BE Automobile Engineering सत्र १ (इंजीनियरिंग) - University of Mumbai Important Questions for Applied Mathematics 1

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. 

Investigate for what values of 𝝁 "𝒂𝒏𝒅" 𝝀 the equations : `2x+3y+5z=9`

`7x+3y-2z=8`

`2x+3y+λz=μ`

Have (i) no solution (ii) unique solution (iii) Infinite value 

Obtain the root of `x^3-x-1=0` by Newton Raphson Method` (upto three decimal places). 

If u= `sin^-1 ((x+y)/(sqrtx+sqrty)), " prove that ""`i.xu_x+yu_y=1/2 tanu`

ii. `x^2uxx+2xyu_xy+y^2u_(y y)=(-sinu.cos2u)/(4cos^3u)`

Show that the matrix A is unitary where A = `[[alpha+igamma,-beta+idel],[beta+idel,alpha-igamma]]` is unitary if `alpha^2+beta^2+gamma^2+del^2=1` 

Using the encoding matrix `[(1,1),(0,1)]` encode and decode the messag I*LOVE*MUMBAI.

Using the matrix A = `[[-1,2],[-1,1]]`decode the message of matrix C= `[[4,11,12,-2],[-4,4,9,-2]]`

If u `=sin^(-1)((x^(1/3)+y^(1/3))/(x^(1/2)-y^(1/2)))`, Prove that 

`x^2(del^2u)/(delx^2)+2xy(del^2u)/(delxdely)+y^2(del^2u)/(dely^2)=tanu/144(tan^2u+13)`

Prove that the following matrix is orthogonal & hence find 𝑨−𝟏.

A`=1/3[(-2,1,2),(2,2,1),(1,-2,2)]`

Investigate for what values of 𝝁 𝒂𝒏𝒅 𝝀 the equation x+y+z=6; x+2y+3z=10; x+2y+𝜆z=𝝁 have
(i)no solution,
(ii) a unique solution,
(iii) infinite no. of solution.

Reduce matrix to PAQ normal form and find 2 non-Singular matrices P & Q.

`[[1,2,-1,2],[2,5,.2,3],[1,2,1,2]]`