Modulus of complex numbers Practice Questions - NDA Mathematics Question Bank

Question 1 of 5 NDA MCQ

What is the modulus of z?

Detailed Explanation

Question 2 of 5 NDA MCQ

What is the modulus of the complex number

Detailed Explanation

Solution

Option (b) is correct.

Explanation:

\[ \frac{\cos \theta + i \sin \theta}{\cos \theta - i \sin \theta} \]

Multiply both numerator and denominator by the conjugate of the denominator:

\[ \frac{(\cos \theta + i \sin \theta)(\cos \theta + i \sin \theta)}{(\cos \theta - i \sin \theta)(\cos \theta + i \sin \theta)} \]

Calculating the denominator:

\[ \cos^2 \theta + \sin^2 \theta = 1 \]

Calculating the numerator:

\[ \cos^2 \theta + 2 i \cos \theta \sin \theta - \sin^2 \theta = \cos^2 \theta - \sin^2 \theta + 2 i \cos \theta \sin \theta \]

The expression simplifies to:

\[ \cos^2 \theta - \sin^2 \theta + 2i \cos \theta \sin \theta \]

Taking modulus:

\[ \sqrt{(\cos^2 \theta - \sin^2 \theta)^2 + (2 \cos \theta \sin \theta)^2} = 1 \]

Thus, the modulus of the complex number is equal to 1.

📌 Hints / Properties Used:

Modulus of a complex number: \(|z| = \sqrt{x^2 + y^2}\)
Conjugate helps in simplifying complex fractions.
Identities: \(\cos^2 \theta + \sin^2 \theta = 1\)

📊 Visual Diagram Suggestion:

Consider a representation of the complex plane where the modulus corresponds to the distance from the origin to the point (\(\cos \theta, \sin \theta\)). A unit circle can illustrate this clearly.

📖 Factual Verification & Reference:

Verified against NCERT Class XI Mathematics, Chapter 12 (Complex Numbers).

Question 3 of 5 NDA MCQ

What is the minimum value of |x - 1|, where x ∈ R ?

Detailed Explanation

Solution

Option (a) is correct.

Explanation:

\[ |x - 1| \geq 0 \]

\[ |x - 1| = 0 \text{ when } x = 1 \]

\[ \text{Therefore, minimum value is } 0 \]

📌 Hints / Properties Used:

Modulus Function: \( |a| \geq 0 \)
Minimum occurs when the expression inside modulus equals zero.

📊 Visual Diagram Suggestion: A number line depicting the point \( x = 1 \), showing that the closest distance from this point to any real number is zero when \( x \) equals 1.

📖 Factual Verification & Reference:

Verified against NCERT Class XI Mathematics, Chapter on Real Numbers.

Question 4 of 5 NDA MCQ

What is the remainder when 2¹²⁰ is divided by 7?

Detailed Explanation

Question 5 of 5 NDA MCQ

What is the remainder when 7ⁿ – 6n is divided by 36 for n = 100?

Detailed Explanation

Modulus of complex numbers

Solution

Solution

Unlock 10,000+ Practice Questions