Step 1 (Formula / Governing Law & Given Data): [1/3 Mark]

Given that $|\vec{a}| = 5$ and the scalar $\lambda$ lies in the interval $-2 \le \lambda \le 1$. We use the property of the magnitude of a scalar multiple of a vector:

$$|\lambda \vec{a}| = |\lambda| \cdot |\vec{a}| = 5|\lambda|$$

Step 2 (Step-by-Step Mathematical Working): [1/3 Mark]

Since $-2 \le \lambda \le 1$, we determine the range of the absolute value $|\lambda|$:

• The values of $\lambda$ range from $-2$ to $1$.
• The absolute value $|\lambda|$ will range from $0$ (since $\lambda$ can be $0$) to the maximum distance from zero, which is $|-2| = 2$.
• Therefore, $0 \le |\lambda| \le 2$.

Now, multiply by $|\vec{a}| = 5$:

$$0 \le 5|\lambda| \le 5(2)$$ $$0 \le |\lambda \vec{a}| \le 10$$

The smallest value is $0$ and the greatest value is $10$.

Step 3 (Final Concluded Answer): [1/3 Mark]

The sum of the greatest and smallest values is:

$$10 + 0 = 10$$

Correct Option: (C) 10

💡 EL10 Concept Insight:

Think of $|\lambda|$ as the "stretching factor" of the vector; since $\lambda$ can be as far as 2 units away from zero (at -2) or as close as 0, the length of your vector can be anything from 0 to 10. Adding the biggest possible length (10) to the smallest possible length (0) gives you 10.