Fundamentals of Mathematics Class 11: Complete Guide & Free PDF Download (JEE & Boards)
Fundamentals of Mathematics — Competishun
Fundamentals of Mathematics Class 11: Complete Guide & Free PDF Download (JEE & Boards)
Number Systems · Logarithms · Inequalities · Modulus · Quadratic Equations
Fundamentals of Mathematics is the foundation of all mathematics. It covers the basic concepts and tools that are essential for understanding every other chapter in JEE and board exams. From number systems to logarithms, inequalities to modulus, these topics form the language in which all of mathematics is written.
This chapter covers number systems (natural, whole, integers, rational, irrational, real, and complex numbers), intervals, algebraic identities, indices and surds, logarithms, modulus (absolute value), ratio and proportion, polynomials, quadratic equations, inequalities (including the wavy-curve method), exponential and logarithmic inequalities, and greatest integer and fractional part functions.
This page gives you the complete guide to Fundamentals of Mathematics with all concepts explained in depth. You will find clear definitions, formulas, examples, and practice questions. Download the free PDF below and keep it handy for quick revision before your JEE Main, board exams, or any other test.
Number SystemsN · W · Z · Q · R · C
LogarithmsLaws & Properties
InequalitiesWavy-Curve Method
Free PDFDownload
Download the Fundamentals of Mathematics Complete Guide PDF
Get all Fundamentals of Mathematics concepts, number systems, logarithms, inequalities, modulus, and quadratic equations in one clean PDF, free. Perfect for JEE and board revision.
Download Free PDFOpens in Google Drive · No sign-up needed
What are the Fundamentals of Mathematics?
Definition: The Fundamentals of Mathematics are the basic concepts and tools that form the foundation of all mathematics. They include number systems, algebraic operations, logarithms, inequalities, and functions.
The fundamentals of mathematics are essential for understanding every other chapter in JEE and board exams. Without a solid grasp of these concepts, it is impossible to solve problems in algebra, calculus, coordinate geometry, and beyond.
Number Systems
The building blocks of mathematics. Understanding the different types of numbers is essential for all of mathematics.
Mathematical Tools
Logarithms, inequalities, modulus, and algebraic identities are the tools used to solve problems across all of mathematics.
Key Insight: The fundamentals of mathematics are the language in which all of mathematics is written. Master these concepts, and you will be able to understand and solve problems in every other chapter.
Glossary of Fundamentals Terms — Complete A to Z
Before diving deep into each topic, let's understand the key terminology used in this chapter:
| Term | Definition |
|---|---|
| Natural Numbers (N) | The set of positive integers: {1, 2, 3, ...} |
| Whole Numbers (W) | The set of non-negative integers: {0, 1, 2, 3, ...} |
| Integers (Z) | The set of positive and negative whole numbers: {..., -2, -1, 0, 1, 2, ...} |
| Rational Numbers (Q) | Numbers that can be expressed as p/q, where p, q ∈ Z, q ≠ 0 |
| Irrational Numbers (Qc) | Numbers that cannot be expressed as p/q. Non-terminating, non-recurring decimals. |
| Real Numbers (R) | The set of all rational and irrational numbers. |
| Complex Numbers (C) | Numbers of the form a + ib, where a, b ∈ R and i² = -1 |
| Logarithm | The inverse of exponentiation. If aˣ = N, then logₐN = x. |
| Modulus (Absolute Value) | The distance of a number from zero on the number line. |x| = x if x ≥ 0, and |x| = -x if x < 0. |
| Inequality | A mathematical statement that compares two expressions using <, >, ≤, or ≥. |
| Wavy-Curve Method | A method for solving inequalities involving products or quotients of linear factors. |
| Greatest Integer Function | [x] = greatest integer ≤ x. Also called the floor function. |
| Fractional Part Function | {x} = x - [x]. It lies in [0, 1). |
| Mastering these terms is essential for understanding the Fundamentals of Mathematics. | |
Number Systems — The Building Blocks
Definition: The number system is the foundation of mathematics. It is a way of representing numbers using a consistent set of symbols and rules.
The number system consists of the following sets:
| Set | Symbol | Definition | Examples |
|---|---|---|---|
| Natural Numbers | N | {1, 2, 3, 4, ...} | 1, 2, 3, 4, 5 |
| Whole Numbers | W | {0, 1, 2, 3, ...} | 0, 1, 2, 3, 4 |
| Integers | Z | {..., -2, -1, 0, 1, 2, ...} | -3, -2, -1, 0, 1, 2 |
| Rational Numbers | Q | {p/q : p, q ∈ Z, q ≠ 0} | 1/2, 3/4, -5/7, 0.333... |
| Irrational Numbers | Qc | Numbers that are not rational | √2, π, e, √3 |
| Real Numbers | R | Q ∪ Qc | All rational and irrational numbers |
| Complex Numbers | C | {a + ib : a, b ∈ R, i² = -1} | 2 + 3i, -1 + i, 4i |
| Chain: N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C | |||
Number System Hierarchy
The number system hierarchy: N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C
Important Facts
- 0 is neither positive nor negative.
- 1 is neither prime nor composite.
- 2 is the only even prime.
- Every integer is rational.
- √(non-perfect square) is irrational.
- The sum/product of a non-zero rational with an irrational is always irrational.
Important: The number system hierarchy is fundamental. Understanding the relationships between different sets of numbers is essential for solving problems in all of mathematics.
Intervals on the Real Line
Definition: A continuous set of real numbers is written as an interval. Intervals can be open, closed, or half-open.
| Interval | Set-Builder Form | Explanation |
|---|---|---|
| (a, b) | {x : a < x < b} | Open interval — both ends excluded |
| [a, b] | {x : a ≤ x ≤ b} | Closed interval — both ends included |
| (a, b] | {x : a < x ≤ b} | Half-open interval |
| [a, b) | {x : a ≤ x < b} | Half-open interval |
| (−∞, a) | {x : x < a} | Unbounded open interval |
| (−∞, a] | {x : x ≤ a} | Unbounded closed interval |
| (a, ∞) | {x : x > a} | Unbounded open interval |
| [a, ∞) | {x : x ≥ a} | Unbounded closed interval |
| (−∞, ∞) | R | All real numbers |
| The symbols ∞ and -∞ are always used with open endpoints. | ||
Key Insight: Intervals are used to represent ranges of values in inequalities, domain and range of functions, and many other mathematical contexts.
Basic Algebraic Identities
Definition: Algebraic identities are equations that are true for all values of the variables. They are used to simplify expressions and solve equations.
Square of Sum
(a + b)² = a² + 2ab + b²
Perfect square
Square of Difference
(a − b)² = a² − 2ab + b²
Perfect square
Difference of Squares
(a + b)(a − b) = a² − b²
Product of sum and difference
Cube of Sum
(a + b)³ = a³ + 3a²b + 3ab² + b³
= a³ + b³ + 3ab(a+b)
Cube of Difference
(a − b)³ = a³ − 3a²b + 3ab² − b³
= a³ − b³ − 3ab(a−b)
Sum of Cubes
a³ + b³ = (a + b)(a² − ab + b²)
Factorization
Difference of Cubes
a³ − b³ = (a − b)(a² + ab + b²)
Factorization
Square of Trinomial
(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
Three-term expansion
Useful Results
- a² + b² = (a + b)² − 2ab
- (a + b)² − (a − b)² = 4ab
- a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
- If a + b + c = 0, then a³ + b³ + c³ = 3abc
Example: If a + 1/a = 3, then a² + 1/a² = (a + 1/a)² − 2 = 9 − 2 = 7. Also, a³ + 1/a³ = (a + 1/a)³ − 3(a + 1/a) = 27 − 9 = 18.
Important: Algebraic identities are the foundation of algebra. They are used extensively in simplifying expressions, solving equations, and factoring polynomials.
Indices and Surds — Laws of Exponents
Definition: Indices (or exponents) are a shorthand way of writing repeated multiplication. Surds are irrational numbers that involve roots.
Laws of Indices
Product Rule
aᵐ · aⁿ = aᵐ⁺ⁿ
Add exponents when multiplying
Quotient Rule
aᵐ / aⁿ = aᵐ⁻ⁿ
Subtract exponents when dividing
Power of a Power
(aᵐ)ⁿ = aᵐⁿ
Multiply exponents
Power of a Product
(ab)ⁿ = aⁿbⁿ
Distribute exponent
Zero Exponent
a⁰ = 1 (a ≠ 0)
Any non-zero number to the power 0 is 1
Negative Exponent
a⁻ⁿ = 1/aⁿ
Reciprocal
Fractional Exponent
aᵐ/ⁿ = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
Roots as exponents
Surds
- √a · √b = √(ab) (for a, b ≥ 0)
- Rationalisation: 1/(√a + √b) = (√a − √b)/(a − b)
- If aˣ = aʸ, then x = y (for a > 0, a ≠ 1)
- If aˣ = bˣ, then a = b (for x ≠ 0)
Example: (2ⁿ⁺¹ · 4ⁿ) / 8ⁿ⁻¹ = (2ⁿ⁺¹ · 2²ⁿ) / 2³ⁿ⁻³ = 2^(n+1+2n-3n+3) = 2⁴ = 16.
Key Insight: The laws of indices are essential for simplifying expressions and solving equations involving exponents. Surds are irrational numbers that appear frequently in mathematics.
Logarithms — The Inverse of Exponents
Definition: A logarithm is the inverse of an exponent. If aˣ = N, then logₐN = x, where a > 0, a ≠ 1, and N > 0.
Laws of Logarithms
Product Law
logₐ(mn) = logₐm + logₐn
Log of a product
Quotient Law
logₐ(m/n) = logₐm − logₐn
Log of a quotient
Power Law
logₐ(mⁿ) = n logₐm
Log of a power
Change of Base
logₐN = log_bN / log_ba
Convert to any base
Reciprocal
logₐb = 1 / log_ba
Log of a base
Special Values
logₐa = 1, logₐ1 = 0
Base and unity
Important Results
- a^(logₐN) = N
- logₐ(mᵏ) = (1/k) logₐm
- Common log: log₁₀x (base 10)
- Natural log: ln x = logₑx (where e ≈ 2.718)
Example: log₂(log₃81) = log₂(log₃3⁴) = log₂4 = 2. Also, 2^(log₂7) = 7, log₅(1/25) = -2, log₉27 = 3/2.
Important: Domain first! A logarithm exists only when its argument is positive and the base is > 0 and ≠ 1. Check this before applying any log law.
Modulus (Absolute Value)
Definition: The modulus (or absolute value) of a number x, denoted by |x|, is the distance of x from zero on the number line.
Definition and Properties
Definition
|x| = {x, x ≥ 0; -x, x < 0}
Distance from zero
Non-negative
|x| ≥ 0
Always non-negative
Square Root
√(x²) = |x|
Important identity
Product
|xy| = |x||y|
Multiplicative
Triangle Inequality
|x + y| ≤ |x| + |y|
Sum property
Reverse Triangle
||x| − |y|| ≤ |x − y|
Difference property
Solving Equations with Modulus
- |x| = a ⇒ x = ±a (for a ≥ 0)
- |x| < a ⇒ −a < x < a (for a > 0)
- |x| > a ⇒ x < −a or x > a (for a > 0)
- If a < 0, |x| > a is true for all x
- If a ≤ 0, |x| < a has no solution
Example: Solve |x − 2| < 3: −3 < x − 2 < 3 ⇒ −1 < x < 5, so x ∈ (−1, 5).
Key Insight: The modulus function represents distance. Understanding its properties is essential for solving inequalities and equations involving absolute values.
Ratio, Proportion & Variation
Definition: A ratio is a comparison of two quantities. A proportion is an equation stating that two ratios are equal.
Properties of Proportion
Invertendo
a/b = c/d ⇒ b/a = d/c
Invert both ratios
Alternendo
a/b = c/d ⇒ a/c = b/d
Alternate terms
Componendo
a/b = c/d ⇒ (a+b)/b = (c+d)/d
Add numerator to denominator
Dividendo
a/b = c/d ⇒ (a−b)/b = (c−d)/d
Subtract numerator from denominator
Componendo-Dividendo
(a+b)/(a−b) = (c+d)/(c−d)
Combined property
Variation
- Direct Variation: y ∝ x ⇒ y = kx
- Inverse Variation: y ∝ 1/x ⇒ xy = k
- If a/b = c/d = e/f, then each ratio = (a + c + e)/(b + d + f)
Example: If a/b = 3/5, then by componendo-dividendo, (a+b)/(a−b) = (3+5)/(3−5) = 8/(−2) = −4.
Important: Componendo and dividendo are very useful for simplifying expressions in ratio and proportion problems.
Polynomials — Remainder & Factor Theorems
Definition: A polynomial of degree n is an expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₀, where aₙ ≠ 0.
Key Theorems
Division Algorithm
p(x) = (x − a)q(x) + r
r is a constant
Remainder Theorem
p(a) = remainder when divided by (x − a)
Evaluate p(a)
Factor Theorem
(x − a) is a factor ⇔ p(a) = 0
Zero of the polynomial
Key Insight: The factor theorem is essential for factorising polynomials. A polynomial of degree n has exactly n roots (counting multiplicity).
Quadratic Equations — The Basics
Definition: A quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0.
Key Formulas
Quadratic Formula
x = (-b ± √D) / 2a
D = b² − 4ac
Discriminant
D = b² − 4ac
Determines nature of roots
Sum of Roots
α + β = −b/a
Vieta's formula
Product of Roots
αβ = c/a
Vieta's formula
Equation from Roots
x² − (α+β)x + αβ = 0
Forming a quadratic
Nature of Roots
- D > 0: Real and distinct roots
- D = 0: Real and equal roots (x = −b/2a)
- D < 0: Complex conjugate roots
Example: x² − 5x + 6 = 0. D = 25 − 24 = 1 > 0 (real, distinct). Roots: x = (5 ± 1)/2 = 2, 3. Sum = 5 = −b/a, Product = 6 = c/a.
Important: A quadratic has real roots iff b² ≥ 4ac. With rational coefficients, roots are rational iff D is a perfect square.
Inequalities and the Wavy-Curve Method
Definition: An inequality is a mathematical statement that compares two expressions using <, >, ≤, or ≥.
Rules for Solving Inequalities
- Adding/subtracting the same number keeps the inequality.
- Multiplying or dividing by a positive number keeps the inequality.
- Multiplying or dividing by a negative number reverses the inequality.
Wavy-Curve Method (Method of Intervals)
The wavy-curve method is used to solve inequalities involving products or quotients of linear factors.
- Step 1: Write the expression as a product/quotient of linear factors.
- Step 2: Mark the roots on a number line.
- Step 3: Draw a curve that starts positive from the far right and alternates sign at each simple root.
- Step 4: At a factor of even multiplicity, the sign does not change.
Sign Scheme of (x + 2)(x − 1)(x − 3)
The sign alternates across the simple roots −2, 1, 3.
Example: (x + 2)(x − 1)(x − 3) > 0. From the scheme, the expression is positive on (−2, 1) ∪ (3, ∞).
Key Insight: The wavy-curve method is the most reliable way to solve polynomial inequalities. Remember: sign changes at simple roots, but not at repeated roots.
AM ≥ GM ≥ HM Inequality
Definition: For positive real numbers a and b, the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM) satisfy AM ≥ GM ≥ HM.
Arithmetic Mean
AM = (a + b) / 2
Average
Geometric Mean
GM = √(ab)
Square root of product
Harmonic Mean
HM = 2ab / (a + b)
Reciprocal of average of reciprocals
AM ≥ GM ≥ HM: (a + b)/2 ≥ √(ab) ≥ 2ab/(a + b). Equality holds iff a = b.
In general, for n positive numbers: (a₁ + a₂ + ... + aₙ)/n ≥ (a₁a₂...aₙ)^(1/n).
- Consequences: a + 1/a ≥ 2 for a > 0.
- If a + b is fixed, ab is maximum when a = b.
Important: The AM-GM inequality is one of the most powerful inequalities in mathematics. It is frequently used in JEE problems.
Exponential and Logarithmic Inequalities
Definition: The direction of exponential and logarithmic inequalities depends entirely on whether the base is greater than 1 or between 0 and 1.
Exponential Inequalities
Base > 1
a^f(x) > a^g(x) ⇒ f(x) > g(x)
Increasing function
0 < Base < 1
a^f(x) > a^g(x) ⇒ f(x) < g(x)
Decreasing function (reverse!)
Logarithmic Inequalities
Base > 1
logₐf(x) > logₐg(x) ⇒ f(x) > g(x) > 0
Increasing
0 < Base < 1
logₐf(x) > logₐg(x) ⇒ 0 < f(x) < g(x)
Decreasing (reverse!)
Example: (1/2)ˣ > 1/8. Base < 1, so reverse: x < 3. log₂x < 3 ⇒ 0 < x < 2³ = 8.
Key Insight: The direction of the inequality reverses when the base is between 0 and 1. Always check the base first!
Greatest Integer and Fractional Part Functions
Definition: The greatest integer function [x] (floor) returns the greatest integer ≤ x. The fractional part {x} = x − [x] lies in [0, 1).
Key Properties
Definition
[x] = greatest integer ≤ x
Floor function
Fractional Part
{x} = x − [x]
0 ≤ {x} < 1
Inequality
[x] ≤ x < [x] + 1
Range of floor
Integer Shift
[x + n] = [x] + n
For any integer n
Fractional Shift
{x + n} = {x}
Periodic with period 1
Equivalence
[x] = n ⇔ n ≤ x < n + 1
Important for solving
Examples: [3.7] = 3, [-1.2] = -2, {-1.2} = 0.8. [√2] = 1, so {√2} = √2 − 1.
Important: The greatest integer function is a step function. It jumps at every integer. Be careful with negative numbers!
Common Values to Memorise
| Value | Approximation |
|---|---|
| √2 | 1.414 |
| √3 | 1.732 |
| e | 2.718 |
| π | 3.141 |
| log₁₀2 | 0.301 |
| log₁₀3 | 0.477 |
| ln 2 | 0.693 |
| Memorise these values for quick calculations. | |
Practice Questions — From JEE and Boards
| Question | Answer |
|---|---|
| Q1: Simplify: √(x²) for x < 0. | √(x²) = |x| = -x (since x < 0). |
| Q2: Solve: |x - 3| = 5. | x - 3 = 5 ⇒ x = 8 or x - 3 = -5 ⇒ x = -2. |
| Q3: Evaluate: log₂(16) + log₃(27). | log₂16 = 4, log₃27 = 3. Sum = 7. |
| Q4: Solve: (x - 1)(x + 2) < 0. | Roots: -2 and 1. Expression is negative between -2 and 1. x ∈ (-2, 1). |
| Q5: If a + b = 5 and ab = 6, find a² + b². | a² + b² = (a+b)² − 2ab = 25 − 12 = 13. |
| Q6: Find the roots of x² - 4x + 3 = 0. | D = 16 - 12 = 4. x = (4 ± 2)/2 = 3, 1. |
| Q7: Evaluate: [3.7] + [-2.3]. | [3.7] = 3, [-2.3] = -3. Sum = 0. |
| Q8: Solve: 2ˣ > 8. | 2ˣ > 2³ ⇒ x > 3. |
| Practise these types of questions to become comfortable with applying Fundamentals of Mathematics concepts in exam scenarios. | |
All Fundamentals of Mathematics Formulas at a Glance
| Category | Formula |
|---|---|
| Number Systems | N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C |
| Algebraic Identities | (a+b)² = a²+2ab+b² · (a−b)² = a²−2ab+b² · a²−b² = (a+b)(a−b) |
| Indices | aᵐ·aⁿ = aᵐ⁺ⁿ · aᵐ/aⁿ = aᵐ⁻ⁿ · (aᵐ)ⁿ = aᵐⁿ · a⁰ = 1 |
| Logarithms | logₐ(mn) = logₐm+logₐn · logₐ(m/n) = logₐm−logₐn · logₐmⁿ = n logₐm |
| Modulus | |x| = x if x≥0, -x if x<0 · |x| ≥ 0 · √(x²) = |x| · |x+y| ≤ |x|+|y| |
| Quadratic Equations | D = b²−4ac · x = (-b±√D)/2a · α+β = −b/a · αβ = c/a |
| AM-GM | (a+b)/2 ≥ √(ab) ≥ 2ab/(a+b) |
| Greatest Integer | [x] ≤ x < [x]+1 · [x+n] = [x]+n · {x} = x−[x] |
| Memorise these formulas for the Fundamentals of Mathematics. They are the key to scoring full marks in this chapter. | |
Common Mistakes in Fundamentals of Mathematics
- √(x²) = x (without absolute value): √(x²) = |x|, not x. This is a very common error.
- Forgetting the domain of logarithms: A log exists only when the argument is positive and the base is > 0 and ≠ 1.
- Not reversing the inequality when multiplying by a negative: Multiplying or dividing an inequality by a negative number reverses the direction.
- 0⁰ is indeterminate: a⁰ = 1 only for a ≠ 0.
- |x| > a for a < 0: If a < 0, |x| > a is true for all x. If a ≤ 0, |x| < a has no solution.
- log(m + n) ≠ log m + log n: The log laws apply to products and quotients, not sums.
Golden Rule: In the Fundamentals of Mathematics, always check the domain first. Pay attention to absolute values, logarithm domains, and inequality signs.
Why Fundamentals of Mathematics Matters for JEE and Boards
- Foundation for all mathematics: The fundamentals of mathematics are the building blocks for every other chapter in JEE and board exams.
- High weightage: Questions based on these concepts appear in almost every exam paper.
- Conceptual clarity: This chapter rewards students who understand the concepts rather than just memorizing formulas.
- Practical relevance: These concepts are used everywhere, from physics to economics to computer science.
Why this guide helps: A comprehensive Fundamentals of Mathematics guide with all concepts, definitions, formulas, and practice questions saves you time during revision and helps you quickly recall everything during the exam. You won't need to look anywhere else.
Get the Complete Fundamentals of Mathematics PDF for Free
Download the full Fundamentals of Mathematics guide with all concepts, definitions, formulas, and practice questions. Perfect for last-minute revision before JEE and board exams.
Download Free PDFOpens in Google Drive · No sign-up needed
Master Class 11 Maths with Competishun
Competishun teaches complete JEE mathematics with structured lectures, DPPs, PYQs and an All India Test Series.
All taught by faculty with 18 to 25 years of experience.
Class 11 · JEE 2028
Pratham
Two-year foundation program that builds strong Class 11 maths, physics and chemistry concepts.
Explore PrathamClass 12 · JEE 2027
Prakhar
Class 12 program balancing boards, JEE Main and Advanced preparation together.
Explore PrakharDroppers · JEE 2027
Praveen & Fast Lane
Dropper program for a focused repeat attempt, including the accelerated Fast Lane batch.
Explore PraveenLive Batch
Pragyaan
Live interactive batch with real-time classes and mentorship for JEE and NEET.
Explore PragyaanDistance Learning
DLP Study Material
Complete printed material with DPPs and PYQs for Class 11, Class 12 and droppers, delivered home.
Explore DLPAll Stages
All India Test Series
Regular JEE and NEET level tests with detailed analysis to track and improve your rank.
Explore Test SeriesFree Maths Lectures on YouTube
Want to see the Fundamentals of Mathematics taught step by step with clear explanations? Competishun's full concept lectures are free on its YouTube channel.
More than 2.2 million+ students follow the channel for free concept lectures, PYQ solutions and exam strategy.
Learn the concepts, then use this Fundamentals of Mathematics guide to revise.
Build Your Maths Foundation
Class 11, Class 12 or dropper, explore every Competishun program.
Or start free with full concept lectures on YouTube.
About Competishun
Competishun is an online JEE and NEET coaching platform with programs for Class 11, Class 12 and droppers.
Complete DLP study material and an All India Test Series covering JEE Main, Advanced and NEET.
Taught by faculty with 18 to 25 years of experience.
More than 2.2 million+ students follow the Competishun YouTube channel.
Free JEE and Boards Concept Lectures
Follow Competishun for free Physics, Chemistry and Maths lectures, PYQ solutions and strategy.
Read These Next
Maths
Trigonometry Class 11: Complete Guide, All Formulas & Free PDF Download
Angle measurement, trigonometric ratios, identities, compound angles, equations, and graphs explained in depth.
Maths
Circles Class 11: Complete Guide, All Formulas & Free PDF Download
Standard and general equations, tangent, normal, chord, family of circles, and power of a point explained in depth.
Maths
Quadratic Equations Class 11: Complete Guide, All Formulas & Free PDF Download
Discriminant, nature of roots, Vieta's formulas, graphing parabolas, and word problems explained in depth.
Frequently Asked Questions — Fundamentals of Mathematics
What is the number system in mathematics?
The number system is the foundation of mathematics. It includes Natural numbers (N), Whole numbers (W), Integers (Z), Rational numbers (Q), Irrational numbers (Qc), Real numbers (R), and Complex numbers (C). The chain is N ⊂ W ⊂ Z ⊂ Q ⊂ R ⊂ C.
What are the laws of logarithms?
The laws of logarithms are: logₐ(mn) = logₐm + logₐn, logₐ(m/n) = logₐm − logₐn, logₐ(mⁿ) = n logₐm, and change of base: logₐN = log_bN / log_ba.
What is the wavy curve method for solving inequalities?
The wavy curve method is used to solve inequalities involving products or quotients of linear factors. Mark the roots on a number line, and draw a curve that starts positive from the far right and alternates sign at each simple root. At a factor of even multiplicity, the sign does not change.
What is the modulus function and its properties?
The modulus function |x| gives the distance of x from 0. Properties include: |x| ≥ 0, √(x²) = |x|, |xy| = |x||y|, and the triangle inequality: |x+y| ≤ |x| + |y|.
Can I download the Fundamentals of Mathematics formula sheet PDF for free?
Yes. You can download the complete Fundamentals of Mathematics formula sheet PDF for free using the download button on this page. It covers number systems, logarithms, inequalities, modulus, and quadratic equations in one comprehensive place for quick revision before JEE and board exams.
Tags
