1. Matrix Transformations and Invariant Lines | Deriving a transformation matrix for a stretch and shear, finding its inverse, and determining the equation of the invariant line.

2. Mathematical Induction Proof | Proving a differentiation identity involving a polynomial function using mathematical induction.

3. Quartic Equations and Sum of Powers of Roots | Finding a new quartic equation from transformed roots and computing sums of specific powers of the roots.

4. Method of Differences in Series Summation | Using the method of differences to sum a given series, determining an unknown constant, and finding a closed-form expression.

5. Polar Coordinates and Curve Properties | Converting a Cartesian equation to polar form, sketching the curve, and finding its enclosed area and maximum distances.

6. Vector Geometry and Planes | Finding the equation of a plane containing a line, calculating the acute angle between two planes, and deriving a vector equation for the shortest distance between skew lines.