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 2nd and 3rd 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) …