Notes, summaries, assignments, exams, and problems for Mathematics

Working Papers

These are the documents in which the auditor records the data and information obtained during their review, as well as the results of the tests performed.

Summary Warrant

A sample of major accounts that are an item.

Worksheet

This is the card that shows the groups or items that comprise the financial statements.

Decree of Detail

The items that comprise an account balance of a major account or any other account.

Certificates of Verification

The work performed to verify the correctness of an item or transaction.

Situations for Audit Work Plan

• Ensure that the service is clearly controlled.
• Be fully aware of what is being offered.

Conditions to Perform the Audit

• Maintain independence of mind regarding the specific client.
• Possess the ability to

Saw-Tooth Diagram for Inventory Management

The Saw-Tooth Diagram is a design used for input control and monitoring of stock. This graph serves to manage inventories, ordering, and the volume of flow itself.

The optimal quantity is determined according to the purchase price, storage costs, and handling volume. The optimal lot (often related to the Economic Order Quantity or EOQ, here referred to as PDID) includes the purchase price, carrying costs, PDID costs, overhead costs, and costs incurred until the item reaches the point of sale or placement.

Henry Gantt and the Gantt Chart

Henry Gantt was an American engineer belonging to the classical school of organization who planned American military transport during World War I, using bar-line graphs....

The Architecture of Basilica San Lorenzo

The Basilica San Lorenzo was built during the Quattrocento by the architect Filippo Brunelleschi, with construction conducted between 1422 and 1470. It is inspired by the Church of the Holy Cross (Gothic). The structure is divided into three naves and lateral chapels. In the middle of the transept is a cupola, which is surrounded by chapels. Opposite the middle of the transept is the main chapel.

Geometric and Mathematical Design

Its plan is a Latin cross with three naves and side chapels, forming a basilica that is almost extremely longitudinal. In the transept, which has little development, he set a dome with a lantern. The space was mathematically and geometrically modulated by the circle inscribed in

The Expectation-Maximization (EM) algorithm is an iterative method used to estimate the parameters of statistical models that involve latent (unobserved) variables, such as missing data or hidden cluster assignments. It is especially useful for fitting Gaussian Mixture Models (GMMs), where the goal is to model data as a mixture of several Gaussian distributions.

How the EM Algorithm Works

The EM algorithm alternates between two steps:

• Expectation Step (E-step): Given the current parameter estimates (means, covariances, and mixing coefficients for GMMs), the algorithm computes the probability (or "responsibility") that each data point belongs to each Gaussian component. This step essentially fills in the missing information about which cluster

Boolean Algebra Fundamentals

Formulating Expressions: SOP and POS

Sum of Products (SOP) Formulation Steps

1. Circle rows in the truth table where the output $Y = 1$.
2. Identify the minterms corresponding to the circled rows.
3. Sum (OR) the selected minterms to form the final expression.

Product of Sums (POS) Formulation Steps

1. Circle rows in the truth table where the output $Y = 0$.
2. Identify the maxterms corresponding to the circled rows.
3. Multiply (AND) the selected maxterms to form the final expression.

Order of Evaluation in Boolean Algebra

Operations are evaluated in the following sequence:

1. Parentheses
2. NOT (Complementation)
3. AND (Multiplication)
4. OR (Addition)

Fundamental Boolean Laws and Theorems

• Identity Laws

  • $A + 0 = A$
  • $A \cdot 1 = A$

• Null Laws

  • $A + 1 = 1$
  • $A \cdot

Lease Accounting: Vehicle Leases and Journal Entries

Today is May 31, X1, and we lease three vehicles to visit our factories for €120,000. The contractual conditions are as follows:

• The term of the lease is 4 years.
• The payments (€35,000 each) are annual and will be made on May 31. An additional €1,000 will be added as a purchase option in the final payment.
• Effective interest rate: 6.75%
• The estimated useful life for each vehicle is 8 years.

Lease Amortization Schedule

Journal Entries for Lease Transactions

May 31, X1: Initial Lease Recognition

Fundamental Logarithm Properties

These are key cancellation properties of logarithms and exponentials:

• e^ln8 = 8
• ln(e⁴) = 4
• log₈(8³) = 3
• 5^log5(2) = 2

Essential Logarithm Rules

• Product Rule: log_a(xy) = log_a(x) + log_a(y)
• Quotient Rule: log_a(x/y) = log_a(x) - log_a(y)
• Power Rule: log_a(x^p) = p log_a(x)
• Change of Base Formula: log_B(D) = log(D) / log(B) (using common log) or ln(D) / ln(B) (using natural log)
• Logarithmic to Exponential Form: log_b(n) = a is equivalent to n = b^a.

Logarithm Definitions and Undefined Cases

• A logarithm answers the question: "What is the power?" For example, log_b(n) asks "To what power must b be raised to get n?"
• Logarithms are undefined for:
  • log(0)
  • log(-#) (logarithm of a negative number)
• The natural logarithm: log_e(x) = ln(x).
• Exponential

Mathematical Aptitude - Average

Question 1

The average age of 30 students is 9 years. If the age of their teacher is included, it becomes 10 years. The age of the teacher (in years) is

_______

(A) 31

(B) 35

(C) 40

(D) 43

Question 2

The average weight of 10 men is decreased by 3 kg when one of them whose weight is 80 kg is replaced by a new person. The weight of the new person is

_______

(A) 50

(B) 60

(C) 70

(D) 73

Question 3

The average of 17 numbers is 45. The average of the first 9 of these numbers is 51 and the last 9 of these numbers is 36. What is the ninth number?

(A) 14

(B) 16

(C) 18

(D) 20

Question 4

The average weight of 9 mangoes increases by 20 gm if one of them weighing 120 gm is replaced by another. The weight of the new mango is

________

(A) 180 gm

(B)...

Simple Random Sample

A sample designed in such a way as to ensure that (1) every member of a population has an equal chance of being included in the sample, and (2) every sample of size n has the same chance of occurring.

Selecting Elements

Subjects are selected at fixed sampling intervals by selecting every kth element thereafter. The interval between selections is determined by the ratio of the population size to the sample size (N/n). For example, if the population size is N=1,000 and a sample size of n=100 is desired, then the sampling interval is 1,000/100 = 10, so every 10th person is selected into the sample.

Contradiction

The reading and the lecture are both about ______. Whereas the author of the reading states that _______, the lecturer suggests that ________. The lecturer casts doubt on the main points made in the reading by providing two/three reasons.

First of all, according to the reading, _____________ .However, the lecturer disputes this point. He/She says that _________. Furthermore, he/she mentions that __________

Secondly, the reading states that _______. Nevertheless, the lecturer refutes this argument. He/She argues that __________. In addition, he/she points out ________.

Finally, the reading claims that _________. On the other hand, the lecturer believe that _____. He/She thinks that _______. Moreover, he/she feels that ________

In conclusion,...