CBSE class 12 commerce Maths fast track revision notes
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CBSE class 12 commerce Maths fast track revision notes
CBSE class 12 commerce Maths fast track revision notesLinear Programming
 A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables (called objective function) subject to the conditions that the variables are nonnegative and satisfy a set of linear inequalities (called linear constraints). Variables are sometimes called decision variables and are nonnegative.
 A few important linear programming problems are:
 Diet problems
 Manufacturing problems
 Transportation problems
 The common region determined by all the constraints including the nonnegative constraints x ≥ 0, y ≥ 0 of a linear programming problem is called the feasible region (or solution region) for the problem.
 Points within and on the boundary of the feasible region represent feasible solutions of the constraints. Any point outside the feasible region is an infeasible solution.
 Any point in the feasible region that gives the optimal value (maximum or minimum) of the objective function is called an optimal solution.
 The following Theorems are fundamental in solving linear programming problems:
 Theorem 1 Let R be the feasible region (convex polygon) for a linear programming problem and let Z ax by = + be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.
 Theorem 2 Let R be the feasible region for a linear programming problem, and let be the objective function. If R is bounded, then the objective function Z has both a maximum and a minimum value on R and each of these occurs at a corner point (vertex) of R.
 If the feasible region is unbounded, then a maximum or a minimum may not exist. However, if it exists, it must occur at a corner point of R.

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 Find the feasible region of the linear programming problem and determine its corner points (vertices).
 Evaluate the objective function Z ax by = + at each corner point. Let M and m respectively be the largest and smallest values at these points.
 If the feasible region is bounded, M and m respectively are the maximum and minimum values of the objective function.
 If the feasible region is unbounded, then,
 M is the maximum value of the objective function, if the open half plane determined by ax by M + > has no point in common with the feasible region. Otherwise, the objective function has no maximum value.
 m is the minimum value of the objective function, if the open half plane determined by ax by m + < has no point in common with the feasible region. Otherwise, the objective function has no minimum value.
 If two corner points of the feasible region are both optimal solutions of the same type, i.e., both produce the same maximum or minimum, then any point on the line segment joining these two points is also an optimal solution of the same type.
Relation and Function
TYPES OF RELATIONS:
 A relation R in a set A is called reflexive if (a, a) ∈ R for every a ∈ A.
 A relation R in a set A is called symmetric if (a1, a2) ∈ R implies that (a2, a1) ∈ R, for all a1, a2 ∈
 A relation R in a set A is called transitive if (a1, a2) ∈ R, and (a2, a3) ∈ R together imply that (a1
 all a1, a2, a3 ∈ A.
EQUIVALENCE RELATION
 A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric and transitive.

Equivalence Classes
 Every arbitrary equivalence relation R in a set X divides X into mutually disjoint subsets (Ai) called partitions or subdivisions of X satisfying the following conditions:
 All elements of Ai are related to each other for all i
 No element of Ai is related to any element of Aj whenever i ≠ j
 Ai ∪ Ai X and Ai = ∩ Ai = Φ, i ≠ j . These subsets ( i (A )) are called equivalence classes.
 For an equivalence relation in a set X, the equivalence class containing a ∈ X, denoted by [a], is the subset of X containing all elements b related to a.
**Function: Arelation f: A → B is said to be a function if every clement of A is correlated to a
Unique element in B.
*Aisdomain
* Biscodomain
Application of Integrals
 The area of the region bounded by the curve y = f (x), xaxis and the lines x = a and x = b (b > a) is given by the formula: b b Area= a a ydx = f(x)dx. ∫ ∫
 The area of the region bounded by the curve x = ϕ ( y) ), yaxis and the lines y = c, y = d is given by the formula: b d Area = c c xdy = θ(y)dy ∫ ∫
 The area of the region enclosed between two curves y = f (x), y = g (x) and the lines x = a, x = b is given by the formula, b Area= a [f(x)g(x)]dx,where, f(x) ≥ g(x) in [a, b] ∫
 If f ( x g x in a ) ≥ ( ) [ , c and ] f ( x)≤ g x in c ( ) [ , , b] a
Determinant
 For any square matrix A, the A satisfy following properties.
 A′ = A, where A′ = transpose of A.
 If we interchange any two rows (or columns), then sign of determinant changes.
 If any two rows or any two columns are identical or proportional, then value of determinant is zero.
 If we multiply each element of a row or a column of a determinant by constant k, then value of determinant is multiplied by k.
 Multiplying a determinant by k means multiply elements of only one row (or one column) by k.
 3 ij 3 3 If A= a then k. A =k ,  A 
 If elements of a row or a column in a determinant can be expressed as sum of two or more elements, then the given determinant can be expressed as sum of two or more determinants.
 If to each element of a row or a column of a determinant the equimultiples of corresponding elements of other rows or columns are added, then value of determinant remains same.
 Area of a triangle with vertices ( x y x y and x y 1, 1 ), ( 2, 2 ) ( 3, 3 ) is given by 1 1 2 2 3 3 x y 1 1 x y 1 2 x y 1 ∆ =
 Minor of an element aij of the determinant of matrix A is the determinant obtained by deleting th i row and thj column and denoted by Mij • Cofactor of aij of given by Aij = (– 1)i+ j Mij
 Value of determinant of a matrix A is obtained by sum of product of elements of a row (or a column) with corresponding cofactors. For example, 11 11 12 12 13 13 A = a A a A a + + A .
 If elements of one row (or column) are multiplied with cofactors of elements of any other row (or column), then their sum is zero. For example, 11 21 12 22 13 23 a A a + A a A + = 0
 A adj A adj A A A I ( ) = ( ) = , where A is square matrix of order n.
 A square matrix A is said to be singular or nonsingular according as A or A = 0  ≠ 0.
 If AB = BA = I, where B is square matrix, then B is called inverse of A. Also ( ) 1 1 1 1 A B or B A and hence A A. − − − − = = =
 A square matrix A has inverse if and only if A is nonsingular.
CBSE class 12 commerce Maths fast track revision notes
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CBSE class 12 commerce Maths fast track revision notes
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