The time to test the preparations has reached. CBSE Class 10 and 12 Term 2 board exams have started. Students are in their full energised state. They are trying to prove their dedication by achieving an excellent score.
June 7, 2022, is the CBSE Class 12 Term 2 Mathematics exam date. It is one of the major subjects for the Class 12 Science and Commerce stream students. Term 2 exams are going to be in subjective mode. Hence, there is a need to focus more on the revision.
Mathematics is an interesting and fun subject. But, students find themselves with an unknown fear towards it. Only practice is the thing that can reduce this fear in you and bring confidence in you. To help you build your confidence, we are here with the NCERT concepts notes for the CBSE Class Term 2 Syllabus Mathematics.
This article will find all the important Mathematics notes of NCERT related to the CBSE Class 12 Term 2 Maths syllabus.
|Table Of Contents|
|Syllabus & Weightage Of Class 12 Term 2 Mathematics Exam 2022|
|Exam Pattern Of Class 12 Term 2 Mathematics Exam 2022|
|Chapter-Wise Notes Of Important Concepts Of Mathematics Based On NCERT For Class 12 Term 2 Board Exam|
Did you know why every expert advises you to prepare from NCERT books? It is because CBSE mainly focuses on the NCERT syllabus, and other than this, NCERT textbooks and NCERT Solutions themselves contain every required information for each topic. But, the student needs to give some extra attention to memorise them.
Before going through the notes of NCERT concepts, it is necessary to understand the syllabus and exam pattern of the Term 2 exam. Let’s take a look at it first.
Syllabus & Weightage Of Class 12 Term 2 Mathematics Exam 2022
The latest updated syllabus of the Term 2 Mathematics exam 2022 for the Class 12 with the marking scheme is as follows:
|Unit Name||Chapter Name||Weightage|
|Vector & 3-D Geometry||
Exam Pattern Of Class 12 Term 2 Mathematics Exam 2022
The exam pattern for the Class 12 Term 2 Mathematics exam 2022 is as follows:
|Section||Type Of Question||Number Of Question||Weightage|
|A||Short Answer Type Questions||6||12|
|B||Short Answer Type Questions||4||12|
|C||Long Answer Type Questions||4||16|
Also See: CBSE Class 12 Answer Key 2022
Chapter-Wise Notes Of Important Concepts Of Mathematics Based On NCERT For Class 12 Term 2 Board Exam
The notes of Cbse Class 12 Maths important concepts of each chapter are as follows:
Chapter – Integral
Integration is the reverse process of differentiation.
- Some standard formulae of integration
- ∫xⁿ dx = (xⁿ⁺¹/n+1) + C, n ≠ -1
- ∫1/x dx = log |x| + C
- ∫ eˣ dx = eˣ + C
- ∫ aˣ dx = (aˣ/logₑa) + C
- ∫sin x dx = -cos x + C
- ∫cos x dx = sin x + C
- ∫sec²x dx = tan x + C
- ∫cosec²x dx = -cot x + C
- ∫sec x tan x dx = sec x + C
- ∫cosec x cot x dx = -cosec x + C
- Methods of integration
There are five ways to solve integration.
- Integration by substitution
- Integration by parts
- Integration by partial fraction
- Integration using trigonometric identities
- Integration by a particular fraction
- Integration using partial fractions
We first resolve the denominator of the given fraction into the simplest factors. Based on these factors, we obtain the corresponding partial fractions.
- Definite integrals
Let F(x) be an antiderivative of f(x), then for any two values of the independent variable x, say a and b, the difference F(b) – F(a) is called the definite integral.
∫f(x)dx = F(b)−F(a), limit a→b
- Properties Of definite integrals
- Definite integral as a limit of a sum
The definite integral ∫ x dx, with a limit a→b of sum, is given by
∫ x dx = [b² – a²]/2, limit a→b
Chapter – Application of Integrals
- Area under curves
Let y = f(x) be a continuous and finite function in [a,b]
Area = ∫f(x)dx = F(b)−F(a), limit a→b
- The area bounded by two curves
Let f(x) and g(x) be continuous in [a,b] interval, then the area bounded by two curves is defined by,
Area = ∫ [f(x) – g(x)] dx, limit a→b
- The area bounded by a curve and a line
If the curve is defined as y = f(x) and line by y = g(x), then the area bounded by a curve and a line is defined as,
Area = ∫ [f(x) – g(x)] dx, limit a→b
Chapter – Differential Equation
An equation containing an independent variable, a dependent variable, and the derivatives of the dependent variable is called a differential equation.
- Differential equation formation
Step 1: Differentiate the equation of the given family of curves n times to get n more equations.
Step 2: Eliminate n constants using these (n+1) equations.
- Linear differential equation
The most general form of a linear differential equation is dy/dx + Py = Q, where P is a constant and Q is a constant or a function of x.
(dy/dx) + Py = Q
- Homogeneous differential equation
An equation of the form dy/dx = f(x,y)/g(x,y), where both f(x,y) and g(x,y) are homogeneous functions of degree n, is called homogeneous differential equation.
dy/dx = (x²-y²)/xy
Chapter – Vectors
A definite magnitude and definite directions specify vector quantities.
- Product of vectors
The scalar product of vectors is solved by using this formula:
We can express the scalar product as:
- Laws of vectors
- Vector addition is commutative.
- Vector addition is associative.
- Vector addition shows the existence of additive identity.
- The additive inverse also exists in vector addition.
Chapter – Three – dimensional Geometry
- Direction cosine and direction ratios
Direction cosines of a line:
If a line makes angles , , with the x-axis, y-axis, and z-axis, respectively,
l = cos , m = cos , n = cos
are called the direction cosines.
Direction ratios of a line:
Any three numbers a, b, c, proportion to the direction cosines l, m, n respectively of a line, are called the direction ratios of the line.
Direction ratios = l/a = m/b = n/c
- Cartesian equation of a line
The equations of a line passing through two given points, A(x₁, y₁, z₁) and B(x₂, y₂, z₂), are given by
(x-x₁)/(x₂-x₁) = (y-y₁)/(y₂-y₁) = (z-z₁)/(z₂-z₁)
- Distance between two lines
If you have two parallel lines
r = a₁ +b and r = a₂ +b
The shortest distance between them is given by [ | a₂ – a₁ | x b] / |b|
- Distance between a point and a plane
If there is a point P (x₀, y₀, z₀) and a plane Q with the equation Ax+By+Cz = D.
The distance between P and Q is given by
Distance = [ |Ax₀ + By₀ +Cz₀ + D| ] / √ (A² + B² + C²)
Chapter – Probability
This unit is based on the multiplication theorem on probability for independent events and conditional probability.
- Multiplication theorem of probability
If A and B are two events of a sample space such that
P(A) ≠ 0 and P(B) ≠ 0
P(A∩B) = P(A) * P(B/A) = P(B) * (PA/B)
- Bayes’ Theorem
Bayes’ theorem is stated mathematically as the following equation:
- Binomial distribution
Conditions for binomial distribution:
- The experiment is performed for a finite and fixed number of trials.
- Each trial must give either a success or a failure.
- The probability of success in each trial is the same.
- Random variable and its probability distribution
The probability distribution of a random variable X is defined only when
- Each pₗ >= 0
- ∑pₗ = 1
Mathematics asks for practice, and it is hard to achieve an excellent score in this exam without practice. Due to the new examination pattern from earlier, you have more chances to perform better than previous batches’ students.
If you find the fear of Mathematics, you only have to do one thing, i.e., go through NCERT Solutions for Class 12 Mathematics and books and revise from there. Do you believe that preparation from NCERT books advances you in scoring best in your board exams and helps you with competitive exams like JEE, NEET, CUET, and other exams syllabus at the same time?
We hope the above-detailed information and concepts notes will help you boost your CBSE Class 12 Term 2 Mathematics exam 2022 score.
Wish you all the best!
1. Why do we use definite integrals?
We use integrals to find areas, volumes, and many useful things. However, often we use it to find the area under the graph of a function like this. We can find this area by adding slices that approach zero in width. Moreover, there are rules of integration that help to get the answer.
2. What tips should I follow for the Class 12 Term 2 Mathematics exam?
You should follow the given below tips:
- Update yourself with the latest syllabus, exam pattern, and marking scheme of the Class 12 Term 2 Mathematics exam.
- Go through all study material, especially NCERT books.
- Revise the notes, formulae, and important concepts of each unit.
- Solve plenty of sample papers, previous years’ papers, and the latest mock papers.
- If you are stuck with any problem, ask for help from your peers, teachers, and experts.
3. What is the internal assessment marking scheme for the Class 12 Term 2 Mathematics exam 2022?
The marking scheme of internal assessment for Class 12 Term 2 Mathematics exam 2022 is as follows:
|Activity file record and Term-end assessment of one activity & Viva||5|
It is necessary to perform well in periodic tests and Mathematics activities conducted in the school to fetch maximum marks. These scores will be added to the Term 2 board exams.
4. What are the important topics of Vectors for Class 12 Term 2 Mathematics exam 2022?
The important topics of Vectors for Term 2 board exam 2022 are as follows:
- Vectors & scalars
- Magnitude & direction of a vector
- Direction cosines & direction ratios of a vector
- Types of vectors
- The position vector of a point
- Negative of a vector
- Components of a vector
- Addition of vectors
- Multiplication of a vector by a scalar
- The position vector of a point dividing a line segment in a given ratio
- Definition & geometrical interpretation of a vector
- Properties & applications of dot product and cross product of vectors
5. Find the general solution of the differential equation x dy – (y + 2×2) . dx = 0
We can write the simplified form of the differential equation as:
(dy/dx) – (y/x) = 2x
Comparing this with the differential equation dy/dx + Py = Q,
We have the values of P = -1/x and Q = 2x
The integration factor IF =
e ∫(−1/x) . dx = e−log x =1/x
We need to solve the equation as:
y 1/x = ∫2x . 1/y . dx + c
y/x = ∫2 . dx + c
y/x = 2x + c
y = 2×2 + xc, which is the required solution of the differential equation.