Ch 5 Maths Class 12 Solutions

Handwritten notes of Ch 5 Maths Class 12 Solutions written by Professor M. Sulman Sherazi Suib. These notes are very helpful in the preparation of ch 5 maths class 12 pdf for the students of Mathematics of the Intermediate and these are according to the paper patterns of all Punjab boards.

Summary and Contents:
Topics which are discussed in the notes are given below:
  • Important ch 5 maths class 12 important questions for Intermediate part-II students.
  • Linear Inequalities in one variable: The inequalities of the form ax+bc, ax+b less then c. ax+b greater than c are called linear inequalities in one variable.
  • Linear Inequalities in two variables: The inequalities of the form ax+by c, ax+by less then c, ax+by greater than c are called linear inequalities in two variables x and y. where a, b and c are constants.
  • Important ch 5 maths class 12 for Intermediate part-II students.
  • Corresponding Equation/Associated Equation: The corresponding equation to any inequality is an equation formed by replacing the inequality symbol with an equal sign. For example
  • (i) Corresponding equation of x+2y26 is x+2y = 6. (ii) Corresponding equations of x70 and 2x+y>2 are x = 0 and 2x+y=2 respectively.
  • Important class 12 maths 5.1 for Intermediate part-II students.
  • Graphing of a Linear Inequality in two variables:
  • (i) The corresponding equation is useful for graphing inequalities, because this equation forms the boundary line to the graph of given inequality.
  • (ii) A vertical line (line Il to y-axis) divides the xy-plane into two regions called half planes. (Left half plane and right half plane) 
  • (iii) A non-vertical line (line Il to x-axis) divides xy-plane into two regions called "half planes". (Upper Half plane and Lower half plane) 
  • (iv) If the inequality is strict (< or>) then we draw dashed or broken boundary line.
  • (v) of the inequality is non-strict (r) then we draw a solid boundary line.
  • Important class 12 maths ch 5 for Intermediate part-II students.
  • Note: x0 means right half plane, X50 means left half planey means upper half plane, y 0 means lower half plane.
  • Procedure for Graphing a linear Inequality in two variables: 
  • (i) Graph the corresponding equation of given inequality. 
  • (ii) Select any test point (not on the graph of corresponding equation of inequality). The origin (0,0) is most convenient point to choose as a test point.
  • (iii) put the coordinates of the test point in the inequality. 
  • (iv) If the test point satisfies the giren inequality, then shade the half plane containing the test point.
  • (v) If the test point does not satisfy the given inequality then the shade the half plane that does not contain the test point.
  • Important class 12 maths ch 5 ex 5.1 for Intermediate part-II students.
  • Associated equation with example
  • What is the corner point or vertex of the solution region
  • Indicate the solution region of the following system of linear inequalities by shading
  • Important class 12 maths ch 5 ex 5.2 for Intermediate part-II students.
  • Conrner Point/ Vertex: A point of the solution region where two of its bountary lines intersect is called the carner point or vertex of the solution region.
  • Important ex 5.1 class 12 maths solutions for Intermediate part-II students.
  • Problem Constraints: The restrictions applied on the every day life problems are called problem constraints.
  • Non-Problem Constraints: The restrictions that are always satisfied are called natural constraints or non-negative constraints.
  • Decision Variables: The variable used in non-negative constraints are decision variables.
  • Important ex 5.2 class 12 maths solutions for Intermediate part-II students.
  • Feasible Region: The solution region which is restricted to the first which is restricted to the first quadrant is called feasible region by non-negative constraints x greater than or equal to 0 and y greater than or equal to 0.
  • Feasible Solution: Each point of feasible region is called feasible solution of the system.
  • Feasible Solution Set: A set consisting of all the feasible solutions of the system is called feasible solution set.
  • Problem constraints
  • Decision variables
  • Definition of feasible solution
  • Definition of feasible solution set
  • Definition of Convex: If the line segment obtained by joining any two points of a region lies entirely within the region, then the region is called Convex.
  • Definition of an objective function
  • Definition of an optimal solution
  • Theorem of linear programming
  • Important class 12 maths exercise 5.1 for Intermediate part-II students.
  • Important class 12 maths exercise 5.2 for Intermediate part-II students.
  • Important class 12 maths exercise 5.3 for Intermediate part-II students.
  • Important class 12 maths exercise 5.4 for Intermediate part-II students.
  • Important class 12 maths exercise 5.5 for Intermediate part-II students.

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