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

Algebra and Trigonometry Problem Set

1. y = -6x³ + 4x² - x + 2
2. Vertical shrink by 1/2, horizontal translation 2 to the left, vertical translation 1 down
3. 4x⁴ - 11x³ + 7x² + 5x - 3
4. $713,476.20
5. Brian is designing a handrail for a staircase and needs to determine the length: 12
6. A meteorologist predicts a 42% chance of rain on Saturday and a 54% chance of rain on Sunday: 27%
7. a_n = 3n - 11
8. (-2, 6) lies on the terminal side of an angle θ in standard position: Sinθ = 3√10 is False, True, True
9. y = log(x - 2)
10. Find the sum of the series: 12
11. The table shows enrollment by gender as an undergraduate or graduate student: 515/2463 = 0.209

Zeros of Functions

12. Find the zeros of the function shown below:
f(x) = x² + 3x - 10: x = 2 and -5

Statistical Distributions

13. Normal... Continue reading "Algebra and Trigonometry Practice Problem Solutions" »

Set Difference Calculation

To find the set difference A - B, we identify all elements present in set A but not in set B.

Step-by-Step Subtraction

• Is 1 ∈ B? No. (Keep 1)
• Is 2 ∈ B? No. (Keep 2)
• Is 3 ∈ B? Yes. (Remove 3)
• Is 5 ∈ B? No. (Keep 5)
• Is 7 ∈ B? Yes. (Remove 7)
• Is 8 ∈ B? No. (Keep 8)

The remaining elements from set A are {1, 2, 5, 8}.

Symbolic Logic

In symbolic logic, the word "but" functions like "and," indicating that both conditions occur simultaneously. To write "He is rich but not generous" in symbolic form:

• p: "He is rich"
• q: "He is generous"
• ¬q: "He is not generous"
• ∧: The conjunction operator

Logic Symbol Reference

Logarithm Calculations

To find the value of log 360,... Continue reading "Step-by-Step Solutions for Mathematical Problems" »

Parametric Inference Fundamentals

The probability distribution of the population under study is known, except for a finite number of parameters. Its goal is to estimate those parameters. Examples include the T-test and ANOVA.

Non-Parametric Inference Basics

The distribution of the population is not known. It is used to test the assumptions of parametric methods, for example, to check if the population distribution is normal.

What is a Statistic?

A random variable function of the sample that does not depend on the unknown parameter.

Understanding Estimators

A statistic whose values are acceptable for estimating an unknown parameter.

Unbiasedness in Estimation

We do not allow systematic overestimation or underestimation of the parameter, which would result... Continue reading "Statistical Inference & Hypothesis Testing Concepts" »

De Morgan's Law

De Morgan's Law: (Flip if the union is true)

, image of set: [min, max]; one-to-one: horizontal line test; Onto: Image must equal domain; Bijective: one-to-one and Onto

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Possible Outcomes and Probability Calculations

• Repetition formula: n^k
  • Example: 5 awards (k) and 30 students (n), with no limit to awards per student.
• Permutation formula: P(n, k) = n! / (n - k)!
  • Example: Each student gets 1 award, so the number of students decreases by one each award.
• No overlap probability: P(n, k) / repetition formula
• Arrangements: a = slots → a! can be multiplied by arrangements within slots
• Die sum probability:
  • List combinations that lead to the sum for each die.
  • If a die is rolled multiple times, each combination has (rolls)! permutations.
  • Add

Probability Distributions for Discrete Events

The following table matches common scenarios to their appropriate probability distributions:

Bar Graph: Annual Sales of Product A and B

The bar chart illustrates the annual dollar sales of Product A and Product B for the years 2015, 2016, and 2017. As can be seen in the graph, between 2015 and 2017 sales of Product A were higher than sales of Product B. In 2015, sales of Product B were slightly lower than Product A, and in 2016 sales of Product A reached 80,000 USD while sales of Product B only reached 50,000 USD. For 2017, both Product A and Product B had a slight growth, increasing their sales by 10,000 USD compared to the previous year. Overall, we can see that sales of both products have grown in the last three years; however, the product that generates the most revenue is Product A.

Line Chart: Six-Year Sales Trend

The graph shows... Continue reading "Annual Sales Trends and Household Water Usage Data" »

Random Variables

A random variable is a numerical quantity that takes on different values depending on chance. There are two primary types:

• Discrete (PMF): A countable set of possible outcomes (e.g., the number of cases in an SRS from the population).
• Continuous (PDF): An unbroken continuum of possible outcomes (e.g., the average weight of an SRS of newborns selected from the population).

Key Statistical Definitions

• Population: The set of all possible values for a random variable.
• Event: An outcome or a set of outcomes.
• Probability: The proportion of times an event is expected to occur in the population.

Note: Ideas about probability are founded on relative frequencies (proportions) in populations.

Probability Calculation Example

In a given year, there... Continue reading "Understanding Random Variables and Probability Distributions" »

AdaBoost: Adaptive Boosting Explained

AdaBoost is one of the simplest and earliest boosting algorithms. The main idea behind AdaBoost is to combine many weak learners (models that do slightly better than random guessing) into one strong learner.

It works by training multiple models one after another. After each model, the algorithm checks which data points were predicted wrong. It then gives more importance (weight) to those wrongly predicted samples so that the next model focuses more on correcting those mistakes.

Each new model tries to fix the errors made by the previous ones. At the end, all models are combined using weighted voting to make the final prediction. This helps improve accuracy and reduces errors.

Key Characteristics of AdaBoost

• Combines

TDS Return Filing Due Dates

A Tax Deducted at Source (TDS) return is a quarterly statement submitted to the Income Tax Department. It is mandatory for deductors to submit these returns on time to avoid late fees, which accrue at ₹200 per day under Section 234E.

The financial year is divided into four quarters. The due date is generally the last day of the month following the end of the quarter, with an exception for the final quarter.

Differences Between PAN and TAN

While both are unique 10-... Continue reading "TDS Return Due Dates and PAN vs TAN Comparison" »

I. Algebra

1. Quadratic Equations

• Standard form: f(x) = ax² + bx + c
• Vertex form: f(x) = a(x - h)² + k (Vertex = (h, k))
• Example: f(x) = x² - 5x + 6 = 0 → (x - 2)(x - 3) = 0 → x = 2 or 3

2. Solving Equations

• Example: (2x/3) + 1 = 5/6
  Subtract 1: 2x/3 = -1/6
  Multiply by 3: 2x = -1/2
  Divide: x = -1/4

3. Function Composition

• (f ∘ g)(x) = f(g(x))
• Example: f(x) = x² + 2x, g(x) = 3x - 1
  (f ∘ g)(2) = f(5) = 25 + 10 = 35

4. Absolute Value

• General form: f(x) = a|x - h| + k
  a = vertical stretch/shrink, h = horizontal shift, k = vertical shift
• Example: g(x) = 2|x - 3| - 1 (Stretch by 2, right 3, down 1)

II. Geometry

5. Parallel Lines & Angles

• Alternate interior angles are equal.
• Example: 5x = 70 → x = 14

6. Pythagorean Theorem

• a² + b² = c²
• Example: leg =