Q: The equation for the ellipsoid of inertia of a solid body is P(x) = 4x₁² + 4x₂² + x₃² − 2x₁x₂. What is the standard form in terms of a new orthogonal set of axes O(y₁, y₂, y₃)?
(a) y₁² − 3y₂² + 3y₃³
(b) y₁² + 5y₂² + 3y₃²
(c) y₁² − 5y₂² + 3y₃²
(d) y₁² − 5y₂² − 3y₃²
Show Answer Q: What is the general solution of a homogeneous differential equation with the characteristic equation λ³(λ + 4)²(λ² + 2λ + 5)² = 0?
(a) y(x) = c₁ + c₂x + c₃x² + c₄e⁻⁴ˣ + c₅xe⁻⁴ˣ + eˣ{c₆ cos 2x + c₇ sin 2x} + c₈x cos 2x + c₉x sin 2x
(b) y(x) = c₁ + c₂x + c₃x² + c₄e⁻⁴ˣ + c₅xe⁻⁴ˣ + e⁻ˣ{c₆ cos 2x + c₇ sin 2x} + eˣ{c₈x cos 2x + c₉x sin 2x}
(c) y(x) = c₁ + c₂x + c₃x² + c₄e⁻⁴ˣ + c₅xe⁻⁴ˣ + eˣ{c₆ cos 2x + c₇ sin 2x} + e⁻ˣ{c₈x cos 2x + c₉x sin 2x}
(d) y(x) = c₁ + c₂x + c₃x² + c₄e⁻⁴ˣ + c₅xe⁻⁴ˣ + e⁻ˣ{c₆ cos 2x + c₇ sin 2x} + xe⁻ˣ{c₈ cos 2x + c₉ sin 2x}
Show Answer Q: What is the initial value if d²y/dx² + 4(dy/dx) + 3y = e⁻ˣ, with y(0) = 2, (dy/dx)|_{x=0} = 1?
(a) y(x) = (13/4 + x/2)e⁻ˣ − (5/4)e⁻³ˣ
(b) y(x) = (13/4 + x/2)e⁻³ˣ − (5/4)e⁻ˣ
(c) y(x) = (13/4 + x/2)e⁻ˣ + (5/4)e⁻³ˣ
(d) y(x) = (13/4 − x/2)e⁻ˣ − (5/4)e⁻³ˣ
Show Answer Q: If L{f(t)} = e⁻³ˢ(1 − 2s)/(2s² − s + 1), then L{f(3t)} = ?
(a) e⁻ˢ(−3 − 2s)/(2s² − 3s + 9)
(b) e⁻ˢ(3 + 2s)/(2s² − 3s + 9)
(c) e⁻ˢ(3 − s)/(2s² − 3s + 9)
(d) e⁻ˢ(3 − 2s)/(2s² − 3s + 9)
Show Answer Q: What is the solution of the equation d²y/dt² + y(t) = ∫₀ᵗ sinτ · y(t − π)dτ, subject to the initial conditions y(0) = 1 and (dy/dt)|_{t=0} = 0?
(a) y(t) = (1/2)(1 − cos√2t), for t > 0
(b) y(t) = (1/2)(1 + cos√2t), for t > 0
(c) y(t) = (1/2)(−1 − cos√2t), for t > 0
(d) y(t) = −(1/2)(1 − cos√2t), for t > 0
Show Answer Q: The nth coefficient of a series is given by aₙ = 1·5·9·13...(4n+1)/2ⁿ. What is the expression aₙ in terms of the gamma function?
(a) aₙ = 2ⁿ⁺² Γ(n + 5/4) / Γ(1/4)
(b) aₙ = 2ⁿ⁺¹ Γ(n + 5/4) / Γ(1/4)
(c) aₙ = 2ⁿ Γ(n + 5/4) / Γ(1/4)
(d) aₙ = 2ⁿ⁺³ Γ(n + 5/4) / Γ(1/4)
Show Answer Q: Fourier series representation of f(x) = x + 1 for −1 ≤ x ≤ 1 is
(a) f(x) = 1 − (2/π) ∑ [(−1)ⁿ⁺¹/n] sin(nπx)
(b) f(x) = −1 − (2/π) ∑ [(−1)ⁿ⁺¹/n] sin(nπx)
(c) f(x) = 1 + (2/π) ∑ [(−1)ⁿ⁺¹/n] sin(nπx)
(d) f(x) = −1 + (2/π) ∑ [(−1)ⁿ⁺¹/n] sin(nπx)
Show Answer Q: Let f(x) = {1, |x| < a; 0, |x| > a} and g(x) = {1, 0 < x < a; 0, otherwise}, then the Fourier transform of 3f(x) − 2g(x) is
(a) √(2/π) {3sinωa/ω + (1 − e^{iωa})/(iω)}
(b) √(2/π) {3sinωa/ω − (1 + e^{−iωa})/(iω)}
(c) √(2/π) {−3sinωa/ω − (1 − e^{−iωa})/(iω)}
(d) √(2/π) {3sinωa/ω − (1 − e^{−iωa})/(iω)}
Show Answer Q: For what values of a and b is the vector field F = (x + z)i + a(y + z)j + b(x + y)k a conservative field?
(a) a = b = 1
(b) a = b = −1
(c) a = 1, b = −1
(d) a = −1, b = 1
Show Answer Q: Let S be the surface of the paraboloid of revolution z = 1 − x² − y² with the domain of definition x² + y² ≤ 1, and let Γ be the boundary of the paraboloid. Given: F = x³i + (x + y − z)j + yzk. What is the value of ∬_S curl F · dS?
Show Answer Q: The fixed point iterative scheme for determining √2 is
(a) x_{n+1} = (1/2)(x_n − 2/x_n)
(b) x_{n+1} = (1/2)(−x_n + 2/x_n)
(c) x_{n+1} = −(1/2)(x_n + 2/x_n)
(d) x_{n+1} = (1/2)(x_n + 2/x_n)
Show Answer Q: The Gauss-Seidel iterative method for the system of equations: −(1/4)x₂ − (1/4)x₃ + x₄ = 1/4, −(1/4)x₁ + x₃ − (1/4)x₄ = 1/4, x₁ − (1/4)x₂ − (1/4)x₃ = 1/2, −(1/4)x₁ + x₂ − (1/4)x₄ = 1/2 is
(a) x₁^(n+1) = 0.5 − 0.25x₂^(n) + 0.25x₃^(n), x₂^(n+1) = 0.5 + 0.25x₁^(n+1) + 0.25x₄^(n), x₃^(n+1) = 0.25 + 0.25x₁^(n+1) + 0.25x₄^(n), x₄^(n+1) = 0.25 − 0.25x₂^(n+1) + 0.25x₃^(n+1)
(b) x₁^(n+1) = 0.5 + 0.25x₂^(n) + 0.25x₃^(n), x₂^(n+1) = 0.5 + 0.25x₁^(n+1) + 0.25x₄^(n), x₃^(n+1) = 0.25 + 0.25x₁^(n+1) + 0.25x₄^(n), x₄^(n+1) = 0.25 + 0.25x₂^(n+1) + 0.25x₃^(n+1)
(c) x₁^(n+1) = 0.5 + 0.25x₂^(n) + 0.25x₃^(n), x₂^(n+1) = 0.5 + 0.25x₁^(n+1) − 0.25x₄^(n), x₃^(n+1) = 0.25 + 0.25x₁^(n+1) − 0.25x₄^(n), x₄^(n+1) = 0.25 + 0.25x₂^(n+1) + 0.25x₃^(n+1)
(d) x₁^(n+1) = 0.5 + 0.25x₂^(n) + 0.25x₃^(n), x₂^(n+1) = 0.5 − 0.25x₁^(n+1) + 0.25x₄^(n), x₃^(n+1) = 0.25 + 0.25x₁^(n+1) + 0.25x₄^(n), x₄^(n+1) = 0.25 + 0.25x₂^(n+1) + 0.25x₃^(n+1)
Show Answer Q: What is the missing figure in the following table?
x: 1, 2, 3, 4, 5
y = f(x): 2, 5, 7, ?, 32
Show Answer Q: What is f'(0.2) from the following tabular data?
x: 0.0, 0.2, 0.4, 0.6, 0.8, 1.0
y(x): 1.00, 1.16, 3.56, 13.96, 41.96, 101.00
Show Answer 