(a) Evaluate the integral: $$\int_{5}^{12} (3x^2 + 2x + 9) \, dx$$

(b) Solve the differential equation: $$\frac{dy}{dx} = 1e^{5x}$$ with the initial condition $y(0) = 5$.

(c) Find the determinant of the matrix: $$\begin{bmatrix} 7 & 1 & 2 \\ 8 & 5 & 10 \\ 4 & 4 & 8 \end{bmatrix}$$

(d) Compute the eigenvalues of the matrix: $$\begin{bmatrix} 2 & 5 \\ 3 & 1 \end{bmatrix}$$

(e) Solve for $x$: $$\log_{5}(x) + \log_{5}(4) = 4$$

(f) Expand using the binomial theorem: $$(7x + 4y)^{3}$$

(g) Find the value of the series: $$\sum_{n=1}^{13} 8n^2$$

(h) Solve the quadratic equation: $$4x^2 + 4x + -5 = 0$$

(i) Compute the Fourier transform of: $$f(x) = 3\sin(7x)$$

(j) Find the probability that $X \sim N(2, 5^2)$ satisfies $X > 4$.

Advanced Mathematics Worksheet