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 coordinate 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
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) …