<div class="container mt-5"> <h1 class="text-center mb-4">Advanced Mathematics Worksheet</h1> <div class="mb-4"> <h5>Solve the following problems:</h5> <div class="mb-3"> <p>(a) Evaluate the integral: $$\int_{§§V0(0,10,1)§§}^{§§V1(10,20,1)§§} (§§V2(1,5,1)§§x^2 + §§V3(1,5,1)§§x + §§V4(0,10,1)§§) \, dx$$</p> </div> <div class="mb-3"> <p>(b) Solve the differential equation: $$\frac{dy}{dx} = §§V5(1,10,1)§§e^{§§V6(1,5,1)§§x}$$ with the initial condition \(y(§§V7(0,2,1)§§) = §§V8(1,5,1)§§\).</p> </div> <div class="mb-3"> <p>(c) Find the determinant of the matrix: $$\begin{bmatrix} §§V9(1,10,1)§§ & §§V10(1,10,1)§§ & §§V11(1,10,1)§§ \\ §§V12(1,10,1)§§ & §§V13(1,10,1)§§ & §§V14(1,10,1)§§ \\ §§V15(1,10,1)§§ & §§V16(1,10,1)§§ & §§V17(1,10,1)§§ \end{bmatrix}$$</p> </div> <div class="mb-3"> <p>(d) Compute the eigenvalues of the matrix: $$\begin{bmatrix} §§V18(1,5,1)§§ & §§V19(1,5,1)§§ \\ §§V20(1,5,1)§§ & §§V21(1,5,1)§§ \end{bmatrix}$$</p> </div> <div class="mb-3"> <p>(e) Solve for \(x\): $$\log_{§§V22(2,10,1)§§}(x) + \log_{§§V23(2,10,1)§§}(§§V24(1,20,1)§§) = §§V25(1,5,1)§§$$</p> </div> <div class="mb-3"> <p>(f) Expand using the binomial theorem: $$(§§V26(1,10,1)§§x + §§V27(1,10,1)§§y)^{§§V28(2,6,1)§§}$$</p> </div> <div class="mb-3"> <p>(g) Find the value of the series: $$\sum_{n=1}^{§§V29(5,15,1)§§} §§V30(1,10,1)§§n^2$$</p> </div> <div class="mb-3"> <p>(h) Solve the quadratic equation: $$§§V31(1,5,1)§§x^2 + §§V32(1,5,1)§§x + §§V33(-10,10,1)§§ = 0$$</p> </div> <div class="mb-3"> <p>(i) Compute the Fourier transform of: $$f(x) = §§V34(1,5,1)§§\sin(§§V35(1,10,1)§§x)$$</p> </div> <div class="mb-3"> <p>(j) Find the probability that \(X \sim N(§§V36(0,10,1)§§, §§V37(1,5,1)§§^2)\) satisfies \(X > §§V38(0,10,1)§§\).</p> </div> </div> </div> <img src="https://www.mathkiss.com/uploads/goose4.jpg" width="400"/>