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Learn MoreLatest Blog: Mastering Calculus for Class 12 Board Exams
Calculus forms a crucial part of the Class 12 CBSE Mathematics curriculum and is essential for engineering and medical entrance exams. Understanding calculus thoroughly can help students solve complex problems in higher education.
In this comprehensive guide, we'll cover differentiation, integration, differential equations, and their applications. We provide step-by-step solutions to typical board exam problems, highlighting common mistakes students make and how to avoid them.
Additionally, we'll discuss the importance of calculus in real-life applications and provide a curated list of practice questions from NCERT textbook and previous year papers. By the end of this guide, you'll have a solid understanding of calculus concepts and be able to solve any calculus-related problem in your board exams.
Read Full Article →Mathematics for Class 12
Chapter 1: Relations and Functions
This chapter introduces types of relations, functions, composition of functions, invertible functions, and binary operations.
Key Concepts:
- Relations: Types of relations - reflexive, symmetric, transitive, equivalence
- Functions: One-one, onto, bijective functions
- Composition: Composition of functions and invertible functions
- Binary Operations: Commutative, associative, identity element, inverse
Example Problem:
Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a - b| is even} is an equivalence relation.
Solution:
To prove R is equivalence relation, we need to show it is:
1. Reflexive: |a - a| = 0 (even) ⇒ (a,a) ∈ R
2. Symmetric: If |a - b| is even ⇒ |b - a| is even ⇒ (b,a) ∈ R
3. Transitive: If |a - b| is even and |b - c| is even ⇒ |a - c| is even ⇒ (a,c) ∈ R
Hence, R is an equivalence relation.
Practice Worksheet (Downloadable PDF)
Download Relations and Functions WorksheetChapter 2: Calculus - Differentiation
This chapter covers continuity, differentiability, derivatives of composite functions, chain rule, and derivatives of inverse trigonometric functions.
Physics for Class 12
Chapter 1: Electric Charges and Fields
This chapter covers electric charge, Coulomb's law, electric field, electric dipole, Gauss's law and its applications.
Key Concepts:
- Coulomb's Law: Force between two point charges
- Electric Field: Due to point charge and dipole
- Gauss's Law: Applications to find field due to infinite plane sheet, line charge etc.
- Dipole: Field and torque on dipole in uniform electric field
Interactive Diagram: Electric Field Lines
Fig 1.1: Electric field lines for different charge configurations
Numerical Problem:
Calculate the electric field intensity due to a dipole of length 10 cm and charge ±5 μC at a point 20 cm from the center of the dipole on its axial line.
Solution:
Given: 2a = 10 cm ⇒ a = 5 cm = 0.05 m
q = 5 μC = 5 × 10⁻⁶ C
r = 20 cm = 0.2 m
Eaxial = (1/4πε₀) (2p/r³)
where p = q × 2a = 5 × 10⁻⁶ × 0.1 = 5 × 10⁻⁷ Cm
Eaxial = 9 × 10⁹ × (2 × 5 × 10⁻⁷)/(0.2)³
= 9 × 10⁹ × 10⁻⁶/8 × 10⁻³
= 1.125 × 10⁶ N/C along the dipole axis
Chemistry for Class 12
Chapter 1: The Solid State
This chapter covers classification of solids, crystal lattices, unit cells, packing efficiency, defects in solids and electrical properties.
| Type of Solid | Constituent Particles | Examples | Properties |
|---|---|---|---|
| Molecular | Molecules | Ice, Iodine | Soft, low melting point |
| Ionic | Ions | NaCl, CaF₂ | Hard, brittle, high melting point |
Numerical Problem:
An element crystallizes in fcc lattice with edge length 400 pm. Calculate the density if 200 g of this element contains 2.5 × 10²⁴ atoms.
Solution:
For fcc, Z = 4
a = 400 pm = 400 × 10⁻¹⁰ cm
Volume = a³ = (400 × 10⁻¹⁰)³ = 6.4 × 10⁻²³ cm³
2.5 × 10²⁴ atoms = 200 g
6.022 × 10²³ atoms = (200 × 6.022 × 10²³)/(2.5 × 10²⁴) = 48.176 g
Density = (Z × M)/(a³ × Nₐ)
= (4 × 48.176)/(6.4 × 10⁻²³ × 6.022 × 10²³)
= 5 g/cm³
Biology for Class 12
Chapter 1: Reproduction in Organisms
This chapter covers asexual and sexual reproduction, pre-fertilization and post-fertilization events, and reproductive health.
Diagram: Human Female Reproductive System
Fig 1.1: Structure of human female reproductive system
Important Questions:
- Differentiate between asexual and sexual reproduction.
- Draw a labeled diagram of a mature embryo sac.
- Explain the events of fertilization in flowering plants.
English for Class 12
Flamingo - Chapter 1: The Last Lesson
This story by Alphonse Daudet highlights the importance of language and the impact of political decisions on education.
"For the last two years all our bad news had come from there - the lost battles, the draft, the orders of the commanding officer... Now it was their turn. The master had put on his fine Sunday clothes... and the old men of the village were sitting quietly at the back of the room."
Important Questions:
- Why was Franz reluctant to go to school that morning?
- What changes did Franz notice in the school that day?
- What was the significance of the last lesson?
Accountancy for Class 12
Chapter 1: Accounting for Partnership Firms
This chapter covers fundamentals of partnership, partnership deed, profit and loss appropriation account, and guarantee of profits.
Example Problem:
A and B are partners sharing profits in the ratio of 3:2. C is admitted as a new partner for 1/5th share which he acquires equally from A and B. Calculate the new profit sharing ratio.
Solution:
C's share = 1/5 (acquired 1/10 from A and 1/10 from B)
A's new share = 3/5 - 1/10 = (6-1)/10 = 5/10
B's new share = 2/5 - 1/10 = (4-1)/10 = 3/10
C's share = 1/5 = 2/10
New ratio = 5:3:2
CBSE Board Exam Preparation
Time Management Strategy
Allocate specific time slots for each subject based on weightage. Follow the 50-10 rule (50 minutes study, 10 minutes break).
Sample Papers
Solve CBSE sample papers and previous 10 years' papers to understand the exam pattern and marking scheme.
Revision Techniques
Create concise notes, mind maps and flashcards for quick revision of important concepts and formulas.
Board Exam Pattern
| Subject | Total Marks | Theory | Practical |
|---|---|---|---|
| Physics | 100 | 70 | 30 |
| Chemistry | 100 | 70 | 30 |
| Mathematics | 100 | 80 | 20 (Internal) |
| Accountancy | 100 | 80 | 20 (Project) |