Optionals IAS Mains Mathematics: Questions 187 - 194 of 283

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Question 187

Appeared in Year: 2009

Describe in Detail

Essay▾

Form the Partial Differential Equation by eliminating the arbitrary function by (CS Paper -2)

Explanation

Given arbitrary function is … eq. (1)

Let … eq. (2)

Then, (1) becomes

… eq. (3)

Differentiating (3) , partially w. r. t. , we have

… eq. (4)

Where

Now from we have and … eq. (5)

Using (5) , (4) reduces to

… eq. (6)

Again, differentiating (3) , partially w. r. t. y, we get

… eq. (7)

Dividing (6) and (7) , we get

… (7 more words) …

Question 188

Appeared in Year: 2004

Describe in Detail

Essay▾

Find the co-ordinate of the points on the sphere , the tangent planes at which are parallel to the plane (CS Paper -1)

Explanation

The equation of the sphere is … eq. (1)

The equation of any plane parallel to is

… eq. (2)

Center of sphere (1) is (2, -1,0)

And radius =

The plane (2) will touch sphere (1) is

Distance of the centre (2, -1,0) from plane (2)

= radius of the sphere

i.e.. Is

Putting in (2) we get … eq. (3)

And … eq. (4)

Which are the two planes

The equation of any strai…

… (37 more words) …

Question 189

Appeared in Year: 2013

Describe in Detail

Essay▾

Form a partial differential equation by eliminating the arbitrary functions f and g from (CS paper -2)

Explanation

Given … eq. (1)

Differentiating (1) partially w. r. t. x and y, we get

… eq. (2)

And … eq. (3)

Differentiating (3) w. r. t.

… eq. (4)

From (2) and (3)

Substituting these value in (4) , we have

… (4 more words) …

Question 190

Appeared in Year: 2011

Describe in Detail

Essay▾

Solve by simplex method, the following LP problem:

Maximize,

Constraints,

(CS paper -2)

Explanation

Given

Maximize,

Constraints,

We introduce slack variables and convert constraints into equation and assign ‘o’

Coefficient to the slack variable in the objective function

Max

Subject to

Where

Simplex Table 1

Simplex Table 2

Table Shows the Simplex Table 2

Simplex Table 3

Since all the

So optimal solution obtained

Max

… (21 more words) …

Question 191

Appeared in Year: 2013

Describe in Detail

Essay▾

Find the general solution of the equation

(CS paper - 1)

Explanation

Given differential equation is

Put

Then given equation becomes

>

Its Auxiliary equation is

Now

Solution is

Question 192

Appeared in Year: 2008

Describe in Detail

Essay▾

Find the Dual of the following linear programming problem:

Max

Such that

(CS paper -2)

Explanation

Given primal problem is

Max

Such that

To make the dual in case of maximisation problem all the constraints should be of type. Hence are multiply III constraint by and divide the constraint no II into 2 parts and rewrite the above problem as:

Max

Subject to

Dual of the above problem is as below.

Min

Subject to

Where

But in primal there are 3 constrai…

… (26 more words) …

Question 193

Appeared in Year: 2009

Describe in Detail

Essay▾

Find the integral surface of

Which passes through the curve: (CS Paper -2)

Explanation

Given … eq. (1)

Given curve is given by … eq. (2)

Here Lagrange՚s auxiliary equations for (1) are … eq. (3)

Taking first and third fractions of (3)

Integrating

… eq. (4)

Taking the second and third fractions of (1)

Integrating

… eq. (5)

Adding (4) and (5)

Using (2) , we get

Substituting the value of from (4) and (5) in (6) , we get

… (7 more words) …

Question 194

Appeared in Year: 2009

Describe in Detail

Essay▾

Show that the differential equation of all cones which have their vertex at the origin is . Verify that this equation is satisfied by the surface (CS Paper -2)

Explanation

The equation of any cone with vertex at origin is

… eq. (1)

Where are Parameters

Differentiating (1) partially w. r. t. ‘x’ and ‘y’ by turn, we have (noting that )

… eq. (2)

And

… eq. (3)

Multiply (2) by x and (3) by y, and adding, we get

- [using (1) ]

-

… eq. (4)

Which is required partial differential equation

Second Part

Given surface is … eq. (5)

Dif…

… (35 more words) …