Algebra

Hello everyone! Take your seats. Today, we are going to dive into one of the most powerful and exciting branches of mathematics: Algebra. When you first heard the word algebra, you probably thought about finding the mysterious value of $x$. But algebra is much more than that. It is the language of patterns, the toolkit for solving complex puzzles, and the backbone of modern science, technology, and finance.

Think of basic arithmetic as learning the alphabet. Arithmetic deals with specific numbers, like adding five and three. Algebra is like learning how to form sentences and stories. Instead of using just numbers, we use symbols and letters to represent numbers and quantities in formulas and equations. This allows us to solve general problems rather than just one specific puzzle at a time.

Today, we will journey through several major chapters of algebra. We will look at how to prove mathematical statements without checking infinite cases, how numbers can have imaginary parts, how to solve inequalities, how to arrange and select items, how to expand massive binomial powers, and how to predict patterns in sequences and series. Grab your notebooks and pens, and let us break these concepts down together, step by step.

Meaning

Mathematical Induction is a method of mathematical proof used to establish that a given statement is true for all natural numbers. Imagine lining up an infinite row of dominoes. If you can prove that knocking over the first domino guarantees the next one falls, and that any arbitrary domino falling will always knock over the one immediately after it, then you know for certain that all the dominoes in the infinite line will fall.

Definitions

• According to Richard Courant and Herbert Robbins in What Is Mathematics?, mathematical induction is a method of proof that allows us to establish the validity of an infinite sequence of propositions by verifying a base case and proving that the truth of the $k$-th proposition implies the truth of the $(k+1)$-th proposition.
• In the words of mathematician George Polya, induction is the process of discovering general laws through the examination of particular cases, which in its rigorous mathematical form, transitions from a finite check to an infinite guarantee.

Simple and Clear Explanation with Key Points

To prove a statement $P(n)$ for all natural numbers $n$:

• Step 1: Base Step: Prove that the statement is true for the smallest possible value of $n$, usually $n = 1$.
• Step 2: Induction Hypothesis: Assume that the statement is true for some arbitrary positive integer $n = k$.
• Step 3: Induction Step: Prove that the statement must also be true for $n = k + 1$, using the assumption made in Step 2.

Key Points

• Induction does not help you find the formula; it only helps you prove that a given formula is correct.
• The base step is non-negotiable. If the first domino does not fall, the chain reaction never starts.

Examples

• Prove that $1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$ for all natural numbers $n$.
• Prove that $2^n > n$ for all positive integers $n$.

Real-Life Uses

• Computer Science Algorithm Verification: Software engineers use induction-style logic to prove that recursive algorithms and loops terminate correctly and produce accurate outputs for inputs of any arbitrary size.
• Digital Security: Cryptographers use induction principles to prove the ongoing integrity and security of looping cryptographic hash functions over infinite data streams.

Common Misconceptions and Their Corrections

• Misconception: People often think you check the statement for every single number.
• Correction: Induction actually lets you prove an infinite number of cases in just two or three steps without checking them individually.

Concise Takeaway

Mathematical induction is the mathematical domino effect: prove the first one works, assume the $k$-th works, and prove the $(k+1)$-th works.

Meaning

Real numbers are great, but they cannot solve equations like $x^2 + 1 = 0$. When you try, you end up needing the square root of a negative number. This led mathematicians to invent the imaginary unit $i$, where $i^2 = -1$. A complex number combines a real number and an imaginary number into a single expression.

Definitions

• According to mathematician Leonhard Euler, a complex number is an expression of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ represents the fundamental square root of negative one.
• In the definition by textbook author James Stewart, complex numbers form an extension of the real number system where every non-zero polynomial equation has a root.

Simple and Clear Explanation with Key Points

• A complex number is written as $z = a + bi$.
• Here, $a$ is the real part and $b$ is the imaginary part.
• If $a = 0$, the number is purely imaginary. If $b = 0$, the number is purely real.

Key Points

• Powers of $i$ cycle in a pattern of four: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$.
• Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.

Examples

• Simplify $(3 + 2i) + (1 - 4i)$.
• Multiply $(2 + 3i)$ by $(4 - i)$.

Real-Life Uses

• Electrical Engineering: Alternating Current (AC) circuits use complex numbers to calculate impedance, phase angles, and voltage drops without dealing with messy trigonometric identities.
• Quantum Mechanics: The fundamental wave equations that describe the behavior of subatomic particles rely entirely on complex numbers.

Common Misconceptions and Their Corrections

• Misconception: Students think "imaginary" numbers mean they do not exist in the real world.
• Correction: Imaginary numbers are just as real in their application to physics and engineering as negative numbers or fractions; the name is just an unfortunate historical label.

Concise Takeaway

Complex numbers expand our mathematical universe by giving solutions to equations that stop dead in their tracks within the real number system.

Meaning

Just as real numbers live on a number line, complex numbers live on a two-dimensional plane called the Argand plane or complex plane. Furthermore, instead of describing a point by its horizontal and vertical coordinates ($a$ and $b$), we can describe it by its distance from the origin and the angle it makes with the positive real axis. This is the polar form.

Definitions

• According to mathematician Jean-Robert Argand, the complex plane is a geometric representation of complex numbers utilizing the horizontal axis for real parts and the vertical axis for imaginary parts.
• Polar representation expresses a complex number in terms of its magnitude (modulus) and direction (argument) relative to the origin.

Simple and Clear Explanation with Key Points

• Argand Plane: The x-axis is the real axis, and the y-axis is the imaginary axis. The number $a + bi$ corresponds to the coordinate point $(a, b)$.
• Modulus ($r$): The distance from the origin, given by $r = |z| = \sqrt{a^2 + b^2}$.
• Argument ($\theta$): The angle with the positive real axis, given by $\tan\theta = \frac{b}{a}$.
• Polar Form: $z = r(\cos\theta + i\sin\theta)$, which can also be written using Euler's formula as $re^{i\theta}$.

Key Points

• The modulus is always a non-negative real number.
• Converting between rectangular form ($a + bi$) and polar form ($r(\cos\theta + i\sin\theta)$) makes multiplication and division much simpler.

Examples

• Find the modulus of the complex number $3 + 4i$.
• Convert the complex number $1 + i$ into polar form.

Real-Life Uses

• Aerospace Navigation: Radar systems track aircraft position and velocity using polar coordinates and complex vectors to determine distance and bearing simultaneously.
• Signal Processing: Audio engineering and telecommunications use polar forms of complex numbers to analyze frequencies and wave phase shifts in sound and radio waves.

Common Misconceptions and Their Corrections

• Misconception: Students forget to check which quadrant the angle $\theta$ lies in when calculating the argument using inverse tangents.
• Correction: Always sketch the point on the Argand plane first to ensure your angle is in the correct quadrant.

Concise Takeaway

The Argand plane and polar form turn complex number arithmetic into simple geometry involving distances and rotations.

Meaning

An equation says two things are equal, using an equals sign. An inequality says two things are not necessarily equal, using symbols like greater than ($>$), less than ($<$), greater than or equal to ($\ge$), or less than or equal to ($\le$). Solving linear inequalities means finding the entire range of values that make the statement true.

Definitions

• According to algebra texts by Hall and Knight, a linear inequality is a mathematical statement expressing a linear relationship between expressions where one quantity is strictly or non-strictly greater or less than another.
• Graphical solution of a linear inequality refers to the half-plane in a Cartesian coordinate system containing all ordered pairs that satisfy the inequality condition.

Simple and Clear Explanation with Key Points

• Algebraic Solution: Isolate the variable on one side, remembering to flip the inequality sign whenever you multiply or divide both sides by a negative number.
• Graphical Solution: Draw the boundary line (solid for $\le$ or $\ge$, dashed for $<$ or $>$), and shade the region that satisfies the inequality test point.

Key Points

• Multiplying or dividing by a negative number reverses the inequality direction.
• Systems of linear inequalities create a shaded region called the feasible region, which is heavily used in business optimization.

Examples

• Solve algebraically: $3x - 5 < 7$.
• Solve the inequality $-2x + 4 \ge 10$.

Real-Life Uses

• Budget Planning and Supply Chain: Businesses use systems of linear inequalities to determine how many products to manufacture given limits on raw materials, labor hours, and warehouse space.
• Dietary Planning: Nutritionists use linear inequalities to find food combinations that meet minimum vitamin requirements without exceeding maximum calorie limits.

Common Misconceptions and Their Corrections

• Misconception: Forgetting to flip the inequality sign when multiplying or dividing by a negative number.
• Correction: Remember that inequality is like a seesaw; multiplying by negatives flips the orientation entirely.

Concise Takeaway

Inequalities define boundaries and ranges rather than single fixed answers, showing all the acceptable possibilities for a real-world constraint.

Meaning

Counting sounds easy, but what happens when you need to find out how many different ways you can arrange a playlist, or how many different poker hands can be dealt from a deck of cards? This is where combinatorics comes in.

Definitions

• According to mathematician and educator Donald Knuth, permutations refer to the arrangement of items where order matters, while combinations refer to the selection of items where order does not matter.
• The Fundamental Principle of Counting states that if there are $a$ ways to do one thing and $b$ ways to do another, there are $a \times b$ ways to do both sequentially.

Simple and Clear Explanation with Key Points

• Factorial ($n!$): The product of all positive integers less than or equal to $n$. For example, $4! = 4 \times 3 \times 2 \times 1 = 24$. Note that $0! = 1$.
• Permutation ($P$): Arrangement where order matters. Formula: $P(n, r) = \frac{n!}{(n-r)!}$.
• Combination ($C$): Selection where order does not matter. Formula: $C(n, r) = \frac{n!}{r!(n-r)!}$.

Key Points

• Permutation deals with passwords, running race finishes, and seating arrangements.
• Combination deals with lottery tickets, selecting committee members, and dealing cards.
• Always ask yourself: "Does the order of the chosen items change the outcome?" If yes, use permutation. If no, use combination.

Examples

• Find the number of ways to arrange 3 books on a shelf out of 5 available books.
• Find the number of ways to choose a committee of 3 students from a group of 10 students.

Real-Life Uses

• Password Generation and Cybersecurity: IT systems calculate the total number of possible password permutations to determine how long it would take hackers to crack a system via brute force.
• Lotto and Gaming: Probability experts use combinations to calculate the exact odds of winning major lottery jackpots or drawing specific hands in card games.

Common Misconceptions and Their Corrections

• Misconception: Confusing when to use permutations versus combinations.
• Correction: Ask yourself: If John comes first and Mary second, is that different from Mary first and John second? If yes, it is a permutation. If it is just a group of two people regardless of order, it is a combination.

Concise Takeaway

Permutations care about the seating order; combinations only care about who is in the room.

Meaning

You already know how to expand $(a + b)^2 = a^2 + 2ab + b^2$. But what happens if you need to expand $(a + b)^{10}$ or $(a + b)^{50}$? Multiplying that out by hand would take hours. The Binomial Theorem provides a lightning-fast shortcut to expand any binomial raised to a positive integer power.

Definitions

• According to Sir Isaac Newton, the Binomial Theorem is an algebraic formula that gives the expansion powers of sums in the form of a polynomial series.
• In algebraic convention, the Binomial Theorem states that $(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^k$.

Simple and Clear Explanation with Key Points

• Pascal's Triangle: A triangular array of numbers where each number is the sum of the two directly above it. The rows of Pascal's triangle give the exact coefficients for binomial expansions.
• General Term: The $(r+1)$-th term in the expansion of $(a+b)^n$ is given by $T_{r+1} = \binom{n}{r} a^{n-r}b^r$.
• Middle Term:
  • If $n$ is even, there is one middle term when $r = \frac{n}{2}$.
  • If $n$ is odd, there are two middle terms when $r = \frac{n-1}{2}$ and $r = \frac{n+1}{2}$.

Key Points

• The sum of the exponents of $a$ and $b$ in every term of the expansion of $(a+b)^n$ is always equal to $n$.
• The number of terms in the expansion of $(a+b)^n$ is always $n + 1$.

Examples

• Expand $(x + 2)^4$ using the Binomial Theorem.
• Find the 4th term in the expansion of $(2x + y)^5$.

Real-Life Uses

• Probability and Statistics: Statisticians use the Binomial Theorem to calculate the exact probabilities of independent success and failure events over many trials, such as quality control testing in manufacturing.
• Financial Compound Interest: Economists use binomial expansion approximations to model complex multi-period investment growth and option pricing models like the binomial options pricing model.

Common Misconceptions and Their Corrections

• Misconception: Students often forget that the index $r$ in the general term formula starts counting from $0$, making the first term $T_1$, where $r = 0$.
• Correction: Always remember that the first term corresponds to $r = 0$, so the $k$-th term has $r = k - 1$.

Concise Takeaway

The Binomial Theorem and Pascal's triangle are your ultimate shortcuts for expanding massive algebraic powers without breaking a sweat.

Meaning

A sequence is an ordered list of numbers following a specific rule, and a series is the sum of the terms of that sequence. We will look at two of the most important types of sequences: Arithmetic Progressions (AP), where terms grow by adding a constant, and Geometric Progressions (GP), where terms grow by multiplying by a constant.

Definitions

• According to mathematician Leonhard Euler, an arithmetic progression is a sequence of numbers such that the difference between the consecutive terms is constant, while a geometric progression has a constant ratio between consecutive terms.
• The Arithmetic Mean-Geometric Mean (AM-GM) inequality states that for any set of non-negative real numbers, the arithmetic mean is always greater than or equal to the geometric mean.

Simple and Clear Explanation with Key Points

• Arithmetic Progression (AP):
  • General term: $a_n = a + (n-1)d$
  • Sum of $n$ terms: $S_n = \frac{n}{2}[2a + (n-1)d]$
• Geometric Progression (GP):
  • General term: $a_n = ar^{n-1}$
  • Sum of $n$ terms: $S_n = \frac{a(r^n - 1)}{r - 1}$ (for $r \neq 1$)
• AM and GM Relationship: For any two positive real numbers $a$ and $b$:
  • $\text{AM} = \frac{a + b}{2}$
  • $\text{GM} = \sqrt{ab}$
  • $\text{AM} \ge \text{GM}$ (Equality holds when $a = b$)

Key Points

• In an AP, the common difference $d = a_{n+1} - a_n$ is constant.
• In a GP, the common ratio $r = \frac{a_{n+1}}{a_n}$ is constant.
• The AM-GM inequality is a powerful tool for proving maximum and minimum values in calculus and algebra problems.

Examples

• Find the 10th term of the AP: $2, 5, 8, 11, \dots$.
• Find the sum of the first 5 terms of the GP: $3, 6, 12, 24, \dots$.

Real-Life Uses

• Finance and Loans: APs model straight-line depreciation of equipment, while GPs model exponential growth in compound savings accounts and mortgage amortization schedules.
• Population and Biology: Biologists use geometric progressions to model bacteria and virus reproduction rates over hourly intervals.

Common Misconceptions and Their Corrections

• Misconception: Confusing whether a sequence adds a constant (AP) or multiplies by a constant (GP).
• Correction: Always check the difference between consecutive terms versus the ratio between consecutive terms. If subtraction is constant, it is AP. If division is constant, it is GP.

Concise Takeaway

APs add your steps evenly; GPs multiply your steps exponentially. Together, they model almost all predictable growth in the universe.

Here is a quick cheat sheet of all the vital formulas we covered today for your exam revision:

• Complex Number: $z = a + bi$
• Modulus of Complex Number: $|z| = \sqrt{a^2 + b^2}$
• Polar Form: $z = r(\cos\theta + i\sin\theta)$
• Permutations: $P(n, r) = \frac{n!}{(n-r)!}$
• Combinations: $C(n, r) = \frac{n!}{r!(n-r)!}$
• Binomial General Term: $T_{r+1} = \binom{n}{r} a^{n-r}b^r$
• AP $n$-th Term: $a_n = a + (n-1)d$
• AP Sum: $S_n = \frac{n}{2}[2a + (n-1)d]$
• GP $n$-th Term: $a_n = ar^{n-1}$
• GP Sum: $S_n = \frac{a(r^n - 1)}{r - 1}$
• AM-GM Inequality: $\frac{a+b}{2} \ge \sqrt{ab}$

When you step outside the classroom, algebra is automated by powerful computational engines. Here are the top modern tools used by professionals:

1. Python (SymPy and NumPy libraries): Data scientists, engineers, and researchers use Python scripts to instantly expand massive binomials, solve complex algebraic equation systems, and compute sequences and series numerically.
2. Wolfram Mathematica: A gold-standard computational software system used in aerospace, physics, and advanced mathematics for symbolic algebraic manipulation, complex number plotting on Argand planes, and automated proof verification.
3. GeoGebra: An educational and professional tool used to visualize linear inequalities, plot complex numbers in the Argand plane, and graph geometric and arithmetic sequences dynamically in real-time.
4. MATLAB: Widely used in electrical engineering and signal processing to perform matrix operations, complex number arithmetic, and polynomial expansions instantly.

Title: Optimizing Tech Startup Resource Allocation and Expansion

Background and Scenario:
Imagine you are the chief operations officer of a growing tech startup based in Seattle. Your company builds high-performance servers, and you are facing three major logistical challenges that require a firm grasp of algebra to solve successfully.

First, your software engineering team needs to verify that a new recursive data indexing algorithm works correctly for any arbitrary database size $n$. They decide to use the Principle of Mathematical Induction to prove the algorithm's time complexity formula $T(n) = \frac{n(n+1)}{2}$ milliseconds. The base case is verified for $1$ server node, where $T(1) = \frac{1(2)}{2} = 1$ millisecond. Assuming the formula holds true for $k$ nodes, the team proves algebraically that for $k+1$ nodes, the execution time matches the predicted polynomial expansion, securing the software architecture against infinite loop failures.

Second, your hardware engineering division is designing a specialized AC circuit filter for server power supplies. The total impedance of the circuit is expressed as a complex number $Z = 3 + 4i$ ohms. To calibrate the phase angle and signal resonance correctly, the engineers convert this rectangular form into polar form. Using the Argand plane equations, they calculate the modulus as $|Z| = \sqrt{3^2 + 4^2} = 5$ ohms, and the argument angle as $\theta = \arctan\left(\frac{4}{3}\right)$. This polar representation allows them to tune the circuit components instantly without cumbersome trigonometric expansions.

Third, your human resources and marketing departments are facing strict operational constraints. HR needs to select a 4-person executive committee out of 12 qualified senior engineers. Because order does not matter in a committee, they use combinations: $C(12, 4) = \frac{12!}{4!8!} = 495$ possible committees. Meanwhile, the marketing team needs to arrange 5 promotional billboards in a single row outside the convention center. Because the physical arrangement order matters deeply for brand visibility, they use permutations: $P(5, 5) = 5! = 120$ possible billboard layouts.

Finally, your finance department models your company revenue growth and server component purchases. Your server part inventory follows a geometric progression where you purchase 3 servers in week one, 6 in week two, 12 in week three, growing exponentially as customer demand scales. To minimize warehouse holding costs while maximizing computing efficiency, you apply the AM-GM inequality to balance two competing hardware resource metrics $a$ and $b$, ensuring that $\frac{a+b}{2} \ge \sqrt{ab}$, which guarantees optimal server operating margins throughout the fiscal year. Through this integrated application of mathematical induction, complex numbers, combinatorics, binomial expansions, and sequences, your startup scales smoothly and profitably.

1. What is the base step requirement in Mathematical Induction?
   A) Proving the statement for $n = k$
   B) Proving the statement for $n = k+1$
   C) Proving the statement for the smallest natural number, usually $n = 1$
   D) Proving the statement for infinity
   Answer: C
   Explanation: The base step requires verifying that the proposition holds true for the initial starting integer, typically $n = 1$, to kickstart the induction chain reaction.


2. What is the value of the imaginary unit $i^3$?
   A) $1$
   B) $-1$
   C) $i$
   D) $-i$
   Answer: D
   Explanation: Since $i^2 = -1$, multiplying by $i$ gives $i^3 = i^2 \cdot i = (-1) \cdot i = -i$.


3. On the Argand plane, what does the horizontal axis represent?
   A) The imaginary part of a complex number
   B) The real part of a complex number
   C) The modulus of a complex number
   D) The argument angle
   Answer: B
   Explanation: In the Argand plane, the horizontal axis corresponds to the real component of the complex number, while the vertical axis represents the imaginary component.


4. When solving a linear inequality, when must you reverse the inequality sign?
   A) When adding a positive number to both sides
   B) When subtracting a number from both sides
   C) When multiplying or dividing both sides by a negative number
   D) When squaring both sides
   Answer: C
   Explanation: Multiplying or dividing an inequality by a negative number flips the direction of the inequality because it reflects the values across zero on the number line.


5. How many ways can a committee of 3 people be chosen from a group of 7 people?
   A) 21
   B) 35
   C) 210
   D) 42
   Answer: B
   Explanation: This is a combination problem where order does not matter: $C(7, 3) = \frac{7!}{3!4!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$.


6. What is the number of terms in the binomial expansion of $(a + b)^n$?
   A) $n$
   B) $n - 1$
   C) $n + 1$
   D) $2n$
   Answer: C
   Explanation: The expansion of $(a+b)^n$ has $n + 1$ terms, starting with $r = 0$ up to $r = n$.


7. In an Arithmetic Progression (AP), what remains constant between consecutive terms?
   A) The sum
   B) The product
   C) The difference
   D) The ratio
   Answer: C
   Explanation: By definition, an AP has a constant common difference $d$ between any term and its preceding term.


8. In a Geometric Progression (GP), if the first term is $3$ and the common ratio is $2$, what is the 3rd term?
   A) 6
   B) 9
   C) 12
   D) 24
   Answer: C
   Explanation: Using $a_n = ar^{n-1}$, the 3rd term is $3 \cdot (2)^{3-1} = 3 \cdot 2^2 = 3 \cdot 4 = 12$.


9. For any two positive real numbers $a$ and $b$, what is the relationship between AM and GM?
   A) $\text{AM} < \text{GM}$
   B) $\text{AM} \ge \text{GM}$
   C) $\text{AM} = \text{GM}$
   D) $\text{AM} + \text{GM} = 0$
   Answer: B
   Explanation: The Arithmetic Mean-Geometric Mean inequality states that the arithmetic mean is always greater than or equal to the geometric mean ($\frac{a+b}{2} \ge \sqrt{ab}$).


10. What is the value of $0!$ (zero factorial)?
    A) $0$
    B) $1$
    C) Undefined
    D) Infinity
    Answer: B
    Explanation: By mathematical convention and definition in permutation formulas, $0! = 1$, representing the number of ways to arrange zero items.

1. Question 1: Use mathematical induction to prove that $1 + 2 + 3 + \dots + n = \frac{n(n+1)}{2}$ for all natural numbers $n$.
   Answer:
   • Step 1 (Base Case): For $n = 1$, Left Hand Side $= 1$. Right Hand Side $= \frac{1(1+1)}{2} = 1$. Since LHS $=$ RHS, the statement is true for $n = 1$.
   • Step 2 (Induction Hypothesis): Assume the statement is true for $n = k$: $1 + 2 + \dots + k = \frac{k(k+1)}{2}$.
   • Step 3 (Induction Step): We must prove it for $n = k+1$.
     $1 + 2 + \dots + k + (k+1) = \frac{k(k+1)}{2} + (k+1)$
     $= (k+1) \left(\frac{k}{2} + 1\right) = \frac{(k+1)(k+2)}{2}$.
     This matches the formula for $n = k+1$. Thus, by induction, the statement is true for all natural numbers $n$.


2. Question 2: Express the complex number $-1 - i\sqrt{3}$ in polar form.
   Answer:
   • First, find the modulus $r = \sqrt{(-1)^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$.
   • Next, find the argument $\theta$. Since both real and imaginary parts are negative, the point lies in the third quadrant. $\tan\theta = \frac{-\sqrt{3}}{-1} = \sqrt{3}$, giving a reference angle of $\frac{\pi}{3}$. In the third quadrant, $\theta = -\frac{2\pi}{3}$.
   • Polar form: $z = 2\left(\cos\left(-\frac{2\pi}{3}\right) + i\sin\left(-\frac{2\pi}{3}\right)\right)$.


3. Question 3: Solve the linear inequality graphically and algebraically: $2x - 3 \ge 5$ for real numbers.
   Answer:
   • Algebraic Solution: $2x \ge 8 \implies x \ge 4$.
   • Graphical Solution: On a number line, draw a solid dot at $4$ and shade the entire ray extending infinitely to the right toward positive infinity.


4. Question 4: Find the number of words that can be formed using all the letters of the word "ALGEBRA" if vowels must always come together.
   Answer:
   • The word "ALGEBRA" has 7 letters, with 'A' repeating twice.
   • Vowels in ALGEBRA are A, E, A (3 vowels, with 'A' repeated twice).
   • Treat the group of vowels as a single entity. We now have 5 entities to arrange: (AEA), L, G, B, R.
   • The number of ways to arrange these 5 entities is $\frac{5!}{1!} = 120$.
   • Within the vowel group, the 3 letters (A, E, A) can be arranged among themselves in $\frac{3!}{2!} = 3$ ways.
   • Total arrangements $= 120 \times 3 = 360$.


5. Question 5: Find the 5th term in the binomial expansion of $(x + 2y)^6$.
   Answer:
   • Using the general term formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$.
   • Here $n = 6$, and for the 5th term, $r = 4$.
   • $T_5 = \binom{6}{4} (x)^{6-4} (2y)^4$
   • $\binom{6}{4} = \frac{6 \times 5}{2 \times 1} = 15$.
   • $T_5 = 15 \cdot x^2 \cdot 16y^4 = 240x^2y^4$.


6. Question 6: Which term of the AP $3, 8, 13, 18, \dots$ is $78$?
   Answer:
   • Here, first term $a = 3$, common difference $d = 8 - 3 = 5$, and $a_n = 78$.
   • Using formula $a_n = a + (n-1)d$:
     $78 = 3 + (n-1)5$
     $75 = 5(n-1)$
     $n - 1 = 15 \implies n = 16$.
   • Thus, the 16th term is $78$.


7. Question 7: Find the sum of the first 6 terms of the Geometric Progression: $2, 6, 18, 54, \dots$.
   Answer:
   • First term $a = 2$, common ratio $r = \frac{6}{2} = 3$, number of terms $n = 6$.
   • Using the sum formula $S_n = \frac{a(r^n - 1)}{r - 1}$:
     $S_6 = \frac{2(3^6 - 1)}{3 - 1} = \frac{2(729 - 1)}{2} = 728$.


8. Question 8: Prove that the arithmetic mean of two positive numbers is always greater than or equal to their geometric mean ($a+b \ge 2\sqrt{ab}$).
   Answer:
   • Consider the square of a real number, which is always non-negative: $(\sqrt{a} - \sqrt{b})^2 \ge 0$.
   • Expand the binomial: $a - 2\sqrt{ab} + b \ge 0$.
   • Add $2\sqrt{ab}$ to both sides: $a + b \ge 2\sqrt{ab}$.
   • Divide by $2$: $\frac{a+b}{2} \ge \sqrt{ab}$. Hence proved.


9. Question 9: Solve the system of inequalities: $x + y \le 5$ and $2x - y > 1$.
   Answer:
   • For the first inequality $x + y \le 5$, graph the solid boundary line $x + y = 5$ and shade below it toward the origin.
   • For the second inequality $2x - y > 1$, graph the dashed boundary line $2x - y = 1$ and shade the half-plane containing the region that satisfies the test point condition.
   • The intersection of both shaded regions represents the feasible region solution.


10. Question 10: Find the coefficient of $x^5$ in the expansion of $\left(x^2 + \frac{1}{x}\right)^7$.
    Answer:
    • General term $T_{r+1} = \binom{7}{r} (x^2)^{7-r} \left(\frac{1}{x}\right)^r = \binom{7}{r} x^{14-2r} x^{-r} = \binom{7}{r} x^{14-3r}$.
    • We want the power of $x$ to be $5$, so set $14 - 3r = 5$.
    • $3r = 9 \implies r = 3$.
    • Coefficient $= \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$.

• Permutations vs. Combinations: Permutation starts with P for Position (order matters). Combination starts with C for Committee (order does not matter).
• AM-GM Inequality: "Arithmetic is Above, Geometric is Grounded" (AM is always greater than or equal to GM).
• AP vs. GP: AP = Addition (common difference), GP = Geometric = Growth by multiplication (common ratio).
• Factorial Rule: Never confuse zero factorial with zero. $0!$ is defined as $1$, whereas $0$ multiplied by anything is $0$.

Mnemonic: Keep formulas close, sketch your Argand planes, and always check your inequality direction when multiplying by negatives!