Class CBSE Class 12 Mathematics Vector Algebra Q #1381
KNOWLEDGE BASED
REMEMBER
2 Marks 2025 AISSCE(Board Exam) VSA
If $\vec{a}$ and $\vec{b}$ are two non-collinear vectors, then find x, such that $\vec{\alpha}=(x-2)\vec{a}+\vec{b}$ and $\vec{\beta}=(3+2x)\vec{a}-2\vec{b}$ are collinear.
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Detailed Solution

Step 1: Understanding Collinearity

Two vectors $\vec{\alpha}$ and $\vec{\beta}$ are collinear if one is a scalar multiple of the other. That is, $\vec{\alpha} = \lambda \vec{\beta}$ for some scalar $\lambda$.

Step 2: Setting up the Equation

Given $\vec{\alpha}=(x-2)\vec{a}+\vec{b}$ and $\vec{\beta}=(3+2x)\vec{a}-2\vec{b}$, we can write: $$(x-2)\vec{a}+\vec{b} = \lambda((3+2x)\vec{a}-2\vec{b})$$

Step 3: Equating Coefficients

Since $\vec{a}$ and $\vec{b}$ are non-collinear, we can equate the coefficients of $\vec{a}$ and $\vec{b}$ on both sides: $$x-2 = \lambda(3+2x) \quad \text{and} \quad 1 = -2\lambda$$

Step 4: Solving for $\lambda$

From the second equation, we can find $\lambda$: $$\lambda = -\frac{1}{2}$$

Step 5: Substituting $\lambda$ into the First Equation

Substitute $\lambda = -\frac{1}{2}$ into the first equation: $$x-2 = -\frac{1}{2}(3+2x)$$

Step 6: Solving for x

Multiply both sides by 2: $$2(x-2) = -(3+2x)$$ $$2x - 4 = -3 - 2x$$ $$4x = 1$$ $$x = \frac{1}{4}$$

Final Answer: 1/4

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Pedagogical Audit
Bloom's Analysis: This is an REMEMBER question because it requires recalling the condition for collinearity of vectors and applying basic algebraic manipulation to solve for the unknown variable.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concept of collinearity of vectors and how it relates to scalar multiples and linear combinations.
Syllabus Audit: In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the student's understanding of vector algebra concepts, specifically collinearity, as covered in the textbook.

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