Find characteristic equation and then using cayley-

Question

1. Using Laplace transform, the following differential equation

(d²y/dt²) + (dy/dt) - 3y = sint, 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.

2. Solve the following inequality

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

Extended reading is required to answer this question.

3. 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^).

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

5. The characte istic equation is given by |A - λl| = 0. By Cayley-Hamilton's theorem, it's well-known that every square matrix A satisfies its characteristic equation. Given that

find its characteristic equation and then using Cayley- Hamilton's theorem

2A⁵ - 3A⁴ + A² - 4I₂ = 0.

6. If A + B = 45°, show that

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

and hence deduce the value of tan 22.1/2^o.