Let `vecp = 3ax^(2) hati - 2(x-1)hatj, q=b(x-1)hati + xhatj` and `ab lt 0`. Then `vecp` and `vecq` are parallel for:

Options

A. atleast one x in (0,1)
B. atleast one x in `(-1,0)`
C. atleast one x in (1,2)
D. None of these

Answer

To determine when the vectors \(\vec{p}\) and \(\vec{q}\) are parallel, we can use the property that two vectors are parallel if their cross product is zero. Let's solve the problem step by step.

### Step 1: Define the vectors
Given:
\[ \vec{p} = 3a x^2 \hat{i} - 2(x-1) \hat{j} \]
\[ \vec{q} = b(x-1) \hat{i} + x \hat{j} \]

### Step 2: Set up the cross product
The cross product \(\vec{p} \times \vec{q}\) can be calculated using the determinant of a matrix formed by the unit vectors and the components of \(\vec{p}\) and \(\vec{q}\):
\[ \vec{p} \times \vec{q} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3a x^2 & -2(x-1) & 0 \\ b(x-1) & x & 0 \end{vmatrix} \]

### Step 3: Calculate the determinant
Calculating the determinant, we focus on the \(\hat{k}\) component since the other components will yield zero:
\[ \vec{p} \times \vec{q} = \hat{k} \left( 3a x^2 \cdot x - (-2(x-1)) \cdot b(x-1) \right) \]
\[ = \hat{k} \left( 3a x^3 + 2b(x-1)^2 \right) \]

### Step 4: Set the cross product equal to zero
For the vectors to be parallel, we set the cross product equal to zero:
\[ 3a x^3 + 2b(x-1)^2 = 0 \]

### Step 5: Analyze the equation
This can be rearranged to:
\[ 3a x^3 + 2b(x^2 - 2x + 1) = 0 \]
\[ 3a x^3 + 2b x^2 - 4b x + 2b = 0 \]

### Step 6: Identify the roots
Let \(f(x) = 3a x^3 + 2b x^2 - 4b x + 2b\). We need to find the conditions under which this cubic polynomial has at least one root.

### Step 7: Evaluate at specific points
Evaluate \(f(0)\) and \(f(1)\):
- \(f(0) = 2b\)
- \(f(1) = 3a + 2b - 4b + 2b = 3a\)

### Step 8: Determine the sign of the product
Since \(ab < 0\), we have:
- If \(b > 0\), then \(a < 0\) which implies \(f(0) > 0\) and \(f(1) < 0\).
- If \(b < 0\), then \(a > 0\) which implies \(f(0) < 0\) and \(f(1) > 0\).

In either case, \(f(0) \cdot f(1) < 0\), indicating that there is at least one root in the interval \( (0, 1) \).

### Conclusion
Thus, the vectors \(\vec{p}\) and \(\vec{q}\) are parallel for at least one value of \(x\) in the interval \( (0, 1) \).

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