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

If y=(x+√x²-1 ,Prove that

`(x^2-1)y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2)y_n=0`

Using De Moivre’s theorem prove that]

`cos^6theta-sin^6theta=1/16(cos6theta+15cos2theta)`

Solve  `x^4-x^3+x^2-x+1=0.`

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`

If y=sin[log(x²+2x+1)] then prove that (x+1)²y_n+2 +(2n +1)(x+ 1)y_n+1 + (n²+4)y_n=0.

By using De Moivre's Theorem obtain tan 5θ in terms of tan θ and show that `1-10 tan^2(pi/10)+5tan^4(pi/10)=0`.

If Z=f(x.y). x=r cos θ, y=r sinθ. prove that `((delz)/(delx))^2+((delz)/(dely))^2=((delz)/(delr))^2+1/r^2((delz)/(delθ))^2`

If 𝒚 satisfies the equation `(dy)/(dx)=x^2y-1` with `x_0=0, y_0=1` using Taylor’s Series Method find 𝒚 𝒂𝒕 𝒙= 𝟎.𝟏 (take h=0.1).

Use Taylor’s series method to find a solution of `(dy)/(dx) =1+y^2, y(0)=0` At x = 0.1 taking h=0.1 correct upto 3 decimal places.

Use Taylor series method to find a solution of `dy/dx=xy+1,y(0)=0` X=0.2 taking h=0.1 correct upto 4 decimal places. 

Expand 2 𝒙³ + 7 𝒙² + 𝒙 – 6 in power of (𝒙 – 2) by using Taylors Theorem.

Show that the roots of x⁵ =1 can be written as 1, `alpha^1,alpha^2,alpha^3,alpha^4` .hence show that `(1-alpha^1) (1-alpha^2) (1-alpha^3)(1-alpha^4)=5.`

If x = u+v+w, y = uv+vw+uw, z = uvw and φ is a function of x, y and z
Prove that

If tan(θ+iφ)=tanα+isecα
Prove that
1)`e^(2varphi)=cot(varphi/2)`
2) `2theta=npi+pi/2+alpha`

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`

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

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

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

State and Prove Euler’s Theorem for three variables.

Show that all roots of `(x+1)^6+(x-1)^6=0` are given by -icot`((2k+1)n)/12`where k=0,1,2,3,4,5.

Find all values of `(1 + i)^(1/3` and show that their continued product is (1+ 𝒊 ).