Reference no: EM131479021
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
![159_Figure.jpg](https://secure.expertsmind.com/CMSImages/159_Figure.jpg)
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 22.1/2o.