JEE MAIN - Mathematics (2022 - 26th July Morning Shift - No. 2)
Let O be the origin and A be the point $${z_1} = 1 + 2i$$. If B is the point $${z_2}$$, $${\mathop{\rm Re}\nolimits} ({z_2}) < 0$$, such that OAB is a right angled isosceles triangle with OB as hypotenuse, then which of the following is NOT true?
$$\arg {z_2} = \pi - {\tan ^{ - 1}}3$$
$$\arg ({z_1} - 2{z_2}) = - {\tan ^{ - 1}}{4 \over 3}$$
$$|{z_2}| = \sqrt {10} $$
$$|2{z_1} - {z_2}| = 5$$
Explanation
_26th_July_Morning_Shift_en_2_1.png)
$${{{z_2} - 0} \over {(1 + 2i) - 0}} = {{|OB|} \over {|OA|}}{e^{{{i\pi } \over 4}}}$$
$$ \Rightarrow {{{z_2}} \over {1 + 2i}} = \sqrt 2 {e^{{{i\pi } \over 4}}}$$
OR $${z_2} = (1 + 2i)(1 + i)$$
$$ = - 1 + 3i$$
$$\arg {z_2} = \pi - {\tan ^{ - 1}}3$$
$$|{z_2}| = \sqrt {10} $$
$${z_1} - 2{z_2} = (1 + 2i) + 2 - 6i = 3 - 4i$$
$$\arg ({z_1} - 2{z_2}) = - {\tan ^{ - 1}}{4 \over 3}$$
$$|2{z_1} - {z_2}| = |2 + 4i + 1 - 3i| = |3 + i| = \sqrt {10} $$
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