Subjective Test 2   Student Name   Institutional Affiliation                                     Provide solution to the following questions: 1. Evaluate the following: This question requires integr

Subjective Test 2

Student Name

Institutional Affiliation

Provide solution to the following questions:

1. Evaluate the following:

This question requires integration by parts

Let u=x and   =sin (3x)

du=dx=1

But

 =  where u=3x

                        = [-Cos (u)]

But u =3x

Substituting u=3x in [-Cos (u)] gives;

=Cos (3X)

Hence

= x

                          =Cos (3x) +

But

 =

                            =  sin (u)

Substituting u=3x in  sin (u) gives

 =  sin (3x)

Therefore,

= Cos (3X) + sin (3x)

2. If, then for what value of α is A an identity matrix?

A=

The first step is calculation of transpose,

=

A=

A=

A=

I =

3. The line y = mx + 1 is a tangent to the curve y2 = 4x .Find the value of m.

Substitute y=mx+1 in =4x

= 4x

Expanding = 4x

+1+2mx-4x=0

+x(2m-4)+1=0

The tangent touches the curve at one point hence the roots are equal. Therefore, discriminant =0

-4((1) = 0

+16-16m-=0

16-16m=0

16=16m

M=1

4. Solve the following differential equation: (x2 − y2) dx + 2 xy dy = 0.

Te integrating factor of the equation is

=  since it is homogenous equation

=  (by simplification)

Multiplying by the initial equation changes to:

  = 0

  = 0

Taking the initial points as ; ( ,

The equation  is of the form ;

 =

Solving  gives

Ln(x)- ln()+ln()-ln( =

If x= then,

 ln()= +ln(x)

Hence

+ = CX

5. Solve system of linear equations, using matrix method.

x − y +2z =73x +4y −5z =−52x − y + 3z = 12

  =  

Find the determinant

Det = 1*det-(-1)*det+2*det

       =  +  +

       = 7+19-22

       =4

Therefore,

The inverse, =

  =  

 =

Hence

X=2

Y=1

Z= 3

Reference

Chirgwin, B., & Plumpton, C. (2016). A course of mathematics for engineers and scientists. Elkins Park: Pergamon Press/Elsevier Science.