1. System of Equations and Geometric Interpretation | Finding values of \\\\( k \\\\) for which a system of linear equations has a unique solution and interpreting the result geometrically.

2. Parametric Differentiation and Second Derivative | Finding the second derivative of \\\\( y \\\\) with respect to \\\\( x \\\\) given parametric equations and expressing it in a specific form.

3. Surface Area of Revolution | Deriving an integral expression for the surface area generated by rotating a given function and evaluating it using hyperbolic substitutions.

4. Eigenvalues and Eigenvectors of a Matrix | Finding eigenvalues and eigenvectors, solving the characteristic equation, and computing the inverse matrix using eigenvalues.

5. Second-Order Differential Equations | Solving a given differential equation with specific initial conditions to find the particular solution.

6. Integral Bounds Using Rectangles | Approximating an integral using upper and lower bounds with rectangles, and finding the least \\\\( N \\\\) for a given error bound.

7. Integrating Factor for Differential Equations | Finding the integrating factor for a first-order differential equation and solving it with an initial condition.

8. De Moivre’s Theorem and Cosine Expansions | Using binomial expansion and complex numbers to express powers of cosine in terms of lower powers.

9. Recursive Integral Evaluation | Deriving and solving recurrence relations for a given integral, applying results to find exact values for specific cases.