# Optionals IAS Mains Mathematics: Questions 179 - 186 of 283

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## Question 179

Appeared in Year: *2014*

### Describe in Detail

Essay▾Examine whether the plane cuts the cone in perpendicular lines. (CS Paper -1)

### Explanation

The given equation are … eq. (1)

… eq. (2)

Let the equation of a line of section be

… eq. (3)

It lines in the plane (1)

It is perpendicular to the Normal to the plane

… eq. (4)

Again as at lines on the cone (2)

… eq. (5)

From (4) ,

Putting this value of n in (5) , we get

. . eq. (6)

Which is quadratic in , giving two values of

If are direction number…

… (53 more words) …

## Question 180

Appeared in Year: *2007*

### Describe in Detail

Essay▾Solve the following by simplex method

Maximize

Subject to

(CS paper -2)

### Explanation

Given problem is

Maximize

Subject to

We introduce slack variables and convert constraints into equation and assign ‘o’ co-efficient to the slack variables in the objective function.

Max

Subject to

Simplex Table 1

Simplex Table 2

Simplex Table 3

Simplex Table 4

Since all

So optimal solution is obtained

… (41 more words) …

## Question 181

Appeared in Year: *2008*

### Describe in Detail

Essay▾Find the general solution of the partial differential equation:

Where (CS paper -2)

### Explanation

Given

Its auxiliary equation is

Being arbitrary function

Now, we find particular integral

General solution is

## Question 182

Appeared in Year: *2008*

### Describe in Detail

Essay▾If represent one of a set of three mutually perpendicular generators of the cone

Find the equations of the other two. (CS paper -2)

### Explanation

The given cone is … eq. (1)

The cone have mutually perpendicular generator is

Here sum of the coefficient of

Cone (1) has three mutually perpendicular generators.

The equation of one of three perpendicular generators are

… eq. (2) , the equation of the plane passing through the vertex. of the cone and perpendicular to line (2) is

… eq. (3)

Now the …

… (70 more words) …

## Question 183

Appeared in Year: *2012*

### Describe in Detail

Essay▾A variable plane is parallel to the plane and meets the axes in A, B, C respectively, prove that the circle ABC lies on the cone

(CS paper -1)

### Explanation

The equation of any plane parallel to the plane is

… eq. (1)

It meets x-axis in A where putting in (1) we get

Similarly B is

The equation of sphere is … eq. (2)

The equation (1) and (2) together represent a circle ABC,

Making (2) Homogeneous by means of (1) , we get

Which is required equation of cone

… (2 more words) …

## Question 184

Appeared in Year: *2013*

### Describe in Detail

Essay▾Minimize

Subject to the constraints

(CS paper -2)

### Explanation

Given

Minimize

Maximize

Subject to the constraints

Table 1

Table 2

Table 3

Since all the values of

So optimal solution obtained

… (35 more words) …

## Question 185

Appeared in Year: *2012*

### Describe in Detail

Essay▾Use Newton-Raphson method to find the real root of equation correct up to four Decimal places (CS paper -2)

### Explanation

The given equation is

Let

Now

Since is nearer to zero, we take as initial approximation

Now

By Newton-Raphson՚s iterative formula, we have

Now

And

The second approximation to the root is given by

Now

And

The third approximation to the root is given by

From 2^{nd} and 3^{rd} iterations, it is clear that there is no change in the successive approximation to …

… (20 more words) …

## Question 186

Appeared in Year: *2013*

### Describe in Detail

Essay▾Maximize

Subjected to

And (CS paper -2)

### Explanation

Given max

Subjected to

And

We introduce surplus and artificial variable and assign O co-efficient to surplus to artificial variable. The resultant objective function and constraint are given as under:

Max

Where

Table 1

Table 2

Table 3

Since all the values of

So optimal solution obtained

… (35 more words) …