Vector Algebra
Vector Algebra
nta_abhyas_2025
Grade 12
Question:
$|\vec{a}| = |4\vec{b} - \lambda^2\vec{c}|$, $|\vec{a}|^2 = 10|\vec{b}|^2 + |\vec{c}|^2 - 8\vec{b}.\vec{c}$, $9 = 16 + \lambda^2 - 8\vec{b}.\vec{c}$, and $\lambda^2 - 8\lambda + 7 = 0$ (as $\vec{b}.\vec{c} = 1$). Find the sum of all values of $\lambda = 8$.
Step-by-Step Solution
Key Concept: Apply Vieta's formulas for sum of roots of a quadratic equation $\lambda^2 - 8\lambda + 7 = 0$ directly.
From $\lambda^2 - 8\lambda + 7 = 0$, by Vieta's formulas, the sum of roots is $\lambda_1 + \lambda_2 = 8$. Alternatively, factoring gives $(\lambda - 1)(\lambda - 7) = 0$, so $\lambda = 1$ or $\lambda = 7$, and their sum is $1 + 7 = 8$.
Correct Answer: 8