The math of 12th grade stresses on the application of the topics that are learned in 11th grade and are required at higher levels. While the math of grade 11 lays a strong basis for higher level mathematics, the 12th grade math stresses more on the application part in addition to the level of complexity.

The basics concepts that are learned in previous grade are taken to advance level of applicative knowledge and importance is given to make use of the topics learned in the real life. 12th grade content of mathematics is divided differently where some of the topics are common to different categories which everything learning math has to study and some are different according to the section been chosen.

The basics concepts that are learned in previous grade are taken to advance level of applicative knowledge and importance is given to make use of the topics learned in the real life. 12th grade content of mathematics is divided differently where some of the topics are common to different categories which everything learning math has to study and some are different according to the section been chosen.

**For Example:**the math students with science study mathematics I and the commerce students study mathematics II. This made it easy for students in not wasting their time and energy on the things they do not require to study unnecessarily.This helps in increased potential of the student in the topics learned. Regular practice and timely revision are the keys for a student to understand easily all the levels may it be easy, medium of hard, that are taught to students.

## Topics

The 12th grade math is categorized in two types as the math of 11th grade, that is, Mathematics I & Mathematics II.

The difference in the math I & math II is because of their use in the studies that a student would have chosen in previous grade. In grade 11 the student decides on whether to take up science or commerce or arts subjects etc. and mathematics is usually one of the common subjects in all the categories of choices given. The difference is in the manner we study.

The common topics that are included in both the categories (commerce and science) of Mathematics are: matrices, determinants, adjoints & inverse of a matrix, differentiability, application of derivatives, indefinite and definite integrals, application of integrals, differential equations and probabilityThe topics that differ in mathematics -I are: sets, relations &functions, vectors and three dimensional geometry.

The topics that differ in mathematics II are: inverse trigonometric functions, application of calculus in commerce & economics, annuities, and linear programming.

## Examples

**Let us take some of the example which is common in both the mathematics I and II:**

**Example 1:**Evaluate the integral given below:

$\int {csc\ x . cot\ x - sec^2x} dx$

**Solution:**

We are given the following:

$\int {csc\ x . cot\ x - sec^2x} dx$

Using the properties of operations on integrals we have,

= $\int{ csc\ x. cot x dx - \int sec^2 x dx

We make use of d/dx (csc x) = -csc x . cot x and d/dx (tan x) = sec^2 x

This implies that

$\int {csc\ x . cot\ x - sec^2x} dx$ = - csc x - tan x + A

Where, A is considered to be the constant of integration.

2 & 5 & 1\\

5 & -3 & 4\\

6 & 5& 0

\end{bmatrix}$

2 & 5 & 1\\

5 & -3 & 4\\

6 & 5& 0

\end{bmatrix}$

det of A = $\begin{vmatrix}

2 & 5 & 1\\

5 & -3 & 4\\

6 & 5& 0

\end{vmatrix}$

= 2(-3 $\times$ 0 - 5$\times$4) - 5(5$\times$0-5$\times$1) + 6(5$\times$4 - (-3)$\times$1)

= 123

Differentiate f(x) with respect to x

f'(x) = 2x - y + (sin x(-sin x) + cos x ( cos x) ) - e$^x$

f'(x) = 2x - y + (cos$^2$x - sin$^2$x) - e$^x$

f'(x) = 2x - y + cos(2x) - e$^x$ (Using identity cos$^2$x - sin$^2$x = cos(2x) )

**Example 2:**Find the determinant of A = $\begin{bmatrix}2 & 5 & 1\\

5 & -3 & 4\\

6 & 5& 0

\end{bmatrix}$

**Solution:**A = $\begin{bmatrix}2 & 5 & 1\\

5 & -3 & 4\\

6 & 5& 0

\end{bmatrix}$

det of A = $\begin{vmatrix}

2 & 5 & 1\\

5 & -3 & 4\\

6 & 5& 0

\end{vmatrix}$

= 2(-3 $\times$ 0 - 5$\times$4) - 5(5$\times$0-5$\times$1) + 6(5$\times$4 - (-3)$\times$1)

= 123

**Example 3:**Determine f'(x) for f(x) = x$^2$ - xy + sin x cos x - e$^x$**Solution:**f(x) = x$^2$ - xy + sin x cos x - e$^x$Differentiate f(x) with respect to x

f'(x) = 2x - y + (sin x(-sin x) + cos x ( cos x) ) - e$^x$

f'(x) = 2x - y + (cos$^2$x - sin$^2$x) - e$^x$

f'(x) = 2x - y + cos(2x) - e$^x$ (Using identity cos$^2$x - sin$^2$x = cos(2x) )