U6FM Hwk

Total: 27 marks. Show all working out. Those not showing mathematical rigour will be penalised.

PRINT OUT FOR YOUR RECORDS

1. Express $\cos 6\theta$ in terms of $\cos^n \theta$

   [3 marks]

2. Express $\cos^6 \theta$ in terms of $\cos (n\theta)$

   [4 marks]

3. Solve, giving your answers in the form $re^{i\theta}$ with $ -\pi < \theta \leq \pi$ , \[ z^6=1-\sqrt3 i \] Represent your solutions on an Argand diagram

   [4 marks]

4. Represent on an Argand diagram the locus of $z$ s.t.
   1. $\arg{(z-3-2i)} = -\frac{5\pi}{6}$

      [3 marks]

   2. $|z+1|=|z-2i|$ also give the cartesian equation

      [3 marks]

   3. $|z+1|=|4z-8i|$ also give the cartesian equation

      [4 marks]

   4. Give the cartesian equation and sketch the locus \[ \arg (\frac{z+3}{z-3}) = 3\pi /4 \]

      [6 marks]

L6FM Hwk 3 Due Mon 26.9.16

Total: 26 marks. Show all working out. Those not showing mathematical rigour will be penalised.

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1. Write in the from $a(x+b)^2 + c$
   i.e. complete the square: \[2x^2-5x-3\]

   [3 marks]

2. Simplify \[ \frac{2}{x^2-3x-10} - \frac{3}{x^2-25} \]

   [4 marks]

3. Simplify as far as possible: \[ \frac{15}{\sqrt {3}} - \frac{11}{2\sqrt{7} + 3\sqrt{3}} \]

   [4 marks]

4. Solve $-3-4x-5x^2>0$ showing how you arrived at your answer.

   [3 marks]

5. \[ f(x)=\frac{2\sqrt x-x^2}{3x\sqrt x} \]
   1. Find the equation of the tangent at $x=4$, giving your answer in the form $ax+by+c=0$ where a,b,c are integers

      [5 marks]

   2. Find the equation of the normal at $x=4$, giving your answer in the form $ax+by+c=0$ where a,b,c are integers

      [4 marks]

   3. Find $f''(4)$

      [3 marks]

L6FM Homework 2 Due Mon 19.9.16

Total: 20 marks. Show all working out. Those not showing mathematical rigour will be penalised.

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1. Write in the from $a(x+b)^2 + c$
   i.e. complete the square: \[3x^2-7x-20\]

   [3 marks]

2. Simplify \[ \frac{3}{x^2-5x +4} - \frac{5}{x^2-16} \]

   [4 marks]

3. Simplify as far as possible: \[ \frac{15}{\sqrt {5}} - \frac{46}{2\sqrt{7} - \sqrt{5}} \]

   [4 marks]

4. Solve $6x^2+x-12>0$

   [3 marks]

5. Solve $5+x+x^2>0$

   [3 marks]

6. Solve \[ x^2 = 4+5xy \\ x=5y +1 \]

   [3 marks]

L6 FM Homework

Total: 10 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. Factorise fully $6x^2-7x-20$

   [2 marks]

2. Simplify as far as possible: \[ \frac{8}{4^{5x-1}}=\sqrt{32^{3+x}} \]

   [4 marks]

3. Simplify as far as possible: \[ \frac{26}{\sqrt {13}} - \frac{8}{\sqrt{13} - \sqrt{11}} \]

   [4 marks]

Further Maths Progress Check due Wed 10.2.16

Total: 51 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. Solve \[ \sin (2x-\frac{\pi}{2}) = -\frac{1}{2}\] for $-\pi \leq x \leq \pi$

   [5 marks]

2. By expressing $\cos x + \sin x$ in the form $R\cos(x- \alpha)$ with $0 \leq \alpha \leq \frac{\pi}{2}$, find the maximum value of \[2 - \cos x - \sin x\] State the smallest positive value of x for which this occurs.

   [5 marks]

3. Prove that \[ \sin 4A + \sin 2A \equiv 2\sin 3A \cos A \]

   [4 marks]

4. Solve \[ \cos \theta + 1 = 2 \sec \theta\] for $-\pi \leq x \leq \pi$

   [4 marks]

5. A is acute and B is obtuse. \[ \text{cosec} A = \frac{5}{3} \\ \sec B = -\frac{13}{5} \] Find $\tan (A+B)$ without a calculator

   [4 marks]

6. \[ f(x) = x^3 - ax^2 + x + b\] $(x-2)$ is a factor of $f(x)$ and the remainder is 5 when $f(x)$ is divided by $(2x+1)$. Find $f(3)$.

   [5 marks]

7. \[ \frac{x^4-x-1}{x^2+2} \equiv ax^2 +bx+c + \frac{dx+e}{x^2+2} \]

   [4 marks]

8. Simply as far as possible \[ 1+ \frac{2x}{x^2-2x-8} - \frac{6}{x^2-16} \]

   [4 marks]

9. Express in partial fractions:
   1. \[ \frac{2x}{(x^2-4)(x+1)} \]
   2. \[ \frac{2-x}{(x^2-4)(x+2)} \]
   3. \[ \frac{3x+2}{(x^2+4)(x+1)} \]
   4. \[ \frac{x^3}{(x^2-1)(x+1)} \]

   [16 marks]

Design your maths camp t shirt

If you are in upper VI and are going to Grinton Maths Camp, why not submit a t-shirt design? Deadline 10th Feb. Email djy

Further Maths Homework due 3.2.16

Total: 27 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. Find the distance between this line and parallel plane: \[ \underline r = \left(\begin{array}{c} -5\\2\\1\\\end{array}\right) + \lambda \left(\begin{array}{c} -3\\1\\2\\\end{array}\right) \\ \underline r \cdot \left(\begin{array}{c} 1\\3\\0\\\end{array}\right) = 4 \]

   [6 marks]

2. Find the equation of the line where these 2 planes meet in the form $ \underline r \times \underline u = \underline v$ \[x+3y-z=2 \\ 2x-y-z = 1 \]

   [4 marks]

3. Find the eigenvalues and assosciated normalised eigenvectors for this linear transformation: \[ A = \left(\begin{array}{ccc} -2&-4&2 \\ -2&1&2\\ 4&2&5 \\ \end{array}\right) \]

   [8 marks]

4. Find the 3x3 matrix for the transformation represented by T. \[ T: \left(\begin{array}{c} x\\y\\z\\\end{array}\right) \rightarrow \left(\begin{array}{c} x+y\\x-2y\\3z\\\end{array}\right) \] Find the image of the line: \[ \underline r = \left(\begin{array}{c} -5\\2\\1\\\end{array}\right) + \lambda \left(\begin{array}{c} -3\\1\\2\\\end{array}\right) \]

   [4 marks]

5. Find the the shortest distance between these lines \[ \underline r = \left(\begin{array}{c} -1\\1\\0\\\end{array}\right) + \lambda \left(\begin{array}{c} 3\\1\\0\\\end{array}\right) \\ \underline r = \left(\begin{array}{c} -2\\3\\-1\\\end{array}\right) + \mu \left(\begin{array}{c} -3\\4\\2\\\end{array}\right) \]

   [5 marks]

Further Maths Homework Due Mon 7th Dec

Total: 30 marks. Show all working out. Those not showing mathematical rigour will be penalised.
A=24, B=21, C=18

1. Find the gradient function for \[y=\frac{x-2}{x^2}\]

   [3 marks]

2. Find the coordinates of any stationary points and determine their nature \[y=\frac{1+x^2}{x}\]

   [5 marks]

3. Find the equations of the tangent and the normal when $x=4$ in the form $ax+by=c$ where a,b and c are integers \[ y=\frac{4\sqrt{x}-x^2}{x^{\frac{3}{2}}}\]

   [7 marks]

4. Find \[ \int{(x-2)^3} dx \]

   [3 marks]

5. Find $y$ if (1,-2) is a boundary condition\[ \frac{dy}{dx}= \frac{x-1}{x^3} \]

   [4 marks]

6. Find \[ \int_0^2{(2x-3)^4} dx \]

   [4 marks]

7. Find the area enclosed by the curve $y=1-x^2$ and the x axis

   [4 marks]

6.6 Fm Answers due 14.10.15

Total: 43 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. The third term of an AP is 24 whilst the 7th is -12. Find the sum of the first 20 terms. \[ a+2d=24 \\ a+6d=-12 \\ 4d=-36 \therefore d=-9 \text{ and }a=42 \\s_{20}=\frac{20}{2}(84+19(-9))=-870 \]

   [3 marks]

2. Find \[ \sum_{r=5}^{30} \frac{5-2r}{3} = -\frac{5}{3} + -\frac{7}{3} + -\frac{9}{3} +... \\ = \frac{26}{2}\left(-\frac{5}{3}+-\frac{55}{3}\right)= -260\]

   [3 marks]

3. Find n s.t. \[ \sum_{r=7}^{n} \frac{4r+1}{3} = 600 \\ n=30\]

   [4 marks]

4. Find the $u_n$ if $S_n = 2n^2+3n+1$ \[u_n= S_n - S_{n-1} \\ = 4n+1 \\ \text{ (after simplification)}\]

   [3 marks]

5. Complete the square: $y=5-4x-2x^2$ and use your answer to sketch y, indicating clearly any points where the curve crosses the axes \[-2(x+1)^2+7\]

   Upside down parabola shape
   y intercept is 5
   x intercepts $-1\pm\sqrt{\frac{7}{2}}$

   [4 marks]

6. Sketch $y=(3-x)^2(x+5)^3$ indicating any intersections with the axes

   y intercept 1125
   x touch at 3
   point of inflexion at x=-5
   basic shape: bottom left to top right

   [4 marks]

7. \[ 4y^2+x^2=36 \\ y+kx=3 \\ \]Find the value(s) of k such that the line is a tangent to the ellipse.

   The only way a line which intersects with the y axis at 3 can be tangent to an ellipse "centred" on the origin is if the tangent is parallel to the x axis so i.e. is of the form y= constant. \[k=0\]

   [4 marks]

8. Find the values of k s.t. $2x^2-x+8=kx$ has no solutions.

   SKETCH THE QUADRATIC \[ (k+1)^2-64<0 \\ \therefore -9\lt k \lt 7 \]

   [3 marks]

9. Find the centre and radius of the circle that passes through A(7,11), B(-10,-6) and C(15,-1)

   Find the equations of 2 of the perpendicular bisectors of AB, AC and BC. Solve these 2 lines simultaneously to find the centre of the circle. Centre (2,-1) and radius 13

   [5 marks]

10. Simplify as far as possible: \[ \frac{4}{\sqrt 3} - \frac{6}{\sqrt 5 - \sqrt 3} \\ = -3\sqrt 5 - \frac{5}{3}\sqrt 3 \]

    [3 marks]

11. Describe the series of transformations which transform $y= \frac{1}{x}$ to the curve \[y=5-\frac{2}{x-2}\]

    Translation of 2 units right
    Stretch in the y direction (from the x axis) scale factor 2
    Reflection in the x axis
    Translation of 5 units up

    [3 marks]

12. $y=3x-4$ is a tangent to a circle at the point (2,2). Given the centre has coords (k,1) find k and hence the equation of the circle.

    The gradient of the tangent is 3 so the gradient of radius is $-\frac{1}{3}$ which implies that, as the centre is 1 unit down from (2,2), it is 3 units right i.e. at (2+3, 2-1) = (5,1). The radius is the distance between these two points $\sqrt{10}$ \[(x-5)^2+(y-1)^2=10\]

    [4 marks]

6.6 FM Progress Homework due 14.10.15

Total: 43 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. The third term of an AP is 24 whilst the 7th is -12. Find the sum of the first 20 terms.

   [3 marks]

2. Find \[ \sum_{r=5}^{30} \frac{5-2r}{3} \]

   [3 marks]

3. Find n s.t. \[ \sum_{r=7}^{n} \frac{4r+1}{3} = 600\]

   [4 marks]

4. Find $u_n$ if $S_n = 2n^2+3n+1$

   [3 marks]

5. Complete the square: $y=5-4x-2x^2$ and use your answer to sketch y, indicating clearly any points where the curve crosses the axes

   [4 marks]

6. Sketch $y=(3-x)^2(x+5)^3$ indicating any intersections with the axes

   [4 marks]

7. \[ 4y^2+x^2=36 \\ y+kx=3 \\ \]Find the value(s) of k such that the line is a tangent to the ellipse.

   [4 marks]

8. Find the values of k s.t. $2x^2-x+8=kx$ has no solutions.

   [3 marks]

9. Find the centre and radius of the circle that passes through A(7,11), B(-10,-6) and C(15,-1)

   [5 marks]

10. Simplify as far as possible: \[ \frac{4}{\sqrt 3} - \frac{6}{\sqrt 5 - \sqrt 3} \]

    [3 marks]

11. Describe the series of transformations which transform $y= \frac{1}{x}$ to the curve \[y=5-\frac{2}{x-2}\]

    [3 marks]

12. $y=3x-4$ is a tangent to a circle at the point (2,2). Given the centre has coords (k,1) find k and hence the equation of the circle.

    [4 marks]

6.6 FM Progress answers 5.10.15

Total: 25 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. Factorise fully $6x^2-7x-20 = (3x+4)(2x-5)$

   [2 marks]

2. Complete the square: \[ y=5-2x-3x^2 = -3(x+\frac{1}{3})^2 + \frac{16}{3} \]

   

   Has max at (-1/3 , 16/3)
   y intercept is 5
   x intercepts are -5/3 and 1

   [4 marks]

3. Sketch $y=(3-x)(2x+5)^2$ indicating any intersections with the axes

   

   
   y intercept is 75
   cross at x=3 with touch x=-5/2
   negative cubic shape

   [4 marks]

4. \[ y^2=x^2+x \\ y=2x+k \\ 4x^2+4kx+k^2 = x^2+x \\ 3x^2+x(4k-1)+k^2=0 \\ \therefore (4k-1)^2-12k^2<0 \\4k^2-8k+1<0 \\ \frac{2-\sqrt 3}{2} < k < \frac{2+\sqrt 3}{2} \]Find the values of k such that the hyperbola and the line do not meet.

   [4 marks]

5. Find the centre and radius of the circle that passes through A(9,3), B(13,-5) and C(-5,-11) \[ (x-3)^2 + (y+5)^2 = 100 \]

   [5 marks]

6. Solve: \[ \frac{3}{27^{5-x}}=\sqrt{81^{5-2x}} \\ \\ 3^{1-3(5-x)}=3^{\frac{4(5-2x)}{2}} \\ \\ 1-15+3x=10-4x \\ x= \frac{24}{7}\]

   [3 marks]

7. Simplify as far as possible: \[ \frac{15}{\sqrt 5} - \frac{6}{\sqrt 7 - \sqrt 5} = -3\sqrt 7 \]

   [3 marks]

4.2 Homework Answers Due 28 Sept 15

Complete neatly in the front of your books. Calculator allowed but working must be shown. Total: 21 marks

1. A teacher has a 3% pay increase to £18,200. What was his previous salary? \[18200 \div 1.03 = 17669.90\]

   [3 marks]

2. £3000 is deposited in a bank account with an interest rate of 3.1% p.a., what will the balance be after 8 years? \[3000 \times 1.031^8 = 3829.93 \]

   [2 marks]

3. Without a calculator, find the value of \[ \frac{5.6 \times 10^{-5}}{8 \times 10^{-8}} = 0.7 \times 10^3 = 7 \times 10^2 \] Give your answer in standard form.

   [2 marks]

4. Solve \[ \text{Start by multiplying both sides by 4} \\ \frac{3x-2}{2}=\frac{4-x}{4} \\ 2(3x-2)=4-x \\ 6x-4=4-x \\ 7x=8 \\ x = \frac{8}{7} \]

   [3 marks]

5. Solve \[ 3(2x+3)-4(5-3x)=1 \\ 6x+9-20+12x=1 \\ 18x-11=1 \\ 18x=12 \\ x=\frac{2}{3} \]

   [3 marks]

6. Find the gradient of the line between A(-1,4) and B(3,-8) and hence find the equation of the line \[ \text{Gradient is} -3 \\ y=-3x+c \\ 4=3 +c \\ c=1 \\ \therefore y=1-3x \]

   [4 marks]

7. Pete's son is a third of his age. 8 years ago, Pete was four times his son's age. How old is Pete today? Show your method clearly. Guesses get nothing!! \[ \begin{array}{|c|c|c|} \hline \ & Pete & Son \\ \hline \ \text{Now} & x & \frac{x}{3} \\ \hline \ \text{8 yrs ago} & x-8 & \frac{x}{3} - 8 \\ \hline \end{array} \\ \text{Let Pete's age be }x \\x-8=4(\frac{x}{3} - 8) \\ x-8 = \frac{4x}{3} - 32 \\ \frac{x}{3} = 24 \\ x=72 \text{ years old}\]

   [4 marks]

5.1 Hwk Answers due 29 Sept 15

To be done neatly in the front of your books
TOTAL 22 marks

1. The plots shows the curve $y=x^2+4x-1$

   

   1. Use the formula to solve the equation $x^2+4x-1=0$ giving your answers to 2 d.p.\[ (x+2)^2 - 5 = 0 \\ x = -2 \pm \sqrt{5} \\ x=0.24 \text{ or } -4.24\]
   2. $x^2+4x-1=k$ has no solutions, where k is an integer. Use the plot above to find the maximum value of k \[k = -6\]

   [4 marks]

2. Complete the square $x^2+8x-4$ and use your answer to solve \[x^2+8x-4=0 \\ (x+4)^2 - 20 = 0 \\ x=-4 \pm \sqrt{20} \\ x=-4 \pm 2\sqrt{5}\]

   [3 marks]

3. Y varies indirectly as the square root of x. If y=7 when x=4, then:
   1. Find a formula for y in terms of x \[y=\frac{k}{\sqrt x} \\ k = 7 \times 2=14 \\ \therefore y=\frac{14}{\sqrt x} \]
   2. Find y when x=10 to 3 s.f. \[ y = 14 \div \sqrt{10} = 4.43 \]
   3. Find x when y=2.2 \[ x=(\frac{14}{y})^2 = (\frac{14}{2.2})^2 = 40.5 \]

   [4 marks]

4. Find the image of the point (-2, -3) under the transformation represented by M \[ \begin{pmatrix} 4 & -3 \\ -5 & -7 \end{pmatrix} \begin{pmatrix} -2 \\ -3 \end{pmatrix} = \begin{pmatrix} 1 \\ 31 \end{pmatrix} \\ \text{Point is } (1,31) \]

   [2 marks]

5. Find the 2x2 matrix which represents a reflection in the x axis. \[ \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \]

   [2 marks]

6. Write $2x^2+6x -3$ in the form $a(x+b)^2 +c$. \[2(x+\frac{3}{2})^2 - \frac{15}{2} \]

   [3 marks]

7. Draw up a table of values for $-2 \leq x \leq 3$ for the function $y=x^3 -2x^2+x-3$ and use it to sketch the curve neatly (and with appropriate scales). \[ \begin{array}{|c|c|c|c|c|c|c|} \hline \ x & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \ y & -21 & -7 & -3 & -3 & -1 & 9 \\ \hline \end{array} \]

   

   [4 marks]

6.6 FM Progress Homework due 5.10.15

Total: 25 marks. Show all working out. Those not showing mathematical rigour will be penalised.

1. Factorise fully $6x^2-7x-20$

   [2 marks]

2. Complete the square: $y=5-2x-3x^2$ and use your answer to sketch y, indicating clearly any points where the curve crosses the axes

   [4 marks]

3. Sketch $y=(3-x)(2x+5)^2$ indicating any intersections with the axes

   [4 marks]

4. \[ y^2=x^2+x \\ y=2x+k \\ \]Find the values of k such that the hyperbola and the line do not meet.

   [4 marks]

5. Find the centre and radius of the circle that passes through A(9,3), B(13,-5) and C(-5,-11)

   [5 marks]

6. Solve: \[ \frac{3}{27^{5-x}}=\sqrt{81^{5-2x}} \]

   [3 marks]

7. Simplify as far as possible: \[ \frac{15}{\sqrt 5} - \frac{6}{\sqrt 7 - \sqrt 5} \]

   [3 marks]

   ANSWERS

4.2 Homework due Mon 28.9.15

Complete neatly in the front of your books. Calculator allowed but working must be shown. Total: 21 marks

1. A teacher has a 3% pay increase to £18,200. What was his previous salary?

   [3 marks]

2. £3000 is deposited in a bank account with an interest rate of 3.1% p.a., what will the balance be after 8 years?

   [2 marks]

3. Without a calculator, find the value of \[ \frac{5.6 \times 10^{-5}}{8 \times 10^{-8}} \] Give your answer in standard form.

   [2 marks]

4. Solve \[\frac{3x-2}{2}=\frac{4-x}{4}\]

   [3 marks]

5. Solve $3(2x+3)-4(5-3x)=1$, leaving your answer as a fraction

   [3 marks]

6. Find the gradient of the line between A(-1,4) and B(3,-8) and hence find the equation of the line

   [4 marks]

7. Pete's son is a third of his age. 8 years ago, Pete was four times his son's age. How old is Pete today? Show your method clearly. Guesses get nothing!!

   [4 marks]

ANSWERS

5.1 Homework due Tue 29.9.15

To be done neatly in the front of your books
TOTAL 22 marks

1. The plots shows the curve $y=x^2+4x-1$

   

   1. Use the formula to solve the equation $x^2+4x-1=0$ giving your answers to 2 d.p.
   2. $x^2+4x-1=k$ has no solutions, where k is an integer. Use the plot above to find the maximum value of k

   [4 marks]

2. Complete the square $x^2+8x-4$ and use your answer to solve $x^2+8x-4=0$, leaving your answers in surd form

   [3 marks]

3. Y varies indirectly as the square root of x. If y=7 when x=4, then:
   1. Find a formula for y in terms of x
   2. Find y when x=10 to 3 s.f.
   3. Find x when y=2.2

   [4 marks]

4. Find the image of the point (-2, -3) under the transformation represented by M $M = \begin{pmatrix} 4 & -3 \\ -5 & -7 \end{pmatrix}$

   [2 marks]

5. Find the 2x2 matrix which represents a reflection in the x axis.

   [2 marks]

6. Write $2x^2+6x -3$ in the form $a(x+b)^2 +c$.

   [3 marks]

7. Draw up a table of values for $-2 \leq x \leq 3$ for the function $y=x^3 -2x^2+x-3$ and use it to sketch the curve neatly (and with appropriate scales).

   [4 marks]

4.2 Homework due 21.9.15

Complete neatly in the front of your books. Calculator allowed but working must be shown. Total: 23 marks

1. A milkman has a 12% pay increase to £15,000. What was his old salary?

   [3 marks]

2. The number of insects on a corpse increases by 25% every hour. At 9am there were 1000 insects on the corpse.
   1. How many insects were there at 10am?
   2. How many insects were there at 11am?
   3. Roughly how many insects will there be by midday?

   [6 marks]

3. A pair of trainers costs £35 AFTER a discount of 45%. How much were they before the discount?

   [3 marks]

4. A paperboy has his wage increased by 15% to £6.65 per hour. What was his wage before it was increased??

   [3 marks]

5. Without a calculator, find the value of \[ \frac{3.2 \times 10^{-4}}{4 \times 10^6} \] Give your answer in standard form.

   [2 marks]

6. Change $\frac{3}{25}$ into standard form showing ALL your working

   [2 marks]

7. Show all your working out for:
   1. $3\dfrac{2}{3} - 1\dfrac{6}{7}$
   2. $1\dfrac{2}{5} \times 4\dfrac{3}{8}$

      [4 marks]

5.1 Homework Due Tue 15.9.15

To be done neatly in the front of your books
TOTAL 20 marks

1. The plots shows the curve $y=-\frac{1}{3}x^2+3x+1$

   

   1. Use the formula to solve the equation $-\frac{1}{3}x^2+3x+1=0$ giving your answers to 2 d.p.
   2. Use the formula to find the solutions to 2 significant figures $-\frac{1}{3}x^2+3x+1=3$
   3. $-\frac{1}{3}x^2+3x+1=k$ has no solutions, where k is an integer. Use the plot above to find the minimum value of k

   [6 marks]

2. By factorising only, solve: $12x^2+23x-24=0$

   [3 marks]

3. 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]

4. Find the image of the point (4, -5) under the transformation represented by M $M = \begin{pmatrix} 3 & -2 \\ 1 & -1 \end{pmatrix}$

   [2 marks]

5. Find the 2x2 matrix which represents a rotation 90 degrees anticlockwise around the origin.

   [2 marks]

6. A line passes through the points A(-1,5) and B(4, -5). By first finding the gradient of AB, write down the equation of the line in the form y=mx+c

   [3 marks]

2.4 Homework 18th March Answers

Complete neatly in the front of your books. Calculator allowed but working must be shown.

1. A postman has a 12% pay increase from £18,000. What is his new salary? \[£18000\times 1.12 = £20160\]

   [3 marks]

2. The number of insects on a corpse increases by 25% every hour. At 9am there were 1000 insects on the corpse.
   1. How many insects were there at 10am? \[1000 \times 1.25 = 1250 \text{ insects}\]
   2. How many insects were there at 11am? \[1250 \times 1.25 = 1562.5 \mbox{ i.e. 1563 insects}\]
   3. Roughly how many insects will there be by midday? \[1562.5 \times 1.25 = 1953.125 \text{ i.e. 1953 insects}\]

   [6 marks]

3. A pair of trainers costs £35 AFTER a discount of 45%. How much were they before the discount? \[? \times 0.55 = 35 \\ ? = 35 \div 0.55 = £63.64 \]

   [3 marks]

4. A paperboy has his wage increased by 15% to £6.65 per hour. What was his wage before it was increased? \[? \times 1.15 = 6.65 \\ ? = 6.65 \div 1.15 = £5.78 \]
5. Express 490 as a product of its prime factors \[ 490 = 2 \times 5 \times 7^2\]

   [2 marks]

6. How many millimetres are there in 150m? \[150m \times 100 = 15000cm \times 10 = 150,000mm\]

   [2 marks]

7. £1 is worth $1.75
   1. How many dollars is £5.85? \[5.85 \times 1.75 = \$10.24\]
   2. How many pounds is 228? \[228 \div 1.75 = £130.29\]

      [3 marks]

3.2 Homework due 17.3.15 ANSWERS

1. Solve \[ \begin{align*} 3&x+2y=13 \\ -5&x+6y=81 \\ &\text{__________} \\ 15&x+10y=65 \\ -15&x+18y=243 \\ 28&y = 308 \\ \therefore &y=11 \\ &x=-3 \end{align*} \]

   [3 marks]

2. For the sequence $-3,2,7,12...$, find $u_n$ and $u_{50}$ \[ u_n=5n-8\\u_{50}=250-8=242 \]

   [3 marks]

3. For the sequence $24,21,18,15...$, find $u_n$ and $u_{50}$ \[ u_n=27-3n\\u_{50}=27-150=-123 \]

   [3 marks]

4. True or false, $-134$ is in the sequence $36,32,28,....$. Give a reason for your answer. \[ u_n=40-4n\\ 40-4n=-134 \\ -4n=-174 \\ n=43.5\] NOT in the sequence as n is not a whole number

   [2 marks]

5. Write down the first five terms of the sequence $u_n=12-5n$ \[ 7,2,-3,-8,-13 \]

   [2 marks]

6. How many green tiles would there be in pattern 120?
   \[ u_n=2n+1\\u_{120}=240+1=241 \]

   [2 marks]

7. Expand $(3x+2)(5x-4)$ \[15x^2-2x-8\]

   [2 marks]

8. Factorise $x^2-3x+28$ \[(x-7)(x+4)\]

   [2 marks]

9. Find $u_n$ for $-1,2,7,14,23,....$ \[u_n=n^2-2 \]

   [2 marks]

10. Find $u_n$ for $2,16,54,128,250....$ \[u_n = 2n^3 \]

    [2 marks]