Complex Numbers jee mains questions | JEE Main Maths Complex Numbers Previous Year Question

Q 1 :
If $z_{1}, z_{2}$ are two distinct complex number such that $| \frac{z_{1} - 2 z_{2}}{\frac{1}{2} - z_{1} {\bar{z}}_{2}} |$ $= 2$ , then [2024]

$z_{1}$ lies on a circle of radius $\frac{1}{2}$ and $z_{2}$ lies on a circle of radius 1.
either $z_{1}$ lies on a circle of radius 1 or $z_{2}$ lies on a circle of radius $\frac{1}{2} .$
both $z_{1}$ and $z_{2}$ lie on the same circle.
either $z_{1}$ lies on a circle of radius $\frac{1}{2}$ or $z_{2}$ lies on a circle of radius 1.

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(B)

We have, $| \frac{z_{1} - 2 z_{2}}{\frac{1}{2} - z_{1} {\bar{z}}_{2}} |$ $= 2$

$\Rightarrow | z_{1} - 2 z_{2} | = | 1 - 2 z_{1} {\bar{z}}_{2} |$

$\Rightarrow | z_{1} - 2 z_{2} |^{2} = | 1 - 2 z_{1} {\bar{z}}_{2} |^{2}$

$\Rightarrow (z_{1} - 2 z_{2}) ({\bar{z}}_{1} - 2 {\bar{z}}_{2}) = (1 - 2 z_{1} {\bar{z}}_{2}) (1 - 2 {\bar{z}}_{1} z_{2})$

$\Rightarrow | z_{1} |^{2} + 4 | z_{2} |^{2} - 2 {\bar{z}}_{1} z_{2} - 2 {\bar{z}}_{2} z_{1}$

$= 1 + 4 | z_{1} |^{2} | z_{2} |^{2} - 2 z_{1} {\bar{z}}_{2} - 2 {\bar{z}}_{1} z_{2}$

$\Rightarrow | z_{1} |^{2} + 4 | z_{2} |^{2} - 4 | z_{1} |^{2} | z_{2} |^{2} - 1 = 0$

$\Rightarrow (| z_{1} |^{2} - 1) (1 - 4 | z_{2} |^{2}) = 0$

$\Rightarrow | z_{1} | = 1, | z_{2} | = \frac{1}{2}$

Q 2 :
Let $r$ and $θ$ respectively be the modulus and amplitude of the complex number $z = 2 - i (2 \tan \frac{5 π}{8}),$ then $(r, θ)$ is equal to [2024]

$(2 \sec \frac{3 π}{8}, \frac{5 π}{8})$
$(2 \sec \frac{3 π}{8}, \frac{3 π}{8})$
$(2 \sec \frac{11 π}{8}, \frac{11 π}{8})$
$(2 \sec \frac{5 π}{8}, \frac{3 π}{8})$

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(B)

We have, $z = 2 - i (2 \tan \frac{5 π}{8})$

$= 2 + 2 i \tan (π - \frac{5 π}{8}) \Rightarrow z = 2 + 2 i \tan \frac{3 π}{8}$

$\Rightarrow z = 2 (\cos \frac{3 π}{8} + i \sin \frac{3 π}{8}) \sec \frac{3 π}{8}$

So, $r = 2 \sec \frac{3 π}{8}, θ = \frac{3 π}{8}$

Q 3 :
Let the complex numbers $α$ and $\frac{1}{\bar{α}}$ lie on the circles $| z - z_{0} |^{2} = 4$ and $| z - z_{0} |^{2} = 16$ respectively, where $z_{0} = 1 + i .$ Then, the value of $100 | α |^{2}$ is ________. [2024]

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(20)

Given $| z - z_{0} |^{2} = 4$ ...(i)

$| z - z_{0} |^{2} = 16$ ...(ii)

$α$ lies on (i), $∴$ $| α - z_{0} |^{2} = 4$

$\Rightarrow (α - z_{0}) (\bar{α} - {\bar{z}}_{0}) = 4 \Rightarrow α \bar{α} - α {\bar{z}}_{0} - z_{0} \bar{α} + | z_{0} |^{2} = 4$

$\Rightarrow | α |^{2} - α {\bar{z}}_{0} - z_{0} \bar{α} + 2 = 4 (∵ | z_{0} | = \sqrt{2})$

$\Rightarrow | α |^{2} - a {\bar{z}}_{0} - z_{0} \bar{α} = 2$ ...(iii)

$\frac{1}{\bar{α}}$ lies on (ii)

$∴$ $| \frac{1}{\bar{α}} - z_{0} |^{2} = 16 \Rightarrow (\frac{1}{\bar{α}} - z_{0}) (\frac{1}{α} - {\bar{z}}_{0}) = 16$

$\Rightarrow (1 - \bar{α} z_{0}) (1 - α {\bar{z}}_{0}) = 16 α \bar{α}$

$\Rightarrow 1 - \bar{α} z_{0} - α {\bar{z}}_{0} + α \bar{α}$ $z_{0} {\bar{z}}_{0} = 16 | α |^{2}$

$\Rightarrow 1 - \bar{α} z_{0} - α {\bar{z}}_{0} = 14 | α |^{2}$ ...(iv)

From (iii) and (iv), we get

$\Rightarrow 15 | α |^{2} = 3 \Rightarrow | α |^{2} = \frac{3}{15} = \frac{1}{5}$

$∴$ $100 | α |^{2} = 100 \times \frac{1}{5} = 20$

Q 4 :
If $α$ denotes the number of solutions of $| 1 - i |^{x}$ $= 2^{x}$ and $β = (\frac{| z |}{\arg (z)}),$ where $z = \frac{π}{4} {(1 + i)}^{4}$ $[\frac{1 - \sqrt{π} i}{\sqrt{π} + i} + \frac{\sqrt{π} - i}{1 + \sqrt{π} i}],$ $i = \sqrt{- 1},$ then the distance of the point $(α, β)$ from the line $4 x - 3 y = 7$ is __________ [2024]

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(3)

We have, $| 1 - i |^{x}$ $= 2^{x}$

$= ({\sqrt{{(1)}^{2} + {(- 1)}^{2}})}^{x} = 2^{x} \Rightarrow {(\sqrt{2})}^{x} = 2^{x}$

$2^{x / 2} = 2^{x} \Rightarrow \frac{x}{2} = x \Rightarrow 2 x - x = 0 \Rightarrow x = 0$

$\Rightarrow α = 1$

and $β = (\frac{| z |}{a r g (z)}),$ where $z = \frac{π}{4} {(1 + i)}^{4}$ $[\frac{1 - \sqrt{π} i}{\sqrt{π} + i} + \frac{\sqrt{π} - i}{1 + \sqrt{π} i}]$

$z = 2 π i \Rightarrow \arg (z) = \frac{π}{2}$

and $| z | = 2 π$

$β = \frac{| z |}{\arg (z)} = \frac{2 π}{π / 2} = 4 \Rightarrow β = 4$

Now, the distance of point $(α, β)$ from the line $4 x - 3 y = 7$ is given by

$d = \frac{| 4 \times 1 - 3 \times 4 - 7 |}{\sqrt{16 + 9}} = \frac{| 4 - 12 - 7 |}{5} = \frac{15}{5} = 3$

Topic Question Set

Q 1 :
If $z_{1}, z_{2}$ are two distinct complex number such that $| \frac{z_{1} - 2 z_{2}}{\frac{1}{2} - z_{1} {\bar{z}}_{2}} |$ $= 2$ , then [2024]

Q 2 :
Let $r$ and $θ$ respectively be the modulus and amplitude of the complex number $z = 2 - i (2 \tan \frac{5 π}{8}),$ then $(r, θ)$ is equal to [2024]

Q 3 :
Let the complex numbers $α$ and $\frac{1}{\bar{α}}$ lie on the circles $| z - z_{0} |^{2} = 4$ and $| z - z_{0} |^{2} = 16$ respectively, where $z_{0} = 1 + i .$ Then, the value of $100 | α |^{2}$ is ________. [2024]

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Topic Question Set

Q 1 : If z1,z2 are two distinct complex number such that |z1-2z212-z1z¯2|=2, then [2024]

Q 2 : Let r and θ respectively be the modulus and amplitude of the complex number z=2-i(2tan5π8), then (r,θ) is equal to [2024]

Q 3 : Let the complex numbers α and 1α¯ lie on the circles |z-z0|2=4 and |z-z0|2=16 respectively, where z0=1+i. Then, the value of 100|α|2 is ________. [2024]

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Q 4 : If α denotes the number of solutions of |1-i|x=2x and β=(|z|arg(z)), where z=π4(1+i)4[1-πiπ+i+π-i1+πi],i=-1, then the distance of the point (α,β) from the line 4x-3y=7 is __________ [2024]

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Q 1 :
If $z_{1}, z_{2}$ are two distinct complex number such that $| \frac{z_{1} - 2 z_{2}}{\frac{1}{2} - z_{1} {\bar{z}}_{2}} |$ $= 2$ , then [2024]

Q 2 :
Let $r$ and $θ$ respectively be the modulus and amplitude of the complex number $z = 2 - i (2 \tan \frac{5 π}{8}),$ then $(r, θ)$ is equal to [2024]

Q 3 :
Let the complex numbers $α$ and $\frac{1}{\bar{α}}$ lie on the circles $| z - z_{0} |^{2} = 4$ and $| z - z_{0} |^{2} = 16$ respectively, where $z_{0} = 1 + i .$ Then, the value of $100 | α |^{2}$ is ________. [2024]