# Matrices, Differentiation and Integration (Math Problem Sample)

This paper was about answering questions on seelcted topics in maths . the topics were matrices, differentiation, and integration. each question required to explain how to reach the answer vivivdly. I used diffrerent and simple formulaes to reach to the answer. I was able to answer the questions correctly and provided releavant solution. again, i gave the client the required references and he was happy about it.

source..Provide solution to the following questions:

1. Evaluate the following:

By employing ILATE Rule:

uv dx=uv dx- dudxv dxdx

Let y = x sin3x dx

Therefore, xxsin 3xdx -ddx(x) . (sin3x dx) dx

= x(-cos3x)3 - 1. -cos3x3 dx

= -xcos3x3 + 13 cos3x dx

= -xcos3x3 +sin3x9 + C

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

Comparing the given matrix A with an identity matrix 1001 ,

We find that cos∝ = 1, - sin∝ = 0 , -cos∝ = 1 and sin∝ = 0

For cos∝ = 1, ∝must be 0° sincecos0° is=1.

For sin∝ = 0, ∝must be 0° sincesin0° is 0

Therefore, the required value of ∝must be 0° for it to be an identity matrix.

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

The tangent and the line passing in the middle of the curve are perpendicular.

Differentiating the equation of the curve we have:

2y dydx = 4

Therefore, dydx = 2y

This means the slope of the curve will be m = 2y and therefore y= 2m

When y= 2m , x = y24 = 4m2 ÷ 4 = 1m2

Hence x = 1m2 and y = 2m will lie on the same line at the point of tangent.

y = mx + 1

2m = m . 1m2 + 1

2m = 1m + 1

2m - 1m = 1

1m = 1

m = 1

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

This is a form of homogenous differential equation;

dydx= - x2-y22xy

Making y = ux , we obtain u+ xdudx=dydx

Utilizing the form u+ xdudx , we have -x2 x2-u22x2u = - 1-u22u

xdudx= -1-u22u+u

xdudx= -1+u22u

2u1+u2 du= -dxx

Finding the integration of both sides gives;

2u1+u2 = -dxx

Introducing logarithms on both sides gives, log1+u2= - logx+logC

logx2+y2x2x=logC

x2+y2x=C

x2+y2=Cx

When y=1, x = 1 and hence 1+1=C;C=2

Therefore x2+y2=2x

=2x

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

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

Making the equation in the matrix form Ax=b

1-1234-52-13 xyz = 7-512

Let A

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