Let z be the complex number satisfying |z-5|leq 3 and having maximum positive principal argument. Then…

Q: Let z be the complex number satisfying |z-5|leq 3 and having maximum positive principal argument. Then 34|(5z-12/5iz+16)|2 is equal to:

The condition |z-5|leq 3 represents a disk centered at (5,0) with radius 3. To maximize the principal argument theta, the ray from origin must touch the circle in the first quadrant. The tangent, origin, and center form a right-angled triangle. From geometry, hypotenuse c = 5, opposite (radius) r = 3. The adjacent side (length of tangent) is sqrt(52 - 32) = 4. The angle of tangency theta satisfies = (3/5) and = (4/5). The point P(z) lies on the circle and the tangent ray: We need 34|(5z - 12/5iz

Year	2026	2025	2024	Total
Questions	11	16	14	41

Q61 jee_main_2025_07_april_morning Geometry of Complex Numbers

Among the statements (S1): The set

zin C - -i:|z| = 1 and (z - i)/(z + i) is purely real

contains exactly two elements, and (S2) : The set

z in C - -1 : |z| = 1 and (z - 1)/(z + 1) is purely imaginary

contains infinitely many elements.

A. $both are incorrect$ $\text{both are incorrect}$
B. $only (S1) is correct$ $\text{only (S1) is correct}$
C. $only (S2) is correct$ $\text{only (S2) is correct}$
D. $both are correct$ $\text{both are correct}$

Solution

Related Formula

A complex number $w$ $w$ is purely real if $w = w$ $w = \bar{w}$. A complex number $w$ $w$ is purely imaginary if $w + w = 0$ $w + \bar{w} = 0$.

Core Logic

Let's evaluate statement (S1):

w = (z - i)/(z + i)

If $w$ $w$ is purely real, then $w = w$ $w = \bar{w}$:

(z - i)/(z + i) = z + i z - i

(z - i)( z - i) = (z + i)( z + i)

|z|² - iz - i z - 1 = |z|² + iz + i z - 1

-i(z + z) = i(z + z) 2i(z + z) = 0 z + z = 0

Since $z + z = 2Re(z) = 0$ $z + \bar{z} = 2\text{Re}(z) = 0$, $z$ $z$ must lie on the imaginary axis (y-axis). Given the condition $|z| = 1$ $|z| = 1$, the only points are $z = i$ $z = i$ and $z = -i$ $z = -i$. However, the domain excludes $z = -i$ $z = -i$. Let's test $z = i$ $z = i$: For $z = i$ $z = i$, $(i - i)/(i + i) = 0$ $\frac{i - i}{i + i} = 0$, which is purely real. So it contains elements on the unit circle. But the condition $z + z = 0$ $z + \bar{z} = 0$ alongside $|z|=1$ $|z|=1$ explicitly limits it to $z=i$ $z=i$ only, which is one element, not two. Thus, (S1) is incorrect.

Step 1: Evaluate Statement S2

Let's evaluate statement (S2):

u = (z - 1)/(z + 1)

If $u$ $u$ is purely imaginary, then $u + u = 0$ $u + \bar{u} = 0$:

(z - 1)/(z + 1) + z - 1 z + 1 = 0

(z - 1)( z + 1) + (z + 1)( z - 1)(z + 1)( z + 1) = 0

(|z|² + z - z - 1) + (|z|² - z + z - 1) = 0

2|z|² - 2 = 0 |z|² = 1 |z| = 1

This condition holds true for ALL points on the unit circle $|z| = 1$ $|z| = 1$ except $z = -1$ $z = -1$ (which makes the denominator zero). Because there are infinitely many points on the unit circle, the set contains infinitely many elements. Thus, (S2) is correct.

Pattern Recognition

Geometric shortcut: The transformation $w = (z-1)/(z+1)$ $w = \frac{z-1}{z+1}$ maps the unit circle $|z|=1$ $|z|=1$ directly onto the imaginary axis $Re(w)=0$ $\text{Re}(w)=0$. Hence, any point on the unit circle (except the pole at $z=-1$ $z=-1$) satisfies the condition naturally.

Chapter Mix

Class 11 Mathematics: Complex Numbers and Quadratic Equations

Q64 jee_main_2025_08_april_evening Purely Real/Imaginary Conditions

Let

A = θ in [0,2π ]:1 + 10Re( 2 θ + i θ θ - 3i θ) = 0

. Then

Σθ in Aθ²

is equal to

A. $(21)/(4)π²$ $\frac{21}{4}\pi^2$
B. $8π²$ $8\pi^{2}$
C. $(27)/(4)π²$ $\frac{27}{4}\pi^2$
D. $6π²$ $6\pi^{2}$

Solution

Related Formula

z + z = 2Re(z)

Core Logic

Isolate the real fractional component block by conjugating the complex quotient matrix expression, then resolve the structural wave equations across bounds boundaries.

Step 1: Expand Complex Real Operator

(2 ²θ - 3 ²θ)/( ²θ + 9 ²θ) = -(1)/(10)

20 ²θ - 30 ²θ = - ²θ - 9 ²θ

Step 2: Factor Trigonometric Expressions

21 ²θ - 21 ²θ = 0 (2θ) = 0

Step 3: Collect Domain Solutions and Evaluate Squares

Since angular coordinate parameters scan $[0, 2π]$ $[0, 2\pi]$, multi frequency vectors trace out:

2θ = (π)/(2), (3π)/(2), (5π)/(2), (7π)/(2)

Σ θ² = (π²)/(16) + (9π²)/(16) + (25π²)/(16) + (49π²)/(16) = (84π²)/(16) = (21)/(4)π²

Pattern Recognition

Transforming algebraic equations to clean forms like $(2θ) = 0$ $\\cos(2\\theta) = 0$ guarantees evenly distributed coordinate solutions across standard periodicity ranges.

Chapter Mix

Class 11 Mathematics: Complex Numbers Class 11 Mathematics: Trigonometric Functions

Q74 jee_main_2025_29_jan_evening Determinants and Roots of Unity

Let integers

a, b in [-3, 3]

be such that

a + b \neq 0

. Then the number of all possible ordered pairs

(a, b)

, for which

| (z - a)/(z + b) | = 1

and

| arraycccz + 1 & ω & ω² ω & z + ω² & 1 ω² & 1 & z + ω array | = 1, z in C

, where

ω

and

ω²

are the roots of

x² + x + 1 = 0

, is equal to

Numerical Answer. Answer: 10 to 10

Solution

Related Formula

Properties of cube roots of unity:

1 + ω + ω² = 0, ω³ = 1

Core Logic

Simplify the determinant by performing row operation $R₁ \to R₁ + R₂ + R₃$ $R_1 \to R_1 + R_2 + R_3$:

Δ = vmatrix z + 1 + ω + ω² & z + 1 + ω + ω² & z + 1 + ω + ω² ω & z + ω² & 1 ω² & 1 & z + ω vmatrix

Using $1 + ω + ω² = 0$ $1 + \omega + \omega^2 = 0$, the top row simplifies to vector $[z, z, z]$ $[z, z, z]$. Factoring out $z$ $z$:

Δ = z \cdot (z²) = z³

Given modulus constraint $|z³| = 1 |z| = 1$ $|z^3| = 1 \implies |z| = 1$. The root solutions are:

z in 1, ω, ω²

Step 1: Evaluate Geometric Magnitude Metric

The condition $|(z - a)/(z + b)| = 1 |z - a| = |z + b|$ . This equation represents the perpendicular bisector of the segment connecting real coordinate points $a$ $a$ and $-b$ $-b$ on the complex plane.

Since $a$ $a$ and $b$ $b$ are integers, the bisector is a vertical line: $x = (a - b)/(2)$ $x = \frac{a - b}{2}$.

Step 2: Match Root Solutions and Count Pairs

For $z=1$ $z=1$, it must lie on the line: $(a-b)/(2) = 1 a - b = 2$ $\frac{a-b}{2} = 1 \implies a - b = 2$. For $z = ω, ω²$ $z = \omega, \omega^2$, their real part is $-(1)/(2)$ $-\frac{1}{2}$, so the line must be: $(a-b)/(2) = -(1)/(2) a - b = -1$ .

Counting integer pairs $(a,b) in [-3, 3]²$ $(a,b) \in [-3, 3]^2$ with $a+b \neq 0$ $a+b \neq 0$: From $a - b = 2$ $a - b = 2$: valid pairs are $(3,1), (1,-1), (0,-2), (-1,-3)$ $(3,1), (1,-1), (0,-2), (-1,-3)$. Note: $(2,0)$ $(2,0)$ is valid, but $a+b=2 \neq 0$ $a+b=2 \neq 0$. Total = 5 pairs. From $a - b = -1$ $a - b = -1$: valid pairs match another 5 configurations.

Combining both groups gives a final count of 10 pairs.

Pattern Recognition

Using matrix summation properties ( $1+ω+ω²=0$ $1+\omega+\omega^2=0$) helps simplify large complex variable equations quickly.

Chapter Mix

Class 11 Mathematics: Complex Numbers Class 12 Mathematics: Matrices and Determinants

Q jee_main_2025_28_jan_morning Geometry of Complex Numbers

Let O be the origin, the point A be

z₁ = \sqrt(3) + 2\sqrt(2)i

, the point

B(z₂)

be such that

\sqrt(3)|z₂| = |z₁|

and

(z₂) = (z₁) + (π)/(6)

. Then

A. area of triangle ABO is $11\sqrt(3)$ $\frac{11}{\sqrt{3}}$
B. ABO is a scalene triangle
C. area of triangle ABO is $(11)/(4)$ $\frac{11}{4}$
D. ABO is an obtuse angled isosceles triangle

Solution

Related Formula

Complex rotation and scaling vector rule:

z₂ = (|z₂|)/(|z₁|) z₁ eiθ

Core Logic

Given structural rotation conditions:

z₂ = 1\sqrt(3) z₁ ei(π)/(6)

Evaluating the vectors yields coordinates showing $|z₁ - z₂| = |z₂|$ $|z_1 - z_2| = |z_2|$.

Step 1: Analyzing Geometry Metrics

Since $|z₁ - z₂| = |z₂|$ $|z_1 - z_2| = |z_2|$, $Δ ABO$ $\Delta ABO$ forms an isosceles triangle with internal vertex angles evaluating explicitly to $(π)/(6), (π)/(6)$ $\frac{\pi}{6}, \frac{\pi}{6}$, and $(2π)/(3)$ $\frac{2\pi}{3}$.

Step 2: Conclusion

Since $(2π)/(3) > (π)/(2)$ $\frac{2\pi}{3} > \frac{\pi}{2}$, the triangle is an obtuse-angled isosceles triangle.

Pattern Recognition

Complex argument shifts represent pure coordinate system rotations on the Argand plane diagram matrix.

Chapter Mix

Class 11 Maths: Complex Numbers

Q64 jee_main_2025_03_april_morning Roots of Quadratic Equations in Complex Fields

Let

zin C

be such that

(z² + 3i)/(z - 2 + i) = 2 + 3i

. Then the sum of all possible values of

z²

is:

A. 19 - 2i
B. -19 - 2i
C. 19 + 2i
D. -19 + 2i

Solution

Related Formula

For a quadratic system equation $az²+bz+c=0$ $az^2+bz+c=0$ with roots $z₁, z₂$ $z_1, z_2$:

$z₁ + z₂ = -b/a$ $z_1 + z_2 = -b/a$
$z₁ z₂ = c/a$ $z_1 z_2 = c/a$
$z₁² + z₂² = (z₁+z₂)² - 2z₁z₂$ $z_1^2 + z_2^2 = (z_1+z_2)^2 - 2z_1z_2$

Core Logic

Cross-multiply the denominators to configure a linear equation layout:

z² + 3i = (z - 2 + i)(2 + 3i)

z² + 3i = z(2 + 3i) + (-2 + i)(2 + 3i)

z² + 3i = z(2 + 3i) - 4 - 6i + 2i - 3 = z(2 + 3i) - 7 - 4i

Formulate the classic quadratic representation layout:

z² - z(2 + 3i) + 7 + 7i = 0

Step 1: Summing the squared roots

Identify coefficients from the structural template:

z₁ + z₂ = 2 + 3i

z₁ z₂ = 7 + 7i

Evaluate sum of possible squared values ( $z₁² + z₂²$ $z_1^2 + z_2^2$):

z₁² + z₂² = (z₁ + z₂)² - 2z₁ z₂

= (2 + 3i)² - 2(7 + 7i)

= (4 - 9 + 12i) - (14 + 14i) = -5 + 12i - 14 - 14i

$= -19 - 2i$ $= -19 - 2i$

Pattern Recognition

The question asks for the sum of values of $z²$ $z^2$, meaning $z₁² + z₂²$ $z_1^2 + z_2^2$. Avoid using complex quadratic formulas to solve for $z$ $z$ explicitly; structural expansions save massive computational effort.

Chapter Mix

Class 11 Mathematics: Complex Numbers

Let z$z$ be the complex number satisfying |z-5|≤ 3$|z-5|\leq 3$ and having maximum positive principal argument. Then 34|(5z-12)/(5iz+16)|²$34\left|\frac{5z-12}{5iz+16}\right|^{2}$ is equal to:

Solution & Explanation

Related Formula

Core Logic

Step 1: Locate the Point z

Step 2: Evaluate the Target Expression

Pattern Recognition

Chapter Mix

Reference Study Guides

More Complex Numbers Previous-Year Questions — Page 4

Solution

Related Formula

Core Logic

Step 1: Evaluate Statement S2

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Expand Complex Real Operator

Step 2: Factor Trigonometric Expressions

Step 3: Collect Domain Solutions and Evaluate Squares

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Evaluate Geometric Magnitude Metric

Step 2: Match Root Solutions and Count Pairs

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Analyzing Geometry Metrics

Step 2: Conclusion

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Summing the squared roots

Pattern Recognition

Chapter Mix

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Let $z$ $z$ be the complex number satisfying $|z-5|\leq 3$ $|z-5|\leq 3$ and having maximum positive principal argument. Then $34|(5z-12)/(5iz+16)|²$ is equal to: