JEE Advanced 2018 Paper 1 Offline | Complex Numbers Question 15 | Mathematics

JEE Advanced 2018 Paper 1 Offline

MCQ (More than One Correct Answer)

-1

For a non-zero complex number z, let arg(z) denote the principal argument with $$-$$ $$\pi $$ < arg(z) $$ \le $$ $$\pi $$. Then, which of the following statement(s) is (are) FALSE?

arg($$-$$1$$-$$i) = $${\pi \over 4}$$, where i = $$\sqrt { - 1} $$

The function f : R $$ \to $$ ($$-$$$$\pi $$, $$\pi $$), defined by f(t) = arg ($$-$$1 + it) for all t $$ \in $$ R, is continuous at all points of R, where i = $$\sqrt { - 1} $$.

For any two non-zero complex numbers z₁ and z₂, arg $$\left( {{{{z_1}} \over {{z_2}}}} \right)$$$$-$$ arg (z₁) + arg(z₂) is an integer multiple of 2$$\pi $$.

For any three given distinct complex numbers z₁, z₂ and z₃, the locus of the point z satisfying the condition arg$$\left( {{{(z - {z_1})({z_2} - {z_3})} \over {(z - {z_3})({z_2} - {z_1})}}} \right) = \pi $$, lies on a straight line.

JEE Advanced 2017 Paper 1 Offline

MCQ (More than One Correct Answer)

-1

Let a, b, x and y be real numbers such that a $$-$$ b = 1 and y $$ \ne $$ 0. If the complex number z = x + iy satisfies $${\mathop{\rm Im}\nolimits} \left( {{{az + b} \over {z + 1}}} \right) = y$$, then which of the following is(are) possible value(s) of x?

$$1 - \sqrt {1 + {y^2}} $$

$$ - 1 - \sqrt {1 - {y^2}} $$

$$1 + \sqrt {1 + {y^2}} $$

$$ - 1 + \sqrt {1 - {y^2}} $$

JEE Advanced 2016 Paper 2 Offline

MCQ (More than One Correct Answer)

-2

Let $$a,\,b \in R\,and\,{a^{2\,}} + {b^2} \ne 0$$. Suppose
$$S = \left\{ {Z \in C:Z = {1 \over {a + ibt}}, + \in R,t \ne 0} \right\}$$, where $$i = \sqrt { - 1} $$. Ifz = x + iy and z $$ \in $$ S, then (x, y) lies on

the circle with radius $${{1 \over {2a}}}$$and centre $$\left\{ {{1 \over {2a}},\,0} \right\}\,for\,a > 0\,,b \ne \,0$$

the circle with radius $$-{{1 \over {2a}}}$$and centre $$\left\{ -{{1 \over {2a}},\,0} \right\}\,for\,a < 0\,,b \ne \,0$$

the x-axis for $$a \ne \,\,0,\,b \ne \,0$$

the y-axis for $$a = \,\,0,\,b \ne \,0$$

JEE Advanced 2013 Paper 2 Offline

MCQ (More than One Correct Answer)

-1

Let $\omega=\frac{\sqrt{3}+i}{2}$ and $P=\left\{\omega^n: n=1,2,3, \ldots\right\}$. Further

$\mathrm{H}_1=\left\{z \in \mathrm{C}: \operatorname{Re} z<\frac{1}{2}\right\}$ and

$\mathrm{H}_2=\left\{z \in \mathrm{C}: \operatorname{Re} z<\frac{-1}{2}\right\}$, where C is the

set of all complex numbers. If $z_1 \in \mathrm{P} \cap \mathrm{H}_1, z_2 \in$ $\mathrm{P} \cap \mathrm{H}_2$ and O

represents the origin, then $\angle z_1 \mathrm{O} z_2=$

$${\pi \over 2}$$

$${\pi \over 6}\,$$

$${{2\pi } \over 3}$$

$${{5\pi } \over 6}$$

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