Optionals IAS Mains Mathematics: Questions 187 - 194 of 283

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

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…

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

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

Simplex Table 3

Since all the

So optimal solution obtained

Max

Given differential equation is

Put

Then given equation becomes

>

Its Auxiliary equation is

Now

Solution is

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…

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

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…