MTHEMATICS QUESTIONS TO BE ANSWERED (Other (Not Listed) Sample)

Provide solution to the following questions: 1. Evaluate the following:   Solution Using Integration by Parts (IBP) udv=uv-vdu Let u=x dv=sin3xdx xsin3xdx=-13xcos3x--13cos3xdx =-13xcos3x-(-19sin3x) =-13xcos3x+19sin3x+C 2. If, then for what value of α is A an identity matrix? An identity matrix is any square matrix where all the elements of the leading diagonal are 1 while all the other elements are 0. We have to equate this matrix to another one which is an identity matrix let’s call it B. A=B cosα-sinαsinα-cosα=1001 Equating the parts cosα=1 α=cos-11 α=0 -sinα=0 α=-sin-10 α=-sin-10 α=0 sinα=0 α=sin-10 α=0 -cosα=1 α=-cos-11 α=-0 The value of α that makes the matrix an identity is 0 3.  The line y = mx + 1 is a tangent to the curve y2 = 4x .Find the value of m. Solution: ∂∂xy2=∂∂x4x 2y×m=4x y1=mx+1 y2=4x y1=y2 4x=mx+1 4x=(mx+1)2 4x=m2x2+2mx+1 0=m2x2+2mx-4x+1 0=m2x2+x(2m-4)+1 We also know that for a tangent, it meets at a point that satisfy the quadratic equation: b2=4ac=0 (2m-4)2=4m2 4m2-16m+16-4m2=0 16=16m m=1 4. Solve the following differential equation: (x2 − y2) dx + 2 xy dy = 0. (x2 − y2) dx + 2 xy dy = 0 Solution M(x,y)= x2y2 N(x,y)=2xy Then using Partial derivatives; ∂M∂x=2y ∂N∂x=2y Since ∂M∂x=∂N∂x ∫x2y2dx=∫2xydx+C x33-xy2=x2y+C When we rearrange the equation, we get: x33-xy2-x2y=C x3-3x2y3-3xy2=C 5. Solve system of linear equations, using matrix method. ...