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

Instructions:

THE client had 5 questions that he required them to be answered correctly and accurately. the answers were provided in as detailed manner as possible with all the methods are shown from the first step to the last one, the questions ranged from different topics including calculus, matrices, statitscs and equations of lines. source..

Content:

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.
...

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