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

By using Regular falsi method solve 2x – 3sin x – 5 = 0.

Expand sec x by McLaurin’s theorem considering up to x⁴ term.

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

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

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). 

Expand `2x^3+7x^2+x-6` in powers of (x-2)

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]]`

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

Prove that 𝒕𝒂𝒏𝒉−𝟏(𝒔𝒊𝒏 𝜽) = 𝒄𝒐𝒔𝒉−𝟏(𝒔𝒆𝒄 𝜽) 

Prove that the matrix `1/sqrt3`  `[[ 1,1+i1],[1-i,-1]]` is unitary. 

`"If"  x=uv & y=u/v "prove that"  jj^1=1`

If u=`f((y-x)/(xy),(z-x)/(xz)),"show that"  x^2 (del_u)/(del_x)+y^2 (del_u)/(del_y)+x^2 del_u/del_z=0`

Find non singular matrices P and Q such that A = `[(1,2,3,2),(2,3,5,1),(1,3,4,5)]`

Solve by Gauss Jacobi Iteration Method: 5x – y + z = 10, 2x + 4y = 12, x + y + 5z = -1. 

Solve using Gauss Jacobi Iteration method 
2𝒙 + 12y + z – 4w = 13
13𝒙 + 5y - 3z + w = 18
2𝒙 + y – 3z + 9w = 31
3𝒙 - 4y + 10z + w = 29

Solve the following equation by Gauss-Seidel method upto four iterations

4x-2y-z=40, x-6y+2y=-28, x-2y+12z=-86.

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)`

Solve the following system of equation by Gauss Siedal Method,20x+y-2z=17
             3x+20y-z =-18
             2x-3y+20z=𝟐𝟓

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)`