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Inverse Trigonometric Functions
Matrices
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Determinants
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Continuity and Differentiability
- Concept of Differentiability
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- Derivatives of Composite Functions
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- Mean Value Theorem
Applications of Derivatives
Integrals
- Introduction of Integrals
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- Methods of Integration> Integration by Substitution
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- Definite Integrals
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- Fundamental Theorem of Integral Calculus
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Applications of the Integrals
- Introduction of Applications of the Integrals
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Differential Equations
- Basic Concepts of Differential Equations
- Order and Degree of a Differential Equation
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- Procedure to Form a Differential Equation that Will Represent a Given Family of Curves
- Methods of Solving First Order, First Degree Differential Equations
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Vectors
- Basic Concepts of Vector Algebra
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- Algebra of Vector Addition
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Three-dimensional Geometry
- Introduction of Three Dimensional Geometry
- Direction Cosines and Direction Ratios of a Line
- Equation of a Line in Space
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- Equation of a Plane in Normal Form
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Linear Programming
Probability
Numbers, Quantification and Numerical Applications
- Modulo Arithmetic
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- Determine the Mean Price of Amixture
- Apply Rule of Allegation
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- Boats and Streams (Entrance Exam)
- Express the Boats and Streams Problem in the Form of an Equation
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Algebra
Calculus
- Second Order Derivative
- Higher Order Derivative
- Derivatives of Functions in Parametric Forms
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Probability Distributions
Index Numbers and Time Based Data
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- Time Series Analysis
- Components of a Time Series
- Time Series Analysis for Uni-variate Data
Financial Mathematics
- Perpetuity Fund
- Sinking Fund
- Calculate Perpetuity
- Differentiate Between Sinking Fund and Saving Account
- Valuation of Bond
- Calculate Value of Bond Using present Value Approach
- Concept of EMI
- Calculation of EMI
- Fixed Instalment Method
- Interpretation Cost, Residual Value and Useful Life of an Asset
Linear Programming
Notes
Let R, S and T be three non collinear points on the plane with position vectors `vec a` , `vec b` and `vec c` respectively in following fig.
The vectors `vec (RS)` and `vec (RT)` are in the given plane. Therefore , the vector `vec (RS) xx vec (RT)` is perpendicular to the plane containing points R,S and T. Let `vec r`be the position vector of any point P in the plane. Therefore, the equation of the plane passing through R and perpendicular to the vector `vec (RS) xx vec (RT)` is
`(vec r - vec a) . (vec (RS) xx vec (RT)) = 0`
or `(vec r - vec a) . [(vec b - vec a) xx (vec c - vec a)] = 0` ...(1)
If the three points were on the same line, then there will be many planes that will contain them Fig.
for example , These planes will resemble the pages of a book where the line containing the points R, S and T are members in the binding of the book.
Cartesian form:
Let `(x_1,y_1,z_1) , (x_2 , y_2 , z_2)` and `(x_3 , y_3 , z_3)` be the coordinates of the points R, S and T respectively. Let (x, y, z) be the coordinates of any point P on the plane with position vector `vec r`. Then
`vec (RP) = (x - x_1) hat i + ( y - y_1) hat j + (z - z_1) hat k`
`vec (RS) = (x_2 - x_1) hat i + ( y_2 - y_1) hat j + (z_2 - z_1) hat k`
`vec (RT) = (x_3 - x_1) hat i + ( y_3 - y_1) hat j + (z_3 - z_1) hat k`
Substituting these values in equation (1) of the vector form and expressing it in the form of a determinant, we have
`|(x - x_1 , y - y_1 , z - z_1),(x_2 - x_1 , y_2 - y_1, z_2 -z_1) ,(x_3 - x_1, y_3 - y_1, z_3 - z_1) | = 0`
which is the equation of the plane in Cartesian form passing through three non collinear points `(x_1, y_1, z_1), (x_2, y_2, z_2)` and `(x_3, y_3, z_3).`
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