£18 - £25 /hr

Kirill M.

Degree: Physics (Masters) - Oxford, St Catherine's College University

#### Subjects offered

SubjectLevelMy prices
Further Mathematics A Level £20 /hr
Maths A Level £20 /hr
Physics A Level £20 /hr
Maths GCSE £18 /hr
Physics GCSE £18 /hr
.PAT. Uni Admissions Test £25 /hr

#### Qualifications

MathsA-LevelA*
Further MathsA-LevelA*
PhysicsA-LevelA*
BiologyA-LevelA*
ChemistryA-LevelA
RussianA-LevelA*
 CRB/DBS Standard No CRB/DBS Enhanced No

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5from 6 customer reviews

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### What is the sum of the series 2/3 − 2/9 + 2/27 − ....? (PAT Q1 2013)

This is the sum of a geometric series with an infinite number of terms. First, find the common ratio: (-2/9)/(2/3) = -1/3 , (2/27)/(-2/9) = -1/3 Therefore, the common ratio is -1/3 -1<(-1/3)<1 therefore the series will converge to a finite number.  The general formula for a sum of an infinit...

This is the sum of a geometric series with an infinite number of terms.

First, find the common ratio:

(-2/9)/(2/3) = -1/3 , (2/27)/(-2/9) = -1/3

Therefore, the common ratio is -1/3

-1<(-1/3)<1 therefore the series will converge to a finite number.

The general formula for a sum of an infinite geometric series is a/(1-r) where a is the first number in the sequence and r is the common ratio.

So, substituting in the numbers, the sum of the series = (2/3)/(1-[-1/3]) = 1/2

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2 years ago

1083 views

### A car is travelling at 10m/s when it brakes and decelerates at 2ms^-2 to a stop. How long does the car take to stop?

This is a question testing the knowledge of the equations of constant acceleration (Suvat equations) First convert the question into a standard form. For example by writing out the variables as follows  s (displacement) = unknown u (initial velocity) = 10m/s v (final velocity) = 0m/s a (ac...

This is a question testing the knowledge of the equations of constant acceleration (Suvat equations)

First convert the question into a standard form. For example by writing out the variables as follows

s (displacement) = unknown

u (initial velocity) = 10m/s

v (final velocity) = 0m/s

a (acceleration) = 2ms^-2

t (time) = unknown

You know v, u and a and you want to calculate t, therefore the equation you need to use is v = u + a*t

Re-arranging t = (v - u) / a

Finally substituting in v, u and a, t = (10 - 0) / 2 = 5s

So the time taken for the car to stop is 5 seconds.

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2 years ago

689 views

### Integrate cos(4x)sin(x)

The easiest way of approaching this question is to use De Moivre's formula:  e^(inx) = cos(nx) + isin(nx)  from which it is simple to show that cos(nx) = (e^(inx) + e^(-inx)) / 2 and sin(nx) = (e^(inx))- e^(-inx)) /2i  therefore, cos(4x)sin(x) = (e^(4ix) + e^(-4ix)) * ((e^(ix)) - (e^(-ix)) / ...

The easiest way of approaching this question is to use De Moivre's formula:

e^(inx) = cos(nx) + isin(nx)

from which it is simple to show that cos(nx) = (e^(inx) + e^(-inx)) / 2 and sin(nx) = (e^(inx))- e^(-inx)) /2i

therefore, cos(4x)sin(x) = (e^(4ix) + e^(-4ix)) * ((e^(ix)) - (e^(-ix)) / 4i

= [e^(5ix) - e^(-5ix) - e^(3ix) + e^(-3ix)] / 4i

= sin(5x)/2 - sin(3x)/2

Finally, integrating, this gives cos(3x)/6 - cos(5x)/10 + integration constant

This can also be done by using various trigonometric identities, however this method is simpler and can continue to be applied to more complex questions.

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2 years ago

1293 views
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