Reference no: EM131475762

APPLICABLE MATHEMATICS ASSIGNMENT

Q1. Using Laplace transform, the following differential equation

d²y/dt² + dy/dt - 3y = sin t, y(0) = 1, y'(0) = 0

reduces to the algebraic equation of the form

y(s) = ((s+1)(s²+1)+1)/(s²+1)(s²+s-3)

Resolve y(s) into partial fractions.

Q2. Solve the following inequality

log(x-1) + log(8 -x) < 1.

Extended reading is required to answer this question.

Q3. Let a^{^}, b^{^}, c^{^} be unit vectors such that a^{^}. b^{^} = a^{^}. c^{^} = 0 and the angle between b^{^} and c^{^} is π/6. Prove that

a^{^} = ±2(b^{^} x c^{^}).

Q4. Let a^{^} and b^{^} be two unit vector such that the magnitude of their difference is √3. Show that the sum of a^{^} and b^{^} is also a unit vector.

Extended reading is required to answer this question.

Q5. The characteristic equation is given by |A - λI| = 0. By Cayley-Hamilton's theorem, it's well-known that every squares matrix A satisfies its characteristic equation. Given that find its characteristic equation and then using Cayley-Hamilton's theorem

2A⁵ - 3A⁴ + A² - 4I₂ = 0

Q6. If A + B = 45^o, show that

(1 + tan A)(1 + tan B) = 2

And hence deduce the value of tan 22½^o.

Q7. Find the coordinates of the orthocenter of the triangle formed by straight lines.

x - y - 5 = 0, 2x - y - 8 = 0 and 3x - y - 9 = 0.

Q8. Given that the equation of the circle

x² + y² + 2gx + 2fy + c = 0

passes through the points (0, 1), (2, 3) and (-2, 5). Find g, f and c using matrix inverse method.

Attachment:- Assignment File.rar