This exam paper covers topics in Vector Calculus, Analytic Geometry, and Abstract Algebra for a third-semester B.Sc course. Here's an outline of the questions provided in Part A, where each question carries 2 marks. Let’s go through each of the questions:

1. Unit Tangent Vector: Calculate the unit tangent vector of the given curve $r(t) = (t \sin t + \cos t) \, \mathbf{i} + (t \cos t - \sin t) \, \mathbf{j}$.
2. Gradient Vector on a Plane: Define the gradient vector of a function on a plane region.
3. Gradient of a Scalar Field: Find the gradient of $f(x, y, z) = x^2 + y^2 + z^2$ at the point $(1, 1, 1)$.
4. Conservative Vector Field Test: Test if the vector field $\mathbf{F} = yzi \, \mathbf{i} + xzj \, \mathbf{j} + xyk \, \mathbf{k}$ is conservative.
5. Curl of a Vector Field: Define the curl of a vector field in space.
6. Divergence of a Vector Field: Find the divergence of $\mathbf{F} = x^3 \, \mathbf{i} + y^3 \, \mathbf{j} + z^3 \, \mathbf{k}$.
7. Parabola Definition: Define a parabola and describe its focus and directrix.
8. Ellipse Equation: Find the equation of an ellipse with foci $(\pm 1, 0)$ and semi-minor axis $b = \sqrt{5}$.
9. Eccentricity of an Ellipse: Determine the eccentricity of the ellipse $2x^2 + y^2 = 4$.
10. Subgroups of $Z_7$: Find the number of subgroups of $Z_7$ under addition modulo 7.

If you need detailed solutions for any of these problems, feel free to specify, and I can help break down each one step-by-step.

Follow-up Questions:

1. What is the process for finding the unit tangent vector of a curve in general?
2. How do you calculate the gradient of a function, and what does it represent geometrically?
3. What does it mean for a vector field to be conservative, and how can you test it?
4. How do you determine the focus and directrix of a parabola given its equation?
5. How can you find the subgroups of a cyclic group like $Z_7$?

Tip: For vector fields, understanding the concepts of gradient, divergence, and curl is crucial as they each describe different properties: direction and rate of change, magnitude of source/sink, and rotational tendency, respectively.