a and b are non-collinear vectors. If `c=(x-2) a+b and d=(2x+1)a-b` are collinear vectors, then the value of x= . . .

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(5)`
D. `(1)/(3)`

Explanation

To solve the problem, we need to find the value of \( x \) such that the vectors \( c \) and \( d \) are collinear. Given the vectors:

\[ c = (x - 2) \mathbf{a} + \mathbf{b} \]
\[ d = (2x + 1) \mathbf{a} - \mathbf{b} \]

### Step 1: Set up the equation for collinearity
Vectors \( c \) and \( d \) are collinear if there exists a scalar \( \lambda \) such that:

\[ c = \lambda d \]

### Step 2: Substitute the expressions for \( c \) and \( d \)
Substituting the expressions for \( c \) and \( d \) into the collinearity condition gives:

\[ (x - 2) \mathbf{a} + \mathbf{b} = \lambda \left( (2x + 1) \mathbf{a} - \mathbf{b} \right) \]

### Step 3: Expand the right-hand side
Expanding the right-hand side, we have:

\[ (x - 2) \mathbf{a} + \mathbf{b} = \lambda (2x + 1) \mathbf{a} - \lambda \mathbf{b} \]

### Step 4: Rearrange the equation
Rearranging the equation, we can separate the coefficients of \( \mathbf{a} \) and \( \mathbf{b} \):

\[ (x - 2) \mathbf{a} + \mathbf{b} = (\lambda (2x + 1)) \mathbf{a} - (\lambda) \mathbf{b} \]

This gives us two equations by comparing coefficients:

1. For \( \mathbf{a} \):
\[ x - 2 = \lambda (2x + 1) \]

2. For \( \mathbf{b} \):
\[ 1 = -\lambda \]

### Step 5: Solve for \( \lambda \)
From the second equation, we find:

\[ \lambda = -1 \]

### Step 6: Substitute \( \lambda \) back into the first equation
Now substituting \( \lambda = -1 \) into the first equation:

\[ x - 2 = -1(2x + 1) \]

### Step 7: Simplify the equation
This simplifies to:

\[ x - 2 = -2x - 1 \]

### Step 8: Rearranging the terms
Rearranging gives:

\[ x + 2x = -1 + 2 \]
\[ 3x = 1 \]

### Step 9: Solve for \( x \)
Dividing both sides by 3:

\[ x = \frac{1}{3} \]

### Final Answer
Thus, the value of \( x \) is:

\[ \boxed{\frac{1}{3}} \]