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

If U = `e^(xyz) f((xy)/z)` prove that `x(delu)/(delx)+z(delu)/(delx)2xyzu` and `y(delu)/(delx)+z(delu)/(delz)=2xyzu` and hence show that `x(del^2u)/(delzdelx)=y(del^2u)/(delzdely)`

If y = log `[tan(pi/4+x/2)]`Prove that

I. tan h`y/2 = tan  pi/2`
II. cos hy cos x = 1

Find the maximum and minimum values of `f(x,y)=x^3+3xy^2-15x^2-15y^2+72x`

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

Obtain tan 5𝜽 in terms of tan 𝜽 & show that `1-10tan^2  x/10+5tan^4  x/10=0`

If y=etan_1x. prove that `(1+x^2)yn+2[2(n+1)x-1]y_n+1+n(n+1)y_n=0`

Find tanhx if 5sinhx-coshx = 5 

Separate into real and imaginary parts of cos`"^-1((3i)/4)` 

Considering only principal values separate into real and imaginary parts

`i^((log)(i+1))`

Find the n^th derivative of `x^3/((x+1)(x-2))`

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

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

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

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)]`

Evaluate `lim_(x->0) sinx log x.`

Find the n^th derivative of y=e^ax cos2 x sin x.

Find nth derivative of `1/(x^2+a^2.`

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

Express `(2x^3+3x^2-8x+7)` in terms of (x-2) using taylor'r series. 

Prove that `tan_1 x=x-x^3/3+x^5/5+.............`