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.
Prev Next

AI Tutor Explanation

Powered by Gemini

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

AI generated content. Review strictly for academic accuracy.

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.

More from this Chapter

SA
During a cricket match, the position of the bowler, the wicket keeper and the leg slip fielder are in a line given by $\vec{B}=2\hat{i}+8\hat{j}$, $\vec{W}=6\hat{i}+12\hat{j}$ and $\vec{F}=12\hat{i}+18\hat{j}$ respectively. Calculate the ratio in which the wicketkeeper divides the line segment joining the bowler and the leg slip fielder.
MCQ_SINGLE
If \(|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|\) for any two vectors, then vectors \(\vec{a}\) and \(\vec{b}\) are:
ASSERTION_REASON
Assertion (A) : If $|\vec{a} \times \vec{b}|^{2}+|\vec{a} \cdot \vec{b}|^{2}=256$ and $|\vec{b}|=8$, then $|\vec{a}|=2$.
VSA
If $\vec{\alpha}$ and $\vec{\beta}$ are position vectors of two points P and Q respectively, then find the position vector of a point R in QP produced such that $QR=\frac{3}{2}QP$.
VSA
Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors $\vec{a}=3\hat{i}+\hat{j}+2\hat{k}$ and $\vec{b}=2\hat{i}-2\hat{j}+4\hat{k}$. Determine the angle formed between the kite strings. Assume there is no slack in the strings.
View All Questions