BE Mechanical Engineering Semester 1 (FE First Year) - University of Mumbai Question Bank Solutions for Applied Mathematics 1

State and prove Euler’s Theorem for three variables.

If z = f (x, y) where x = e^u +e^-v, y = e^-u - e^v then prove that `(delz)/(delu)-(delz)/(delv)=x(delz)/(delx)-y(delz)/(dely).`

If U `=sin^(-1)[(x^(1/3)+y^(1/3))/(x^(1/2)+y^(1/2))]`prove that `x^2(del^2u)/(del^2x)+2xy(del^2u)/(delxdely)+y^2(del^2u)/(del^2y)=(tanu)/144[tan^2"U"+13].`

Using encoding matrix `[[1,1],[0,1]]` ,encode & decode the message "MUMBAI" 

Show that `sin(e^x-1)=x^1+x^2/2-(5x^4)/24+`...................

Find the roots of the equation `x^4+x^3 -7x^2-x+5 = 0` which lies between 2 and 2.1 correct to 3 places of decimals using Regula Falsi method.

Using the matrix A = `[[-1,2],[-1,1]]`decode the message of matrix C= `[[4,11,12,-2],[-4,4,9,-2]]`

Using Newton Raphson method solve 3x – cosx – 1 = 0. Correct upto 3 decimal places. 

Show that xcosecx = `1+x^2/6+(7x^4)/360+......` 

If y= cos (msin_1 x).Prove that `(1-x^2)y_n+2-(2n+1)xy_(n+1)+(m^2-n^2)y_n=0`

Show that the following equations: -2x + y + z = a, x - 2y + z = b, x + y - 2z = c have no solutions unless a +b + c = 0 in which case they have infinitely many solutions. Find these solutions when a=1, b=1, c=-2. 

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

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

Obtain the root of 𝒙^𝟑−𝒙−𝟏=𝟎 by Regula Falsi Method
(Take three iteration).

If coshx = secθ prove that (i) x = log(secθ+tanθ). (ii) `θ=pi/2tan^-1(e^-x)`

Prove that `cos^-1tanh(log x)+ = π – 2(x-x^3/3+x^5/5.........)`

If` y= e^2x sin  x/2 cos   x/2 sin3x. "find"  y_n`

Evaluate `Lim _(x→0) (cot x)^sinx.`

Prove that log `[sin(x+iy)/sin(x-iy)]=2tan^-1 (cot x tanhy)`

`"If" sin^4θcos^3θ = acosθ + bcos3θ + ccos5θ + dcos7θ "then find"  a,b,c,d.`