Test: Generating Functions - Civil Engineering (CE) MCQ

Concept:

Electric Generator:

• A generator is a machine that converts mechanical energy into electrical power for use in an external circuit. ​
• An electromechanical energy conversion device converts electrical energy into mechanical energy or mechanical to electrical energy
• An electromechanical energy conversion device converts electrical energy into mechanical energy or mechanical to electrical energy. The conversion can take place via any medium, electrical, or magnetic medium.
• Generally, the magnetic field is used as the coupling medium between electrical and mechanical mediums because the energy-storing capacity of the magnetic field is much higher than the electric field. It is a reversible process
• A prime mover is a machine that converts energy into work and examples of such machines are the gas turbine, steam turbine, reciprocating internal combustion engine, and hydraulic turbine

Concept:
It is given that positive and negative values are equally likely to occur, Binomial distribution can be adopted.
The probability of ‘r’ number of successes in ‘n’ trials is given by
p(x) = nC_r.p^r.q^n−r
p - Probability of getting negative value
q - Probability of getting positive value

Calculation:
Given:
n = 5 trials
Positive and negative values are equally likely to occur,
p = 1/2 , q = 1/2
At most one negative value so it can be no negative value or 1 negative value
p (At most one negative) = p(r ≤ 1) =  p(r = 0) + p (r = 1)

p (At most one negative) = 6 / 32

Concept:
Recurrence Relation:
A recurrence relation relates the nth term of a sequence to its predecessors. These relations are related to recursive algorithms.

Definition:
A recurrence relation for a sequence a₀, a₁, a₂,.... is a formula (equation) that relates each term an to certain of its predecessors a₀, a₁, a₂,...., a_n-1. The initial conditions for such a recurrence relation specify the values of a₀, a₁, a₂,...., a_n-1. For example, the recursive formula for the sequence 3, 8, 13, 18, 23 is
a₁ = 3, a_n = a_n-1 + 1, 2 ≤ n < ∞

Analysis:
T(n) = 2T(n - 1) + n;  n ≥ 2
put, n = 2
T(2) = 2T(1) + 2 = 2 + 2 = 4
put, n = 3
T(3) = 2T(2) + 3 = 8 + 3 = 11
put, n = 4
T(4) = 2T(3) + 4 = 22 + 4 = 26
Similarly, T(2) = 4, T(3) = 11, T(4) = 26, ......
T(2) = 2²⁺¹ - 2 - 2 = 4
T(3) = 2³⁺¹ - 3 - 2 = 11
T(4) = 2⁴⁺¹ - 4 - 2 = 26
In general,
T(n) = 2ⁿ⁺¹ - n - 2

Consider the coefficients of each xⁿ term. So a₀ = 4, since the coefficient of x₀ is 4 (x₀=1 so this is the constant term). Since 15 is the coefficient of x², so 15 is the term a₂ of the sequence. To find a₁ check the coefficient of x₁ which in this case is 0. So a₁ = 0. Continuing with these we have a₂ = 15, a₃=10, a₄ = 25, and a₅ = 16. So we have the sequence 4, 0, 15, 10, 25, 16,…

The first constant term is 1⋅1, next term will be 1⋅3 + 2⋅1 = 5, the next term: 1⋅5 + 2⋅3 + 3⋅1 = 14, another one: 1⋅7 + 2⋅5 + 3⋅3 + 4⋅1 = 30. The resulting sequence is 1, 5, 14, 30,…

For the given sequence after evaluating the formula the generating formula will be (4/1−7x)+(6/1+2x).

Based on the given generating function, the sequence will be 0, 0, 1, 1, 2, 3, 5, 8,… which is generated by 1/1−x²−x⁴.

The recurrence relation for the sequence 1, 3, 7, 15, 31, 63,… should be a_n = 3a_n-1−2a_n-2. The solution for A: A=1/1 − 3x + 2x².

The sequence should be 0, 4, 8, 12, 16, 20,…for the generating function 4x/(1-x)², when basic generating function: 1/(1-x).