Linear Programming Test

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Linear Programming Test - 18

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Question 1
1 / -0

The region represented by the inequation system $$x, y\ge 0, y\le 6, x+y\le 3$$ is

A
Unbounded in first quadrant

B
Unbounded in first and second quadrants

C
Bounded in first quadrant

D
None of the above

Solution

The given region is bounded in first quadrant.
Question 2
1 / -0

$$z=30x+20y,x+y\le 8,x+2y\ge 4,6x+4y\ge 12,x\ge 0,y\ge 0$$ has

A
Unique solution

B
Infinitely many solution

C
Minimum at $$\left( 4,0 \right) $$

D
Minimum 60 at point $$\left( 0,3 \right) $$

Solution

Since, $$x+y=8$$ ....(i)
This line meets axes at $$\left( 8,0 \right) $$ and $$\left( 0,8 \right) $$ respectively.
$$x+2y=4$$ .....(ii)
$$\Rightarrow \dfrac { x }{ 4 } +\dfrac { y }{ 2 } =1$$
This line meet axes at $$\left( 4,0 \right) $$ and $$\left( 0,2 \right) $$.
And $$6x+4y=12$$ ......(iii)
$$\Rightarrow \dfrac { x }{ 2 } +\dfrac { y }{ 3 } =1$$
This line meets axes at $$\left( 2,0 \right) $$ and $$\left( 0,3 \right) $$
The point of intersection of equations (ii) and (iii) is $$F\left( 1,\dfrac { 3 }{ 2 } \right) $$
Now, at $$A\left( 4,0 \right) ,z=30\times 4=120$$
$$B\left( 8,0 \right) ,z=30\times 8=240$$
$$C\left( 0,8 \right) ,z=20\times 8=160$$
$$D\left( 0,3 \right) ,z=20\times 3=60$$
and $$F\left( 1,\dfrac { 3 }{ 2 } \right) ,z=30\times 1+20\times \dfrac { 3 }{ 2 } =60$$
It is clear that $$z$$ is minimum 60 at points $$D\left( 0,3 \right) $$ and $$F\left( 1,\dfrac { 3 }{ 2 } \right) $$
Hence, option (D) is correct.
Question 3
1 / -0

An aeroplane can carry a maximum of $$200$$ passengers. A profit of Rs.$$1000$$ is made on each executive class ticket and a profit of Rs.$$600$$ is made on each economy class ticket. The airline reserves at least $$20$$ seats for executive class. However, at least $$4$$ times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximize the profit for the airline. What is the maximum profit?

A
$$136000$$

B
$$1360000$$

C
$$13600$$

D
$$1360$$

Solution

Let the airline sell $$x$$ tickets of executive class and $$y$$ tickets of economy class.
The mathematical formulation of the given problem is as follows.
Maximize $$z=1000x+600y .......(1)$$
subject to the constraints,
$$x+y\le 200 ......(2)$$
$$x\ge 20 ..........(3)$$
$$y-4x\ge 0 .......(4)$$
$$x,y\ge 0 ..........(5)$$
The feasible region determined by the constraints is as shown
The corner points of the feasible region are $$A(20,80), B(40,160)$$ and $$C(20,180)$$
The values of $$z$$ at these corner points are as follows.
The maximum value of $$z$$ is $$136000$$ at $$(40,160)$$
Thus, $$40$$ tickets of executive class and $$160$$ tickets of economy class should be sold to maximize the profit and the maximum profit is Rs.$$136000$$.
Question 4
1 / -0

For the LPP $$Minz={ x }_{ 1 }+{ x }_{ 2 }$$ such that inequalities
$$5{ x }_{ 1 }+10{ x }_{ 2 }\ge 0, { x }_{ 1 }+{ x }_{ 2 }\le 1, { x }_{ 2 }\le 4$$ and $${ x }_{ 1 },{ x }_{ 2 }\ge 0$$

A
There is a bounded solution

B
There is no solution

C
There are infinite solutions

D
None of the above

Solution

It is clear from the graph that it is bounded solution.
Question 5
1 / -0

Which inequality is represented by the graph at the right?

A
$$y< 2x+1$$

B
$$y< -2x+1$$

C
$$y<\cfrac{1}{2}x+1$$

D
$$y< -\cfrac{1}{2}x+1$$

Solution
Question 6
1 / -0

Observe the given chart and graph and answer the following: a girl is $$17$$ years old and $$160$$ cms tall. At the end of the growth period she is likely to be how tall

A
$$160$$ cm

B
$$151$$ cm

C
$$160$$ m

D
$$151$$ m

Solution

According to given chart 17 years is the end of the growth period of girls.

$$\therefore $$ answer is 160 cm
Question 7
1 / -0

A wholesale merchant wants to start the business of cereal with Rs. $$24000$$. Wheat is Rs. $$400$$ per quintal and rice is Rs. $$600$$ per quintal. He has capacity to store $$200$$ quintal cereal. He earns the profit Rs.$$25$$ per quintal on wheat and Rs. $$40$$ per quintal on rice. If he stores $$x$$ quintal rice and $$y$$ quintal wheat, then for maximum profit the objective function is

A
$$25x+40y$$

B
$$40x+25y$$

C
$$400x+600y$$

D
$$\dfrac{400}{40}x + \dfrac{600}{25}y$$

Solution
Question 8
1 / -0

The fessible region of LPP is a convex polygon and its two consecutive vertices gives optimum solution the LPP has

A
Only one and finite solution

B
No solution

C
Two solutions

D
None of the above

Solution

The feasible religion of PP convex polygon
The feasible region of LLP is convex pa polygon and its two consecutive verities gives optimum solution the LLP has infinity many solutions,
For examples,
If points P and Q gives optimum solution then all the points on the line segments OQ also give optimum solution.
so is has infinity many solutions.
$$\therefore D is correct.$$
Question 9
1 / -0

A LPP means

A
Only objective function is linear

B
Only constraints are linear

C
Either objective function or constraints are linear

D
All objective function and constraints are linear

Solution
Question 10
1 / -0

Vikas printing company takes fee of Rs. $$28$$ to print a oversized poster and Rs. $$7$$ for each colour of ink used. Raaj printing company does the same work and charges poster for Rs. $$34$$ and Rs. $$5.50$$ for each colour of ink used. If $$z$$ is the colours of ink used, find the values of $$z$$ such that Vikas printing company would charge more to print a poster than Raaj printing company.

A
$$z < 4$$

B
$$2 \leq z \leq 4$$

C
$$4 \leq z \leq 7$$

D
$$z > 4$$

Solution

$$28+7z>34+5.50z$$
$$\rightarrow 1.50z>6$$
$$\rightarrow z>6/1.5$$
$$z>4$$