# Optionals IAS Mains Mathematics: Questions 167 - 170 of 283

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## Question 167

Appeared in Year: *2010*

### Describe in Detail

Essay▾Determine an optimal transportation cost of 340 tons of a certain type of material from three factories to five warehouses is minimized. The warehouses must receive 40 tons, 50 tons, 70 tons, 90 tons and 90 tons respectively. The availability of the material at is 100 tons, 120 tons, 120 tons respectively. The transportation costs per ton from factories to warehouse are given in the table below:

Use Vogel՚s approximation method to obtain the initial basic feasible solution. (CS paper -2)

### Explanation

Given Transportation problem is given below:

Availability | ||||||

Requirement |

By using VAM-Initial feasible solution we get

Noted that there are allocations which are necessary to proceed further.

Total transportation cost = ₹ 1430

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## Question 168

Appeared in Year: *2014*

### Describe in Detail

Essay▾Solve the initial value problem By using Laplace-transform (CS Paper -1)

### Explanation

Taking Laplace Transform, we get

Taking inverse Laplace Transform, we get

Now

And

By using convolution theorem

Now

Put

And

## Question 169

Appeared in Year: *2013*

### Describe in Detail

Essay▾Find the equation of the plane which passes through the points and is parallel to the line joining the points . Find also the distance between the line and the plane. (CS paper - 1)

### Explanation

The equation of the line passing through is

… eq. (1)

The equation of a plane passing through is

… eq. (2)

This passes through the point

… eq. (3)

The plane (3) is parallel to the line

Therefore, normal to the plane is perpendicular to the line

… eq. (4)

By using (3) and (4)

By cross- Multiplication, we get

Substituting the value of in (2) , we get

……

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## Question 170

Appeared in Year: *2012*

### Describe in Detail

Essay▾For each hour per day that Ashok studies mathematics, it yields him 10 marks and for each hour that he studies physics, it yields him 5 marks. He can study at most 14 hours a day and he must get at least 40 marks in each. Determine graphically how many hours a day he should study mathematics and physics each, in order to maximize his marks? (CS paper -2)

### Explanation

Let be the number of hour he study mathematics and y be the number of hour he study physics

Max (marks function)

Subject to

Where

Now, in order to plot the graph, we convert the inequalities into equation.

… eq. (i)

And

… eq. (ii)

And 3) … eq. (iii)

Any Combination of value of x and y which satisfies the given constr…

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