LPP:Linear Programming Problem : Class 12th :DPP

LPP:Linear Programming Problem :
 
Q1.A company makes two products (X and Y) using two machines (A and B). Each unit of X that is produced requires 50 minutes processing time on machine A and 30 minutes processing time on machine B. Each unit of Y that is produced requires 24 minutes processing time on machine A and 33 minutes processing time on machine B.
At the start of the current week there are 30 units of X and 90 units of Y in stock. Available processing time on machine A is forecast to be 40 hours and on machine B is forecast to be 35 hours.
The demand for X in the current week is forecast to be 75 units and for Y is forecast to be 95 units. Company policy is to maximise the combined sum of the units of X and the units of Y in stock at the end of the week.

• Formulate the problem of deciding how much of each product to make in the current week as a linear program.
• Solve this linear program graphically.(4M)

Q2.
A company manufactures two products (pencil and pen) and the profit per unit sold is Rs.3 and Rs.5 respectively. Each product has to be assembled on a particular machine, each unit of product A taking 12 minutes of assembly time and each unit of product (pen) 25 minutes of assembly time. The company estimates that the machine used for assembly has an effective working week of only 30 hours (due to maintenance/breakdown).
Technological constraints mean that for every five units of product(pencil) produced at least two units of product(pen) must be produced.

• Formulate the problem of how much of each product to produce as a linear program.
• Solve this linear program graphically.
• The company has been offered the chance to hire an extra machine, thereby doubling the effective assembly time available. What is the maximum amount you would be prepared to pay (per week) for the hire of this machine and why? (3M)
•  

Q3. Solve the following linear program:    (2M)
maximise 5x₁ + 6x₂
subject to
x₁ + x₂ <= 10
x₁ - x₂ >= 3
5x₁ + 4x₂ <= 35
x₁ >= 0
x₂ >= 0
Q4. A carpenter makes tables and chairs. Each table can be sold for a profit of Rs.30 and each chair for a profit of Rs.10. The carpenter can afford to spend up to 40 hours per week working and takes six hours to make a table and three hours to make a chair. Customer demand requires that he makes at least three times as many chairs as tables. Tables take up four times as much storage space as chairs and there is room for at most four tables each week.
Formulate this problem as a linear programming problem (3M).
Q5.
The production manager of a chemical plant is attempting to devise a shift pattern for his workforce. Each day of every working week is divided into three eight-hour shift periods (00:01-08:00, 08:01-16:00, 16:01-24:00) denoted by night, day and late respectively. The plant must be manned at all times and the minimum number of workers required for each of these shifts over any working week is as below:

          Mon   Tues     Wed     Thur    Fri      Sat     Sun
Night     5     3        2       4       3        2       2
Day       7     8        9       5       7        2       5
Late      9     10       10      7       11       2       2

The union agreement governing acceptable shifts for workers is as follows:

1. Each worker is assigned to work either a night shift or a day shift or a late shift and once a worker has been assigned to a shift they must remain on the same shift every day that they work.
2. Each worker works four consecutive days during any seven day period.In total there are currently 60 workers.Formulate the production manager's problem as a linear program (4m).

Q6.

A store has requested a manufacturer to produce pants and sports jackets.

For materials, the manufacturer has 750 m² of cotton textile and 1,000 m² of polyester. Every pair of pants (1 unit) needs 1 m² of cotton and 2 m² of polyester. Every jacket needs 1.5 m² of cotton and 1 m² of polyester.

The price of the pants is fixed at Rs. 550 and the jacket,Rs. 640.

What is the number of pants and jackets that the manufacturer must give to the stores so that these items obtain a maximum sale? (4m)

 

Plane: Class 12th : Dpp

Plane: Class 12th : Dpp

Q1. Equation of a plane in normal form. (2)
Q2. Find the vector equation of the plane which is at a distance 5 units from origin and which is normal to vector2i-j+2k.      (2)                                  Q3. Find the Coordinates of the foot of perpendicular drawn from the origin to the plane x+y+3z-4=0(2)

Q4. The foot of perpendicular drawn  from the origin to the plane is (1,2,3).
Find the equation of a plane.(2)

Q5. Find the equation of the plane passing through the intersection of the plane r.(i+j+k) =2 and r.( 2i+3j+k)-4 = 0     (3)     
Q6.Find  the angle between the two planes 3x-6y+2z=8 and 2x+2y-2z=10                            (3)

Q7. Condition of coplanarity of two lines.    (1)

Line:Class 12th :Dpp

Line:Class 12th :Dpp

Q1. Vector equation of a line passing through given point and parallel to given vector. (2)
Q2. Find the vector equation of the line passing through the point with position vector 2i+j-k and parallel to line joining the points -i+j+4k and i+2j+2k Also find the Cartesian form of this equation. (3)                                                                
Q3. Find the Cartesian equation of line passing through the points A(4,2,1), B(2,-1,3). Also reduce it to vector form. (3)
Q4.Find  the angle between the lines.                            (3)

(x-1)/4= (y-3)/1=(z)/8 and (x-2)/2=(y+1)/2=(z-4)/1   

Q5. Find equation of line passing through the point (3,1,2) and perpendicular to (3)

  (x-1)/1=(y-2)/2=(z-3/3) and (x)/-3=(y)/2=(z)/5

Q6. Find the foot of perpendicular from point (0,2,3) on the line  

     (x+3)/5=(y-1)/2=(z+4)/3   
 Also find the length of the perpendicular.  (3)

Q7. Find the shortest distance between the lines r= (2i-j)+λ(2i+j-3k) and r= (i-j+2k)+μ(2i+j-5k) (3)

3D GEOMETRY: CLASS 12TH:DIRECTION RATIOS AND DIRECTION COSINES

3D GEOMETRY:
Q1. Define Direction Ratios and direction cosines                      (2)
Q2. Find the Direction Ratios and direction cosines of line passing through the points A(-4,2,3)and B(1,3,-2)                                        (2)
Q3. Show that the points A(-7,4,-2), B(-2,1,0), C(3,-2,2) are collinear.  (2)

Q4. If a line makes angles 90⁰, 135⁰ and 45⁰ with X,Y and Z axis respectively.
      find its direction cosines        (2)       
Q5. If the line makes angles α,β,γ with coordinate axes Prove that

 i)Sin²α+sin²β+sin²γ=2
 ii)Cos2α+cos2β+cos2γ = -1                  (4)

Q6. If l,m,n are directions cosines of a line then l²+m²+n² = 1                          (3)

Q7. Find the direction cosines of the line which is perpendicular to the lines with direction rarios-1,2,2 and 0,2,1.                           (3)

Q8. Show that the angle between any two diagonals of a cube is cos^-1(1/3).                                                (2)