A

To find the magnitude and angle of the given vectors, we will break down each vector into its components (x and y) and then use trigonometric functions to find the angle with respect to the x-axis.

**1. For vector (i):** \( \mathbf{a} = (3\hat{i} + 4\hat{j}) - (1\hat{i} + 2\hat{j}) \)
Substituting values: \( \mathbf{a} = (3 - 1)\hat{i} + (4 - 2)\hat{j} = 2\hat{i} + 2\hat{j} \)
Magnitude: \( ||\mathbf{a}|| = \sqrt{(2^2 + 2^2)} = \sqrt{8} = 2\sqrt{2} \)
Angle: \( \theta = \tan^{-1}\left(\frac{2}{2}\right) = \tan^{-1}(1) = 45^{\circ} \)

**2. For vector (ii):** \( \mathbf{b} = 12\hat{i} + 4\hat{j} \)
Magnitude: \( ||\mathbf{b}|| = \sqrt{(12^2 + 4^2)} = \sqrt{160} = 4\sqrt{10} \)
Angle: \( \theta = \tan^{-1}\left(\frac{4}{12}\right) = \tan^{-1}(\frac{1}{3}) \) (approximately \(18.43^{\circ}\))

**3. For vector (iii):** \( \mathbf{c} = -(3\hat{i} + 4\hat{j}) - (5\hat{i} + 6\hat{j}) = -8\hat{i} - 10\hat{j} \)
Magnitude: \( ||\mathbf{c}|| = \sqrt{((-8)^2 + (-10)^2)} = \sqrt{164} \)
Angle: \( \theta = \tan^{-1}\left(\frac{-10}{-8}\right) = \tan^{-1}(\frac{5}{4}) \) (approximately \(51.34^{\circ}\) with respect to x-axis but in quadrant III)

**4. For vector (iv):** \( \mathbf{d} = -2\hat{j} \)
Magnitude: \( ||\mathbf{d}|| = 2 \)
Angle: \(\theta = 270^{\circ} \)

**5. For vector (v):** \( \mathbf{e} = 3\hat{i} + 4\hat{j} \)
Magnitude: \( ||\mathbf{e}|| = \sqrt{(3^2 + 4^2)} = 5 \)
Angle: \( \theta = 53.13^{\circ} \)

**6. For vector (vi):** \( \mathbf{f} = 5\hat{i} + 9\hat{j} - 12\hat{j} = 5\hat{i} - 3\hat{j} \)
Magnitude: \( ||\mathbf{f}|| = \sqrt{(5^2 + (-3)^2)} = \sqrt{34} \)
Angle: \( \theta = \tan^{-1}\left(\frac{-3}{5}\right) \) (approximately \( -30.96^{\circ}\) with respect to x-axis)

**In summary:**
- Vector (i): Magnitude = 2\sqrt{2}, Angle = 45^{\circ}
- Vector (ii): Magnitude = 4\sqrt{10}, Angle ≈ 18.43^{\circ}
- Vector (iii): Magnitude = √164, Angle ≈ 226.34^{\circ}
- Vector (iv): Magnitude = 2, Angle = 270^{\circ}
- Vector (v): Magnitude = 5, Angle = 53.13^{\circ}
- Vector (vi): Magnitude = √34, Angle ≈ -30.96^{\circ} \}