BE Automobile Engineering Semester 1 (FE First Year) - University of Mumbai Question Bank Solutions

Evaluate : `Lim_(x→0) (x)^(1/(1-x))`

If x = uv, y `=(u+v)/(u-v).`find `(del(u,v))/(del(x,y))`.

If `y=2^xsin^2x cosx` find `y_n`

Prove that log `[tan(pi/4+(ix)/2)]=i.tan^-1(sinhx)`

If `Z=x^2 tan-1y /x-y^2 tan -1 x/y del` 

Prove that `(del^z z)/(del_ydel_x)=(x^2-y^2)/(x^2+y^2)`

If `u=x^2+y^2+z^2` where `x=e^t, y=e^tsint,z=e^tcost`

Prove that `(du)/(dt)=4e^(2t)`

If `z=tan(y-ax)+(y-ax)^(3/2)` then show that `(del^2z)/(delx^2)= a^2 (del^2z)/(dely^2)`

Show that `ilog((x-i)/(x+i))=pi-2tan6-1x`

If `x=uv, y=u/v."prove that"  jj,=1`

Find the maxima and minima of `x^3 y^2(1-x-y)`

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).`

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`

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

Show that sec h^-1(sin θ) =log cot (`theta/2` ).

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

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)`