Advanced Mathematics Worksheet

(a) Evaluate the integral: $$\int_{3}^{20} (1x^2 + 3x + 2) \, dx$$

(b) Solve the differential equation: $$\frac{dy}{dx} = 10e^{5x}$$ with the initial condition \(y(2) = 4\).

(c) Find the determinant of the matrix: $$\begin{bmatrix} 7 & 2 & 10 \ 5 & 10 & 6 \ 9 & 6 & 2 \end{bmatrix}$$

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

(e) Solve for \(x\): $$\log_{4}(x) + \log_{2}(3) = 5$$

(f) Expand using the binomial theorem: $$(9x + 7y)^{5}$$

(g) Find the value of the series: $$\sum_{n=1}^{9} 8n^2$$

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

(i) Compute the Fourier transform of: $$f(x) = 3\sin(9x)$$

(j) Find the probability that \(X \sim N(7, 4^2)\) satisfies \(X > 0\).