Class 12 Maths: These Are The Most Important Topics For Board Exams, JEE Main

Mathematics Class 12 Weightage

The CBSE Class 12 board exam in Mathematics has no practical component and the weightage for each topic is fixed. However, in the Joint Entrance Examination (JEE) Main for engineering, there is no fixed distribution.

According to the marks distribution across the Class 12 Maths syllabus, Calculus, Vectors and 3D Geometry, and Algebra have more weightage than other units. Given below are the marks that each unit and sub-unit carries in the descending order of weightage.

Calculus: 35 Marks

Calculus questions together account for 35 out of 80 marks in the Class 12 board examinations. So, you should begin your studies with calculus because it covers almost 50 percent of the total marks. There are five topics in calculus but Continuity and Differentiability and Integrals carry more weight than the other three topics. Both short-answer and long-answer questions have been asked from these units.

Topics and marks:

• Continuity and Differentiability: 9 marks

• Integrals: 9 marks

• Differential Equations: 7 marks

• Application of Derivative and Application of Integrals: 10 marks

Vectors and 3D Geometry: 14 Marks

It has the second-highest weightage after calculus. Most of the time long answer type questions have been asked in the previous examination from this unit.

Topics and marks:

• Vectors: 7 marks

• 3D Geometry: 7 marks

Algebra: 10 Marks

This is the third-highest weighted unit.

These are the marks each topic carries:

• Matrices: 5 Marks

• Determinant: 5 marks

Probability: 8 Marks

This is a small unit from which short-answer questions are the most common. Most questions that have been asked so far are application-based or real-life situations based – card or dice games, choosing a colour, and more.

Relations and Functions: 8 Marks

This unit includes inverse trigonometry (4 marks ) and relations and functions (4 marks). Formula-based short-answer questions are most commonly asked from the trigonometry section.

Linear Programming: 5 Marks

This is a small unit bearing the least weightage. Typically, a long answer type question has been asked from this unit in previous board exams.

Also Read | Class 12 Board Exams - Top Maths Formulae You Should Know

JEE Main And Maths

Analysis of previous years’ JEE Main question papers has shown that questions from some topics are asked repeatedly in the engineering entrance exam. Careers360’s analysis could be a useful tool in developing an effective preparation strategy. These are the most important.

Linear Differential Equation

There are several important concepts that you will need to understand in order to solve linear differential equations in the JEE Main. These include:

• Order and degree of a differential equation: The order of a differential equation is the highest derivative that appears in the equation, and the degree is the highest power of the derivative. Understanding these concepts is important because they determine the general form of the solution to the differential equation.

• Linearity: A differential equation is linear if it is a linear combination of the unknown function and its derivatives, with constant coefficients. Solving linear differential equations is generally easier than solving nonlinear differential equations.

• Separable equations: A differential equation is separable if it can be written in the form dy/dx = f(y)g(x), where f(y) and g(x) are functions of y and x, respectively. These types of differential equations can be solved by separating the variables and integrating them.

• Homogeneous equations: A differential equation is homogeneous if it is of the form y' = f(y/x)" or y" + f(y/x) = 0. These types of differential equations can be solved by making a substitution and then solving the resulting equation.

• Exact equations: A differential equation is exact if it can be written in the form M(x,y)dx + N(x,y)dy = 0, where M and N are functions of x and y. These types of differential equations can be solved by finding a function that satisfies the equation.

• First-order linear equations: A differential equation is first-order linear if it is of the form y' + p(x)y = g(x). These types of differential equations can be solved using techniques such as integrating factors or variation of parameters.

Applications of Permutation

To understand the applications of permutations, you will need to understand the following concepts:

• Permutations: A permutation is an arrangement of objects in a specific order. The number of permutations of n objects is given by n!.

• Permutations with repetition: A permutation with repetition is an arrangement of objects in which some objects may be repeated. The number of permutations with repetition of n objects, where the objects can be repeated r times, is given by n^r.

• Permutations with restrictions: A permutation with restrictions is an arrangement of objects in which certain objects must be placed in specific positions. The number of permutations with restrictions can be calculated using the principle of inclusion-exclusion.

• Combinations: A combination is a selection of objects in which the order does not matter. The number of combinations of n objects taken r at a time is given by n!/(r!(n-r)!).

• Permutations and combinations in counting problems: Permutations and combinations can be used to solve counting problems in which you need to determine the number of ways in which a certain event can occur.

Also Read | CBSE Class 12 Biology: Marks Allotted To Important Topics, Paper Pattern, Theory And Practicals

General Term of Binomial Expansion

The general term of the binomial expansion of a binomial raised to a power n can be expressed using the binomial coefficient and the terms of the binomial:

(a+b)ⁿ = ⁿC₀ aⁿ b⁰ + ⁿC₁ a^n-1 b¹ + ⁿC₂ a^n-2 b² + ............+ ⁿC_n-1 a¹ b^n-1 + ⁿC_n a⁰ bⁿ

where ⁿC_k represents the binomial coefficient, which is calculated as ⁿC_k = n! / k! (n-k)! here n! represents the factorial of n (n factorial).

Cramer’s Law

Cramer's rule is a method for solving systems of linear equations. It can be used to find the values of the variables in a system of n linear equations with n unknowns.

The general form of Cramer's rule for a system of n linear equations with n unknowns is:

x₁ = (D₁/D) (a₁₁x₁ + a₁₂x₂ +......+ a_1nx_n)

x₂ = (D₂/D) (a₂₁x₁ + a₂₂x₂ +......+ a_2nx_n)

……

x_n = (D_n/D)(a_n1x₁ + a_n2x₂ + ......+ a_nnx_n)

where D is the determinant of the coefficient matrix A, and D_i is the determinant of the matrix obtained by replacing the ith column of A with the column vector b.

Cramer's rule can be used to solve systems of linear equations, but it can be computationally expensive and may not be the most efficient method in all cases.

Area Bounded by Two Curves

To calculate the area bounded by two curves, you can use the following steps:

1. Identify the two curves and the region they are bound. The region should be between the two curves and above the x-axis.

2. Determine the limits of integration. These will be the values of x where the two curves intersect.

3. Choose one of the curves as the base and the other as the top curve. The base curve will be used to set up the integral, and the top curve will be used to calculate the height at each point along the base curve.

4. Set up the integral using the base curve as the lower limit of integration and the top curve as the upper limit of integration. The integral should be set up with respect to x.

5. Evaluate the integral to calculate the area bounded by the two curves.

Piecewise Definite Integration

It is a method for integrating a function that is defined piecewise, or, as a combination of several functions defined at different intervals.

To perform piecewise definite integration, you need to first identify the different intervals on which the function is defined, and then evaluate the definite integral of the function on each interval separately.

Multiplication of Two Matrices

To multiply two matrices, you need to follow these steps:

• The product of two matrices is defined only when the number of columns in the first matrix is equal to the number of rows in other matrices therefore students should first ensure that the number of columns in the first matrix is the same as the number of rows in the second matrix.

• To calculate each element in the result matrix, take the dot product of each row in the first matrix with each column in the second matrix. The dot product is calculated by multiplying the corresponding elements of the row and column and adding them together.

Also Read | Class 12 Board Exams - Important Topics In Physics You Must Never Omit

Dispersion (Variance and Standard Deviation)

Dispersion is a measure of how spread out a set of data is. There are two commonly used measures of dispersion: variance and standard deviation.

Variance is a measure of the average squared deviation of a set of data from the mean. It is calculated as follows:

• Calculate the mean of the data.

• For each data point, subtract the mean from the data point and square the result.

• Calculate the average of all the squared deviations.

The formula for variance is:

Variance = S = √{(x - μ)² / (n-1)}

where x is a data point, is the mean of the data, and n is the number of data points.

Standard deviation is a measure of the average deviation of a set of data from the mean. It is calculated as the square root of the variance. The formula for standard deviation is:

Standard Deviation= σ = √ (Variance)

Algebra of Statements

In the algebra of statements, we use letters to represent statements and we use logical operations to combine these statements. The most common logical operations are negation, conjunction, and disjunction.

Family of Planes

In mathematics, a plane is a flat, two-dimensional surface that extends infinitely in all directions. It is defined by a set of three points that are not all on the same line, known as the vertices of the plane. In Class 12 math includes topics such as the distance between a point and a plane, the angle between a plane and a line, and the intersection of planes.

In three dimensions space equation of a plane is written as

ax+by+cz+d=0

Where a, b, and c are the coefficient of the normal vector to the plane and d is the distance from the origin to the plane.

A family of planes is a collection of planes that are all related. There are several different types of families of planes that you might encounter like parallel planes, orthogonal planes, perpendicular planes, tangent planes, and more.