Optionals IAS Mains Mathematics: Questions 26 - 35 of 283

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Question 26

Row and Column Reduction

Appeared in Year: 2016

Describe in Detail

Essay▾

If then find dim and dim [CS (Main) Paper 1]

Explanation

Given

Now

i.e.. is the solution space of equation

Now, we shall find at basis.

Hence a bases of solution space

and dim

Now given

i.e.. is the solution space of the equation

Now we shall find it basis

Here we have 3 unknown out of which an independent.

Hence a basis of solution space

and dim

Now

Now, we find to find its basis

Here

Here, we have thre…

… (38 more words) …

Question 27

Appeared in Year: 2015

Describe in Detail

Essay▾

Obtain Laplace Inverse Transform of

[CS (Main) Paper 1]

Explanation

  • Given is

  • Now

  • Where

    Now

    Let

  • Now

  • But

  • Now

  • We know

    By second shifting theorem

Question 28

Appeared in Year: 2015

Describe in Detail

Essay▾

Using Laplace transform, solve [CS (Main) Paper 1]

Explanation

  • Given
  • Apply Laplace transform on both sides, we get

  • using given condition we get

  • Taking inverse Laplace Transform, we get

  • Now

  • Putting these value in we get

Question 29

Appeared in Year: 2015

Describe in Detail

Essay▾

Evaluate , where is the reactangle with vertices [CS (Main) Paper 1]

Explanation

  • By Green՚s Theorem in plane,

  • Here

Green՚S Theorem

… (4 more words) …

Question 30

Row and Column Reduction

Appeared in Year: 2015

Describe in Detail

Essay▾

If Matrix then find [CS (Main) Paper 1]

Explanation

Given

where and

Now

Now

Question 31

Appeared in Year: 2015

Describe in Detail

Essay▾

Find the constant so that is the integrating factor of

and hence solve the differential equation [CS (Main) Paper 1]

Explanation

  • Given differentia; equation is

    … eq. (1)

    If is I. F. of

  • Then

    is exact

  • Comparing with we get

    Now

    is Exact

  • Comparing coefficient both sides, adges

  • and

  • and

  • and

  • So solution is

    (treating as constant.)

… (1 more words) …

Question 32

Appeared in Year: 2016

Describe in Detail

Essay▾

Solve [CS (Main) Paper 1]

Explanation

  • Given Differential equation is

  • Now compare it with

    Linear differential equation.

    We get

  • Thus, the solution of this differential equation is

    i.e.. .

    Now

  • Then Solution is

    Put

Question 33

Appeared in Year: 2016

Describe in Detail

Essay▾

Determine the characteristic of the equation and find the integral surface which passes through the parabola [CS (Main) Paper 2]

Explanation

  • The given differential equation is

  • The initial condition for are

  • The initial condition of are determined by

    and

    and

    and

    and

    and

  • The characteristic equations are

  • Integrating, we have

  • Appling Initial condition

    we get

  • Also

  • On integrating, we have

  • Appling initial condition

    We have

    Also

  • On integrating, we get

  • Apply initial condition

    Now

  • on integrating both …

… (45 more words) …

Question 34

Solution of System of Linear Equations

Appeared in Year: 2016

Describe in Detail

Essay▾

Using elementary two operations, find the condition that the linear equations.

Have a solution [CS (Main) ]

Explanation

  • Given linear equation is

    Here

    Now

    Operate:

    Operate:

  • Now, here
  • The linear equation have a solution if

    Now iff

    (Answer)

  • If then given linear equation will not have any solution.

Question 35

Appeared in Year: 2015

Describe in Detail

Essay▾

For what positive value of , the plane touches the sphere and hence find the point of contact. [CS (Main) Paper 1]

Explanation

  • The equation of sphere is

  • Its centre is

    and radius

  • The Equation of plane is

  • The plane will touch the sphere is perpendicular distance of centre from plane

    if

    if

    if

    if

    if

    if

    if

  • To find point of contact
  • The direction ratio՚s of the normal to the plane are
  • The equation of line through the centre of the sphere plane are

    Any point on it is if it lies …

… (9 more words) …