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 = 45o, show that
(1 + tan A)(1 + tan B) = 2
And hence deduce the value of tan 22½o.
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