Let integers a, b [-3, 3] be such that a + b neq 0. Then the number of all possible ordered pairs (a, b), for which |…

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More Complex Numbers Previous-Year Questions — Page 4

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

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

[cite: 627, 629]. Then the sum of all possible values of

z²

is[cite: 630]:

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 [cite: 1355]: $z² + 3i = (z - 2 + i)(2 + 3i)$ $$z^2 + 3i = (z - 2 + i)(2 + 3i)$$ [cite: 1355] $z² + 3i = z(2 + 3i) + (-2 + i)(2 + 3i)$ [cite: 1355] $z² + 3i = z(2 + 3i) - 4 - 6i + 2i - 3 = z(2 + 3i) - 7 - 4i$ [cite: 1355]

Formulate the classic quadratic representation layout [cite: 1356]: $z² - z(2 + 3i) + 7 + 7i = 0$ $$z^2 - z(2 + 3i) + 7 + 7i = 0$$ [cite: 1356]

Step 1: Summing the squared roots

Identify coefficients from the structural template [cite: 1357]:

z₁ + z₂ = 2 + 3i

z₁ z₂ = 7 + 7i

Evaluate sum of possible squared values ( $z₁² + z₂²$ $z_1^2 + z_2^2$) [cite: 1357]: $z₁² + z₂² = (z₁ + z₂)² - 2z₁ z₂$ [cite: 1357] $= (2 + 3i)² - 2(7 + 7i)$ $$= (2 + 3i)^2 - 2(7 + 7i)$$ [cite: 1357] $= (4 - 9 + 12i) - (14 + 14i) = -5 + 12i - 14 - 14i$ [cite: 1357] $= -19 - 2i$ $= -19 - 2i$ [cite: 1358]

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

Solution & Explanation

Related Formula

Core Logic

Step 1: Evaluate Geometric Magnitude Metric

Step 2: Match Root Solutions and Count Pairs

Pattern Recognition

Chapter Mix

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: 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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