MCQ Questions for Class 12 Commerce Maths Linear Programming Quiz 4

Q.1.

The maximum profit that can be made in a day is:

0%
Rs. $$450$$
0%
Rs. $$800$$
0%
Rs. $$650$$
0%
Rs. $$525$$

Explanation

Q.2.

The Convex Polygon Theorem states that the optimum (maximum or minimum) solution of a LPP is attained at atleast one of the ______ of the convex set over which the solution is feasible.

0%
origin
0%
corner points
0%
centre
0%
edge

Explanation

Q.3.

In order to obtain maximum profit, the quantity of normal and scientific calculators to be manufactured daily is:

0%
$$(100,170)$$
0%
$$(200,170)$$
0%
$$(100,80)$$
0%
$$(200,80)$$

Explanation

Q.4.

The region on the graph sheet with satisfies the constraints including the non- negativity restrictions is called the _______ space.

0%
solution
0%
interval
0%
concave
0%
convex

Explanation

Q.5.

In order to maximize the profit of the company, the optimal solution of which of the following equations is required?

0%
$$P = x+y-200$$
0%
$$P = 5y-2x$$
0%
$$P = y - 80$$
0%
$$P = 200-x$$

Explanation

Q.6.

How many acres of each (wheat and rye) should the farmer plant in order to get maximum profit?

0%
$$(5,5)$$
0%
$$(4,4)$$
0%
$$(4,5)$$
0%
$$(4,3)$$

Explanation

let $$x$$ be the acres of wheat planted and

$$y$$ be the acres of rye planted

Given that there are a total of $$10$$ acres of land to plant.

Atleast $$7$$ acres is to be planted i.e., $$x+y\geq 7$$

Given that the cost to plant one acre of wheat is $$\$200$$

Therefore, the cost for $$x$$ acres of wheat is $$200x$$

Given that the cost to plant one acre of rye is $$\$100$$

Therefore, the cost for $$y$$ acres of rye is $$100y$$

Given that, amount for planting wheat and rye is $$\$1200$$

Therefore the total cost to plant wheat and rye is $$200x+100y\leq 1200\implies 2x+y\leq 12$$

Given that, the time taken to plant one acre of wheat is $$1$$ hr

Therefore, the time taken to plant $$x$$ acres of wheat is $$x$$ hrs

Given that, the time taken to plant one acre of rye is $$2$$ hrs

Therefore, the time taken to plant $$y$$ acres of rye is $$2y$$ hrs

Given that, the total time for planting is $$12$$ hrs

Therefore, the total time to plant wheat and rye is $$x+2y\leq 12$$

Given that, one acre of wheat yields a profit of $$\$500$$

Therefore, the profit from $$x$$ acres of wheat is $$500x$$

Given that, one acre of rye yields a profit of $$\$300$$

Therefore, the profit from $$y$$ acres of wheat is $$300y$$

therefore the total profit from the wheat and rye is $$P=500x+300y$$

Now substituting the options in the profit expression and verifying

Substituting option A $$(x,y)=(5,5)$$

$$x+y\geq 0\implies 5+5\geq7\implies 10\geq 7$$ True

$$2x+y\leq 12\implies 2(5)+5\leq12\implies 15\leq 12$$ False

Substituting option B $$(x,y)=(4,4)$$

$$x+y\geq 0\implies 4+4\geq7\implies 8\geq 7$$ True

$$2x+y\leq 12\implies 2(4)+4\leq12\implies 12\leq 12$$ True

$$x+2y\leq 12\implies 4+2(4)\leq12\implies 12\leq 12$$ True

Substituting option C $$(x,y)=(4,5)$$

$$x+y\geq 0\implies 4+5\geq7\implies 9\geq 7$$ True

$$2x+y\leq 12\implies 2(4)+5\leq12\implies 13\leq 12$$ False

Substituting option D $$(x,y)=(4,3)$$

$$x+y\geq 0\implies 4+3\geq7\implies 7\geq 7$$ True

$$2x+y\leq 12\implies 2(4)+3\leq12\implies 11\leq 12$$ True

$$x+2y\leq 12\implies 4+2(3)\leq12\implies 10\leq 12$$ True

$$(4,4), (4,3)$$ satisfies the constraints. Therefore finding the profit

for $$(4,4)$$, $$P=500x+300y=500(4)+300(4)=3200$$

for $$(4,3)$$, $$P=500x+300y=500(4)+300(3)=2900$$

Therefore the maximum profit is attained at $$(4,4)$$

Q.7.

How many acres of land will be left unplanted if the farmer aims for a maximum profit?

0%
$$0$$
0%
$$1$$
0%
$$2$$
0%
$$3$$

Explanation

Q.8.

If $$a,b,c \in +R$$ such that $$\lambda abc$$ is the minimum value of $$a(b^2+c^2)+b(c^2+a^2)+c(a^2+b^2)$$, then $$\lambda=$$

0%
$$1$$
0%
$$3$$
0%
$$4$$
0%
None of the above.

Explanation

Q.9.

Consider the objective function $$Z = 40x + 50y$$ The minimum number of constraints that are required to maximize $$Z$$ are

0%
$$4$$
0%
$$2$$
0%
$$3$$
0%
$$1$$

Explanation

Q.10.

Maximum value of $$z=3x+4y$$ subject to $$x-y\le -1,-x+y\le 0,x,y\ge 0$$ is given by?

0%
$$1$$
0%
$$4$$
0%
$$6$$
0%
None of the these

Explanation

Q.11.

The objective function $$z={ x }_{ 1 }+{ x }_{ 2 }$$, subject to $${ x }_{ 1 }+{ x }_{ 2 }\le 10,{ -2x }_{ 1 }+3{ x }_{ 2 }\le 15,{ x }_{ 1 }\le 6$$, $${ x }_{ 1 },{ x }_{ 2 }\ge 0$$ has maximum value .................. of the feasible region.

0%
at only one point
0%
at only two points
0%
at every point of the segment joining two points
0%
at every point of the line joining two points

Explanation

Q.12.

The objective function $$z = 4x_{1} + 5x_{2}$$, subject to $$2x_{1} + x_{2}\geq 7, 2x_{1} + 3x_{2} \leq 15, x_{2}\leq 3, x_{1}, x_{2} \geq 0$$ has minimum value at the point.

0%
On x-axis
0%
On y-axis
0%
At the origin
0%
On the parallel to x-axis

Explanation

Q.13.

The objective function of LPP defined over the convex set attains its optimum value at

0%
atleast two of the corner points
0%
all the corner points
0%
atleast one of the corner points
0%
none of the corner points

Explanation

Q.14.

The constraints
$$-{ x }_{ 1 }+{ x }_{ 2 }\le 1$$
$$-{ x }_{ 1 }+3{ x }_{ 2 }\le 9$$
$${x}_{1},{x}_{2}\ge 0$$ defines on

0%
Bounded feasible space
0%
Unbounded feasible space
0%
Both bounded and unbounded feasible
0%
None of the above

Explanation

Q.15.

An article manufactured by a company consists of two parts $$X$$ and $$Y$$. In the process of manufacture of the part $$X$$. $$9$$ out of $$100$$ parts may be defective. Similarly $$5$$ out of $$100$$ are likely to be defective in part $$Y$$. Calculate the probability that the assembled product will not be defective.

0%
$$0.86$$
0%
$$0.864$$
0%
$$0.8456$$
0%
$$0.8645$$

Explanation

Q.16.

The corner points of the feasible region determined by the system of linear constraints are $$(0, 10), (5, 5), (15, 15), (0, 20)$$. Let $$z=px+qy$$ where $$p, q > 0$$. Condition on p and q so that the maximum of z occurs at both the points $$(15, 15)$$ and $$(0, 20)$$ is __________.

0%
$$q=2p$$
0%
$$p=2p$$
0%
$$p=q$$
0%
$$q=3p$$

Explanation

Q.17.

Corner points of the bounded feasible region for an LP problem are $$A(0,5) B(0,3) C(1,0) D(6,0)$$. Let $$z = -50x + 20y$$ be the objective function. Minimum value of z occurs at ______ center point.

0%
$$(0,5)$$
0%
$$(1,0)$$
0%
$$(6,0)$$
0%
$$(0,3)$$

Explanation

Q.18.

The corner points of the feasible region are $$A(0,0),B(16,0),C(8,16)$$ and $$D(0,24)$$. The minimum value of the objective function $$z=300x+190y$$ is _______

0%
$$5440$$
0%
$$4800$$
0%
$$4560$$
0%
$$0$$

Explanation

Q.19.

The given table shows the number of cars manufactured in four different colours on a particular day. Study it carefully and answer the question.

Colour	Number of cars manufactured
Colour	Vento	Creta	WagonR
Red	65	88	93
White	54	42	80
Black	66	52	88
Sliver	37	49	74

Which car was twice the number of silver Vento?

0%
Silver WagonR
0%
Red WagonR
0%
Red Vento
0%
White Creta

Explanation

Q.20.

A firm manufactures three products $$A,B$$ and $$C$$. Time to manufacture product $$A$$ is twice that for $$B$$ and thrice that for $$C$$ and if the entire labour is engaged in making product $$A,1600$$ units of this product can be produced.These products are to be produced in the ratio $$3:4:5.$$ There is demand for at least $$300,250$$ and $$200$$ units of products $$A,B$$ and $$C$$ and the profit earned per unit is Rs.$$90,$$ Rs$$40$$ and Rs.$$30$$ respectively.

Raw material	Requirement per unit product(Kg) A	Requirement per unit product(Kg) B	Requirement per unit product(Kg) C	Total availability (kg)
$$P$$	$$6$$	$$5$$	$$2$$	$$5,000$$
$$Q$$	$$4$$	$$7$$	$$3$$	$$6,000$$

Formulate the problem as a linear programming problem and find all the constraints for the above product mix problem.

0%
$$3{x}_{1}-4{x}_{2}=0$$ and $$5{x}_{2}-4{x}_{3}=0$$ where $${x}_{1},{x}_{2},{x}_{3}\ge0$$
0%
$$4{x}_{1}-3{x}_{2}=0$$ and $$5{x}_{2}-4{x}_{3}=0$$ where $${x}_{1},{x}_{2},{x}_{3}\ge0$$
0%
$$4{x}_{1}-3{x}_{2}=0$$ and $$4{x}_{2}-5{x}_{3}=0$$ where $${x}_{1},{x}_{2},{x}_{3}\ge0$$
0%
$$4{x}_{1}-3{x}_{2}=0$$ and $$5{x}_{2}-4{x}_{3}=0$$ where $${x}_{1},{x}_{2},{x}_{3}\le0$$

Explanation

Q.21.

Equation of normal drawn to the graph of the function defined as $$f(x)=\dfrac{\sin x^2}{x}$$, $$x\neq 0$$ and $$f(0)=0$$ at the origin is?

0%
$$x+y=0$$
0%
$$x-y=0$$
0%
$$y=0$$
0%
$$x=0$$

Explanation

Q.22.

The taxi fare in a city is as follows. For the first km the fare is $$Rs.10$$ and subsequent distance is $$Rs.6 / km.$$ Taking the distance covered as $$x \ km$$ and fare as $$Rs\ y$$ ,write a linear equation.

0%
$$y=4+6x$$
0%
$$y=4+5x$$
0%
$$y=3+6x$$
0%
$$y=3+5x$$

Explanation

Q.23.

The feasible region for anLPP is shown shaded in the figure. Find the maximum value of the objective function $$z=11x+7y$$.

0%
35
0%
47
0%
21
0%
14

Explanation

Q.24.

" the relation S is defined on $$N\times N\left ( a,b \right )S\left ( c,d \right )\Leftrightarrow bc\left ( a+d \right )=ad$$ in an equivalance " that statement is ?

0%
True
0%
False

Q.25.

The inequalities $$f(-1)\le -4,f(1) \le 0$$ & $$f(3) \le 5$$ are known to hold for $$f(x)=ax^{2}+bx+c$$ then the least value of $$'a'$$ is:

0%
$$-1/4$$
0%
$$-1/3$$
0%
$$1/4$$
0%
$$1/8$$

Q.26.

The problem associated with $$ LPP$$ is

0%
single objective function
0%
Double objective function
0%
No any objective function
0%
None

Explanation

Q.27.

Shade region is represented by

0%
$$2x+5y \ge 80,x+y \le 20, x \ge 0,y \le 0$$
0%
$$2x+5y \ge 80,x+y \ge 20, x \ge 0,y \ge 0$$
0%
$$2x+5y \le 80,x+y \le 20, x \ge 0,y \ge 0$$
0%
$$2x+5y \le 80,x+y \le 20, x \ge 0,y \le 0$$

Q.28.

The maximum value of $$P=3x+4y$$ subject to the constraints $$x +y \le 40,x+2y \le 60,x \ge 0$$ and $$y \ge 0$$ is

0%
$$120$$
0%
$$140$$
0%
$$100$$
0%
$$160$$

Q.29.

The maximum value of $$4x+5y$$ subject to the constraints $$x+y \le 20,x+2y \le 35,x-3y \le 12$$ is

0%
$$84$$
0%
$$95$$
0%
$$100$$
0%
$$96$$

Q.30.

If $$x$$ is any real number, then which of the following is correct?

0%
$$\dfrac { x ^ { 2 } } { 1 + x ^ { 4 } } \geq \dfrac { 1 } { 2 }$$
0%
$$\dfrac { x ^ { 2 } } { 1 + x ^ { 4 } } \geq \dfrac { 1 } { 4 }$$
0%
$$\dfrac { x ^ { 2 } } { 1 + x ^ { 4 } } \leq \dfrac { 1 } { 2 }$$
0%
none of these

Explanation

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