SURYAMG - Complete Math Formulas (Class 6 to 12)

Chapter 1: Knowing Our Numbers

Indian System: Ones, Tens, Hundreds, Thousands, Ten Thousand, Lakh, Ten Lakh, Crore, Ten Crore.
International System: Ones, Tens, Hundreds, Thousands, Ten Thousand, Hundred Thousand, Million, Ten Million, Hundred Million, Billion.
Estimation / Rounding off: Nearest tens (look at digits < 5 or >= 5), Nearest hundreds, Nearest thousands.

Chapter 2: Whole Numbers

Predecessor: Number - 1 (except for 0 in whole numbers).
Successor: Number + 1.
Closure Property: Whole numbers are closed under addition and multiplication (a + b and a x b are whole numbers).
Commutative Property: a + b = b + a and a x b = b x a.
Associative Property: (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c).
Distributive Property: a x (b + c) = (a x b) + (a x c).

Chapter 3: Playing with Numbers

Factors: Exact divisor of a number. Every number is a factor of itself; 1 is a factor of every number.
Multiples: Number multiplied by integers. Every multiple is greater than or equal to the number.
Divisibility Rules: By 2 (last digit even), By 3 (sum of digits divisible by 3), By 4 (last two digits divisible by 4), By 5 (ends in 0 or 5), By 6 (divisible by both 2 and 3), By 8 (last three digits divisible by 8), By 9 (sum of digits divisible by 9), By 10 (ends in 0), By 11 (difference of sum of alternating digits is 0 or multiple of 11).
Relationship: HCF x LCM = Product of two given numbers.

Chapter 4: Basic Geometrical Ideas

Line: Extends indefinitely in both directions (no endpoints).
Line Segment: Part of a line with two definite endpoints.
Ray: Part of a line with one starting point and extending endlessly in one direction.
Polygon: Simple closed curve made entirely of line segments (Triangle = 3, Quadrilateral = 4, Pentagon = 5, Hexagon = 6).

Chapter 5: Understanding Elementary Shapes

Types of Angles: Acute (< 90 degrees), Right (= 90 degrees), Obtuse (90 to 180 degrees), Straight (= 180 degrees), Reflex (> 180 and < 360 degrees), Complete (= 360 degrees).
Perpendicular Lines: Lines intersecting at 90 degrees.
Triangle Classification: By sides (Scalene, Isosceles, Equilateral); By angles (Acute-angled, Right-angled, Obtuse-angled).

Chapter 6: Integers

Addition of Integers: Same signs add and keep sign; Opposite signs subtract and take the sign of the larger absolute value.
Subtraction of Integers: a - b = a + (additive inverse of b).

Chapter 7: Fractions

Fraction Form: Numerator / Denominator.
Addition/Subtraction (Like): (a +- b) / c.
Addition/Subtraction (Unlike): Convert denominators to LCM first, then add/subtract.
Multiplication: (Numerator x Numerator) / (Denominator x Denominator).
Division: Fraction A / Fraction B = Fraction A x Reciprocal of Fraction B.

Chapter 8: Decimals

Place Value in Decimals: Tenths (1/10), Hundredths (1/100), Thousandths (1/1000).
Conversions: Money, length, weight (e.g., 100 paise = 1 rupee, 1000g = 1kg, 100cm = 1m).

Chapter 9: Data Handling

Tally Marks: Grouping data in sets of five (four vertical lines crossed by a fifth).
Pictograph & Bar Graph: Visual representation of numerical data using symbols or bars of uniform width.

Chapter 10: Mensuration

Perimeter of Rectangle: 2 x (Length + Breadth)
Area of Rectangle: Length x Breadth
Perimeter of Square: 4 x Side
Area of Square: Side x Side
Perimeter of Regular Pentagon: 5 x Side
Perimeter of Equilateral Triangle: 3 x Side

Chapter 11: Algebra

Variable: Quantity that can take various numerical values, represented by letters (x, y, z, etc.).
Expression Formation: Combining variables and constants with arithmetic operations (e.g., 4x + 5).

Chapter 12: Ratio and Proportion

Ratio: a : b = a / b (Units must be the same).
Proportion: a : b = c : d means a x d = b x c (Product of extremes = Product of means).
Unitary Method: Find the value of one unit first, then multiply to find the value of required units.

Chapter 1: Integers

Multiplication Rules: (+) x (+) = (+), (-) x (-) = (+), (+) x (-) = (-), (-) x (+) = (-).
Division Rules: (+) / (+) = (+), (-) / (-) = (+), (+) / (-) = (-), (-) / (+) = (-).
Properties: Closure, Commutative, Associative, Distributive properties of integers.

Chapter 2: Fractions and Decimals

Fraction Operations: Full rules for improper, mixed fractions, addition, subtraction, multiplication, and division.
Decimal Multiplication/Division: Shifting the decimal point to the right (multiplication by powers of 10) or left (division by powers of 10).

Chapter 3: Data Handling

Arithmetic Mean: Sum of all observations / Total number of observations
Median: Middle-most value of data arranged in ascending/descending order.
Mode: Observation that occurs most frequently.
Range: Highest observation - Lowest observation.

Chapter 4: Simple Equations

Equation Balance Rule: Performing the same mathematical operation on both sides keeps the equation balanced.
Transposition: Moving a term from one side to the other with a change of sign (+ to -, - to +, x to /, / to x).

Chapter 5: Lines and Angles

Complementary Angles: Sum of two angles = 90 degrees.
Supplementary Angles: Sum of two angles = 180 degrees.
Adjacent Angles: Common vertex, common arm, non-common arms on opposite sides.
Linear Pair: Adjacent supplementary angles forming a straight line (sum = 180 degrees).
Vertically Opposite Angles: Equal in measure when two lines intersect.
Parallel Lines Transversal: Corresponding angles equal, Alternate interior angles equal, Co-interior angles supplementary.

Chapter 6: The Triangle and its Properties

Median of Triangle: Line segment joining vertex to the midpoint of the opposite side.
Altitude of Triangle: Perpendicular line segment from vertex to opposite side.
Angle Sum Property: Sum of interior angles of a triangle = 180 degrees.
Exterior Angle Property: Exterior angle = Sum of two opposite interior angles.
Triangle Inequality Property: Sum of lengths of any two sides is always greater than the third side (a + b > c).
Pythagoras Theorem (Right Triangle): Hypotenuse squared = Base squared + Perpendicular squared (c^2 = a^2 + b^2).

Chapter 7: Comparing Quantities

Percentage: (Part / Total) x 100
Profit: Selling Price (SP) - Cost Price (CP), when SP > CP
Loss: Cost Price (CP) - Selling Price (SP), when CP > SP
Profit Percentage: (Profit / CP) x 100
Loss Percentage: (Loss / CP) x 100
Simple Interest: SI = (P x R x T) / 100 (where P = Principal, R = Rate percent per annum, T = Time in years).
Total Amount (Simple Interest): A = Principal (P) + Simple Interest (SI)

Chapter 8: Rational Numbers

Rational Number Form: p/q where p and q are integers and q is not equal to 0.
Standard Form: q is positive and HCF of |p| and |q| is 1.

Chapter 9: Perimeter and Area

Area of Rectangle: Length x Breadth
Area of Square: Side x Side
Area of Parallelogram: Base x Height
Area of Triangle: 1/2 x Base x Height
Circumference of Circle: 2 x pi x r (where pi approx 22/7 or 3.14)
Area of Circle: pi x r^2

Chapter 10: Algebraic Expressions

Terms & Factors: Parts of an expression added together are terms; components of terms are factors.
Like/Unlike Terms: Terms with the same algebraic factors are like terms; otherwise unlike terms.

Chapter 11: Exponents and Powers

Product Law: a^m x a^n = a^(m+n)
Quotient Law: a^m / a^n = a^(m-n) (for m > n)
Power of a Power Law: (a^m)^n = a^(m x n)
Power of a Product: (a x b)^m = a^m x b^m
Power of a Quotient: (a / b)^m = a^m / b^m
Zero Exponent Law: a^0 = 1 (where a is not 0)

Chapter 12: Symmetry & Visualising Solid Shapes

Lines of Symmetry: Axis along which a figure can be folded into two identical halves.
Rotational Symmetry: Order of rotation, angle of rotation.
Euler's Formula for Polyhedrons: Faces (F) + Vertices (V) - Edges (E) = 2

Chapter 1: Rational Numbers

Additive Identity: 0 (a + 0 = a)
Multiplicative Identity: 1 (a x 1 = a)
Additive Inverse: For a/b, additive inverse is -a/b
Multiplicative Inverse (Reciprocal): For a/b, reciprocal is b/a (a x b/a = 1)

Chapter 2: Linear Equations in One Variable

General Form: ax + b = 0 or ax + b = cx + d
Solution Method: Isolating the variable on one side by transposition and inverse operations.

Chapter 3: Understanding Quadrilaterals

Sum of Interior Angles of n-sided Polygon: (n - 2) x 180 degrees
Sum of Exterior Angles of any Polygon: 360 degrees
Parallelogram Properties: Opposite sides equal, opposite angles equal, diagonals bisect each other.

Chapter 4: Data Handling

Grouped Data Frequency Distribution: Class intervals, class size (upper limit - lower limit), class mark = (Upper Limit + Lower Limit) / 2.
Probability: P(E) = Number of favorable outcomes / Total number of possible outcomes.

Chapter 5: Squares and Square Roots

Square Properties: Numbers ending in 2, 3, 7, 8 are never perfect squares.
Square Root Methods: Prime factorization method and Long division method.
Pythagorean Triplets: For any natural number m > 1, (2m, m^2 - 1, m^2 + 1) forms a Pythagorean triplet.

Chapter 6: Cubes and Cube Roots

Cube: Number multiplied by itself three times (x^3).
Cube Root: Prime factorization method for finding cubic roots (denoted as cube root of x).

Chapter 7: Comparing Quantities

Discount Percentage: (Discount / Marked Price) x 100
Selling Price after Discount: Marked Price - Discount
GST (Goods and Services Tax): Added to the selling price. GST amount = (GST % / 100) x SP.
Compound Interest Amount: A = P x (1 + R / 100)^n
Compound Interest: CI = A - P
Compounded Half-Yearly: Rate becomes R/2, Time period becomes 2n.
Population Growth Formula: P = P_0 x (1 + R/100)^n

Chapter 8: Algebraic Expressions and Identities

Identity 1: (a + b)^2 = a^2 + 2ab + b^2
Identity 2: (a - b)^2 = a^2 - 2ab + b^2
Identity 3: (a + b)(a - b) = a^2 - b^2
Identity 4: (x + a)(x + b) = x^2 + (a + b)x + ab

Chapter 9: Mensuration

Area of Trapezium: 1/2 x (Sum of parallel sides) x Height
Area of General Quadrilateral: 1/2 x Diagonal x (Sum of offsets/perpendiculars on diagonal)
Total Surface Area of Cuboid: 2(lb + bh + hl)
Lateral Surface Area of Cuboid: 2h(l + b)
Total Surface Area of Cube: 6 x Side^2
Lateral Surface Area of Cube: 4 x Side^2
Curved Surface Area of Cylinder: 2 x pi x r x h
Total Surface Area of Cylinder: 2 x pi x r x (r + h)
Volume of Cuboid: Length x Breadth x Height
Volume of Cube: Side^3
Volume of Cylinder: pi x r^2 x h

Chapter 10: Exponents and Powers

Negative Exponent Law: a^(-m) = 1 / a^m
Standard Form of Numbers: Expressing a number as k x 10^n where 1 <= k < 10 and n is an integer.

Chapter 11: Direct and Inverse Proportions

Direct Proportion: x / y = k (or x1/y1 = x2/y2), meaning as x increases, y increases proportionally.
Inverse Proportion: x x y = k (or x1 x y1 = x2 x y2), meaning as x increases, y decreases proportionally.

Chapter 12: Factorisation

Common Factors Method: Factorizing by grouping terms and pulling out common monomial factors.
Using Identities: Applying a^2 - b^2 = (a-b)(a+b) or (a+b)^2 patterns.
Splitting the Middle Term: Factorizing quadratic polynomials x^2 + (a+b)x + ab into (x+a)(x+b).

Chapter 13: Introduction to Graphs

Cartesian Coordinates: Ordered pair (x, y) where x is the x-coordinate (abscissa) and y is the y-coordinate (ordinate).
Linear Graphs: Graphs that are straight lines, indicating a direct relationship between variables.

Chapter 1: Number Systems

Rational Exponents: a^(m/n) = nth root of (a^m) = (nth root of a)^m
Laws of Radicals: sqrt(a x b) = sqrt(a) x sqrt(b), sqrt(a/b) = sqrt(a) / sqrt(b).
Rationalizing Factor: Converting denominator with surds into a rational number by multiplying numerator and denominator by its conjugate.

Chapter 2: Polynomials

Remainder Theorem: If p(x) is divided by linear divisor (x - a), the remainder is p(a).
Factor Theorem: (x - a) is a factor of polynomial p(x) if and only if p(a) = 0.
Algebraic Identities: (x + y + z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx, (x + y)^3 = x^3 + y^3 + 3xy(x + y), (x - y)^3 = x^3 - y^3 - 3xy(x - y), x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx).

Chapter 3: Coordinate Geometry

Axes: Horizontal axis (x-axis), Vertical axis (y-axis), meeting at Origin (0,0).
Quadrants: Four quadrants (I: +,+; II: -,+; III: -,-; IV: +,-).

Chapter 4: Linear Equations in Two Variables

Standard Form: ax + by + c = 0 (where a, b, c are real numbers and a, b are not both zero).
Solution Set: Infinite number of solutions represented as a straight line on the Cartesian plane.

Chapter 5: Introduction to Euclid's Geometry

Axioms & Postulates: Self-evident assumptions (e.g., Postulate 5: Parallel postulate regarding lines intersecting when interior angle sum is less than 180 degrees).

Chapter 6: Lines and Angles

Parallel Line Theorems: Alternate interior angles theorem, corresponding angles axiom, consecutive interior angles supplement theorem.

Chapter 7: Triangles

Congruence Criteria: SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS, SSS (Side-Side-Side), RHS (Right angle-Hypotenuse-Side).
Triangle Inequalities: Angle opposite to the longer side is greater; Side opposite to the larger angle is longer.

Chapter 8: Quadrilaterals

Mid-Point Theorem: Line segment joining the midpoints of two sides of a triangle is parallel to the third side and half of it.

Chapter 9: Circles

Chord Properties: Equal chords subtend equal angles at the centre; perpendicular from centre bisects the chord.
Angle Subtended Theorem: Angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
Cyclic Quadrilateral: Sum of opposite angles of a cyclic quadrilateral is 180 degrees.

Chapter 10: Heron's Formula

Semi-perimeter (s): s = (a + b + c) / 2
Area of Triangle: Area = sqrt(s x (s - a) x (s - b) x (s - c))

Chapter 11: Surface Areas and Volumes

Curved Surface Area of Cylinder: 2 x pi x r x h
Total Surface Area of Cylinder: 2 x pi x r x (r + h)
Volume of Cylinder: pi x r^2 x h
Curved Surface Area of Cone: pi x r x l (where l = slant height = sqrt(r^2 + h^2))
Total Surface Area of Cone: pi x r x (l + r)
Volume of Cone: (1/3) x pi x r^2 x h
Surface Area of Sphere: 4 x pi x r^2
Volume of Sphere: (4/3) x pi x r^3
Curved Surface Area of Hemisphere: 2 x pi x r^2
Total Surface Area of Hemisphere: 3 x pi x r^2
Volume of Hemisphere: (2/3) x pi x r^3

Chapter 12: Statistics

Mean for Ungrouped Data: sum(x_i) / n
Median Formulas: If n is odd, median = value of ((n+1)/2)th term. If n is even, median = average of (n/2)th and ((n/2)+1)th terms.
Range & Class Width: Standard statistical measures.

Chapter 1: Real Numbers

Euclid's Division Lemma: For any positive integers a and b, there exist unique integers q and r such that a = bq + r (0 <= r < b).
Fundamental Theorem of Arithmetic: Every composite number can be factored as a product of primes, and this factorization is unique.
HCF and LCM Relation: HCF(a, b) x LCM(a, b) = a x b.

Chapter 2: Polynomials

Relationship between Zeroes and Coefficients (Quadratic ax^2 + bx + c): Sum of zeroes (alpha + beta) = -b/a; Product of zeroes (alpha x beta) = c/a.
Cubic Polynomial Relations (ax^3 + bx^2 + cx + d): Sum of roots = -b/a, Sum of product of roots taken two at a time = c/a, Product of roots = -d/a.

Chapter 3: Pair of Linear Equations in Two Variables

General Form: a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0.
Consistency Conditions: Intersecting lines (Unique Solution): a1/a2 not equal to b1/b2; Coincident lines (Infinitely Many Solutions): a1/a2 = b1/b2 = c1/c2; Parallel lines (No Solution): a1/a2 = b1/b2 not equal to c1/c2.
Algebraic Methods: Substitution method, Elimination method, Cross-multiplication method.

Chapter 4: Quadratic Equations

Standard Form: ax^2 + bx + c = 0 (where a not equal to 0).
Quadratic Formula (Sridharacharya Formula): x = (-b +- sqrt(b^2 - 4ac)) / (2a)
Discriminant (D): D = b^2 - 4ac
Nature of Roots: If D > 0 (Two distinct real roots), If D = 0 (Two equal real roots, x = -b/2a), If D < 0 (No real roots / complex roots).

Chapter 5: Arithmetic Progressions (AP)

General Form of AP: a, a+d, a+2d, a+3d, ...
Nth Term of AP: a_n = a + (n - 1)d
Sum of First n Terms: S_n = (n/2) x [2a + (n - 1)d] or S_n = (n/2) x (a + l) where l is the last term.

Chapter 6: Triangles

Thales Theorem (Basic Proportionality Theorem): If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.
Similarity Criteria: AAA (Angle-Angle-Angle), SSS (Side-Side-Side), SAS (Side-Angle-Side).
Area Ratio Theorem: Ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Pythagoras Theorem: In a right triangle, square of hypotenuse equals sum of squares of other two sides.

Chapter 7: Coordinate Geometry

Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Section Formula: Coordinates = ((m1x2 + m2x1)/(m1 + m2), (m1y2 + m2y1)/(m1 + m2))
Mid-point Formula: ((x1 + x2)/2, (y1 + y2)/2)
Area of Triangle: 1/2 x absolute value of [x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)]

Chapter 8: Introduction to Trigonometry

Trigonometric Ratios: sin(theta) = Perpendicular/Hypotenuse, cos(theta) = Base/Hypotenuse, tan(theta) = Perpendicular/Base, cosec(theta) = 1/sin, sec(theta) = 1/cos, cot(theta) = 1/tan.
Trigonometric Identities: sin^2(theta) + cos^2(theta) = 1, 1 + tan^2(theta) = sec^2(theta), 1 + cot^2(theta) = cosec^2(theta).

Chapter 9: Some Applications of Trigonometry

Line of Sight & Angles: Angle of Elevation (looking up from horizontal), Angle of Depression (looking down from horizontal).

Chapter 10: Circles

Tangent Property: Tangent at any point of a circle is perpendicular to the radius through the point of contact.
Tangent Length Theorem: Lengths of tangents drawn from an external point to a circle are equal.

Chapter 11: Areas Related to Circles

Area of Circle: pi x r^2
Circumference of Circle: 2 x pi x r
Area of Sector: (theta / 360) x pi x r^2
Length of Arc: (theta / 360) x 2 x pi x r
Area of Segment: Area of corresponding sector - Area of corresponding triangle.

Chapter 12: Surface Areas and Volumes

Frustum of Cone Surface Areas: CSA = pi x (r1 + r2) x l (where slant height l = sqrt(h^2 + (r1 - r2)^2)), TSA = pi x (r1 + r2) x l + pi x r1^2 + pi x r2^2.
Volume of Frustum: (1/3) x pi x h x (r1^2 + r2^2 + (r1 x r2)).

Chapter 13: Statistics

Mean (Assumed Mean Method): Mean = a + (sum(f_i x d_i) / sum(f_i)) where d_i = x_i - a.
Mean (Step-Deviation Method): Mean = a + (sum(f_i x u_i) / sum(f_i)) x h.
Median Formula: Median = L + [ ((n/2 - cf) / f) ] x h
Mode Formula: Mode = L + [ ((f1 - f0) / (2f1 - f0 - f2)) ] x h
Empirical Relationship: 3 Median = Mode + 2 Mean.

Chapter 14: Probability

Classical Probability: P(E) = Number of outcomes favorable to E / Total number of possible outcomes of the experiment.
Complementary Event: P(E) + P(not E) = 1.

Chapter 1: Sets

Set Operations: Union (A union B), Intersection (A intersection B), Difference (A - B).
De Morgan's Laws: (A union B)' = A' intersection B'; (A intersection B)' = A' union B'.
Cardinality Formula: n(A union B) = n(A) + n(B) - n(A intersection B).
Three Set Cardinality: n(A union B union C) = n(A) + n(B) + n(C) - n(A int B) - n(B int C) - n(C int A) + n(A int B int C).

Chapter 2: Relations and Functions

Cartesian Product: A x B = {(a, b) : a in A and b in B}. If n(A)=p, n(B)=q, then n(A x B) = p x q.
Relation: Subset of A x B. Domain, Codomain, and Range.
Functions: Special relations where every element in domain has a unique image in codomain.

Chapter 3: Trigonometric Functions

Degree-Radian Conversion: Radian = Degree x (pi / 180).
Compound Angle Formulas: sin(x + y) = sin x cos y + cos x sin y, sin(x - y) = sin x cos y - cos x sin y, cos(x + y) = cos x cos y - sin x sin y, cos(x - y) = cos x cos y + sin x sin y.
Transformation Formulas: cos x + cos y = 2 cos((x+y)/2) cos((x-y)/2), sin x + sin y = 2 sin((x+y)/2) cos((x-y)/2).
Sine Rule & Cosine Rule (Properties of Triangles): a/sin A = b/sin B = c/sin C and cos A = (b^2 + c^2 - a^2)/(2bc).

Chapter 4: Complex Numbers and Quadratic Equations

Complex Number Standard Form: z = x + iy (where i^2 = -1).
Modulus & Conjugate: |z| = sqrt(x^2 + y^2), Conjugate (z bar) = x - iy.
Multiplicative Inverse: z^(-1) = (z bar) / |z|^2.
Polar Form: z = r(cos(theta) + i sin(theta)) where r = |z| and theta = argument.

Chapter 5: Linear Inequalities

Rules of Inequalities: Multiplying or dividing both sides by a negative number reverses the inequality sign. Graphing solution regions on the Cartesian plane.

Chapter 6: Permutations and Combinations

Factorial Notation: n! = n x (n-1) x ... x 1.
Permutation Formula (Arrangements): nPr = n! / (n - r)!
Combination Formula (Selections): nCr = n! / (r! x (n - r)!)
Key Identities: nCr = nC(n-r), nCr + nC(r-1) = (n+1)Cr.

Chapter 7: Binomial Theorem

Binomial Expansion: (a + b)^n = sum(nCr x a^(n-r) x b^r) from r=0 to n.
General Term: T_(r+1) = nCr x a^(n-r) x b^r
Middle Term(s): Depends on whether n is even or odd.

Chapter 8: Sequences and Series

Arithmetic Progression (AP): a_n = a + (n-1)d, S_n = (n/2)[2a + (n-1)d].
Geometric Progression (GP): a_n = a x r^(n-1), S_n = a(r^n - 1)/(r - 1) (for r not equal to 1).
Sum to Infinity of GP: S_inf = a / (1 - r) (for |r| < 1).
Means: AM = (a+b)/2, GM = sqrt(ab). Relationship: AM >= GM.

Chapter 9: Straight Lines

Slope (m): m = tan(theta) = (y2 - y1) / (x2 - x1).
Equation Forms: Slope-intercept form: y = mx + c; Point-slope form: (y - y1) = m(x - x1); Two-point form: (y - y1) = ((y2 - y1)/(x2 - x1)) x (x - x1); Intercept form: x/a + y/b = 1; Normal form: x cos(alpha) + y sin(alpha) = p.
Distance of a Point from a Line: d = |ax1 + by1 + c| / sqrt(a^2 + b^2).

Chapter 10: Conic Sections

Circle: (x - h)^2 + (y - k)^2 = r^2.
Parabola Standard Equations: y^2 = 4ax, y^2 = -4ax, x^2 = 4ay, x^2 = -4ay.
Ellipse: (x^2 / a^2) + (y^2 / b^2) = 1 (where a > b). Eccentricity e = sqrt(1 - b^2/a^2).
Hyperbola: (x^2 / a^2) - (y^2 / b^2) = 1. Eccentricity e = sqrt(1 + b^2/a^2).

Chapter 11: Introduction to Three Dimensional Geometry

Distance Formula in 3D: d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).
Section Formula in 3D: Coordinates = ((m1x2 + m2x1)/(m1+m2), (m1y2 + m2y1)/(m1+m2), (m1z2 + m2z1)/(m1+m2)).

Chapter 12: Limits and Derivatives

Standard Limits: limit (x->a) (x^n - a^n)/(x - a) = n x a^(n-1), limit (x->0) sin(x)/x = 1, limit (x->0) (e^x - 1)/x = 1.
Derivative Definition (First Principle): f'(x) = limit (h->0) [f(x + h) - f(x)] / h.
Derivative Rules: Product rule (uv)' = u'v + uv', Quotient rule (u/v)' = (u'v - uv')/v^2.

Chapter 13: Statistics

Mean Deviation: MD from mean or median = sum(|x_i - x_bar|) / n.
Variance (sigma^2): sum((x_i - mean)^2) / n.
Standard Deviation (sigma): Square root of variance.

Chapter 14: Probability

Addition Theorem: P(A union B) = P(A) + P(B) - P(A intersection B).
Mutually Exclusive Events: P(A intersection B) = 0.

Chapter 1: Relations and Functions

Types of Relations: Reflexive, Symmetric, Transitive, Equivalence Relation.
Types of Functions: One-one (Injective), Onto (Surjective), Bijective (One-one and Onto).
Composition: (g o f)(x) = g(f(x)).

Chapter 2: Inverse Trigonometric Functions

Principal Value Branches: Ranges of sin^(-1), cos^(-1), tan^(-1), etc.
Properties: sin^(-1)(1/x) = cosec^(-1)(x), sin^(-1)(-x) = -sin^(-1)(x), cos^(-1)(-x) = pi - cos^(-1)(x), sin^(-1)(x) + cos^(-1)(x) = pi/2.

Chapter 3: Matrices

Matrix Operations: Addition, Scalar multiplication, Multiplication of matrices (A x B).
Transpose: (A')' = A, (A + B)' = A' + B', (AB)' = B'A'.
Symmetric & Skew-Symmetric: A' = A (Symmetric), A' = -A (Skew-Symmetric). Every square matrix can be expressed as sum of a symmetric and skew-symmetric matrix.

Chapter 4: Determinants

Determinant Properties: |AB| = |A| x |B|, |adj A| = |A|^(n-1).
Inverse Matrix: A^(-1) = (adj A) / |A| (exists only if |A| not equal to 0).
Area of Triangle: 1/2 x determinant of coordinate matrix.
Solving Linear Equations: AX = B implies X = A^(-1)B.

Chapter 5: Continuity and Differentiability

Continuity Condition: Function f(x) is continuous at x = a if Left Hand Limit = Right Hand Limit = f(a).
Chain Rule: dy/dx = (dy/dt) x (dt/dx).
Logarithmic Differentiation: Taking log on both sides to differentiate functions of the form f(x)^g(x).

Chapter 6: Application of Derivatives

Rate of Change: dy/dx represents rate of change of y with respect to x.
Slope of Tangent / Normal: Slope of tangent = dy/dx; Slope of normal = -1 / (dy/dx).
Increasing/Decreasing Functions: f'(x) > 0 (strictly increasing), f'(x) < 0 (strictly decreasing).
Maxima and Minima: First derivative test and Second derivative test (f''(x) < 0 for local maxima, f''(x) > 0 for local minima).

Chapter 7: Integrals

Standard Integrals: integral x^n dx = x^(n+1)/(n+1) + C, integral e^x dx = e^x + C, integral (1/x) dx = log|x| + C, integral sin(x) dx = -cos(x) + C.
Integration by Parts: integral u v dx = u x integral v dx - integral [ (du/dx) x integral v dx ] dx.
Partial Fractions: Decomposing complex rational expressions into simpler fractions for integration.
Definite Integrals: Fundamental Theorem of Calculus: integral from a to b of f(x)dx = F(b) - F(a). Properties include integral from a to b of f(x)dx = integral from a to b of f(a+b-x)dx.

Chapter 8: Application of Integrals

Area under a Curve: Area = integral from a to b of y dx = integral from a to b of f(x) dx.
Area between Two Curves: Area = integral from a to b of [f(x) - g(x)] dx.

Chapter 9: Differential Equations

Order & Degree: Order is the highest derivative present; Degree is the power of the highest-order derivative.
Variable Separable Method: Rewriting equation as f(x)dx = g(y)dy and integrating both sides.
Linear Differential Equations: dy/dx + Py = Q. Integrating Factor (IF) = e^(integral P dx). Solution: y x IF = integral (Q x IF) dx + C.

Chapter 10: Vector Algebra

Magnitude of Vector: |a| = sqrt(x^2 + y^2 + z^2).
Scalar (Dot) Product: a . b = |a| |b| cos(theta). cos(theta) = (a . b) / (|a| |b|).
Vector (Cross) Product: a x b = |a| |b| sin(theta) n^hat. Area of triangle = 1/2 |a x b|.

Chapter 11: Three Dimensional Geometry

Direction Cosines & Ratios: l, m, n satisfying l^2 + m^2 + n^2 = 1.
Equation of a Line: Vector form: r = a + lambda b. Cartesian form: (x - x1)/a = (y - y1)/b = (z - z1)/c.
Shortest Distance between Two Skew Lines: Formula involving vector cross products and vector differences.
Equation of a Plane: Normal form (r . n = d), Cartesian form (Ax + By + Cz = D).

Chapter 12: Linear Programming

Objective Function: Linear function Z = ax + by to be maximized or minimized subject to constraints expressed as linear inequalities. Corner point method for optimization.

Chapter 13: Probability

Conditional Probability: P(A | B) = P(A intersection B) / P(B).
Multiplication Theorem: P(A intersection B) = P(A) x P(B | A).
Bayes' Theorem: P(E_i | A) = [P(E_i) x P(A | E_i)] / sum[P(E_j) x P(A | E_j)].
Probability Distribution: Mean (Expectation) mu = sum(x_i x P(x_i)), Variance sigma^2 = sum(x_i^2 x P(x_i)) - mu^2.