JEE Advanced Maths – Complex Numbers & Quadratic Equations PYQ with Detailed Solutions

Q 1 :

If $\frac{w - \bar{w} z}{1 - z}$ is purely real where $w = α + i β, β \neq 0$ and $z \neq 1$ , then the set of the values of $z$ is [2006]

${z : | z | = 1}$
${z : z = \bar{z}}$
${z : z \neq 1}$
${z : | z | = 1, z \neq 1}$

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(4)

$∵$ $\frac{w - \bar{w} z}{1 - z} is purely real$

$∴$ $\bar{(\frac{w - \bar{w} z}{1 - z})} = (\frac{w - \bar{w} z}{1 - z}) \Rightarrow \frac{\bar{w} - w \bar{z}}{1 - \bar{z}} = \frac{w - \bar{w} z}{1 - z}$

$\Rightarrow \bar{w} - w \bar{z} - \bar{w} z + w z \bar{z} = w - \bar{w} z - w \bar{z} + \bar{w} z \bar{z}$

$\Rightarrow w - \bar{w} = (w - \bar{w}) | z |^{2}$

$\Rightarrow | z |^{2} = 1$ $(∵ w = α + i β and β \neq 0)$

$\Rightarrow$ $| z |$ $= 1 and also given that z \neq 1$

$∴$ $The required set is {z : | z | = 1, z \neq 1}$

Q 2 :

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that

$| z_{1} |$ $=$ $| z_{2} |$ $=$ $| z_{3} |$ $=$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $= 1,$ then $| z_{1} + z_{2} + z_{3} |$ is [2000]

equal to 1
less than 1
greater than 3
equal to 3

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(1)

$Given :$ $| z_{1} |$ $=$ $| z_{2} |$ $=$ $| z_{3} |$ $= 1$

$Now,$ $| z_{1} |$ $= 1 \Rightarrow$ $| z_{1} |^{2}$ $= 1 \Rightarrow z_{1} \bar{z_{1}} = 1$

$Similarly z_{2} \bar{z_{2}} = 1, z_{3} \bar{z_{3}} = 1$

$Now,$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $= 1 \Rightarrow$ $| \bar{z_{1}} + \bar{z_{2}} + \bar{z_{3}} |$ $= 1$

$\Rightarrow$ $| \bar{z_{1} + z_{2} + z_{3}} |$ $= 1 \Rightarrow$ $| z_{1} + z_{2} + z_{3} |$ $= 1$

Q 3 :

For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} |$ $= 12$ and $| z_{2} - 3 - 4 i |$ $= 5$ , the minimum value of $| z_{1} - z_{2} |$ is [2002]

0
2
7
17

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(2)

$| z_{1} |$ $= 12 \Rightarrow z_{1} lies on a circle with centre (0, 0) and radius 12 units.$

$And$ $| z_{2} - 3 - 4 i |$ $= 5 \Rightarrow z_{2} lies on a circle with centre (3, 4) and radius 5 units.$

[IMAGE 5]

$From figure, it is clear that$ $| z_{1} - z_{2} |$ $i.e., distance between z_{1} and z_{2} will be minimum when they lie at A and B respectively,$

$i.e., O, C, B, A are collinear as shown.$

$Then$ $| z_{1} - z_{2} |$ $= A B = O A - O B = 12 - 2 (5) = 2 .$

$As above is the minimum value, we must have$ $| z_{1} - z_{2} |$ $\geq 2 .$

Q 4 :

If $\arg (z) < 0$ , then $\arg (- z) - \arg (z) =$ [2000]

$π$
$- π$
$- \frac{π}{2}$
$\frac{π}{2}$

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(1)

$Given: \arg (z) < 0 (given) \Rightarrow \arg (z) = - θ$

[IMAGE 6]

$Now, z = r \cos (- θ) + i \sin (- θ) = r [\cos θ - i \sin θ]$

$Again, - z = - r [\cos θ - i \sin θ] = r [\cos (π - θ) + i \sin (π - θ)]$

$∴$ $\arg (- z) = π - θ$

$Thus \arg (- z) - \arg (z) = (π - θ) - (- θ) = π - θ + θ = π$

Q 5 :

Let $A = {\frac{1967 + 1686 i \sin θ}{7 - 3 i \cos θ} : θ \in ℝ} .$ If A contains exactly one positive integer $n$ , then the value of $n$ is [2023]

0%

(281)

$\frac{281 (49 + 18 \sin θ \cos θ + i (21 \cos θ + 42 \sin θ))}{49 + 9 \cos^{2} θ}$

is a positive integer. For positive integer $Im (z) = 0$

$\Rightarrow 21 \cos θ + 42 \sin θ = 0$

$\Rightarrow \tan θ = - \frac{1}{2}, \sin 2 θ = - \frac{4}{5}, \cos^{2} θ = \frac{4}{5}$

$Now R e (2) = \frac{281 (49 - 9 \sin 2 θ)}{49 + 9 \cos^{2} θ}$

$= \frac{281 (49 - 9 \times (- \frac{4}{5}))}{49 + 9 \times \frac{4}{5}}$

$= \frac{281 (49 + \frac{36}{5})}{49 + \frac{36}{5}}$

$= 281$

Q 6 :

For any integer $k$ , let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}),$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||}$ is [2015]

0%

(4)

$Given: α_{k} = \cos \frac{k π}{7} + i \sin \frac{k π}{7} = e^{\frac{i π k}{7}}$

$α_{k + 1} - α_{k} = e^{\frac{i π (k + 1)}{7}} - e^{\frac{i π k}{7}} = e^{\frac{i π k}{7}} (e^{\frac{i π}{7}} - 1)$

$| α_{k + 1} - α_{k} |$ $=$ $| e^{\frac{i π}{7}} - 1 |$

$\Rightarrow \sum_{k = 1}^{12}$ $| α_{k + 1} - α_{k} |$ $= 12$ $| e^{\frac{i π}{7}} - 1 |$

$Similarly, \sum_{k = 1}^{3}$ $| α_{4 k - 1} - α_{4 k - 2} |$ $= 3$ $| e^{\frac{i π}{7}} - 1 |$

$∴$ $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||} = 4$

Q 7 :

If $z$ is any complex number satisfying $| z - 3 - 2 i |$ $\leq 2$ , then the minimum value of $| 2 z - 6 + 5 i |$ is [2011]

0%

(5)

[IMAGE 7]

$Given:$ $| z - 3 - 2 i |$ $< 2,$

which represents a circular region with centre $(3, 2)$ and radius 2.

$Now,$ $| 2 z - 6 + 5 i |$ $= 2$ $| z - (3 - \frac{5}{2} i) |$

$= 2 \times distance of z from P (where z lies in or on the circle)$

$Also min distance of z from P = \frac{5}{2}$

$∴$ $Minimum value of$ $| 2 z - 6 + 5 i |$ $= 5$

Q 8 :

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of $| z |^{2}$ is _______. [2022]

0%

(0.50)

$Let X = \frac{4 z^{2} + 3 z + 2}{4 z^{2} - 3 z + 2}$

$It can be written as = 1 + \frac{6 z}{4 z^{2} - 3 z + 2}; Now X = 1 + \frac{6}{2 (2 z + \frac{1}{z}) - 3}$

$∵$ $X \in ℝ, then 2 z + \frac{1}{z} \in ℝ$

$\Rightarrow 2 z + \frac{1}{z} = 2 \bar{z} + \frac{1}{\bar{z}}$ $\Rightarrow 2 (z - \bar{z}) - \frac{z - \bar{z}}{| z |^{2}} = 0$

$∵$ $(z - \bar{z}) (2 - \frac{1}{| z |^{2}}) = 0$

$∵$ $z \neq \bar{z} (given), So, | z |^{2} = \frac{1}{2}$

Q 9 :

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is _______ . [2022]

0%

(4)

$Given, \bar{z} - z^{2} = i (\bar{z} + z^{2})$

$It can be written as \bar{z} (1 - i) = z^{2} (1 + i)$

$So | \bar{z} | | 1 - i | = | z |^{2} | 1 + i |$

$| z | = | z |^{2} \Rightarrow | z | = 0 or | z | = 1$

$Let \arg (z) = α . So from (i), we get$

$2 n π - α - \frac{π}{4} = 2 α + \frac{π}{4}$

$\Rightarrow α = \frac{1}{3} (\frac{4 n - 1}{2}) π = \frac{(4 n - 1) π}{6}$

So we will get 3 distinct values of $α$ . Hence there will be total 4 possible values of complex number $z$ .

Q 10 :

Let $S = {a + b \sqrt{2} : a, b \in ℤ}, T_{1} = {{(- 1 + \sqrt{2})}^{n} : n \in ℕ}$ and $T_{2} = {{(1 + \sqrt{2})}^{n} : n \in ℕ} .$ Then which of the following statements is (are) TRUE? [2024]

$ℤ \cup T_{1} \cup T_{2} \subset S$
$T_{1} \cap (0, \frac{1}{2024}) = ϕ$ , where $ϕ$ denotes the empty set.
$T_{2} \cap (2024, \infty) \neq ϕ$
For any given $a, b \in ℤ$ , $\cos (π (a + b \sqrt{2})) + i \sin (π (a + b \sqrt{2})) \in ℤ$ if and only if $b = 0$ , where $i = \sqrt{- 1}$ .

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Select one or more options

(1, 3, 4)

$(a) S = {a + b \sqrt{2} : a, b \in ℤ}$

$For b = 0; ℤ \subset S$

$T_{1} = {(- 1 + \sqrt{2})}^{n} = m + \sqrt{2} n, m, n \in ℤ$

$T_{2} = {(1 + \sqrt{2})}^{n} = m_{1} + \sqrt{2} n_{1}, m_{1}, n_{1} \in ℤ$

$For n \in ℕ, elements of T_{1} and T_{2} are of the form a + b \sqrt{2}$

$Hence ℤ \cup T_{1} \cup T_{2} \subset S$

$(b) Now, - 1 + \sqrt{2} < 1 and its higher powers decrease$

$\Rightarrow {(- 1 + \sqrt{2})}^{n} < 1 and can be made in (0, \frac{1}{2024}) for some higher n$

$(c) 1 + \sqrt{2} > 1 and its higher power increases$

$\Rightarrow {(1 + \sqrt{2})}^{n} can be made in (2024, \infty) for some higher n$

$(d) \cos π (a + b \sqrt{2}) + i \sin π (a + b \sqrt{2}) \in ℤ$

$a + b \sqrt{2}$ is an integer $\Rightarrow b = 0$

Topic Question Set

Q 1 :

If $\frac{w - \bar{w} z}{1 - z}$ is purely real where $w = α + i β, β \neq 0$ and $z \neq 1$ , then the set of the values of $z$ is [2006]

Q 2 :

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that

$| z_{1} |$ $=$ $| z_{2} |$ $=$ $| z_{3} |$ $=$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $= 1,$ then $| z_{1} + z_{2} + z_{3} |$ is [2000]

Q 3 :

For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} |$ $= 12$ and $| z_{2} - 3 - 4 i |$ $= 5$ , the minimum value of $| z_{1} - z_{2} |$ is [2002]

Q 4 :

If $\arg (z) < 0$ , then $\arg (- z) - \arg (z) =$ [2000]

Q 5 :

Let $A = {\frac{1967 + 1686 i \sin θ}{7 - 3 i \cos θ} : θ \in ℝ} .$ If A contains exactly one positive integer $n$ , then the value of $n$ is [2023]

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Q 6 :

For any integer $k$ , let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}),$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||}$ is [2015]

0%

Q 7 :

If $z$ is any complex number satisfying $| z - 3 - 2 i |$ $\leq 2$ , then the minimum value of $| 2 z - 6 + 5 i |$ is [2011]

0%

Q 8 :

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of $| z |^{2}$ is _______. [2022]

0%

Q 9 :

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is _______ . [2022]

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Q 10 :

Let $S = {a + b \sqrt{2} : a, b \in ℤ}, T_{1} = {{(- 1 + \sqrt{2})}^{n} : n \in ℕ}$ and $T_{2} = {{(1 + \sqrt{2})}^{n} : n \in ℕ} .$ Then which of the following statements is (are) TRUE? [2024]

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Topic Question Set

Q 1 : If w-w¯z1-z is purely real where w=α+iβ, β≠0 and z≠1, then the set of the values of z is [2006]

Q 2 : If z1,z2 and z3 are complex numbers such that |z1|=|z2|=|z3|=|1z1+1z2+1z3|=1, then |z1+z2+z3| is [2000]

Q 3 : For all complex numbers z1,z2 satisfying |z1|=12 and |z2-3-4i|=5, the minimum value of |z1-z2| is [2002]

Q 4 : If arg(z)<0, then arg(-z)-arg(z)= [2000]

Q 5 : Let A={1967+1686 isinθ7-3icosθ:θ∈ℝ}. If A contains exactly one positive integer n, then the value of n is [2023]

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Q 6 : For any integer k, let αk=cos(kπ7)+isin(kπ7), where i=-1. The value of the expression ∑k=112|αk+1-αk|∑k=13|α4k-1-α4k-2| is [2015]

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Q 7 : If z is any complex number satisfying |z-3-2i|≤2, then the minimum value of |2z-6+5i| is [2011]

0%

Q 8 : Let z be a complex number with non-zero imaginary part. If 2+3z+4z22-3z+4z2 is a real number, then the value of |z|2 is _______. [2022]

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Q 9 : Let z¯ denote the complex conjugate of a complex number z and let i=-1. In the set of complex numbers, the number of distinct roots of the equation z¯-z2=i(z¯+z2)is _______ . [2022]

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Q 10 : Let S={a+b2:a,b∈ℤ}, T1={(-1+2)n:n∈ℕ} and T2={(1+2)n:n∈ℕ}. Then which of the following statements is (are) TRUE? [2024]

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Q 1 :

If $\frac{w - \bar{w} z}{1 - z}$ is purely real where $w = α + i β, β \neq 0$ and $z \neq 1$ , then the set of the values of $z$ is [2006]

Q 2 :

If $z_{1}, z_{2}$ and $z_{3}$ are complex numbers such that

$| z_{1} |$ $=$ $| z_{2} |$ $=$ $| z_{3} |$ $=$ $| \frac{1}{z_{1}} + \frac{1}{z_{2}} + \frac{1}{z_{3}} |$ $= 1,$ then $| z_{1} + z_{2} + z_{3} |$ is [2000]

Q 3 :

For all complex numbers $z_{1}, z_{2}$ satisfying $| z_{1} |$ $= 12$ and $| z_{2} - 3 - 4 i |$ $= 5$ , the minimum value of $| z_{1} - z_{2} |$ is [2002]

Q 4 :

If $\arg (z) < 0$ , then $\arg (- z) - \arg (z) =$ [2000]

Q 5 :

Let $A = {\frac{1967 + 1686 i \sin θ}{7 - 3 i \cos θ} : θ \in ℝ} .$ If A contains exactly one positive integer $n$ , then the value of $n$ is [2023]

Q 6 :

For any integer $k$ , let $α_{k} = \cos (\frac{k π}{7}) + i \sin (\frac{k π}{7}),$ where $i = \sqrt{- 1}$ . The value of the expression $\frac{\sum_{k = 1}^{12} || α_{k + 1} - α_{k} ||}{\sum_{k = 1}^{3} || α_{4 k - 1} - α_{4 k - 2} ||}$ is [2015]

Q 7 :

If $z$ is any complex number satisfying $| z - 3 - 2 i |$ $\leq 2$ , then the minimum value of $| 2 z - 6 + 5 i |$ is [2011]

Q 8 :

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2 + 3 z + 4 z^{2}}{2 - 3 z + 4 z^{2}}$ is a real number, then the value of $| z |^{2}$ is _______. [2022]

Q 9 :

Let $\bar{z}$ denote the complex conjugate of a complex number $z$ and let $i = \sqrt{- 1}$ . In the set of complex numbers, the number of distinct roots of the equation $\bar{z} - z^{2} = i (\bar{z} + z^{2})$ is _______ . [2022]

Q 10 :

Let $S = {a + b \sqrt{2} : a, b \in ℤ}, T_{1} = {{(- 1 + \sqrt{2})}^{n} : n \in ℕ}$ and $T_{2} = {{(1 + \sqrt{2})}^{n} : n \in ℕ} .$ Then which of the following statements is (are) TRUE? [2024]