Class CBSE Class 12 Mathematics Vector Algebra Q #1722
COMPETENCY BASED
APPLY
1 Marks 2026 AISSCE(Board Exam) MCQ SINGLE
If $|\vec{a}|=5$ and $-2 \le \lambda \le 1$ then the sum of greatest and the smallest value of $|\lambda\vec{a}|$ is
(A) -5
(B) 5
(C) 10
(D) 15
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Correct Answer: C

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Detailed Solution

Step 1: Understand the expression

We are given the vector magnitude $|\vec{a}| = 5$ and the scalar range $-2 \le \lambda \le 1$. We need to find the range of the magnitude of the vector $\lambda\vec{a}$, which is given by $|\lambda\vec{a}| = |\lambda| |\vec{a}|$.

Step 2: Determine the range of $|\lambda|$

Since $-2 \le \lambda \le 1$, the absolute value $|\lambda|$ represents the distance of $\lambda$ from zero on the number line. The values of $\lambda$ range from $-2$ to $1$. The maximum distance from zero is at $\lambda = -2$, where $|\lambda| = 2$. The minimum distance from zero is at $\lambda = 0$, where $|\lambda| = 0$. Thus, $0 \le |\lambda| \le 2$.

Step 3: Calculate the range of $|\lambda\vec{a}|$

Multiplying the inequality by $|\vec{a}| = 5$: $$0 \times 5 \le |\lambda| |\vec{a}| \le 2 \times 5$$ $$0 \le |\lambda\vec{a}| \le 10$$

Step 4: Find the sum of greatest and smallest values

The greatest value is $10$ and the smallest value is $0$. The sum is: $$10 + 0 = 10$$

Final Answer: 10

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Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply the properties of vector magnitude and scalar multiplication to a specific interval constraint.
Knowledge Dimension: PROCEDURAL
Justification: The student must follow a sequence of steps involving inequality manipulation and absolute value properties to reach the solution.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the conceptual understanding of vector algebra beyond rote memorization of formulas.

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