A linear programming problem is one that is concerned with

**A. ** finding the upper limits of a linear function of several variables

**B. ** finding the lower limit of a linear function of several variables

**C. ** finding the limiting values of a linear function of several variables

**D. ** finding the optimal value (maximum or minimum) of a linear function of several variables

**Answer : ****Option D**

**Explaination / Solution: **

A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables .

A linear programming problem is one that is concerned with finding the optimal value (maximum or minimum) of a linear function of several variables .

Workspace

Report

In Corner point method for solving a linear programming problem the second step after finding the feasible region of the linear programming problem and determining its corner points is

**A. ** Evaluate the objective function Z = ax + by at the mid points

**B. ** None of these

**C. ** Evaluate the objective function Z = ax + by at each corner point.

**D. ** Evaluate the objective function Z = ax + by at the center point

**Answer : ****Option C**

**Explaination / Solution: **

In Corner point method for solving a linear programming problem the second step after finding the feasible region of the linear programming problem and determining its corner points is : To evaluate the objective function Z = ax + by at each corner point.

In Corner point method for solving a linear programming problem the second step after finding the feasible region of the linear programming problem and determining its corner points is : To evaluate the objective function Z = ax + by at each corner point.

Workspace

Report

Maximise Z = 3x + 4y subject to the constraints: x + y ≤ 4, x ≥ 0, y ≥ 0.

**A. ** Maximum Z = 16 at (0, 4)

**B. ** Maximum Z = 19 at (1, 5)

**C. ** Maximum Z = 17 at (0, 5)

**D. ** Maximum Z = 18 at (1, 4)

**Answer : ****Option A**

**Explaination / Solution: **

Objective function is Z = 3x + 4 y ……(1).

The given constraints are : x + y ≤ 4, x ≥ 0, y ≥ 0.

The corner points obtained by constructing the line x+ y= 4, are (0,0),(0,4) and (4,0).

Corner points | Z = 3x +4y |

O ( 0 ,0 ) | Z = 3(0)+4(0) = 0 |

A ( 4 , 0 ) | Z = 3(4) + 4 (0) = 12 |

B ( 0 , 4 ) | Z = 3(0) + 4 ( 4) = 16 …( Max. ) |

therefore Z = 16 is maximum at ( 0 , 4 ) .

Workspace

Report

Maximize Z = – x + 2y, subject to the constraints: x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.

**A. ** Maximum Z = 12 at (2, 6)

**B. ** Maximum Z = 14 at (2, 6)

**C. ** Z has no maximum value

**D. ** Maximum Z = 10 at (2, 6)

**Answer : ****Option C**

**Explaination / Solution: **

Objective function is Z = - x + 2 y ……………………(1).

The given constraints are : x ≥ 3, x + y ≥ 5, x + 2y ≥ 6, y ≥ 0.

Corner points | Z = - x + 2y |

D(6,0 ) | -6 |

A(4,1) | -2 |

B(3,2) | 1 |

Here , the open half plane has points in common with the feasible region .

Therefore , Z has no maximum value.

Workspace

Report

There are two types of fertilizers F1 and F2. F1 consists of 10% nitrogen and 6% phosphoric acid and F2 consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, a farmer finds that she needs atleast 14 kg of nitrogen and 14 kg of phosphoric acid for her crop. If F1 costs Rs 6/kg and F2 costsRs 5/kg, determine how much of each type of fertiliser should be used so that nutrient requirements are met at a minimum cost. What is the minimum cost?

**A. ** 130 kg of fertilizer F1 and 80 kg of fertilizer F2; Minimum cost = Rs 1300

**B. ** 120 kg of fertilizer F1 and 80 kg of fertilizer F2; Minimum cost = Rs 1200

**C. ** 100 kg of fertilizer F1 and 80 kg of fertilizer F2; Minimum cost = Rs 1000

**D. ** 110 kg of fertilizer F1 and 80 kg of fertilizer F2; Minimum cost = Rs 1100

**Answer : ****Option C**

**Explaination / Solution: **

Let number of kgs. of fertilizer F1 = x

And number of kgs. of fertilizer F2 = y

Therefore , the above L.P.P. is given as :

Minimise , Z = 6x +5y , subject to the constraints : 10/100 x + 5/100y ≥ 14 and 6/100x + 10/100y ≥ 14, i.e. 2 x + y ≥ 280 and 3x + 5y ≥ 700, x,y ≥ 0.,

Corner points | Z =6x +5 y |

A ( 0 , 280 ) | 1400 |

D(700/3,0 ) | 1400 |

B(100,80) | 1000………….(Min.) |

Corner points Z =6x +5 y A ( 0 , 280 ) 1400 D(700/3,0 ) 1400 B(100,80) 1000………….(Min.) Here Z = 1000 is minimum.

i.e. 100 kg of fertilizer F1 and 80 kg of fertilizer F2; Minimum cost = Rs 1000.

Workspace

Report

The feasible region for a LPP is shown in Figure. Find the minimum value of Z = 11x + 7y.

Corner points | Z = 11x +7 y |

(0, 5 ) | 35 |

(0,3) | 21…………………(min.) |

(3,2 ) | 47 |

Hence the minimum value is 21

Workspace

Report

Vector equation of a line that passes through the given point whose position vector is and parallel to a given vector is

**A. **

**B. **

**C. **

**D. **

**Answer : ****Option A**

**Explaination / Solution: **

Vector equation of a line that passes through the given point whose position vector is and parallel to a given vector is given by :

Vector equation of a line that passes through the given point whose position vector is and parallel to a given vector is given by :

Workspace

Report

Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector.

The equation of the line which passes through the point (1, 2, 3) and is parallel to the vector

, let vector and vector ,

the equation of line is :

.

Workspace

Report

Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is.

**A. ** – lx + my + nz = d

**B. ** lx – my + nz = d

**C. ** lx + my + nz = – d

**D. ** lx + my + nz = d

**Answer : ****Option D**

**Explaination / Solution: **

In Cartesian co – ordinate system Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is given by : lx + my + nz = d.

In Cartesian co – ordinate system Equation of a plane which is at a distance d from the origin and the direction cosines of the normal to the plane are l, m, n is given by : lx + my + nz = d.

Workspace

Report

Find the angle between the planes whose vector equations are

**A. ** cos−1(15731√)
**B. ** sin−1(15731√)
**C. ** tan−1(15731√)

**D. ** cot−1(15731√)
**Answer : ****Option A**

**Explaination / Solution: **

Workspace

Report