MCQMathematics

CBSE · Class 11 · Mathematics · Complex Numbers and Quadratic EquationsIf $z1 = 2 - i$ and $z2 = 1 + i$, what is the value of $\left|\frac{z1 + z2 + 1}{z1 - z2 + i}\right|$?

Q: If $z1 = 2 - i$ and $z2 = 1 + i$, what is the value of $\left|\frac{z1 + z2 + 1}{z1 - z2 + i}\right|$?

Calculate numerator: $z1 + z2 + 1 = (2 - i) + (1 + i) + 1 = 4$. Calculate denominator: $z1 - z2 + i = (2 - i) - (1 + i) + i = 1 - i$. Thus, $\frac{4}{1 - i} = \frac{4(1 + i)}{2} = 2(1 + i)$. The modulus is $|2(1 + i)| = 2 \sqrt{1^2 + 1^2} = 2\sqrt{2}$.

Step-by-Step Solution

Calculate numerator: $z_1 + z_2 + 1 = (2 - i) + (1 + i) + 1 = 4$. Calculate denominator: $z_1 - z_2 + i = (2 - i) - (1 + i) + i = 1 - i$. Thus, $\frac{4}{1 - i} = \frac{4(1 + i)}{2} = 2(1 + i)$. The modulus is $|2(1 + i)| = 2 \sqrt{1^2 + 1^2} = 2\sqrt{2}$.

Detailed Options Breakdown

Option : \sqrt{2}

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Complex Numbers and Quadratic Equations.

Option 1: 2

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Complex Numbers and Quadratic Equations.

Option 2: 2\sqrt{2} (Correct Answer)

Correct choice. Refer to the step-by-step verified solution guidelines above for details.

Option 3: 4

Incorrect choice. This distractor represents a common misunderstanding of the core principles of Complex Numbers and Quadratic Equations.

💡 Study Guide: This question tests core syllabus concepts from Complex Numbers and Quadratic Equations. For formulas, key summaries, and mock exam reference guides, read the full Complex Numbers and Quadratic Equations Revision Notes.

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