Question 1. The output response, f(t) of a feedback control system is given by:

f(t) = k/1+k [1 -e(1+k)t], where is a constant.

Determine:

a) The value of (3), taking k= 5. Give your answer accurate to 2d.p.
b) The inverse function f^-1(), where is a constant.

Question 2. The functions g and h are defined by:

g(x) = 6/3-x; x ≠ 3

h(x) = 5x + 2

Determine:

(a) The function g(h(x).
(b) The function h(g(x)).

Question 3. Solve the equations correct to 2 decimal places accuracy.

(a) Estimate using the bisection method, the positive real root of:
ln(x⁴) = 0.7

Take the initial guess as x_L= 0.5, x_U= 2

(b) Estimate using Newton Raphson method, one root of:
2x³ - 11.7 x² + 17.7 x - 5 =0

Take the initial guess as x₀ = 3.

Question 4. Evaluate ₀∫^Π/3 √sin(x)dx correct to 3 decimal places using n= 6 equal intervals, applying

(a) Trapezoidal rule

(b) Mid-ordinate rule

(c) Simpsons rule

Question 5. Determine the general solution of the differential equations. Write out the solution y explicitly as a function of x.

(a) 3x²y² dy/dx = 2x - 1

(b) 2dy/dx + 3y = e^-2x - 5

Question 6. Determine the particular solution of the equation:

d²y/dx² + 3 dy/dx + 2y = 10 cos(2x) satisfying the initial conditions

y(0) = 1, y'(0) = 0.

Question 7.  Find the inverse Laplace transform of:

a) 3-2s/(s²-4s+9)

b) (s+2)/(s²+9)(s-2)

Question 8. Solve using the Laplace transform method:

d²y/dt² -2dy/dt + y =  5e^t and y(0) = 1, y'(0) = 1