Kirill M.

£30 /hr

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

5.0

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78 completed lessons

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#### Ratings & Reviews

5from 9 customer reviews

N (Parent from Sutton Coldfield)

June 28 2017

Was great help with answering all my questions and any content queries I had too

Maggy (Parent from Y Felinheli)

July 27 2017

Maggy (Parent from Y Felinheli)

July 20 2017

N (Parent from Sutton Coldfield)

April 2 2017

#### Qualifications

MathsA-level (A2)A*
Further MathsA-level (A2)A*
PhysicsA-level (A2)A*
BiologyA-level (A2)A*
ChemistryA-level (A2)A
RussianA-level (A2)A*

#### Subjects offered

SubjectQualificationPrices
Further MathematicsA Level£30 /hr
MathsA Level£30 /hr
PhysicsA Level£30 /hr
MathsGCSE£30 /hr
PhysicsGCSE£30 /hr

### 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 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

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

3 years ago

1994 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 (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.

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.

3 years ago

1015 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)) / 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.

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.

3 years ago

2321 views

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