JEE Advanced 2021 Paper 1 Online | Complex Numbers Question 7 | Mathematics

JEE Advanced 2021 Paper 1 Online

MCQ (More than One Correct Answer)

-2

For any complex number w = c + id, let $$\arg (w) \in ( - \pi ,\pi ]$$, where $$i = \sqrt { - 1} $$. Let $$\alpha$$ and $$\beta$$ be real numbers such that for all complex numbers z = x + iy satisfying $$\arg \left( {{{z + \alpha } \over {z + \beta }}} \right) = {\pi \over 4}$$, the ordered pair (x, y) lies on the circle $${x^2} + {y^2} + 5x - 3y + 4 = 0$$, Then which of the following statements is (are) TRUE?

$$\alpha$$ = $$-$$1

$$\alpha$$$$\beta$$ = 4

$$\alpha$$$$\beta$$ = $$-$$4

$$\beta$$ = 4

JEE Advanced 2020 Paper 1 Offline

MCQ (More than One Correct Answer)

-2

Let S be the set of all complex numbers z
satisfying |z² + z + 1| = 1. Then which of the following statements is/are TRUE?

$$\left| {z + {1 \over 2}} \right|$$ $$ \le $$ $${{1 \over 2}}$$ for all z$$ \in $$S

|z| $$ \le $$ 2 for all z$$ \in $$S

$$\left| {z + {1 \over 2}} \right|\, \ge {1 \over 2}$$ for all z$$ \in $$S

The set S has exactly four elements

JEE Advanced 2018 Paper 2 Offline

MCQ (More than One Correct Answer)

-1

Let s, t, r be non-zero complex numbers and L be the set of solutions $$z = x + iy(x,y \in R,\,i = \sqrt { - 1} )$$ of the equation $$sz + t\overline z + r = 0$$ where $$\overline z $$ = x $$-$$ iy. Then, which of the following statement(s) is(are) TRUE?

If L has exactly one element, then |s|$$ \ne $$|t|

If |s| = |t|, then L has infinitely many elements

The number of elements in $$L \cap \{ z:|z - 1 + i| = 5\} $$ is at most 2

If L has more than one element, then L has infinitely many elements

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.

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