Preliminaries We were introduced to the trigonometric functions in class $latex {XI}&s=1$ with their general definitions for any angle $latex {\theta}&s=1$. The specialty of these functions is that their values get repeated after an interval. This is because after every interval of $latex {2 \pi^c}&s=1$, the angles have same initial and terminal arms. Basics :… Continue reading Trigonometric Functions

# Category: XII

## Indeterminate Forms

Standard Limits $latex {\lim \limits_{x \to 0} \frac {sin(x)}{x} = 1, \lim \limits_{x \to 0} \frac {tan(x)}{x} = 1, \lim \limits_{x \to 0} \frac {1 - cos(x)}{x^2} = \frac 1 2}&s=2$ $latex {\lim \limits_{x \to a } \frac {x^n - a^n}{x-a} = n \cdot x^{n-1}, \lim \limits_{x \to 0} \frac {a^x -1 }{x} = ln(a),… Continue reading Indeterminate Forms

## Planes

Definition of Plane Let there be three non-collinear points. There exist 3 distinct lines, which pass through these points taken 2 at a time. The triangle so formed is a planar surface. A line joining any 2 points on a plane always lies on the plane. Equation of Plane : Normal Form Consider a plane.… Continue reading Planes

## Differentiation

Introduction Let $latex {y = f(x)}&s=1$ be a real-valued function of $latex {x}&s=1$. Consider the following limit : $latex {\lim \limits_{h \to 0} \frac {f(x+h)-f(x)}{(x+h) - (x)} = \frac {f(x+h)-f(x)}{h}}&s=2$ It asks the following question : If $latex {h}&s=1$ is the change in value of $latex {x}&s=1$, what will be the fractional change in $latex… Continue reading Differentiation

## Linear Programming Problems

Linear Programming is a technique of obtaining optimum solution to a problem out of many solutions. It has a specific objective such as maximizing profit or minimizing the expenses. This technique is typically applied to solve decision making problems in business; but its origin is in WWII. It was developed to make the best (optimum) use… Continue reading Linear Programming Problems

## Matrices III

Inverse of a Matrix Unlike numbers, the division operation is not defined for matrices. A similar (but not same) operation is to find the inverse of a matrix. Note that only square matrices can have inverses. When a square matrix $latex A$ is non-singular ($latex |A| \ne 0$), its inverse exists ($latex = A^{-1}$) and is… Continue reading Matrices III

## Three Dimensional Coordinate Geometry II

Lines A line is uniquely specified, when coordinates of 2 distinct points are known. In other words, only 1 line passes through 2 distinct points. Let $latex P (x_1,y_1,z_1)$ and $latex Q(x_2,y_2,z_2)$ be those 2 points. Let $latex R(x,y,z)$ be any point on that line. Then, $latex {\vec {PR} = t (\vec {PQ})}&s=2$ since… Continue reading Three Dimensional Coordinate Geometry II