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

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`

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)]`

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

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

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

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 non singular matrices P & Q such that PAQ is in normal form where A `[[2,-2,3],[3,-1,2],[1,2,-1]]`

Express the matrix as the sum of symmetric and skew symmetric matrices.

Using encoding matrix `[[1,1],[0,1]]` ,encode & decode the message "MUMBAI" 

Reduce the following matrix to its normal form and hence find its rank.

Investigate for what values of μ and λ the equations x+y+z=6, x+2y+3z=10, x+2y+λz=μ has
1) No solution
2) A unique solution
3) Infinite number of solutions.