Advanced Mathematics Worksheet

(a) Evaluate the integral: $$\int_{6}^{14} (4x^2 + 4x + 4) \, dx$$

(b) Solve the differential equation: $$\frac{dy}{dx} = 4e^{3x}$$ with the initial condition \(y(1) = 2\).

(c) Find the determinant of the matrix: $$\begin{bmatrix} 10 & 1 & 1 \\ 3 & 9 & 9 \\ 10 & 5 & 10 \end{bmatrix}$$

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

(e) Solve for \(x\): $$\log_{6}(x) + \log_{9}(18) = 5$$

(f) Expand using the binomial theorem: $$(6x + 6y)^{3}$$

(g) Find the value of the series: $$\sum_{n=1}^{8} 6n^2$$

(h) Solve the quadratic equation: $$5x^2 + 3x + -2 = 0$$

(i) Compute the Fourier transform of: $$f(x) = 4\sin(7x)$$

(j) Find the probability that \(X \sim N(9, 3^2)\) satisfies \(X > 8\).