Straight line Summary

Equations of Straight Lines4.1 THIS IS NOT HOMEWORK

1. Find the equation of the line parallel to the given line which passes through the given point. Give your answer in the form $ay+bx=c$

1. $3x-4y=3$ through (2,-1)
2. $5y-4x=12$ through (-3,-2)
3. $3x+4y=4$ through (1,1)
4. $5x+2y=8$ through (1,0)

[8 marks]

2. Find the equation of the line perpendicular to the given line which passes through the given point. Give your answer in the form $ay+bx+c=0$

1. $3x+2y=6$ through (5,1)
2. $5y-4x=2$ through (-3,3)
3. $y+3x=6$ through (5,0)
4. $4x+2y=5$ through (14,2)

[8 marks]

3.

1. Find the gradient of the line
2. Using the previous part and the coordinates of one of the points on the line, work out the equation of the line

[6 marks]

4. Find the equation of the line which passes through (-4,5) and (4,-7). Give your answer in the form $ax+by+c=0$

[3 marks]

5. Find the gradient and y intercept of the line $4x-3y=6$

[2 marks]

6. Find the equation of the line which is perpendicular to the line $2y-3x=10$ and passes through the point (2,-5). Give your answer in the form $ay+bx+c=0$

[4 marks]

Maths day out in London?

Anyone interested in day in London with a Maths Legend?

Could do a 1 day trip to London to include travel by train for hopefully £50-100. Please let me know personally if you are interested.

Books on proof

• Proof
• "Are you sure? Learning about proof

4.1 Homework 3

Homework 3: To be done neatly in the front of your books.
TOTAL 24 marks

1. Jon is twice the age of his son. 9 years ago he was three times the age of his son. How old is Jon's son currently?

[3 marks]

2. Show all your working out and leave answers as mixed numbers where needed:

1. $2\dfrac{4}{5} \times 3\dfrac{1}{3}$
2. $2\dfrac{1}{6} \div 4\dfrac{1}{3}$

[4 marks]

3.

The graph shows a plot of $y=x^3+x^2-6x-1$.

1. Find the values of x for which the gradient is 0
2. Use the plot to estimate the gradient when $x=0$

[4 marks]

4. Expand $(x+h)^4$ and use the result to find the gradient function for $y=x^4$

[5 marks]

5. Find the equation of a line which is parallel to the line $4x-3y=6$ but goes through the point (-2,7). Give your answer in the form $ay+bx=c$

[4 marks]

6. Find the equation of the line which is perpendicular to the line $3y-2x=12$ and passes through the point (-2,5). Give your answer in the form $ay+bx+c=0$

[4 marks]

4.1 Gradients Worksheet

A* Gradients of Lines and Curves 40 Marks

1. Find the equation of the line parallel to the given line which passes through the given point. Give your answer in the form $ay+bx=c$

1. $3x-4y=12$ through (4,-1)
2. $5y-2x=10$ through (-4,2)
3. $3x+5y=12$ through (5,0)
4. $4x+2y=5$ through (1,1)

[8 marks]

2. Find the equation of the line perpendicular to the given line which passes through the given point. Give your answer in the form $ay+bx+c=0$

1. $2x+3y=6$ through (6,1)
2. $3y-4x=5$ through (-4,-3)
3. $5y+3x=8$ through (2,0)
4. $4x-2y=3$ through (1,0)

[8 marks]

3.

The graph shows a plot of a cubic function.

1. Find the values of x for which the gradient is 0
2. Draw a tangent line at x=1 and use this to estimate the gradient

[4 marks]

4.

The graph shows a plot of a quadratic function.

1. Find the value of x for which the gradient is 0
2. Draw a tangent line at x=2 and use this to estimate the gradient

[4 marks]

5. Start by expanding $(x+h)^2$ and subsequently multiplying by $(x+h)$ to find an expression for $(x+h)^5$. Use the result to find the gradient function for $y=x^5$

[5 marks]

6. Find the gradient funtion for $y=x^2+x$ by looking at the limit of the following as $h\rightarrow 0$ \[ \dfrac{(x+h)^2+(x+h)-(x^2+x)}{h}\]

[5 marks]

7. Can you conjecture a result about the gradient function for $y=x^n$? Can you prove this result?

[6 marks]

Revise GCSE

http://www.mathsmadeeasy.myzen.co.uk/gcse-maths-revision-papers/

What do you think?

Can you prove that if $p$ is prime and $n$ is a natural number: \[ {n \choose p} \equiv \left[ \frac{n}{p} \right](\text{mod } p) \]

Note that $\left[ \frac{n}{p} \right]$ denotes the floor of $\frac{n}{p}$ which is the integer part of it.

Also note that $a \equiv b (\text{mod }n)$ means that $(a-b)$ is a multiple of $n$

4.1 Blog Homework 2

Homework 1: To be done neatly in the front of your books. No Calculator
TOTAL 30 marks

1. In an aptitude test of 100 questions, you score 7 points for a correct answer but lose 3 for an incorrect answer. If Jon scored 210 points, how many questions did he get right? SHOW ALL YOUR WORKING

[4 marks]

2. Show all your working out and leave answers as mixed numbers where needed:

1. $7\dfrac{2}{3} - 3\dfrac{3}{4}$
2. $3\dfrac{4}{5} \times 2\dfrac{1}{4}$
3. $4\dfrac{5}{6} \div 2\dfrac{3}{7}$

[6 marks]

3.

1. Find the gradient of the line
2. Using the previous part and the coordinates of one of the points on the line, work out the equation of the line

[6 marks]

4. Find the equation of the line which passes through (-2,5) and (4,-7). Give your answer in the form $ax+by+c=0$

[3 marks]

5. Find the gradient and y intercept of the line $4x-3y=6$

[2 marks]

6. Factorise $6x^2+11x-10$

[2 marks]

7. Make $x$ the subject: $y=\dfrac{x^2-3}{5+x^2}$

[3 marks]

8. Find the equation of the line which is perpendicular to the line $3y-2x=12$ and passes through the point (-2,5). Give your answer in the form $ay+bx+c=0$

[4 marks]

L6FM Homework 2

Homework/Other 2

A lift has mass 300Kg empty. Bert and Ernie are in the lift accelerating up and the tension in the cable is 5000N. The floor is exerting forces of 350 and 450 N on Bert and Ernie respectively. Find

1. The acceleration of the lift
2. The mass of Bert
3. The mass of Ernie

[5 marks]

5th Year Homework 3

Homework 3: To be done neatly in the front of your books
TOTAL 17 marks

1. Copy and complete the table for $y=-\frac{1}{3}x^2+3x+1$, \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline y & & & 3\frac{2}{3} & & & & & \\ \hline \end{array}

1. Plot the points on a set of axes (use 1 square for 1 unit on x and y axes) and join them to form a smooth curve, your curve should look like this:

   

2. Use your graph to solve the equation $-\frac{1}{3}x^2+3x+1=0$
3. Use your graph to find the solutions to $-\frac{1}{3}x^2+3x+1=3$

[6 marks]

2. The volume of a balloon is directly proportional to the cube of its radius. A balloon of radius 5cm has volume $525cm^3$:

1. Find a formula for v in terms of r
2. Find the volume of the balloon when the radius is 15cm
3. What is the radius of a balloon with volume $2560cm^3$?

[4 marks]

3. The force between 2 masses varies indirectly as the square of the distance between them. Copy and complete the table \begin{array}{|c|c|c|c|} \hline d & 2 & 5 & \\ \hline F & 8 & & 17 \\ \hline \end{array}

[4 marks]

4. A line passes through the point (-1,5) and has gradient 3. Write down the equation of the line in the form y=mx+c

[3 marks]

One to read...

How to Think Like a Mathematician - Dr K Houston

2nd Year Blog Homework 1

To be completed neatly in your books due on Monday 22nd Sept. TOTAL 22 marks

1. Write down all the prime numbers less than 30

[3 marks]

2. True or false: There are no even prime numbers. Explain your answer

[2 marks]

3. Draw prime factor trees for

1. 120
2. 51
3. 216

[6 marks]

4. Which primes make up the composite number 78?

[2 marks]

5.

1. Write down the first 10 multiples of 15
2. Write down the first 10 multiples of 12
3. Using the parts above find the lowest common multiple of 15 and 12

[3 marks]

6.

1. Write down all the factors of 96
2. Write down all the factors of 144
3. Using the parts above find the highest common factor of 96 and 144

[3 marks]

7. A number is written down but the last digit has been blurred and cannot be read. The number is 1,275,03x where x is the missing digit. We know that the number is divisible by three, what could the value of x be (there are several possibilities)?

[2 marks]

8. Explain why 1 is not a prime number

[1 marks]

U6FM Homework 1 Due Wed 30.9.15

Homework on paper. 27 MARKS

1. Find the values of $x$ for which $2\sinh x + 3\cosh x = 4$

[3 marks]

2. Find, simplifying your answer as much as possible!

1. \[\int_{0}^{2} \dfrac {2x^3}{1-x^2} dx\]
2. \[\int_{0}^{2} \dfrac {2x}{1-x^2} dx\]
3. \[\int_0^2 \dfrac {2x}{\sqrt {1-x^2}} dx\]
4. \[\int_0^2 \dfrac {2}{\sqrt {1-x^2}} dx\]

   Note that the last 2 parts have complex (not real) solutions. Bonus points if you can state what they are!! In fact the last one is \[ -\pi(1+2n)-2i\ln{ (2 \pm \sqrt3)} \]

[12 marks]

3. Find \[ \int \text{sech}^2 x\tanh^7 x dx \]

[4 marks]

4. Find $\dfrac{d}{dx} (\text{sech} x)$ from the definitions

[4 marks]

5. Find\[ \int_1^2 \dfrac {1}{\sinh x + 2 \cosh x} dx\]

[4 marks]

4.1 Homework 1

Homework 1: To be done neatly in the front of your books
TOTAL 26 marks

1. It takes 6 lawnmowers 13 hours to mow all the Wimbledon tennis courts.

1. How many hours would it take 17 mowers?
2. What is the minimum number of mowers it would it take to do the job in under 3 hours?

[3 marks]

2. If u is 15 percent more than v and v is 63 percent less than w, by what percentage is u less than w?

[3 marks]

3. If the ratio of A:B is 3:5 and the ratio of B:C is 7:2, what is the ratio of A:C?

[2 marks]

4. A microchip measures $6.1\times10^{-4}mm$ by $4\times10^{-3}mm$, find (show all your working)

1. The perimeter of the chip giving your answer in standard form
2. The area of the chip giving your answer in standard form

[3 marks]

5. Show all your working out and leave answers as mixed numbers where needed:

1. $3\dfrac{2}{5} - 1\dfrac{3}{4}$
2. $5\dfrac{4}{7} \times 1\dfrac{1}{6}$
3. $2\dfrac{2}{5} \div 4\dfrac{3}{4}$

[6 marks]

6. The population of carp in a fishing lake increases by 12 percent each year. If there are 200 carp to begin with, after how many years will there be more than 1000 carp (calculator allowed)?

[3 marks]

7. Make $x$ the subject:

1. $y=\dfrac{3x-2}{5}$
2. $y=\dfrac{x-2}{1-x}$
3. $y^2=\dfrac{5x^2-2}{3}$

[6 marks]

Homework 2

Homework 2: To be done neatly in the front of your books
TOTAL 25 marks

1. Copy and complete the table for $y=5-3x-2x^2$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 & 4 \\ \hline y & -4 & & & & & & & \\ \hline \end{array}

1. Plot the points on a set of axes and join to form a smooth curve
2. Use your graph to solve the equation $5-3x-2x^2=0$
3. Use your graph to find the solutions to $5-3x-2x^2=1$

[6 marks]

2. Find the missing angles

[4 marks]

3. Find x, w and y (the angle marked is 63 degrees!)

[3 marks]

4. A line passes through the point (3,-2) and has gradient 2. Write down the equation of the line in the form y=mx+c

[3 marks]

5. Convert 0.127272727.... to a fraction and simplify. Show all your working.

[3 marks]

6. It takes 4 window cleaners 7 days to clean an office building. How many days would it take 11 window cleaners to the nearest day? Show ALL your working out.

[3 marks]

7. Force (F) is proportional to acceleration (a). A force of 12 Newtons gives an acceleration of $3.6m/s^2$.

1. Find a formula for F in terms of a
2. Use the previous part to calculate the Force required to produce an acceleration of $2.4m/s^2$
3. What is the acceleration given by a 7.1 Newton force?

[3 marks]

L6FM M1 Homework 1

To be handed in on paper.

1. A rocket is propelled vertically up from the ground with speed $8.1 ms^{-1}$. Find

1. The maximum height reached by the rocket
2. The total time the rocket is in the air
3. State one assumption you made

[6 marks]

2. A car is travelling between 2 sets of traffic lights 510m apart. It accelerates from the first set of lights with constant acceleration $3ms^{-2}$ from rest until it reaches a constant speed of $15.6ms^{-1}$. It travels at this speed for T seconds and then decelerates uniformly at $6ms^{-2}$, coming to a stop at the second set of traffic lights.

1. Draw a (t,v) graph
2. Calculate the value of T

[6 marks]

3. A lemming projects himself down from the top of a 200m cliff at $2ms^{-1}$. At the same instant another lemming is fired up from directly below with a speed of $73ms^{-1}$. The two lemmings collide. Calculate after how long this happens and how far from the bottom of the cliff this is. (Sorry this question has been changed due to a typo in the speed of the second lemming)

[6 marks]

4. Travelling along the motorway at 67mph, I notice a car approaching in the rearview mirror. It is approximately quarter of a mile behing me. After 6 minutes it passes me. What speed it the car travelling at?

[2 marks]

5. A firework lets out a green spark from the ground which travels upwards at $10.6ms^{-1}$. After 0.2 seconds it emits a red spark which travel upwards at $15ms^{-1}$. After how long from the green spark being released do the red and green sparks collide?

[4 marks]