4th Year Fri P6

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4th Year Homework 3

Homework 3: To be done neatly in the front of your books for Wed 2nd April
TOTAL 37 marks

1. Solve this equation \[\frac{2x+1}{5}-\frac{3(1-2x)}{3}=x\]

[4 marks]

2. Copy and complete this table for a pie chart \begin{array}{|c|c|c|} \hline \text{Fav. TV Channel} & \text{Frequency} & \text{Angle (1 d.p.)} \\ \hline \text{BBC} & 5 & \\ \hline \text{ITV} & 12 & \\ \hline \text{CH4} & 23 & \\ \hline \text{SKY} & 11 & \\ \hline \text{Other} & 2 & \\ \hline \text{CH5} & 5 & \\ \hline \end{array}

[3 marks]

3. Solve these simultaneous equations algebraically:\[3x+2y=-13\\-x+4y=23\]

[4 marks]

4. True or false the point (3, -2) is on the line $3x-4y=16$? Justify your answer

[2 marks]

5. Draw a Venn diagram and shade the region which represents AnBnC'

[2 marks]

6. Find the gradient of this line: $5y+2x=20$ and use this value to find the gradient of a line which is perpendicular (at right angles) to this line

[4 marks]

7. Copy and complete the table. \begin{array}{|c|c|c|} \hline \text{Score on dice} & \text{Frequency} & \text{Score $\times$ Frequency} \\ \hline \text{1} & 4 & \\ \hline \text{2} & 13 & \\ \hline \text{3} & 31 & \\ \hline \text{4} & 29 & \\ \hline \text{5} & 17 & \\ \hline \text{6} & 6 & \\ \hline \end{array} Use it to:
a. Find the mean
b. Find the median

[4 marks]

8. Copy and complete the table. \begin{array}{|c|c|c|} \hline \text{Weight (w) in Kg} & \text{Frequency} & \text{M.P. $\times$ Frequency} \\ \hline \text{10 $\leq$ w < 15} & 4 & \\ \hline \text{15 $\leq$ w < 20} & 22 & \\ \hline \text{20 $\leq$ w < 25} & 38 & \\ \hline \text{25 $\leq$ w < 30} & 16 & \\ \hline \end{array} Use it to:
a. Find an estimate of the mean
b. Find an estimate of the median
c. Draw a a fully labelled frequency polygon

[8 marks]

9. HARD: Find the length of AC by splitting the triangle

[4 marks]

10. Work out $1.2\times 10^3 + 3.41\times 10^4$ showing your working

[2 marks]

3rd Year Homework 2

Homework 2: To be done neatly in the front of your books for Monday 31st March
TOTAL 28 marks

1. Solve this equation \[\frac{2x+1}{3}-\frac{2(4-3x)}{7}=2\]

[4 marks]

2. A wheelchair ramp of length 10m provides an alternative route to steps which climb 86cm. What angle does the wheelchair ramp make with the ground?

[3 marks]

3. Rob puts £500 in a bank account with interest rate r for 5 years? When he check his balance it is £864.63. What was the interest rate (r)?

[4 marks]

4. By showing your working and not using a calculator, find the value of $ \left( \frac {27}{125} \right)^{-\frac{2}{3}}$

[2 marks]

5. Find the area of a sector of a circle with radius 24cm and angle 35 degrees to 2 significant figures

[3 marks]

6. Find the gradient of this line: $7x-2y=7$ and use this value to find the equation of a line perpendicular to this line which passes through (0,-1)

[4 marks]

7. Solve the simultaneous equations: \[4x+3y=1\\-2x+5y=8\]

[3 marks]

8. HARD: A ship sails 6 km on a bearing of 141 degrees and then 5km on a bearing of 15 degrees. Draw a diagram and work out how far it must travel to return to the starting position. (HINT: one way to do this would be take a look at the Cosine Rule)

[5 marks]

Competition

For what values of x does \[(x^2-5x+5)^{(x^2-12x+35)}=1?\]

Email Mr Rye with your response.

Pre university reading

Link to Cambridge Reading list for Maths

1. How to Think Like a Mathematician - K Houston
2. The Man Who Loved Only Numbers - Paul Hoffman 
3. Mathematics: The New Golden Age - Keith Devlin 
4. Fermat's Last Theorem - Simon Singh, Andrew Wiles 
5. The Mathematical Universe - William Dunham 
6. The Music of Primes - Marcus du Sautoy 
7. Finding Moonshine - Marcus du Sautoy 
8. A History of Pi - Petr Beckmann 
9. The Simpsons and their Mathematical Secrets - S Singh
10. A Mathematician's Apology - G H Hardy

Technical Books:

1. Introduction to Complex Analysis - H A Priestley
2. Mathematical Methods for Science Students - G Stephenson
3. Guide to Analysis - M Hart
4. Linear Algebra and Geometry - David Smart
5. Vector Analysis and Cartesian Tensors - Kendall and Bourne
6. Lebesgue Integration and Measure - A Weir

Physics/meta physics

1. A Brief History of time - Stephen Hawking 
2. The Universe in a Nutshell - Stephen Hawking 
3. Schrödinger's Kittens - John Gribbin 
4. A Stubbornly Persistent Illusion: The essential scientific writings of Albert Einstein - Stephen Hawking 
5. The Elegant Universe - Brian Greene 
6. Fabric of the Cosmos - Brian Greene 
7. Principia Mathematica - Isaac Newton!

Some videos

Courtesy of Mr Rye. 

http://web.mat.bham.ac.uk/C.J.Sangwin/howroundcom/contents.html

http://blog.stevemould.com/50p-coins-make-great-wheels/

4th Year Homework 2

Homework 2: To be done neatly in the front of your books for Wed 26th March
TOTAL 28 marks

1. Solve this equation \[\frac{2x+1}{3}-\frac{3(2-2x)}{5}=2\]

[4 marks]

2. An observer stands 100m from the foot of a pine tree and uses a clinometer to measure the angle of elevation (angle from the horizontal) to the top of the tree which comes out as $27^O$. How high is the tree?

[3 marks]

3. Solve these simultaneous equations algebraically:\[4x+3y=1\\-2x+5y=8\]

[4 marks]

4. In a survey of 45 people, 17 answered that their favourite colour was red. What angle would this be on a pie chart?

[2 marks]

5. Find the mean median mode and range of: 3, 5, -11, 7, 12, 10, 8, 5

[4 marks]

6. Find the gradient of this line: $4y-5x=20$ and use this value to find the gradient of a line which is perpendicular (at right angles) to this line

[4 marks]

7. Copy and complete the table. Use it to find an estimate of the mean height from this data: \begin{array}{|c|c|c|} \hline \text{Height in cm (h)} & \text{Frequency} & \text{M.P. $\times$ Frequency} \\ \hline \text{100 $\leq$ h < 120} & 3 & \\ \hline \text{120 $\leq$ h < 140} & 25 & \\ \hline \text{140 $\leq$ h < 160} & 17 & \\ \hline \end{array}

[3 marks]

8. HARD: Find the length of YZ

[4 marks]

Everyone in the UK is the same age

Everyone in the UK is the same age!!

Proof – which step does this proof go wrong in?

Statement S(n): In any group of n people, everyone in that group has the same age. E.g. All the people in the UK have the same age.

1. In any group that consists of just one person, everybody in the group has the same age, because after all there is only one person!
2. Therefore, statement S(1) is true.
3. The next stage in the induction argument is to prove that, whenever S(n) is true for one number (say n=k), it is also true for the next number (that is, n = k+1).
4. We can do this by (1) assuming that, in every group of k people, everyone has the same age; then (2) deducing from it that, in every group of k+1 people, everyone has the same age.
5. Let G be an arbitrary group of k+1 people; we just need to show that every member of G has the same age.
6. To do this, we just need to show that, if P and Q are any members of G, then they have the same age.
7. Consider everybody in G except P. These people form a group of k people, so they must all have the same age (since we are assuming that, in any group of k people, everyone has the same age).
8. Consider everybody in G except Q. Again, they form a group of k people, so they must all have the same age.
9. Let R be someone else in G other than P or Q.
10. Since Q and R each belong to the group considered in step 7, they are the same age.
11. Since P and R each belong to the group considered in step 8, they are the same age.
12. Since Q and R are the same age, and P and R are the same age, it follows that P and Q are the same age.
13. We have now seen that, if we consider any two people P and Q in G, they have the same age. It follows that everyone in G has the same age.
14. The proof is now complete: we have shown that the statement is true for n=1, and we have shown that whenever it is true for n=k it is also true for n=k+1, so by induction it is true for all n.

Catenary

In deriving the equation of a catenary we turn to the Euler-Lagrange equation: 

\[\frac{d}{dx}\left(\frac {\partial L}{\partial y\prime}\right )= \frac{\partial L}{\partial y}\]

We use this to minimise the potential energy of the hanging object which is an integral. 

The proof that a chain or rope hangs in a cosh shape ($\dfrac{e^x + e^{-x}}{2}$) will follow soon. 

In fact here it is. Write up courtesy of the man Ste. https://docs.google.com/file/d/0B4GMQrQwHpNuXzZ6RzBnMkdnNTQ/edit?usp=docslist_api

How to use mathematical language in your posts

Use $ to start and end any mathematical part to your post. 

Whithin the dollar signs the following can be used:

"^" will give "to the power of"

"_" will give a subscript

"\frac{1}{2}" will display $\frac{1}{2}$

"\sum" will display a sigma sign

For example 

"\sum_{r=1}^{10} \frac{d^r y}{dx^r} = x^2" will display 

$\sum_{r=1}^{10} \frac{d^r y}{dx^r} = x^2$

3rd Year Homework 1

Homework 1: To be done neatly in the front of your books for Monday 24th March
TOTAL 28 marks

1. Solve this equation \[\frac{x+1}{5}-\frac{2(3-x)}{4}=1\]

[4 marks]

2. A 3m ladder is leaning up a wall at an angle to the ground of $60^o$. How far up the wall can it reach?

[3 marks]

3. Pete decided to buy a new car. After 4 years depreciation at 20% from new, Pete decides to sell the car. It does not sell and so Pete reduces the price by 15%. The car still does not sell and he reduces it by a further 7% to £3,200. It sells after 3 days. What did the car cost new?

[5 marks]

4. Expand $(3x-2)(4x+3)$

[2 marks]

5. Factorise $6x^2-11x-35$

[3 marks]

6. Find the gradient of this line: $3x-2y=7$ and use this value to find the equation of a line parallel to this line which passes through (0,4)

[4 marks]

7. In a triangle ABC, AB=10cm, BC=6cm and angle C is a right angle. Find angle A.

[3 marks]

8. A ship sails 9.2km from a lighthouse on a bearing of $135^o$ and arrives due east of a pier which happens to be due south of the lighthouse. Draw a diagram and calculate how far east of the pier the boat is.

[4 marks]

4th Year Homework 1

Homework 1: To be done neatly in the front of your books for Thu 20th March
TOTAL 28 marks

1. Solve this equation \[\frac{x+1}{5}-\frac{2(3-x)}{4}=1\]

[4 marks]

2. A 3m ladder is leaning up a wall at an angle to the ground of $60^o$. How far up the wall can it reach?

[3 marks]

3. Solve these simultaneous equations algebraically:\[3x+2y=1\]\[-2x-y=1\]

[4 marks]

4. If n(E)=20, n(A)=7 and n(B)=17, find n(AnB)

[2 marks]

5. Solve these simultaneous equations algebraically:\[2x+5y=4\]\[3x-5y=2\]

[4 marks]

6. Find the gradient of this line: $3x-2y=7$ and use this value to find the gradient of a line which is perpendicular (at right angles) to this line

[4 marks]

7. In a triangle ABC, AB=10cm, BC=6cm and angle C is a right angle. Find angle A.

[3 marks]

8. A ship sails 9.2km from a lighthouse on a bearing of $135^o$ and arrives due east of a pier which happens to be due south of the lighthouse. Draw a diagram and calculate how far east of the pier the boat is.

[4 marks]

FP1 Homework Review

FP1 Homework

1.Find the inverse of $A=\begin{pmatrix} \\1 & 3 \\4 & -1 \end{pmatrix}$
Use your answer to solve $x+3y=-2$ and $4x-y=5$

A triangle with area 5 is transformed using A, what is the area of the image of the triangle?

2. Find and simplify in terms of n\[\sum_{r=1}^{n} r(r+1)(r+2)\] Use this to find $\sum_{r=10}^{20} r(r+1)(r+2)$

3. \[S=\begin{pmatrix} \\0 & -1 \\-1 & 0 \end{pmatrix}\]S represents a linear transformation. Give a geometrical interpretation of S and show $S^2=I$. Give a geometrical interpretation of $S^{-1}$

4. Use the result \[\sum_{r=1}^{n}r(r+2)=\frac{n}{6}(n+1)(2n+7)\] to find: \[ 3\log2+4\log2^2+5\log{2^3}+....+(n+2)\log{2^n} \]