5.3 Homework 27.11.14

Due Monday 1st December, neatly in front of book. 14 Marks

1. Solve by factorising $3x^2+14x-24=0$

   [3 marks]

2. The length of a rectangle is 6cm longer than the width (call this x). If the area of the rectangle is 50 square cm, write down an equation for x, solve it and use it to find the perimeter of the rectangle to 1 decimal place. You may need to use the formula.

   [4 marks]

3. Solve this inequality with a sketch $2x^2-5x-3<0$

   [4 marks]

4. Evaluate, showing all of your working:\[ \left(\dfrac{125}{8}\right)^{-\frac{2}{3}} + \left(\dfrac{81}{64}\right)^{-\frac{3}{2}} \]

   [3 marks]

4.1 Homework 26.11.14

To be done in the front of your books neatly for Tue 2nd Dec. 24 Marks

1. Simplify
   1. \[\frac{3}{x-3}-\frac{4}{(x-3)^2}\]
   2. \[\frac{3}{x-2}+\frac{2}{x+5}\]

   [6 marks]

2. Simplify \[\frac{3x}{4x^2-25}-\frac{2}{2x^2-3x-5}\]

   [4 marks]

3. Solve, giving answers to 3 significant figures \[ \sqrt{ \frac{(\sqrt{x}+3)^2-5}{3} -3} = 6 \]

   [4 marks]

4. Solve \[ \begin{align*} 5&x+4y=-49 \\ -3&x+8y=45 \end{align*} \]

   [3 marks]

5. Solve, showing the solution on a number line \[ -3<\frac{2x-5}{3}\leq5 \]

   [3 marks]

6. Shade the region which satisfies these inequalities on a pair of x,y axes: \[ -3 < x \leq 2 \\ -4 \leq y < 5 \\ x+y > 1 \]

   [4 marks]

FMSP Revision Video: Edexcel M2 - Collisions

FMSP Revision Video: Edexcel M2 - Work, Energy and Power

FMSP Revision Video: Edexcel M2 - Centre of Mass/Statics

FMSP Revision Video: Edexcel M2 - General Motion

FMSP Revision Video: Edexcel M2 - Projectiles

FP3 Revision Videos

https://www.youtube.com/watch?v=XO12Nq9O34o
https://www.youtube.com/watch?v=dDbMq-ZrCxM
https://www.youtube.com/watch?v=v6FrzyNvB30
https://www.youtube.com/watch?v=qW5xmvdIPOI

Links to M1 Revision Videos

Kinematics
Newton's Laws
Impulse and Momentum
Statics
Moments
Vectors

4.1 Homework 19.11.14

To be done in the front of your books neatly for Tue 25th Nov. 26 Marks

1. Simplify
   1. \[\frac{a}{5}+\frac{b}{25}\]
   2. \[\frac{2}{x}-\frac{5}{x^2}\]
   3. \[\frac{1}{5+x}+\frac{2}{3+x}\]
   4. \[\frac{24x^2y}{5z}\times \frac{75z^3}{96(xy)^2}\]
   5. \[\frac{39a^2b^3c}{24xyz}\div \frac{26ab^2c^4}{96x(yz)^3}\]

   [15 marks]

2. Simplify \[\frac{3x}{x^2-9}-\frac{5}{x^2-x-6}\]

   [4 marks]

3. Prove that if $p$ and $q$ are odd numbers their product is odd

   [3 marks]

4. Solve \[ \begin{align*} 3&x+4y=-19 \\ -5&x+6y=38 \end{align*} \]

   [3 marks]

5.3 Homework 20.11.14

To be done in books neatly showing all working out for Monday 24th Nov. 38 Marks

1. Evaluate:
   1. $4^{-2}$
   2. $\left(\dfrac{8}{27}\right)^{-\dfrac{2}{3}}$
   3. $\left(\dfrac{25}{16}\right)^{-\dfrac{3}{2}}$
   4. $\left(\dfrac{1}{4}\right)^{-\dfrac{1}{2}}$

   [9 marks]

2. Simplify
   1. $2x^3 \times 3x^{-2}$
   2. $8x^{-3} \div 2x^{-\frac{1}{2}}$

   [4 marks]

3. Solve by factorising:
   1. $x^2-9=0$
   2. $x^2+3x=0$
   3. $2x-7x^2=0$
   4. $x^2+3x+2=0$
   5. $x^2-x-12=0$
   6. $6x^2+11x-10=0$

   [12 marks]

4. Solve by using the formula \[ax^2+bx+c=0 \\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]
   1. $x^2-9x-2=0$
   2. $x^2+3x=14$
   3. $2x^2-7x=11$
   4. $4x-5x^2=-11$

   [8 marks]

5. Make y the subject and use the result to find the gradient and intercept \[3x+5y=12\]

   [3 marks]

6. Factorise $2x^2-5x-3$ Help here

   [2 marks]

4th Year Extension Test, 30 minutes, Calculator Allowed. 30 Marks

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1. It takes 6 decorators 35 hours to decorate the interior of a large house.

1. How many hours would it take 4 decorators?
2. What is the minimum number of decorators it would it take to do the job in under 3 hours?

[3 marks]

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2. If u is 15 percent less than v and v is 12 percent less than w, by what percentage is u less than w?

[3 marks]

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3. If the ratio of A:B is 4:9 and the ratio of B:C is 5:3, what is the ratio of A:C?

[2 marks]

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4. The population of insects in a waste bin increases by 19% every hour. If there are 150 insects to begin with, after how many hours will there be more than 2000 insects?

[3 marks]

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5. Make $x$ the subject:

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

[6 marks]

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6. In a class of $(11x+2)$ people, there are 3 music tastes: Pop (P), Classical (C) and Dance (D). Each student likes at least one type of music.
There are $(6x+3)$ students who like pop.
There are $(5x+4)$ students who like classical.
There are $6x$ students who like dance.
There are $4x$ students who like pop and classical.
There are $(3x-1)$ students who like pop and dance.
There are $(4x-1)$ students who like dance and classical.
There are $(x+1)$ students who like dance, pop and classical
Draw a Venn Diagram and use it find x and hence the number of students in the class

[5 marks]

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7. In a 100 question test I get 5 points for every correct answer but lose 2 points for every incorrect answer score was 297. Let $x$ be the number of questions I got correct. Write down an equation for $x$ and solve it.

[4 marks]

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8. Find the equation of a line which is parallel to the line $5x-4y=6$ but goes through the point (-1,3). Give your answer in the form $ay+bx=c$

6.4 M1 Homework

To Be Completed for Wednesday

1. a. A box of mass 5Kg is being pulled up a rough slope inclined at 30 degrees. The rope is inclined at an angle of 10 degrees to the slope. When the tension in the rope is 50 Newtons, the box is in equilibrium, about to slip up the plane. Calculate the coefficient of friction.
   
   

   b. If the tension in the rope is reduced to 30 Newtons, calculate the magnitude and direction of the friction, checking that it is not exceeding limiting friction.

   [7 marks]

2. A uniform plank of weight W Newtons is sticking out over a cliff. The plank is in a position so that 80% of the plank is on the land. When Alf (weight 50N) stands at the dangerous end of the plank, the plank is about to deliver him to his destiny. Alf gets off and the plank is then moved so that 65% of its length is on land. Bert (weight Y Newtons) then stands on the dangerous end and is about to be delivered to destiny. Find Y and W.
   

   [5 marks]

Query 7.2

I've been asked how to differentiate $y=3\times2^x+1$ by someone in 7.2. Remember you can only differentiate exponential functions of the form $e^{f(x)}$. \[ y=3 \times e^{(\ln 2 \times x)}+1 \\ \frac{dy}{dx}=3\ln 2 \times e^{(\ln 2\times x)}+1 \\= 3 \ln 2 \times2^x+1 \] So when $x=0$, $\dfrac{dy}{dx}=3\ln 2 \times 2^0 +1= 3\ln 2+1=\ln 8+1=\ln {8e}$

Joke from Rach in 2.4

What do you get if you divide the circumference of a pumpkin by it's diameter?

...Pumpkin Pi

4th Extension answers

Answers to 4th year extension

1. \[ \\13000 \times 0.85^? = 1000 \\0.85^? = \frac{1}{13} \\?=log_{0.85}{\frac{1}{13}}= 15.8 \]

   i.e. 16 years is needed (or you can do it by trial and error on the calculator) YOU DO NOT NEED TO KNOW THE LOG METHOD, trial and error using a calculator is sufficient at this stage

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2. a. \begin{align*} \\y^2 &= 5-3x^2 \\x^2 &= 5-y^2 \\x &=\sqrt{\frac{5-y^2}{3}} \end{align*}

   b. \begin{align*} \\ y(2x^2-1) &= 2x^2+3 && \textit{multiply by denominator} \\ 2x^2y-y&=2x^2+3 && \textit{expand the brackets} \\ 2x^2y-2x^2&=y+3 && \textit{get all the }x^2 \textit{ stuff on the left} \\ x^2(2y-2)&=y+3 && \textit{take out common factor of }x^2 \\ x^2&=\frac{y+3}{2y-2} \\ x&=\sqrt{\frac{y+3}{2y-2}} \end{align*}

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3. \[ \\3x -2(30-x)=55 \\3x -60+2x=55 \\5x -60=55 \\x =23 \]

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4. Rearranging gives \[y=\frac{3}{2}x + \frac{5}{2} \] Gradient is $\dfrac{3}{2}$, so for the new line: \[ y=\frac{3}{2}x+c \] Now use (2,-4) in here to get c ie \begin{align*} -4 = \frac{3}{2}\times 2 + c \\ \therefore c = -7 \\ \therefore y=\frac{3}{2}x-7 && \times \textit{both sides by 2 and subtract 3}x \\2y -3x = -14 && \textit{to get the required format} \\3x-2y = 14 && \textit{or this, either is fine} \end{align*}

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5. 
   

4th Year Extension Test Practice

Revision - 21 marks

1. A car depreciates (decreases) by 15% of its value every year. After how many years does a car bought for £13000 depreicate to less then £1000?

[3 marks]

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2. Make $x$ the subject:

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

[6 marks]

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3. A slug is climbing a wall. On an "upper" day he climbs upward 3m. On a "dozy" day he slides 2m downward. If after 30 days the slug is 55m up the wall, how many "upper" days did he have?

[4 marks]

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

[4 marks]

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5. Shade regions

[4 marks]

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Additional topics include: Sets, ratio, percentages

Link to the answers

Promoting higher level thinking

John Mason and friends: ‘Questions and Prompts for Mathematical Thinking’ (ATM publication):http://www.atm.org.uk/Shop/Questions-and-Prompts-for-Mathematical-Thinking/dis002  

The ATM book ‘Thinkers’ is very good too.

Roger Nelsen’s ‘Proofs Without Words’ books: http://www.amazon.co.uk/Proofs-without-Words-Exercises-Classroom/dp/0883857006  

and http://www.amazon.co.uk/Proofs-without-Words-Classroom-Materials/dp/0883857219/ref=pd_bxgy_b_img_y