NCERT Maths Class XI Summary

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Chapter 1: Sets 1.1 Introduction Concept of Set theory developed by Georg Cantor. 1.2 Sets and their Representations Definition: A well-defined collection of objects. Objects, elements, members are synonymous. Sets denoted by capital letters (A, B, C). Elements by small letters (a, b, c). $a \in A$: 'a belongs to A'. $b \notin A$: 'b does not belong to A'. Roster Form: List all elements separated by commas, enclosed in braces. Order is immaterial. Elements are not generally repeated. Example: $\{1, 2, 3\}$, $\{a, e, i, o, u\}$ Set-Builder Form: Describe a property common to all elements. Example: $V = \{x : x \text{ is a vowel in English alphabet}\}$ Special Sets: $N$: Natural numbers $Z$: Integers $Q$: Rational numbers $R$: Real numbers $Z^+$: Positive integers $Q^+$: Positive rational numbers $R^+$: Positive real numbers 1.3 The Empty Set Definition: A set containing no elements. Also called Null Set or Void Set. Denoted by $\phi$ or $\{ \}$. Example: $\{x : 1 1.4 Finite and Infinite Sets Definition: A set is finite if it is empty or consists of a definite number of elements. Otherwise, it is infinite. Number of elements in set S is $n(S)$. Example: Days of the week (finite), points on a line (infinite). Infinite sets cannot always be described in roster form (e.g., real numbers). 1.5 Equal Sets Definition: Two sets A and B are equal ($A=B$) if they have exactly the same elements. Order and repetition of elements do not change a set. 1.6 Subsets Definition: Set A is a subset of set B ($A \subset B$) if every element of A is also an element of B. $A \subset B \Leftrightarrow (a \in A \Rightarrow a \in B)$. Every set is a subset of itself ($A \subset A$). The empty set is a subset of every set ($\phi \subset A$). If $A \subset B$ and $B \subset A$, then $A=B$. Proper Subset: If $A \subset B$ and $A \neq B$, then A is a proper subset of B. B is a superset of A. Singleton Set: A set with only one element. Subsets of R: $N \subset Z \subset Q \subset R$. Irrational numbers $T = \{x : x \in R \text{ and } x \notin Q\}$. Intervals as Subsets of R: Let $a, b \in R$ and $a Open interval: $(a, b) = \{y : a Closed interval: $[a, b] = \{x : a \le x \le b\}$ Half-open/closed: $[a, b) = \{x : a \le x Length of interval: $b-a$. 1.7 Power Set Definition: The collection of all subsets of a set A is called the power set of A, denoted by $P(A)$. Every element in $P(A)$ is a set. If $n(A) = m$, then $n(P(A)) = 2^m$. 1.8 Universal Set Definition: A basic set relevant to a particular context, containing all elements under consideration, denoted by U. All other sets in that context are subsets of U. 1.9 Venn Diagrams Graphical representation of relationships between sets using rectangles (for U) and circles (for subsets). 1.10 Operations on Sets Union of Sets ($A \cup B$): The set of all elements which are either in A or in B (including those in both). $A \cup B = \{x : x \in A \text{ or } x \in B\}$ Properties: Commutative ($A \cup B = B \cup A$), Associative ($(A \cup B) \cup C = A \cup (B \cup C)$), Identity ($A \cup \phi = A$), Idempotent ($A \cup A = A$), Law of U ($U \cup A = U$). Intersection of Sets ($A \cap B$): The set of all elements which are common to both A and B. $A \cap B = \{x : x \in A \text{ and } x \in B\}$ Disjoint Sets: If $A \cap B = \phi$. Properties: Commutative ($A \cap B = B \cap A$), Associative ($(A \cap B) \cap C = A \cap (B \cap C)$), Law of $\phi$ and U ($\phi \cap A = \phi, U \cap A = A$), Idempotent ($A \cap A = A$), Distributive ($A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$). Difference of Sets ($A - B$): The set of elements which belong to A but not to B. $A - B = \{x : x \in A \text{ and } x \notin B\}$ Note: $A - B \neq B - A$. Sets $A-B$, $A \cap B$, and $B-A$ are mutually disjoint. 1.11 Complement of a Set ($A'$) Definition: Let U be the universal set and A a subset of U. The complement of A is the set of all elements of U which are not elements of A. $A' = \{x : x \in U \text{ and } x \notin A\} = U - A$. Properties: Complement laws: $A \cup A' = U$, $A \cap A' = \phi$. De Morgan's laws: $(A \cup B)' = A' \cap B'$, $(A \cap B)' = A' \cup B'$. Law of double complementation: $(A')' = A$. Laws of empty set and universal set: $\phi' = U$, $U' = \phi$. 1.12 Practical Problems on Union and Intersection of Two Sets For finite sets A and B: If $A \cap B = \phi$, then $n(A \cup B) = n(A) + n(B)$. $n(A \cup B) = n(A) + n(B) - n(A \cap B)$. $n(A - B) = n(A) - n(A \cap B)$. For finite sets A, B, and C: $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$. Chapter 2: Relations and Functions 2.1 Introduction Relations involve pairs of objects in a certain order. Functions are special relations. 2.2 Cartesian Product of Sets Definition: For non-empty sets P and Q, the Cartesian product $P \times Q$ is the set of all ordered pairs $(p, q)$ where $p \in P$ and $q \in Q$. $P \times Q = \{(p, q) : p \in P, q \in Q\}$. If $P$ or $Q$ is empty, then $P \times Q = \phi$. Two ordered pairs $(a, b)$ and $(x, y)$ are equal iff $a=x$ and $b=y$. If $n(A) = p$ and $n(B) = q$, then $n(A \times B) = pq$. If A or B is infinite, then $A \times B$ is infinite. $A \times A \times A = \{(a, b, c) : a, b, c \in A\}$ is an ordered triplet. Generally, $A \times B \neq B \times A$. $R \times R = \{(x, y) : x, y \in R\}$ represents points in 2D plane. $R \times R \times R = \{(x, y, z) : x, y, z \in R\}$ represents points in 3D space. 2.3 Relations Definition: A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product $A \times B$. The subset is derived by describing a relationship between the first and second elements of ordered pairs. The second element is the image of the first. Domain of R: Set of all first elements of the ordered pairs in R. Range of R: Set of all second elements of the ordered pairs in R. Codomain of R: The entire set B. (Range $\subset$ Codomain). Relations can be represented by Roster form, Set-builder form, or Arrow diagrams. If $n(A) = p$ and $n(B) = q$, the total number of relations from A to B is $2^{pq}$. 2.4 Functions Definition: A relation $f$ from set A to set B is a function if every element of set A has one and only one image in set B. In other words, domain of $f$ is A, and no two distinct ordered pairs in $f$ have the same first element. If $(a, b) \in f$, then $f(a) = b$. $b$ is the image of $a$, $a$ is the pre-image of $b$. Notation: $f: A \to B$. Real Valued Function: A function whose range is R or a subset of R. Real Function: A real valued function whose domain is R or a subset of R. Graphs of some standard functions: Identity Function: $f(x) = x$. Domain = R, Range = R. (Straight line through origin) Constant Function: $f(x) = c$. Domain = R, Range = $\{c\}$. (Line parallel to x-axis) Polynomial Function: $f(x) = a_0 + a_1x + \dots + a_nx^n$. Domain = R. Rational Function: $f(x) = \frac{g(x)}{h(x)}$, where $g(x), h(x)$ are polynomials and $h(x) \neq 0$. Modulus Function: $f(x) = |x| = \begin{cases} x & x \ge 0 \\ -x & x Signum Function: $f(x) = \begin{cases} 1 & x > 0 \\ 0 & x = 0 \\ -1 & x Greatest Integer Function: $f(x) = [x]$ (greatest integer less than or equal to x). Domain = R, Range = Z. Algebra of Real Functions: Let $f, g: X \to R$. $(f+g)(x) = f(x) + g(x)$ $(f-g)(x) = f(x) - g(x)$ $(\alpha f)(x) = \alpha f(x)$ for scalar $\alpha$. $(fg)(x) = f(x)g(x)$ $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, provided $g(x) \neq 0$. Chapter 3: Trigonometric Functions 3.1 Introduction Trigonometry deals with measuring sides of a triangle. Generalisation of trigonometric ratios to functions. 3.2 Angles Definition: Measure of rotation of a ray from initial side to terminal side. Positive angle: Anticlockwise rotation. Negative angle: Clockwise rotation. Degree Measure: $1^\circ = 60'$, $1' = 60''$. Radian Measure: Angle subtended at the centre by an arc of length 1 unit in a unit circle. Relation: $\pi \text{ radian} = 180^\circ$. Radian measure $= \frac{\pi}{180} \times \text{Degree measure}$. Degree measure $= \frac{180}{\pi} \times \text{Radian measure}$. Arc length $l = r\theta$, where $\theta$ is in radians. 3.3 Trigonometric Functions Unit circle definition: For point $P(a, b)$ on unit circle with angle $x$ (radians), $\cos x = a$, $\sin x = b$. $\sin^2 x + \cos^2 x = 1$. Other functions: $\tan x = \frac{\sin x}{\cos x}$, $\csc x = \frac{1}{\sin x}$, $\sec x = \frac{1}{\cos x}$, $\cot x = \frac{\cos x}{\sin x}$. $1 + \tan^2 x = \sec^2 x$, $1 + \cot^2 x = \csc^2 x$. Signs of functions in quadrants (All Sin Tan Cos rule): Quadrant I (0 to $\pi/2$): All positive. Quadrant II ($\pi/2$ to $\pi$): $\sin x, \csc x$ positive. Quadrant III ($\pi$ to $3\pi/2$): $\tan x, \cot x$ positive. Quadrant IV ($3\pi/2$ to $2\pi$): $\cos x, \sec x$ positive. $\sin(-x) = -\sin x$, $\cos(-x) = \cos x$. Domain and Range: $\sin x, \cos x$: Domain R, Range $[-1, 1]$. $\tan x$: Domain $R - \{(2n+1)\frac{\pi}{2} : n \in Z\}$, Range R. $\cot x$: Domain $R - \{n\pi : n \in Z\}$, Range R. $\sec x$: Domain $R - \{(2n+1)\frac{\pi}{2} : n \in Z\}$, Range $(-\infty, -1] \cup [1, \infty)$. $\csc x$: Domain $R - \{n\pi : n \in Z\}$, Range $(-\infty, -1] \cup [1, \infty)$. Trigonometric functions are periodic: $\sin(2n\pi + x) = \sin x$, $\cos(2n\pi + x) = \cos x$. $\tan(\pi + x) = \tan x$, $\cot(\pi + x) = \cot x$. 3.4 Trigonometric Functions of Sum and Difference of Two Angles $\cos(x+y) = \cos x \cos y - \sin x \sin y$ $\cos(x-y) = \cos x \cos y + \sin x \sin y$ $\sin(x+y) = \sin x \cos y + \cos x \sin y$ $\sin(x-y) = \sin x \cos y - \cos x \sin y$ $\tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$ $\tan(x-y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$ $\cot(x+y) = \frac{\cot x \cot y - 1}{\cot y + \cot x}$ $\cot(x-y) = \frac{\cot x \cot y + 1}{\cot y - \cot x}$ Double Angle Formulas: $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x = \frac{1-\tan^2 x}{1+\tan^2 x}$ $\sin 2x = 2\sin x \cos x = \frac{2\tan x}{1+\tan^2 x}$ $\tan 2x = \frac{2\tan x}{1-\tan^2 x}$ Triple Angle Formulas: $\sin 3x = 3\sin x - 4\sin^3 x$ $\cos 3x = 4\cos^3 x - 3\cos x$ $\tan 3x = \frac{3\tan x - \tan^3 x}{1 - 3\tan^2 x}$ Sum-to-Product Formulas: $\cos x + \cos y = 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$ $\cos x - \cos y = -2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$ $\sin x + \sin y = 2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$ $\sin x - \sin y = 2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$ Product-to-Sum Formulas: $2\cos x \cos y = \cos(x+y) + \cos(x-y)$ $-2\sin x \sin y = \cos(x+y) - \cos(x-y)$ $2\sin x \cos y = \sin(x+y) + \sin(x-y)$ $2\cos x \sin y = \sin(x+y) - \sin(x-y)$ 3.5 Trigonometric Equations Principal Solution: Solutions for $0 \le x General Solution: Expression involving integer 'n' that gives all solutions. $\sin x = 0 \Leftrightarrow x = n\pi, n \in Z$. $\cos x = 0 \Leftrightarrow x = (2n+1)\frac{\pi}{2}, n \in Z$. $\sin x = \sin y \Leftrightarrow x = n\pi + (-1)^n y, n \in Z$. $\cos x = \cos y \Leftrightarrow x = 2n\pi \pm y, n \in Z$. $\tan x = \tan y \Leftrightarrow x = n\pi + y, n \in Z$. Chapter 4: Principle of Mathematical Induction 4.1 Introduction Deductive Reasoning: From general to specific. (e.g., Socrates is mortal). Inductive Reasoning: From specific observations to general conjecture. Mathematical Induction is a form of complete induction for statements about natural numbers. 4.3 The Principle of Mathematical Induction Let P(n) be a statement involving the natural number n. To prove P(n) is true for all natural numbers n: Base Step: Show P(1) is true. (Or P($n_0$) for $n \ge n_0$). Inductive Step: Assume P(k) is true for some positive integer k (Inductive Hypothesis). Prove that P(k+1) is true based on the assumption that P(k) is true. If both steps are satisfied, then P(n) is true for all natural numbers n. Chapter 5: Complex Numbers and Quadratic Equations 5.1 Introduction Need to extend real numbers to solve $x^2 + 1 = 0$. 5.2 Complex Numbers $i = \sqrt{-1}$, so $i^2 = -1$. Definition: A number of the form $a+ib$, where $a, b \in R$. $a = \text{Re}(z)$ (real part), $b = \text{Im}(z)$ (imaginary part). Equality: $a+ib = c+id \Leftrightarrow a=c \text{ and } b=d$. 5.3 Algebra of Complex Numbers Addition: $(a+ib) + (c+id) = (a+c) + i(b+d)$. Properties: Closure, Commutative, Associative, Additive Identity ($0+i0$), Additive Inverse ($-z = -a-ib$). Difference: $(a+ib) - (c+id) = (a-c) + i(b-d)$. Multiplication: $(a+ib)(c+id) = (ac-bd) + i(ad+bc)$. Properties: Closure, Commutative, Associative, Multiplicative Identity ($1+i0$), Multiplicative Inverse ($z^{-1} = \frac{a}{a^2+b^2} - i\frac{b}{a^2+b^2}$ for $z \neq 0$), Distributive. Powers of $i$: $i^1=i, i^2=-1, i^3=-i, i^4=1$. $i^{4k}=1, i^{4k+1}=i, i^{4k+2}=-1, i^{4k+3}=-i$. Square Roots of Negative Real Numbers: $\sqrt{-a} = i\sqrt{a}$ for $a>0$. $\sqrt{a}\sqrt{b} = \sqrt{ab}$ is NOT true if both $a,b$ are negative. 5.4 The Modulus and the Conjugate of a Complex Number Conjugate ($\bar{z}$): If $z=a+ib$, then $\bar{z}=a-ib$. Properties: $\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}$, $\overline{z_1 z_2} = \bar{z_1} \bar{z_2}$, $\overline{(\frac{z_1}{z_2})} = \frac{\bar{z_1}}{\bar{z_2}}$. Modulus ($|z|$): If $z=a+ib$, then $|z|=\sqrt{a^2+b^2}$ (non-negative real number). Properties: $|z_1 z_2| = |z_1||z_2|$, $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$. $z \bar{z} = |z|^2$. Multiplicative inverse: $z^{-1} = \frac{\bar{z}}{|z|^2}$. 5.5 Argand Plane and Polar Representation A complex number $z=x+iy$ can be represented as a point $(x, y)$ in the Argand plane (complex plane). $|z|$ is the distance from the origin to $(x, y)$. $\bar{z}$ is the mirror image of $z$ on the real axis. Polar Form: $z = r(\cos \theta + i \sin \theta)$, where $r = |z| = \sqrt{x^2+y^2}$ and $\theta$ is the argument of $z$. $x = r \cos \theta$, $y = r \sin \theta$. Principal argument: $-\pi 5.6 Quadratic Equations For $ax^2+bx+c=0$ with $D = b^2-4ac Fundamental Theorem of Algebra: A polynomial of degree $n$ has $n$ roots (counting multiplicity). Chapter 6: Linear Inequalities 6.1 Introduction Inequalities describe relations where quantities are not equal. 6.2 Inequalities Definition: Two real numbers or algebraic expressions related by $ , \le, \ge$. Strict inequalities: $ $. Slack inequalities: $\le, \ge$. Linear inequalities in one variable: $ax+b Linear inequalities in two variables: $ax+by 6.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation Solution of an inequality: A value of the variable that makes the inequality a true statement. Solution Set: The set of all solutions. Rules for solving inequalities: Adding/subtracting same number on both sides does not change inequality sign. Multiplying/dividing by a positive number on both sides does not change inequality sign. Multiplying/dividing by a negative number on both sides reverses inequality sign. Graphical representation on number line: Open circle for strict inequality ($ $). Closed circle for slack inequality ($\le, \ge$). 6.4 Graphical Solution of Linear Inequalities in Two Variables A line $ax+by=c$ divides the Cartesian plane into two half-planes. Points satisfying $ax+by>c$ (or $ Points satisfying $ax+by \ge c$ (or $\le c$) include points on the line. Procedure: Draw the line $ax+by=c$. Choose a test point (e.g., $(0,0)$ if not on the line). Substitute the test point into the inequality. If it satisfies, shade the half-plane containing the point. Otherwise, shade the other half-plane. Use a dashed line for strict inequalities and a solid line for slack inequalities. Solution Region: The region containing all solutions, usually shaded. 6.5 Solution of System of Linear Inequalities in Two Variables Graph each inequality on the same coordinate plane. The common shaded region (intersection of all solution regions) is the solution to the system. If $x \ge 0, y \ge 0$, the solution region is restricted to the first quadrant. Chapter 7: Permutations and Combinations 7.1 Introduction Counting techniques for arranging and selecting objects. 7.2 Fundamental Principle of Counting If an event can occur in $m$ different ways, and following which another event can occur in $n$ different ways, then the total number of occurrences of the events in the given order is $m \times n$. Generalises to any finite number of events. 7.3 Permutations Definition: An arrangement in a definite order of a number of objects taken some or all at a time. Order matters. Factorial Notation ($n!$): Product of first $n$ natural numbers. $n! = 1 \times 2 \times \dots \times n$. $0! = 1$. $n! = n(n-1)!$. Permutations of n different objects taken r at a time ($^nP_r$): Without repetition: $^nP_r = \frac{n!}{(n-r)!}$, where $0 \le r \le n$. With repetition allowed: $n^r$. Permutations of n objects when not all are distinct: If $p$ objects are of one kind and rest are different: $\frac{n!}{p!}$. If $p_1$ objects are of first kind, $p_2$ of second kind, ..., $p_k$ of k-th kind: $\frac{n!}{p_1! p_2! \dots p_k!}$. 7.4 Combinations Definition: A selection of objects where the order does not matter. Combinations of n different objects taken r at a time ($^nC_r$): $^nC_r = \frac{n!}{r!(n-r)!}$, where $0 \le r \le n$. Properties: $^nC_r = ^nC_{n-r}$. $^nC_0 = 1$. $^nC_n = 1$. $^nC_a = ^nC_b \Rightarrow a=b \text{ or } a+b=n$. $^nC_r + ^nC_{r-1} = ^{n+1}C_r$. Chapter 8: Binomial Theorem 8.1 Introduction For expanding $(a+b)^n$ for positive integral indices. 8.2 Binomial Theorem for Positive Integral Indices For any positive integer $n$: $(a+b)^n = ^nC_0 a^n + ^nC_1 a^{n-1}b + ^nC_2 a^{n-2}b^2 + \dots + ^nC_r a^{n-r}b^r + \dots + ^nC_n b^n$. This can be written as $\sum_{k=0}^{n} ^nC_k a^{n-k}b^k$. Observations: Number of terms is $n+1$. Powers of $a$ decrease, powers of $b$ increase. Sum of indices of $a$ and $b$ in each term is $n$. Coefficients $^nC_k$ are called binomial coefficients. Pascal's Triangle: Triangular array of binomial coefficients. Each number is the sum of the two numbers directly above it. Special Cases: $(x-y)^n = ^nC_0 x^n - ^nC_1 x^{n-1}y + ^nC_2 x^{n-2}y^2 - \dots + (-1)^n ^nC_n y^n$. $(1+x)^n = ^nC_0 + ^nC_1 x + ^nC_2 x^2 + \dots + ^nC_n x^n$. $(1-x)^n = ^nC_0 - ^nC_1 x + ^nC_2 x^2 - \dots + (-1)^n ^nC_n x^n$. Sum of binomial coefficients: $^nC_0 + ^nC_1 + \dots + ^nC_n = 2^n$. 8.3 General and Middle Terms General Term ($T_{r+1}$): In the expansion of $(a+b)^n$, the $(r+1)$-th term is given by $T_{r+1} = ^nC_r a^{n-r}b^r$. Middle Term(s): If $n$ is even, there is one middle term: $T_{n/2 + 1}$. If $n$ is odd, there are two middle terms: $T_{(n+1)/2}$ and $T_{(n+3)/2}$. Term Independent of x (Constant Term): Term where the power of x is 0. Chapter 9: Sequences and Series 9.1 Introduction Sequence: An ordered collection of objects/numbers. Progression: Sequence following specific patterns. 9.2 Sequences Definition: A sequence is an arrangement of numbers in a definite order according to some rule. Terms: $a_1, a_2, \dots, a_n, \dots$. $a_n$ is the $n$-th term or general term. Finite Sequence: Contains a finite number of terms. Infinite Sequence: Not finite. Can be defined by an algebraic formula for $a_n$ or by recurrence relation (e.g., Fibonacci sequence). 9.3 Series Definition: The expression $a_1 + a_2 + \dots + a_n + \dots$ associated with a sequence. Finite or infinite series based on the sequence. Sigma notation: $\sum_{k=1}^{n} a_k$. 9.4 Arithmetic Progression (A.P.) Definition: A sequence where the difference between consecutive terms is constant (common difference, $d$). Standard form: $a, a+d, a+2d, \dots$. $n$-th term: $a_n = a + (n-1)d$. Sum of first $n$ terms ($S_n$): $S_n = \frac{n}{2}[2a + (n-1)d]$ or $S_n = \frac{n}{2}[a+l]$, where $l=a_n$. Properties: Adding/subtracting/multiplying/dividing a constant to each term results in an A.P. Arithmetic Mean (A.M.): For $a, b$, the A.M. is $A = \frac{a+b}{2}$. Inserting $n$ numbers $A_1, \dots, A_n$ between $a$ and $b$ to form an A.P.: $d = \frac{b-a}{n+1}$. 9.5 Geometric Progression (G.P.) Definition: A sequence where the ratio of any term to its preceding term is constant (common ratio, $r$). All terms are non-zero. Standard form: $a, ar, ar^2, \dots$. $n$-th term: $a_n = ar^{n-1}$. Sum of first $n$ terms ($S_n$): If $r=1$, $S_n = na$. If $r \neq 1$, $S_n = \frac{a(r^n-1)}{r-1}$ or $S_n = \frac{a(1-r^n)}{1-r}$. Geometric Mean (G.M.): For positive $a, b$, the G.M. is $G = \sqrt{ab}$. Inserting $n$ numbers $G_1, \dots, G_n$ between $a$ and $b$ to form a G.P.: $r = \left(\frac{b}{a}\right)^{\frac{1}{n+1}}$. 9.6 Relationship Between A.M. and G.M. For two positive numbers $a, b$: $A = \frac{a+b}{2}$, $G = \sqrt{ab}$. $A \ge G$. 9.7 Sum to n Terms of Special Series Sum of first $n$ natural numbers: $\sum_{k=1}^{n} k = \frac{n(n+1)}{2}$. Sum of squares of first $n$ natural numbers: $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$. Sum of cubes of first $n$ natural numbers: $\sum_{k=1}^{n} k^3 = \left[\frac{n(n+1)}{2}\right]^2$. Chapter 10: Straight Lines 10.1 Introduction Recall of 2D coordinate geometry basics (distance, section formula, area of triangle). 10.2 Slope of a Line Inclination ($\theta$): Angle line makes with positive x-axis (anticlockwise, $0^\circ \le \theta Slope ($m$): $m = \tan \theta$, $\theta \neq 90^\circ$. Slope of x-axis (horizontal line) is 0. Slope of y-axis (vertical line) is undefined. Slope through $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2-y_1}{x_2-x_1}$, $x_1 \neq x_2$. Parallel Lines: $l_1 || l_2 \Leftrightarrow m_1 = m_2$. Perpendicular Lines: $l_1 \perp l_2 \Leftrightarrow m_1 m_2 = -1$. Angle between two lines: If $\theta$ is acute angle between lines with slopes $m_1, m_2$: $\tan \theta = \left|\frac{m_2-m_1}{1+m_1m_2}\right|$, $1+m_1m_2 \neq 0$. Collinearity of three points: A, B, C are collinear iff slope of AB = slope of BC. 10.3 Various Forms of the Equation of a Line Horizontal Line: $y=a$ (distance $a$ from x-axis). Vertical Line: $x=b$ (distance $b$ from y-axis). Point-Slope Form: $y-y_0 = m(x-x_0)$ (line with slope $m$ through $(x_0, y_0)$). Two-Point Form: $\frac{y-y_1}{y_2-y_1} = \frac{x-x_1}{x_2-x_1}$ or $y-y_1 = \left(\frac{y_2-y_1}{x_2-x_1}\right)(x-x_1)$ (line through $(x_1, y_1)$ and $(x_2, y_2)$). Slope-Intercept Form: $y = mx+c$ (slope $m$, y-intercept $c$). $y = m(x-d)$ (slope $m$, x-intercept $d$). Intercept Form: $\frac{x}{a} + \frac{y}{b} = 1$ (x-intercept $a$, y-intercept $b$). Normal Form: $x \cos \omega + y \sin \omega = p$ (perpendicular distance $p$ from origin, angle $\omega$ normal makes with x-axis). 10.4 General Equation of a Line General linear equation: $Ax+By+C=0$ (A, B not simultaneously zero). Conversion to other forms: Slope-intercept: $y = (-\frac{A}{B})x + (-\frac{C}{B})$ (if $B \neq 0$). Slope $m = -A/B$, y-intercept $c = -C/B$. Intercept form: $\frac{x}{(-C/A)} + \frac{y}{(-C/B)} = 1$ (if $C \neq 0, A \neq 0, B \neq 0$). x-intercept $a = -C/A$, y-intercept $b = -C/B$. Normal form: $x \cos \omega + y \sin \omega = p$, where $\cos \omega = \frac{A}{\pm\sqrt{A^2+B^2}}$, $\sin \omega = \frac{B}{\pm\sqrt{A^2+B^2}}$, $p = \frac{-C}{\pm\sqrt{A^2+B^2}}$ (sign chosen to make $p>0$). 10.5 Distance of a Point From a Line Distance $d$ of point $(x_1, y_1)$ from line $Ax+By+C=0$: $d = \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$. Distance between two parallel lines $Ax+By+C_1=0$ and $Ax+By+C_2=0$: $d = \frac{|C_1-C_2|}{\sqrt{A^2+B^2}}$. Chapter 11: Conic Sections 11.1 Introduction Conic sections (conics) are curves formed by intersection of a plane with a double-napped right circular cone. 11.2 Sections of a Cone Let $\alpha$ be angle of generator with axis, $\beta$ be angle of plane with axis. Circle: $\beta = 90^\circ$. Ellipse: $\alpha Parabola: $\beta = \alpha$. Hyperbola: $0 \le \beta Degenerated Conics (plane through vertex): Point: $\alpha Straight line: $\beta = \alpha$. Pair of intersecting straight lines: $0 \le \beta 11.3 Circle Definition: Set of all points equidistant from a fixed point (centre). Equation of circle with centre $(h, k)$ and radius $r$: $(x-h)^2 + (y-k)^2 = r^2$. 11.4 Parabola Definition: Set of all points equidistant from a fixed line (directrix) and a fixed point (focus) not on the line. Axis: Line through focus and perpendicular to directrix. Vertex: Point of intersection of parabola with its axis. Standard Equations (vertex at origin): $y^2 = 4ax$ (focus $(a, 0)$, directrix $x=-a$, opens right) $y^2 = -4ax$ (focus $(-a, 0)$, directrix $x=a$, opens left) $x^2 = 4ay$ (focus $(0, a)$, directrix $y=-a$, opens up) $x^2 = -4ay$ (focus $(0, -a)$, directrix $y=a$, opens down) Latus Rectum: Line segment through focus, perpendicular to axis, with endpoints on parabola. Length $= 4a$. 11.5 Ellipse Definition: Set of all points where sum of distances from two fixed points (foci) is constant ($2a$). Centre: Midpoint of segment joining foci. Major Axis: Line segment through foci, length $2a$. Endpoints are vertices. Minor Axis: Line segment through centre, perpendicular to major axis, length $2b$. Relation: $c^2 = a^2 - b^2$, where $2c$ is distance between foci. Eccentricity ($e$): $e = \frac{c}{a}$, ($0 Standard Equations (centre at origin): $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (foci on x-axis, major axis along x-axis) $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$ (foci on y-axis, major axis along y-axis) Latus Rectum: Line segment through focus, perpendicular to major axis, with endpoints on ellipse. Length $= \frac{2b^2}{a}$. 11.6 Hyperbola Definition: Set of all points where difference of distances from two fixed points (foci) is constant ($2a$). Centre: Midpoint of segment joining foci. Transverse Axis: Line through foci, length $2a$. Endpoints are vertices. Conjugate Axis: Line through centre, perpendicular to transverse axis, length $2b$. Relation: $c^2 = a^2 + b^2$, where $2c$ is distance between foci. Eccentricity ($e$): $e = \frac{c}{a}$, ($e > 1$). Standard Equations (centre at origin): $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (foci on x-axis, transverse axis along x-axis) $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (foci on y-axis, transverse axis along y-axis) Latus Rectum: Line segment through focus, perpendicular to transverse axis, with endpoints on hyperbola. Length $= \frac{2b^2}{a}$. Equilateral Hyperbola: Where $a=b$. Chapter 12: Introduction to Three Dimensional Geometry 12.1 Introduction Need three coordinates $(x, y, z)$ to locate a point in space. 12.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space Three mutually perpendicular lines: x-axis, y-axis, z-axis. Three coordinate planes: XY-plane, YZ-plane, ZX-plane. Origin (O): Intersection of axes and planes. Coordinates $(0,0,0)$. Octants: Space divided into 8 parts by coordinate planes. Signs of $(x,y,z)$ define octant. Point on x-axis: $(x,0,0)$. Point in YZ-plane: $(0,y,z)$. 12.3 Coordinates of a Point in Space A point P in space is represented by an ordered triplet $(x, y, z)$. $x, y, z$ are perpendicular distances from YZ, ZX, XY planes respectively. 12.4 Distance between Two Points Distance between $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$: $PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$. Distance from origin to $Q(x,y,z)$: $OQ = \sqrt{x^2+y^2+z^2}$. Collinearity: Points A, B, C are collinear if $AB+BC=AC$ (or any permutation). 12.5 Section Formula Coordinates of point $R(x,y,z)$ dividing line segment joining $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in ratio $m:n$: Internally: $\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}, \frac{mz_2+nz_1}{m+n}\right)$. Externally: $\left(\frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n}, \frac{mz_2-nz_1}{m-n}\right)$. Mid-point: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$. Centroid of triangle: For vertices $(x_1,y_1,z_1), (x_2,y_2,z_2), (x_3,y_3,z_3)$: $\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3}\right)$. Chapter 13: Limits and Derivatives 13.1 Introduction Calculus studies change in function values. 13.2 Intuitive Idea of Derivatives Derivative represents instantaneous rate of change (e.g., instantaneous velocity). Geometrically, it is the slope of the tangent to the curve. 13.3 Limits Limit of $f(x)$ as $x \to a$: Value $f(x)$ approaches as $x$ gets arbitrarily close to $a$ (but not equal to $a$). Denoted $\lim_{x \to a} f(x)$. Left Hand Limit ($\lim_{x \to a^-} f(x)$): Value approached as $x$ tends to $a$ from values less than $a$. Right Hand Limit ($\lim_{x \to a^+} f(x)$): Value approached as $x$ tends to $a$ from values greater than $a$. $\lim_{x \to a} f(x)$ exists iff $\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x)$. The limit $\lim_{x \to a} f(x)$ may exist even if $f(a)$ is undefined or $f(a) \neq \lim_{x \to a} f(x)$. Algebra of Limits: If $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist: $\lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$. $\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$. $\lim_{x \to a} [\lambda f(x)] = \lambda \lim_{x \to a} f(x)$. $\lim_{x \to a} \left[\frac{f(x)}{g(x)}\right] = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$, if $\lim_{x \to a} g(x) \neq 0$. Limits of Polynomials: If $f(x)$ is a polynomial, $\lim_{x \to a} f(x) = f(a)$. Limits of Rational Functions: If $f(x) = \frac{g(x)}{h(x)}$ and $h(a) \neq 0$, then $\lim_{x \to a} f(x) = \frac{g(a)}{h(a)}$. If $\lim_{x \to a} f(x) = \frac{0}{0}$ form, factorize and cancel common factors. Important Limit: $\lim_{x \to a} \frac{x^n - a^n}{x-a} = na^{n-1}$ (for any rational $n$ and positive $a$). 13.4 Limits of Trigonometric Functions Sandwich Theorem: If $f(x) \le g(x) \le h(x)$ and $\lim_{x \to a} f(x) = L = \lim_{x \to a} h(x)$, then $\lim_{x \to a} g(x) = L$. Important Limits: $\lim_{x \to 0} \frac{\sin x}{x} = 1$. $\lim_{x \to 0} \frac{1-\cos x}{x} = 0$. $\lim_{x \to 0} \frac{\tan x}{x} = 1$. 13.5 Derivatives Definition (at a point $a$): $f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$, if the limit exists. Definition (as a function): $f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$, if the limit exists. This is the first principle of derivative. Notations: $f'(x)$, $\frac{df}{dx}$, $\frac{dy}{dx}$, $D(f(x))$. Geometrically, $f'(a)$ is the slope of the tangent to $y=f(x)$ at $(a, f(a))$. Algebra of Derivatives: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$. Product Rule: $\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)$. (Leibnitz rule) Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$, if $g(x) \neq 0$. Standard Derivatives: $\frac{d}{dx}(c) = 0$ (constant). $\frac{d}{dx}(x^n) = nx^{n-1}$. $\frac{d}{dx}(\sin x) = \cos x$. $\frac{d}{dx}(\cos x) = -\sin x$. $\frac{d}{dx}(\tan x) = \sec^2 x$. $\frac{d}{dx}(\cot x) = -\csc^2 x$. $\frac{d}{dx}(\sec x) = \sec x \tan x$. $\frac{d}{dx}(\csc x) = -\csc x \cot x$. Chapter 14: Mathematical Reasoning 14.1 Introduction Basic ideas of mathematical reasoning (deductive). 14.2 Statements Definition: A sentence is a mathematically acceptable statement if it is either true or false, but not both. Statements are denoted by $p, q, r, \dots$. Sentences that are subjective, exclamations, orders, questions, or involve variable time/place/pronouns are not statements. 14.3 New Statements from Old Negation of a Statement ($\sim p$): The denial of a statement $p$. Read as 'not p'. Phrases like "It is not the case that" or "It is false that" can be used. Compound Statement: A statement made up of two or more statements (component statements) using connecting words. 14.4 Special Words/Phrases (Connectives and Quantifiers) The word "And": Compound statement "p and q" is true iff both $p$ and $q$ are true. "p and q" is false if any component is false. The word "Or": Exclusive "Or": One or the other, but not both. Inclusive "Or": One or the other, or both. (Common in mathematics) Compound statement "p or q" is true if at least one component is true. "p or q" is false iff both components are false. Quantifiers: "There exists": Means "at least one". "For every" (or "For all"): Means "for all members of a set". 14.5 Implications "If-then" statements ("If p then q"): Denoted $p \Rightarrow q$ (p implies q). Ways to interpret "If p then q": p implies q. p is a sufficient condition for q. p only if q. q is a necessary condition for p. $\sim q$ implies $\sim p$. Contrapositive: Of "If p then q" is "If $\sim q$ then $\sim p$". (Convey same meaning). Converse: Of "If p then q" is "If q then p". "If and only if" ($p \Leftrightarrow q$): Means "If p then q" AND "If q then p". "p is necessary and sufficient condition for q". 14.6 Validating Statements Rules for checking if a statement is true: "p and q" is true if $p$ is true AND $q$ is true. "p or q" is true if $p$ is true OR $q$ is true (or both). "If p then q": Direct method: Assume $p$ is true, prove $q$ is true. Contrapositive method: Assume $\sim q$ is true, prove $\sim p$ is true. "p if and only if q": Prove "If p then q" AND "If q then p". Method of Contradiction: To prove $p$ is true, assume $\sim p$ is true and derive a contradiction. Counter Example: To show a statement is false, provide one example where the statement does not hold. Chapter 15: Statistics 15.1 Introduction Statistics deals with data analysis and interpretation. Measures of central tendency (mean, median, mode) give central value. Measures of dispersion (variability) describe how data is spread out. 15.2 Measures of Dispersion Types: Range, Quartile deviation, Mean deviation, Standard deviation. 15.3 Range Definition: Maximum value - Minimum value. Gives a rough idea of spread, but not about dispersion from central tendency. 15.4 Mean Deviation Definition: Mean of the absolute values of the deviations of observations from a central value 'a'. Denoted M.D.(a). M.D.(a) = $\frac{\sum |x_i - a|}{n}$. Commonly used: Mean deviation about mean (M.D.($\bar{x}$)) and Median (M.D.(M)). For ungrouped data: Calculate $\bar{x}$ or M. Find $|x_i - \bar{x}|$ or $|x_i - M|$. Calculate the mean of these absolute deviations. For discrete frequency distribution: M.D.($\bar{x}$) = $\frac{\sum f_i|x_i - \bar{x}|}{N}$, where $\bar{x} = \frac{\sum f_ix_i}{N}$ and $N = \sum f_i$. M.D.(M) = $\frac{\sum f_i|x_i - M|}{N}$. For continuous frequency distribution: Find mid-points ($x_i$) for each class. Proceed as for discrete frequency distribution. For median, find median class and use formula: Median $= L + \frac{\frac{N}{2}-C}{f} \times h$. Limitations: Not suitable for highly variable data, not amenable to further algebraic treatment. 15.5 Variance and Standard Deviation Overcomes limitations of mean deviation by squaring deviations. Variance ($\sigma^2$): Mean of the squares of the deviations from the mean. For ungrouped data: $\sigma^2 = \frac{\sum (x_i - \bar{x})^2}{n}$. For discrete frequency distribution: $\sigma^2 = \frac{\sum f_i(x_i - \bar{x})^2}{N}$. For continuous frequency distribution: Use mid-points $x_i$ and proceed as discrete. Alternative formula: $\sigma^2 = \frac{\sum f_ix_i^2}{N} - \left(\frac{\sum f_ix_i}{N}\right)^2$. Standard Deviation ($\sigma$): Positive square root of variance. $\sigma = \sqrt{\sigma^2}$. Shortcut Method (Step-deviation method) for Variance and S.D.: Let $y_i = \frac{x_i - A}{h}$, where A is assumed mean, $h$ is class width. $\bar{x} = A + h\bar{y}$, where $\bar{y} = \frac{\sum f_iy_i}{N}$. $\sigma^2 = h^2 \left[\frac{\sum f_iy_i^2}{N} - \left(\frac{\sum f_iy_i}{N}\right)^2\right] = h^2 (\sigma_y^2)$. $\sigma = h \sigma_y$. Effect of changing observations: If each $x_i$ is increased/decreased by 'a', variance remains unchanged. If each $x_i$ is multiplied by 'k', variance becomes $k^2$ times original variance. 15.6 Analysis of Frequency Distributions Coefficient of Variation (C.V.): A relative measure of dispersion, independent of units. C.V. $= \frac{\sigma}{\bar{x}} \times 100$, $\bar{x} \neq 0$. Comparison of two series: The series with greater C.V. is more variable (or less consistent). The series with lesser C.V. is more consistent (or less variable). If means are equal, compare $\sigma$. Larger $\sigma$ means more variable. Chapter 16: Probability 16.1 Introduction Classical Theory: Probability = $\frac{\text{Number of favorable outcomes}}{\text{Total number of equally likely outcomes}}$. Statistical Approach: Based on observations/data. Axiomatic Approach: Based on axioms. 16.2 Random Experiments Definition: An experiment with more than one possible outcome, where the outcome cannot be predicted in advance. Outcome: A possible result of a random experiment. Sample Space (S): The set of all possible outcomes of a random experiment. Each element is a sample point. 16.3 Event Definition: Any subset E of a sample space S. Occurrence of an event: Event E occurs if the outcome $\omega$ of the experiment is such that $\omega \in E$. Types of events: Impossible event ($\phi$): Event that cannot occur. Sure event (S): Event that is certain to occur. Simple (Elementary) event: An event with only one sample point. Compound event: An event with more than one sample point. Algebra of events: Complementary Event ($A'$): Event 'not A'. $A' = S-A$. Event 'A or B' ($A \cup B$): Either A or B or both occur. Event 'A and B' ($A \cap B$): Both A and B occur. Event 'A but not B' ($A-B$): A occurs but B does not. $A-B = A \cap B'$. Mutually Exclusive Events: Two events A and B are mutually exclusive if they cannot occur simultaneously ($A \cap B = \phi$). Exhaustive Events: Events $E_1, E_2, \dots, E_n$ are exhaustive if at least one of them necessarily occurs ($E_1 \cup E_2 \cup \dots \cup E_n = S$). Mutually Exclusive and Exhaustive Events: If $E_i \cap E_j = \phi$ for $i \neq j$ and $\bigcup_{i=1}^{n} E_i = S$. 16.4 Axiomatic Approach to Probability Probability $P$ is a real-valued function with domain as power set of $S$ and range $[0,1]$. Axioms: For any event E, $P(E) \ge 0$. $P(S) = 1$. If E and F are mutually exclusive events, then $P(E \cup F) = P(E) + P(F)$. From axioms: $P(\phi) = 0$. If $S = \{\omega_1, \omega_2, \dots, \omega_n\}$: $0 \le P(\omega_i) \le 1$. $\sum_{i=1}^{n} P(\omega_i) = 1$. For any event A, $P(A) = \sum_{\omega_i \in A} P(\omega_i)$. Probabilities of Equally Likely Outcomes: If all $n$ outcomes in $S$ are equally likely, then $P(\omega_i) = \frac{1}{n}$ for each $\omega_i$. $P(E) = \frac{\text{Number of outcomes favourable to E}}{\text{Total number of possible outcomes}} = \frac{n(E)}{n(S)}$. Probability of $A \cup B$: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. If A and B are mutually exclusive, $P(A \cup B) = P(A) + P(B)$. Probability of 'not A': $P(A') = 1 - P(A)$.