BE Computer Engineering Semester 1 (FE First Year) - University of Mumbai Important Questions

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

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

Find the stationary points of the function x3+3xy2-3x2-3y2+4 & also find maximum and minimum values of the function.

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

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`

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 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).

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

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. 

Find maximum and minimum values of x³ +3xy² -15x²-15y²+72x.

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

Expand sec x by McLaurin’s theorem considering up to x⁴ term.

Evaluat `lim_(x->0) (e^(2x)-(1+x)^2)/(xlog(1+x)`

Solve the following equation by Gauss-Seidel method upto four iterations

4x-2y-z=40, x-6y+2y=-28, x-2y+12z=-86.