2017-2018 June

(1) Question no. 1 is compulsory.

(2) Attempt any 3 questions from remaining five questions.

If `tan(x/2)=tanh(u/2),"show that" u = log[(tan(pi/4+x/2))] `

Concept: Inverse Hyperbolic Functions

Prove that the following matrix is orthogonal & hence find 𝑨−𝟏.

A`=1/3[(-2,1,2),(2,2,1),(1,-2,2)]`

Concept: Transpose of a Matrix

State Euler’s theorem on homogeneous function of two variables and if `u=(x+y)/(x^2+y^2)` then evaluate `x(delu)/(delx)+y(delu)/(dely`

Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)

If `u=r^2cos2theta, v=r^2sin2theta. "find"(del(u,v))/(del(r,theta))`

Concept: Jacobian

Find the nth derivative of cos 5x.cos 3x.cos x.

Concept: nth derivative of standard functions

Evaluate : `lim_(x->0)((2x+1)/(x+1))^(1/x)`

Concept: L‐ Hospital Rule

If `y=e^(tan^(-1)x)`.Prove that

`(1+x^2)y_(n+2)+[2(n+1)x-1]y_(n+1)+n(n+1)y_n=0`

Concept: Leibnitz’S Theorem (Without Proof) and Problems

Examine the function `f(x,y)=xy(3-x-y)` for extreme values & find maximum and minimum values of `f(x,y).`

Concept: Maxima and Minima of a Function of Two Independent Variables

Investigate for what values of 𝝁 𝒂𝒏𝒅 𝝀 the equation x+y+z=6; x+2y+3z=10; x+2y+𝜆z=𝝁 have

(i)no solution,

(ii) a unique solution,

(iii) infinite no. of solution.

Concept: consistency and solutions of homogeneous and non – homogeneous equations

If u =`f((y-x)/(xy),(z-x)/(xz)),` show that `x^2(delu)/(delx)+y^2(delu)/(dely)+z^2(delu)/(delz)=0`.

Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)

Prove that `log((a+ib)/(a-ib))=2itan^(-1) b/a & cos[ilog((a+ib)/(a-ib))=(a^2-b^2)/(a^2+b^2)]`

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions

If `u=sin^(-1)((x+y)/(sqrtx+sqrty))`,Prove that

`x^2u_(x x)+2xyu_(xy)+y^2u_(yy)=(-sinu.cos2u)/(4cos^3u)`

Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof)

Using encoding matrix `[(1,1),(0,1)]`encode and decode the message

“ALL IS WELL” .

Concept: Application of Inverse of a Matrix to Coding Theory

Solve the following equation by Gauss Seidal method:

`10x_1+x_2+x_3=12`

`2x_1+10x_2+x_3-13`

`2x_1+2x_2+10x_3=14`

Concept: Gauss Seidal Iteration Method

If `u=e^(xyz)f((xy)/z)` where `f((xy)/z)` is an arbitrary function of `(xy)/z.`

Prove that: `x(delu)/(delx)+z(delu)/(delz)=y(delu)/(dely)+z(delu)/(delz)=2xyz.u`

Concept: Partial Derivatives of First and Higher Order

Prove that `sin^5theta=1/16[sin5theta-5sin3theta+10sintheta]`

Concept: Expansion of sinnθ, cosnθ in powers of sinθ, cosθ

Prove that `log(secx)=1/2x^2+1/12x^4+.........`

Concept: Logarithmic Functions

Expand `2x^3+7x^2+x-1` in powers of x - 2

Concept: Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ

Prove that `sin^(-1)(cosec theta)=pi/2+i.log(cot theta/2)`

Concept: Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ

Prove that `sin^(-1)(cosec theta)=pi/2+i.log(cot theta/2)`

Concept: Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ

Find non singular matrices P and Q such that A = `[(1,2,3,2),(2,3,5,1),(1,3,4,5)]`

Concept: PAQ in normal form

Obtain the root of 𝒙^{𝟑}−𝒙−𝟏=𝟎 by Regula Falsi Method

(Take three iteration).

Concept: Regula – Falsi Equation

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