$$ $$\omega$$2) is
WB JEE 2016

Subjective

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