BE Marine Engineering Semester 1 (FE First Year) - University of Mumbai Important Questions for Applied Mathematics 1

Prove that `sin^5theta=1/16[sin5theta-5sin3theta+10sintheta]`

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

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

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 y=(x+√x²-1 ,Prove that

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

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

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

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`

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

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=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.

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

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

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`

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)`

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))`

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