Mathematics Quiz – 2 0 12345678910 Mathematics Mathematics Quiz – 2 1 / 10 Question If [math]f(x) = x^3 – 3x^2 + 4[/math], find [math]f'(1)[/math] 3 -3 2 1 Explanation: The derivative of [math]f(x)[/math] is [math]f'(x) = 3x^2 – 6x[/math]. So, [math]f'(1) = 3(1)^2 – 6(1) = -3[/math] 2 / 10 Question The inverse Laplace transform of [math] rac{1}{s+2}[/math] is: [math]e^{2t}[/math] [math]e^{-t}[/math] [math]e^{t}[/math] [math]e^{-2t}[/math] Explanation: The inverse Laplace transform is [math]e^{-2t}[/math]. 3 / 10 Question If [math]A[/math] is a skew-symmetric matrix, then [math]A^T[/math] is: [math]A[/math] [math]-A[/math] [math]0[/math] [math]I[/math] Explanation: By definition, the transpose of a skew-symmetric matrix is its negative, i.e., [math]A^T = -A[/math]. 4 / 10 Question The partial derivative of [math]f(x, y) = x^2 y + y^2[/math] with respect to [math]x[/math] is: [math]2xy + 2y[/math] [math]xy^2 + x[/math] [math]2xy[/math] [math]xy + 2x[/math] Explanation: The partial derivative of [math]f(x, y)[/math] with respect to [math]x[/math] is [math] rac{partial f}{partial x} = 2x y[/math]. 5 / 10 Question The value of [math]int e^x dx[/math] is: [math]e^x + C[/math] [math]e^x – C[/math] [math]x e^x[/math] [math]e^{x+1}[/math] Explanation: The integral of [math]e^x[/math] is [math]e^x + C[/math]. 6 / 10 Question For a real symmetric matrix [math]A[/math], the eigenvalues are: Complex Real Imaginary None of the above Explanation: The eigenvalues of a real symmetric matrix are always real. 7 / 10 Question The general solution of the differential equation [math]y” + y = 0[/math] is: [math]C_1 e^x + C_2 e^{-x}[/math] [math]C_1 e^{ix} + C_2 e^{-ix}[/math] [math]C_1 cosh x + C_2 sinh x[/math] [math]C_1 cos x + C_2 sin x[/math] Explanation: The characteristic equation is [math]r^2 + 1 = 0[/math], which gives complex roots [math]r = pm i[/math]. Thus, the general solution is [math]y = C_1 cos x + C_2 sin x[/math]. 8 / 10 Question If [math]f(x)[/math] is continuous and differentiable, and [math]f'(x) = 2f(x)[/math], then [math]f(x)[/math] is: [math]Ce^{2x}[/math] [math]e^{x}[/math] [math]Ce^{-2x}[/math] [math]e^{2x}[/math] Explanation: The general solution is [math]f(x) = Ce^{2x}[/math], where [math]C[/math] is a constant. 9 / 10 Question The general solution to [math] rac{d^2y}{dx^2} – 4y = 0[/math] is: [math]C_1 e^{2x} + C_2 e^{-2x}[/math] [math]C_1 e^{4x} + C_2 e^{-4x}[/math] [math]C_1 e^{2x}[/math] [math]C_2 e^{-2x}[/math] Explanation: The solution is [math]y = C_1 e^{2x} + C_2 e^{-2x}[/math]. 10 / 10 Question If [math]A[/math] and [math]B[/math] are [math]2 imes 2[/math] matrices, then [math](A+B)^2[/math] is: [math]A^2 + B^2[/math] [math]A^2 + 2AB[/math] [math]2A^2 + 2B^2[/math] [math]A^2 + B^2 + 2AB[/math] Explanation: By matrix multiplication, [math](A+B)^2 = A^2 + B^2 + 2AB[/math]. Your score is The average score is 0% LinkedIn Facebook Twitter VKontakte Restart quiz
