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Question #41613

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{ z \in \mathbb{C} : |z - 2| \le 1 \}$ . If $\lambda$ is a real parameter such that the maximum value of $|\lambda z + 3i|$ for $z \in S$ is $7$ , then the sum of all possible values of $\lambda$ is

Options

A.
$0$
B.
$-\frac{14}{3}$
C.
$\frac{14}{3}$
D.
$-4$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Scalar Multiplication Geometric Scaling|Triangle Inequality Mind Games|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Trap' in the geometric scaling of complex numbers.

Students often incorrectly simplify the scalar multiplication of a vector $|\lambda(z-2)|$ as $\lambda|z-2|$ instead of the rigorous $|\lambda||z-2|$ .

If the absolute value on the real parameter $\lambda$ is omitted, the squaring step yields an extraneous negative root ( $\lambda = -20/3$ ), leading to an incorrect sum of $-14/3$ .

The problem forces adherence to the Triangle Inequality boundary conditions and proper magnitude transformation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Scalar Multiplication Geometric Scaling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free