NCERT Solutions For Class 12 Mathematics Chapter 12: Linear Programming

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NCERT Solutions for class 12 Mathematics chapter 12 Linear Programming are provided in the article. Linear Programming Problem (or LPP) is the problem that’s concerned with achieving the most effective optimal (maximum or minimum) value of a linear function with several variables (called objective function). LLP has some conditions like the variables are non-negative and satisfy a collection of linear inequalities (also referred to as linear constraints).

Download: NCERT Solutions for Class 12 Mathematics Chapter 12 pdf

Class 12 Maths NCERT Solutions Chapter 12 Linear Programming

NCERT Solutions

Important Topics in Class 12 Mathematics Chapter 8 Applications of Integrals

Class 12 Mathematics Chapter 12 has the following important concepts:

Linear Programming Problem and Mathematical Formulation

Mathematical formulation of the problem

Mathematical formulation section explains the formulation of mathematical problems. It defines the non-negative constraints, objective function, decision variables.

Key Points:

A linear programming problem is finding the optimal value [maximum or minimum] of a linear function of variables, which are subjected to certain conditions and satisfying a set of linear constraints.
Optimal value of a linear function is called an objective function.
Variables involved in the objective function are called decision variables.
The constraints are the restrictions on the variables.

Graphical method of solving linear programming problems

Graphical method of solving linear programming problems section covers the definition of feasible region, feasible solution and infeasible solution, optimal solution, bounded and unbounded region of feasible solution. It also includes the Corner Point Method that helps to solve linear programming problems with solved examples.

Key Points:

Region obtained by constraints [and non-negative constraints] can be called the feasible region.
Points within the feasible region are called feasible solutions.
Points outside the feasible region are called infeasible solutions.
Point in the feasible region which gives optimal value of the objective function is called optimal solution.

Different Types of Linear Programming Problems

Here are the different Types of Linear Programming Problems:

Manufacturing Problems

Manufacturing problems can be seen in the manufacturing sector in to optimise production by maximising profits. Profits can be a function of the number of workers, working hours, materials required, the value of the product in the market, the demand for the product, the supply of the product etc.

Diet problems

Diet problems involve optimisation of the amount of intake of different types of foods required by the body to obtain necessary nutrients. Agenda of a diet problem will be selecting those foods with the required nutrient at a lesser cost.

Transportation problems

Transportation problems involve transportation of manufactured goods to different places so that transportation cost is reduced. For big companies, analysis of transportation cost is very important as it caters to a widespread area.

NCERT Solutions For Class 12 Maths Chapter 12 Exercises

Detailed solutions for all NCERT Solutions for Chapter 12 Linear Programming under different exercises are as follows:

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CBSE CLASS XII Related Questions

1.
A rectangle of perimeter \(24\) cm is revolved along one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.
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2.
The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that
(i) target is hit.
(ii) at least one shot misses the target.
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3.
If \[ P = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{bmatrix} \quad \text{and} \quad Q = \begin{bmatrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 1 & -1 & 5 \end{bmatrix} \] find \( QP \) and hence solve the following system of equations using matrix method:
\[ x - y = 3,\quad 2x + 3y + 4z = 13,\quad y + 2z = 7 \]
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4.
Obtain the value of \[ \Delta = \begin{vmatrix} 1 + x & 1 & 1 \\ 1 & 1 + y & 1 \\ 1 & 1 & 1 + z \end{vmatrix} \] in terms of \(x, y, z\). Further, if \(\Delta = 0\) and \(x, y, z\) are non–zero real numbers, prove that \[ x^{-1} + y^{-1} + z^{-1} = -1 \]
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5.
Evaluate : \[ \int_{\frac{1}{12}}^{\frac{5}{12}} \frac{dx}{1+\sqrt{\cot x}} \]
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6.
A racing track is built around an elliptical ground whose equation is given by \[ 9x^2 + 16y^2 = 144 \] The width of the track is \(3\) m as shown. Based on the given information answer the following:

(i) Express \(y\) as a function of \(x\) from the given equation of ellipse.
(ii) Integrate the function obtained in (i) with respect to \(x\).
(iii)(a) Find the area of the region enclosed within the elliptical ground excluding the track using integration.
OR
(iii)(b) Write the coordinates of the points \(P\) and \(Q\) where the outer edge of the track cuts \(x\)-axis and \(y\)-axis in first quadrant and find the area of triangle formed by points \(P,O,Q\).
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