Fundamental of Algebra - JEE Grade 11 Mathematics Session 9

Session Overview and Educational Context

Course: JEE Grade 11 - Mathematics.
Session Number: 9.
Topic: Fundamental of Algebra.
Educational Provider: ALLEN ONLINE.
Results Record (JEE Advanced 2025): 1475 total selections from full-year paid courses. Highlights include: - Arka Banerjee (West Bengal): AIR 395. - Aritro Ray (West Bengal): AIR 50. - Chirag Singh (Uttar Pradesh): AIR 516.
Results Record (JEE Main 2025): - AIR 1: Devdutta Majhi (1-Year Online Test Series Student, West Bengal). - AIR 7: Aayush Chaudhari (6 Months Online Test Series Student, Maharashtra). - AIR 15: Harssh A Gupta (1-Year Online Test Series Student, Telangana). - AIR 23: Harsh Jha (1-Year Online Test Series Student, Jharkhand). - AIR 51: Chirag Singh (2-Year LIVE Course Student, Uttar Pradesh).
Results Record (NEET-UG 2025): - AIR 74: Tanmay Jagga (Online Classroom Course). - AIR 247: Debarghya Bag (Online Classroom Course). - AIR 405: Adyasha A. Jena (Online Classroom Course).

Review of Previous Concepts (Let's Rewind)

The following topics serve as the foundation for Algebra in JEE 11: - Factorization. - Indices and Surds. - Reducible to quadratic forms. - Modulus and its graph. - Wavy Curve method. - Sets and Intervals (Trivial/Non-trivial). - Arithmetic Mean - Geometric Mean (A.M.-G.M.). - Polynomials and Remainder-Factor (R-F) Theorem. - Ratio and Proportion. - Determinants. - System of Equations.

Definition and Properties of Polynomials

General Form: A polynomial function in $x$ is expressed as: - $p(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \dots + a_1 x + a_0$
Conditions: For a valid polynomial, the powers of $x$ must be non-negative integers ( $n \in \{0, 1, 2, \dots\}$ ).
Key Terminology: - Coefficients: The set of real numbers ${a_n, a_{n-1}, \dots, a_0}$ . - Degree: The highest power of $x$ in the polynomial, provided $a_n \neq 0$ . - Leading Term: The term containing the highest power of $x$ , which is $a_n x^n$ . - Leading Coefficient: The coefficient of the leading term, $a_n$ . - Monic Polynomial: A polynomial where the leading coefficient is equal to one ( $a_n = 1$ ). - Zeroes: A polynomial function of degree $n$ (where $n \geq 1$ ) has exactly $n$ zeroes.

Identification and Classification of Polynomials

Identification Exercise: - $x^2 + 7x + 2\sqrt{x}$ : No, because the power of $x$ is $\frac{1}{2}$ , which is not an integer. - $2x + 3$ : Yes, degree 1. - $7x - \frac{9}{x}$ : No, because it includes $x^{-1}$ . - $x^2 + \pi x$ : Yes, coefficients can be irrational numbers like $\pi$ . - $13x + \sin(2x)$ : No, because it contains a transcendental (trigonometric) function. - $3x^3 + \sqrt{2}x$ : Yes, degree 3.
Classification by Degree: - Undefined: The zero polynomial ( $p(x) = 0$ ). - Degree 0: Constant Polynomial (e.g., $p(x) = c$ ). - Degree 1: Linear Polynomial (e.g., $ax + b$ ). - Degree 2: Quadratic Polynomial (e.g., $ax^2 + bx + c$ ). - Degree 3: Cubic Polynomial.
Classification by Number of Terms: - 1 Term: Monomial. - 2 Terms: Binomial. - 3 Terms: Trinomial.
Multiplication Rule: If $f(x)$ is a polynomial of degree $m$ and $g(x)$ is a polynomial of degree $n$ , the degree of the product $f(x) \cdot g(x)$ is $m + n$ . - Example: If $f(x) = x^4 + 3x^3 - x^2 + 7x + 13$ (degree 4) and $g(x) = -x^4 + 5x^3 + 3x^2 + 9x + 25$ (degree 4), the degree of $f(x) \cdot g(x)$ is $4 + 4 = 8$ .

Euclid's Lemma and Division of Polynomials

Division Algorithm Equation: - $P(x) = Q(x) \cdot d(x) + r(x)$ - Where: - $P(x)$ is the Dividend. - $d(x)$ is the Divisor. - $Q(x)$ is the Quotient. - $r(x)$ is the Remainder.
Constraints on Degrees: - The degree of the divisor $d(x)$ must be less than or equal to the degree of the dividend $P(x)$ . - The degree of the remainder $r(x)$ must be strictly less than the degree of the divisor $d(x)$ . - If the degree of $d(x)$ is 3, the possible degree of $r(x)$ is 2, 1, or 0. - If the degree of $d(x)$ is 2, the possible degree of $r(x)$ is 1 or 0.

Remainder Theorem

Statement: Let $P(x)$ be a polynomial of degree $\geq 1$ and $a$ be any real number. If $P(x)$ is divided by $(x - a)$ , then the remainder is $P(a)$ .
Application Examples: - Find remainder for $x^3 + 2x^2 + 3x - 7$ divided by $(x + 1)$ : - Let $x + 1 = 0 \rightarrow x = -1$ . - Remainder = $P(-1) = (-1)^3 + 2(-1)^2 + 3(-1) - 7$ - $= -1 + 2 - 3 - 7 = -9$ . - Find remainder for $p(x) = x^4 - 3x^3 + 2x^2 + 5x + 1$ divided by $x + 1$ : - Remainder = $P(-1) = 1 - 3(-1) + 2(1) + 5(-1) + 1 = 1 + 3 + 2 - 5 + 1 = 2$ . - Find remainder for $p(x) = x^5 - 3x^3 + 2x^2 + 3x + 1$ divided by $x^2 - 1$ : - The divisor is degree 2, so let the remainder be $ax + b$ . - $p(x) = (x^2 - 1)Q(x) + (ax + b)$ . - At $x = 1$ : $1 - 3 + 2 + 3 + 1 = a + b \rightarrow 4 = a + b$ . - At $x = -1$ : $-1 + 3 + 2 - 3 + 1 = -a + b \rightarrow 2 = -a + b$ . - Solving the system: $2b = 6 \rightarrow b = 3$ and $a = 1$ . - Remainder = $x + 3$ .

Factor Theorem

Statement: Let $f(x)$ be a polynomial of degree $\geq 1$ and $a$ be any real constant. - If $f(a) = 0$ , then $(x - a)$ is a factor of $f(x)$ . - Conversely, if $(x - a)$ is a factor of $f(x)$ , then $f(a) = 0$ .
Application Examples: - Is $(x - 1)$ a factor of $2\sqrt{2}x^3 + 5\sqrt{2}x^2 - 7\sqrt{2}$ ? - $f(1) = 2\sqrt{2}(1)^3 + 5\sqrt{2}(1)^2 - 7\sqrt{2} = 2\sqrt{2} + 5\sqrt{2} - 7\sqrt{2} = 0$ . - Result: Yes. - If $(x - 2)$ is a factor of $x^5 - 4x^3 + x + k$ , find $k$ : - $f(2) = 2^5 - 4(2^3) + 2 + k = 0$ - $32 - 32 + 2 + k = 0 \rightarrow k = -2$ . - If $f(x) = 2x^3 + ax^2 + bx + c$ has factors $(x - 1)$ , $(x + 1)$ , and $(x - 2)$ , find $a, b, c$ : - Factors imply zeroes at $1, -1, 2$ . - $f(x) = 2(x - 1)(x + 1)(x - 2) = 2(x^2 - 1)(x - 2) = 2(x^3 - 2x^2 - x + 2)$ . - $f(x) = 2x^3 - 4x^2 - 2x + 4$ . - Result: $a = -4, b = -2, c = 4$ .

Arithmetic Mean - Geometric Mean (A.M.-G.M.) Inequality

Definitions: - Arithmetic Mean (A.M.): For two positive real numbers $x, y$ , $A.M. = \frac{x + y}{2}$ . - Geometric Mean (G.M.): For two positive real numbers $x, y$ , $G.M. = \sqrt{xy}$ .
Inequality Statement: $A.M. \geq G.M.$ for positive real numbers.
Equality Condition: The equality $A.M. = G.M.$ holds if and only if $x = y$ .
Proof: - We know $(\sqrt{x} - \sqrt{y})^2 \geq 0$ - $(\sqrt{x})^2 + (\sqrt{y})^2 - 2\sqrt{x}\sqrt{y} \geq 0$ - $x + y \geq 2\sqrt{xy}$ - $\frac{x + y}{2} \geq \sqrt{xy}$ .
Standard Lower/Upper Bounds: - For all x > 0, $x + \frac{1}{x} \geq 2$ . - For all x < 0, $x + \frac{1}{x} \leq -2$ .
Optimization Examples: - Minimum value of $\frac{x^2 + 16}{x}$ when x > 0: - Let the terms be $x$ and $\frac{16}{x}$ . - $A.M. \geq G.M. \rightarrow \frac{x + \frac{16}{x}}{2} \geq \sqrt{x \cdot \frac{16}{x}}$ - $\frac{x + \frac{16}{x}}{2} \geq 4 \rightarrow x + \frac{16}{x} \geq 8$ . - Result: Minimum value is 8. - Maximum value of $\frac{3x^2 + 12}{x}$ for x < 0: - Let $x = -t$ where t > 0. - Expression becomes $\frac{3t^2 + 12}{-t} = -(3t + \frac{12}{t})$ . - For $3t$ and $\frac{12}{t}$ , $A.M. \geq G.M. \rightarrow \frac{3t + \frac{12}{t}}{2} \geq \sqrt{3t \cdot \frac{12}{t}} = \sqrt{36} = 6$ . - $3t + \frac{12}{t} \geq 12 \rightarrow -(3t + \frac{12}{t}) \leq -12$ . - Result: Maximum value is -12. - Minimum value of $(a + b)\left(\frac{1}{a} + \frac{1}{b}\right)$ for $a, b \in \mathbb{R}^+$ : - $(a + b)\left(\frac{1}{a} + \frac{1}{b}\right) = 1 + \frac{a}{b} + \frac{b}{a} + 1 = 2 + \left(\frac{a}{b} + \frac{b}{a}\right)$ . - Since $\frac{a}{b} + \frac{b}{a} \geq 2$ , the full expression $\geq 2 + 2 = 4$ . - Result: 4.

Ratio and Proportion

Ratio: Given two quantities $a, b$ of the same kind, the ratio is $a:b$ or $\frac{a}{b}$ .
Comparison of Ratios: For positive integers $a, b, c, d$ : - a:b > c:d if ad > bc. - $a:b = c:d$ if $ad = bc$ . - a:b < c:d if ad < bc.
Proportion: If two ratios are equal ( $\frac{a}{b} = \frac{c}{d}$ ), then $a, b, c, d$ are proportional ( $a:b :: c:d$ ). - $a, d$ are called the extremes. - $b, c$ are called the means.
Properties: - If $\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$ , then each ratio is equal to $\frac{a + c + e}{b + d + f}$ . - Componendo & Dividendo: If $\frac{a}{b} = \frac{c}{d}$ , then $\frac{a + b}{a - b} = \frac{c + d}{c - d}$ .
Application Problems: - Mixture Problem: 60 litres total, Milk:Water = 2:1. (Milk = 40L, Water = 20L). To make ratio 1:2, add water $x$ . - $\frac{40}{20 + x} = \frac{1}{2} \rightarrow 80 = 20 + x \rightarrow x = 60 \, L$ . - Money Distribution: Ratio A:B:C:D = 5:2:4:3. If C gets 1000 more than D: - Let shares be $5k, 2k, 4k, 3k$ . - $4k - 3k = 1000 \rightarrow k = 1000$ . - B's share = $2k = 2000$ .

Determinants

Evaluation of 3x3 Determinant: - Example 1: $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 1 \end{vmatrix}$ - $= 1(5 \cdot 1 - 8 \cdot 6) - 2(4 \cdot 1 - 7 \cdot 6) + 3(4 \cdot 8 - 7 \cdot 5)$ - $= 1(5 - 48) - 2(4 - 42) + 3(32 - 35) = -43 + 76 - 9 = 24$ . - Example 2: $\begin{vmatrix} 2 & 3 & 2 \\ 1 & 5 & 4 \\ 2 & 7 & 2 \end{vmatrix}$ - $= 2(10 - 28) - 3(2 - 8) + 2(7 - 10) = 2(-18) - 3(-6) + 2(-3) = -36 + 18 - 6 = -24$ .
Minors and Cofactors: - Minor ( $M_{ij}$ ): Determinant obtained by deleting the $i^{th}$ row and $j^{th}$ column. - Cofactor ( $C_{ij}$ or $A_{ij}$ ): Calculated as $(-1)^{i+j} M_{ij}$ . - Example: In $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{vmatrix}$ , for element $a_{23}$ (row 2, column 3): - $M_{23} = \begin{vmatrix} 1 & 2 \\ 7 & 8 \end{vmatrix} = 8 - 14 = -6$ . - $C_{23} = (-1)^{2+3} M_{23} = -1(-6) = 6$ .