Mathematics 9709,22 Oct Nov 2024 | Cambridge AS - A Level Past Papers With Mark Scheme

1. Logarithmic Transformations & Exponential Equations | Expressing \\\\(5(6x - 8)^7\\\\) in the form \\\\(y = kx\\\\) using logarithms and finding \\\\(k\\\\).

2. Differentiation & Gradient Analysis | Finding the derivative of \\\\(f(x) = 4\\\\sin^3 x\\\\), computing its inverse derivative, and determining the gradient of a given curve.

3. Exponential & Logarithmic Equations | Sketching graphs of \\\\(y = 1 + e^{2x}\\\\) and \\\\(y = 3x - 4\\\\), finding intersection points, and solving using an iterative method.

4. Polynomials & Factorization | Finding unknown coefficients in \\\\(p(x) = ax^3 + bx^2 + ax - 8\\\\) given a factor and remainder conditions, and proving a single real root exists.

5. Definite Integrals & Area Approximation | Using the trapezium rule for area estimation, finding exact areas using integration, and determining estimation accuracy.

6. Trigonometric Identities & Equations | Expressing \\\\(\\\\sin 4\\\\theta + \\\\sin(60^\\\\circ - \\\\theta)\\\\) in the form \\\\(R\\\\sin(\\\\theta - \\\\alpha)\\\\) and solving for the smallest positive \\\\(\\\\theta\\\\).