# Class 12 Maths

Class 12 maths is highly crucial for students to perform well in various competitive exams like IIT JEE and others. In class 12 maths students continue to consolidate their understanding of secondary-level math skills. Thus, learning class 12 maths requires students to have a deep conceptual knowledge of important skills acquired in previous grades.

## Class 12 Maths Syllabus

Students often find class 12 maths ambiguous due to the improper understanding of concepts and topics. To understand the various concepts and topics it is necessary to have a clear understanding of the complete syllabus covered under 12th maths syllabus. Here are the chapters in 12th maths along with a clear list of subtopics.

**Functions**

**Relations and functions**- Types of relations: empty relation, universal relation, reflexive relation, symmetric relation, transitive relation, equivalence relation
- Types of functions: one-one function, onto function, bijective function
- Composition of functions
- Invertible function and Inverse function
- Binary operation: identity element, invertible binary operation, commutative and associative properties

**Inverse trigonometric functions**- Inverse trigonometric functions: sin
^{-1}x, cos^{-1}x, tan^{-1}x, csc^{-1}x, sec^{-1}x, cot^{-1}x - Domain and range of inverse trigonometric functions
- Formulas related to inverse trigonometric functions

- Inverse trigonometric functions: sin

**Algebra**

**Matrices**- Definition of matrices
- Types of matrices: column matrix, row matrix, square matrix, diagonal matrix, scalar matrix, identity matrix, equal matrices
- Scalar multiplication of matrix
- Properties of matrices
- Operations on matrices: addition, subtraction, multiplication
- Symmetric and skew-symmetric matrices
- Elementary operations on matrices
- Inverse of a matrix

**Determinants**

**Three Dimensional Geometry**

- Direction cosines of a line
- Direction ratios of a line
- Skew lines and the angle between them
- Angle between two lines
- Equation of a line in 3D in different forms
- Shortest distance two lines
- Distance between parallel lines
- Equation of plane in different forms
- Angle between two planes
- The distance from a point to a plane

**Vector Algebra**

- Position vector of a point
- Magnitude of vector
- Types of vectors
- Addition of vectors
- Multiplication of vectors by a scalar
- Components of vector
- Vector joining two points
- Section formula of vectors
- Product of two vectors: scalar (dot) product, vector (cross) product
- Projection of vector

**Linear Programming**

- Objective function
- Few linear programming problems like diet problems, manufacturing problems, and transportation problems
- Feasible region and feasible solution
- Optimal solution
- Theorems on linear programming:
- Theorem 1: The optimal value should occur at a corner point of the feasible region
- Theorem 2: If the feasible region is bounded then the objective function has both maximum and minimum value.

- Solving linear programming problems by corner point method.

**Probability**

- Conditional probability
- Independent events
- Baye's theorem
- Mean, variance, and standard deviation of a random variable
- Bernoulli trials

**Calculus**

**Continuity and differentiability**- Definition and properties of a continuous function
- Differentiable functions
- Chain rule
- Derivatives of inverse trigonometric functions
- Derivatives of e^x and log x
- Logarithmic differentiation
- Rolle's theorem
- Mean value theorem

**Applications of derivatives**- Rate of change
- Increasing and decreasing functions
- Tangents and normals
- Critical points
- First derivative test and second derivative test
- Maxima and minima: local maxima and minima absolute maxima and minima

**Integrals****Applications of integrals****Differential equations**- Definition of differential equations
- Order and degree of differential equations
- Solution of differential equation and arbitrary constants
- Variable separable method
- Homogeneous differential equation
- First-order linear differential equation

## Develop class 12 Maths Skills

Class 12 maths is outlined to impart the knowledge of analytical and practical skills in students. It develops a basic understanding of applying maths, for instance, using trigonometric functions to calculate the height or applying algebraic formulas to solve real-life problems. Students can acquire an understanding of these class 12 maths concepts through various interactive activities and visualization. Here are some key class 12 maths skills that students need to develop.

**Inverse Trigonometric Functions: **Inverse trigonometric functions and their applications are important topics covered under mathematics class 12 that students should understand and apply in solving various problems.

**Calculus:** A large amount of calculus is involved in 12th maths like applications of derivatives, integrals (the reverse process of differentiation) and their applications and differential equations. Students must need in-depth understanding of these concepts which would help them in their career-related courses going forward.

**Vector Algebra:**Vectors is completely new for mathematics class 12 students and hence the concept is introduced from basics and the students feel easy to study this chapter. Hence, they can score marks very easily in this chapter during their 12th maths exam.

## Class 12 Maths Tips and Tricks

**Practice Daily:** Class 12 maths is crucial for developing higher-level maths skills thus students must practice daily and track their progress sincerely to improvise their skills.

**Practice Past Years Question Papers:** Practicing question papers from past years will help students to get an idea of the complexity level of the exam. It will also benefit students in enhancing their speed and accuracy

**Cover Full Syllabus:** Covering the full syllabus of class 12 maths is a must for students to clarify all their doubts before exams.

**Refer to Reliable Study material:** Refer reliable study material like NCERT Solutions and NCERT textbooks for the preparation of class 12 maths. These are the best preparation resources as the class 12 maths exam is set according to them.

## Class 12 Maths Formulas

Students of class 12 maths need to memorize so many formulas related to inverse trigonometric functions, vectors, different parts of calculus, vector algebra, etc. Here are some important formulas of mathematics class 12.

- sin
^{-}^{1}x + cos^{-1}x = π / 2 - tan
^{-}^{1}x + cot^{-}^{1}x = π / 2 - sin
^{-}^{1}(-x) = -sin^{-}^{1}x - cos
^{-}^{1}(-x) = π - cos^{-}^{1}x - tan
^{-}^{1}(-x) = -tan^{-}^{1}x - Dot Product: (A • B )= |P| |Q| cos θ
- Cross Product: (A × B )= |P| |Q| sin θ
- P(A ∩ B) = P(A) P(B | A)
- ∫ x
^{n}dx = (x^{n+1}) / (n+1) + C - ∫ e
^{x}dx = e^{x}+ C - ∫ a
^{x}dx = a^{x}/ ln(a) + C - ∫ (1/x) dx = ln x + C
- Distance between two points P(x
_{1}, y_{1}, z_{1}) and Q(x_{2}, y_{2}, z_{2}) in 3D**:**PQ = √ ((x_{1}- x_{2})^{2}+ (y_{1}- y_{2})^{2}+ (z_{1}- z_{2})^{2})

For a complete and detailed list of class 12 maths formulas, click here.

## Solved Examples on Mathematics Class 12

**Example 1:** What is the derivative of y = sin 2x?

**Solution:**

We know that the derivative of sin x is cos x. But we have sin 2x here. So we apply chain rule of differentiation here. Then we get

y'= d/dx (sin 2x)

= cos 2x · d/dx (2x)

= 2 cos 2x

**Answer:** The derivative of sin 2x is 2 cos 2x.

**Example 2:** What is the value of sin^{-1}(-√3/2)?

**Solution:**

By the rules of inverse trigonometric functions, sin^{-1}(-x) = - sin^{-1}x. Then we get

sin^{-1}(-√3/2) = -sin^{-1}(√3/2)

We know from the trigonometric table that sin π/3 = √3/2. This gives sin^{-1}(√3/2) = π/3. Then the above equation becomes:

sin^{-1}(-√3/2) = -π/3

**Answer:** The value of the given inverse trigonometric ratio is -π/3.

## FAQs on Class 12 Maths

### What are Class 12 Maths Chapters?

It is very important for class 12 maths students to keep the list of chapters handy. Here is the list of all chapters. Subtopics are also listed in detail on the same page above.

- Relations and functions
- Inverse trigonometric functions
- Matrices
- Determinants
- Three Dimensional Geometry
- Vector Algebra
- Linear Programming
- Probability
- Continuity and differentiability
- Applications of derivatives
- Integrals
- Applications of integrals
- Differential equations

### How are Class 12 Maths Solutions Important?

12th maths solutions provide step-by-step guidance for solving the problems of each chapter. Hence, class 12 maths students can use them and also keep them in handy to make themselves perfect in all chapters.

### How to Get a Good Score in 12th Maths?

To get very good scores in class 12 maths, try to grasp the concepts first and then move on to solving the questions. Try to clear doubts on a regular basis. Make notes of everything during the lessons. Practice previous year's papers and sample papers. Prepare a timetable to finish the syllabus on time and stick to it. Solve as many worksheets as possible.

### Which is the Hardest Chapter in Class 12 Maths?

Class 12 maths has many concepts that are newly introduced this year. Most of the students feel some chapters like determinants, integrals, differential equations, etc difficult. The hardest chapter is integrals for most of them.

### Is Class 11 Maths Easier or Class 12 Class 12 Maths?

Of course, class 11 maths is easier than that of class 12. In class 12, some concepts of class 11 are advanced and along with that many new concepts like vectors, integrals, etc are newly introduced.

### Is Mathematics Class 12 Hard?

It is just that there are many topics in class 12 maths that are new to students. But class 12 maths is not hard as long as the student tries to understand the concepts and pay attention during the lessons.

### How can I get Full Marks in Class 12 Maths?

To get full marks in class 12 maths, one should be thorough with all the topics. Focus more on new topics like integrals, applications of integrals, differential equations, vectors, etc. Once solving the textbook problems is done, solve extra problems from worksheets. Always check your answers and marks your mistakes for revision. Keep revising the topics and formulas. With these techniques, one can definitely achieve full marks in 12th maths.

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