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A stadium sells tickets at 10 different prices for a sporting event. The table shows the number of tickets sold at each price.
[Table_1]
Ticket price ($x) 22 23 35 40 53 55 58 61 69 73
Number of tickets sold (y) 8600 9100 7000 7600 5200 6000 4800 4500 2600 3000
(a) What type of correlation is shown by the data? ............................ [1]
(b) Find the mean of the 10 ticket prices. $ ............................ [1]
(c) (i) Find the equation of the regression line for $y$ in terms of $x$.
$$y = ext{............................}$$ [2]
(ii) The stadium decides to sell some tickets at a price of $45$.
Use your answer to part (i) to estimate the number of tickets it will sell at this price.
............................ [1]
(a) Translate triangle $A$ with vector $\begin{pmatrix} -3 \\ -5 \end{pmatrix}$. Label the image $B$. [2]
(b) Describe fully the single transformation that maps triangle $B$ onto triangle $A$. ................................................................................................................................................................ [2]
(c) Rotate triangle $A$ through $90^\circ$ clockwise about $(0, 0)$. Label the image $C$. [2]
(d) Reflect triangle $A$ in the line $y = x$. Label the image $D$. [2]
(e) Describe fully the single transformation that maps triangle $C$ onto triangle $D$. ...................................................................................................................................................................................... [2]
Find the next term and the nth term in each of the following sequences.
(a) 13, 18, 23, 28, 33, ...
next term = ...............................................
n\text{th} term = ............................................... [3]
(b) -9, -6, -1, 6, 15, ...
next term = ...............................................
n\text{th} term = ............................................... [3]
(c) 1089, 2178, 3267, 4356, 5445, ...
next term = ...............................................
n\text{th} term = ............................................... [3]
(d) 2, -4, 8, -16, 32, ...
next term = ...............................................
n\text{th} term = ............................................... [2]
The marks, $x$, of 300 students in a chemistry test are shown in the table.
[Table_1]
(a) Calculate an estimate of the mean mark.
.................................................. [2]
(b) Complete the cumulative frequency table.
[Table_2]
(c) On the grid, draw a cumulative frequency curve. [3]
(d) Use your curve in part (c) to find an estimate for
(i) the median mark,
.................................................. [1]
(ii) the interquartile range.
.................................................. [2]
(e) 35% of the students pass the test.
Use your curve in part (c) to find an estimate of the minimum mark needed to pass.
.................................................. [2]
f(x) = 2x - 1 \quad g(x) = 3 - x \quad h(x) = x^2
(a) Find
(i) f(-2), ............................................... [1]
(ii) h(g(-2)). ............................................... [2]
(b) Solve f(x) = 7.
x = ............................................... [2]
(c) Find f(g(x)).
............................................... [1]
(d) Solve f(x) \times g(x) + 2h(x) = 0.
x = ............................................... [3]
(e) Find g^{-1}(x).
g^{-1}(x) = ............................................... [2]
(f)
(i) On the diagram, sketch the graph of $y = h(x)$ for values of $x$ between $-3$ and $3$. [2]
(ii) Write down the equation of the line of symmetry of the graph of $y = h(x)$. ............................................... [1]
(iii) On the diagram, sketch the graph of $y = g(x)$ for values of $x$ between $-3$ and $3$. [1]
(iv) Solve $g(x) > h(x)$. ............................................... [2]
Piero invests $5000 in Bank $A$ and $5000 in Bank $B$.
(a) Bank $A$ pays simple interest at a rate of 6.5\% each year.
(i) Find the total amount Piero has in Bank $A$ at the end of 4 years.
$ \text{...........................................} $ [3]
(ii) Find the number of complete years it takes for the total amount that Piero has in Bank $A$ to be greater than $10000$.
$ \text{...........................................} $ [3]
(b) Bank $B$ pays compound interest at a rate of 4\% each year.
(i) Find the total amount Piero has in Bank $B$ at the end of 4 years.
$ \text{...........................................} $ [2]
(ii) Find the number of complete years it takes for the total amount that Piero has in Bank $B$ to be greater than $10000$.
$ \text{...........................................} $ [4]
(c) By sketching suitable graphs, find the number of complete years it takes for the total amount that Piero has in Bank $B$ to be greater than the total amount in Bank $A$.
$ \text{...........................................} $ [4]
(a) Solve the simultaneous equations. You must show all your working.
\( 7x + 2y = 8 \)
\( 2x - 3y = 13 \)
\( x = \text{.............................................} \)
\( y = \text{.............................................} \) [4]
(b) Solve.
(i) \( 3x - 4 = -19 \)
\( x = \text{.............................................} \) [2]
(ii) \( 15 - 5x = 7 - 3x \)
\( x = \text{.............................................} \) [2]
(iii) \( \frac{28}{(x + 1)} = -4 \)
\( x = \text{.............................................} \) [2]
(c) \( 3 \log p - \log q - \log 8 = 2 \log x \)
Find \( x \) in terms of \( p \) and \( q \).
\( x = \text{.............................................} \) [3]
Spinner A is numbered 2, 3, 4, 5, 6, 7.
Spinner B is numbered 2, 3, 4, 5.
Each spinner is equally likely to stop on any of its numbers.
The two spinners are each spun once and the number that each spinner stops on is recorded.
Find the probability that
(a) spinner A stops on a number less than 4, ................................................ [1]
(b) spinner B stops on 6, ................................................ [1]
(c) spinner A and spinner B both stop on the same number, ................................................ [2]
(d) one number is prime and one number is not prime, ................................................ [3]
(e) the sum of the numbers is a multiple of 3. ................................................ [2]
The diagram shows rectangle $ABCD$ and two right-angled isosceles triangles, $ABF$ and $BCE$.
(a) Find the perimeter of the quadrilateral $CDFE$. ............................................ cm [3]
(b) (i) Find the area of the quadrilateral $CDFE$. ............................................ cm$^2$ [3]
(ii) Quadrilateral $Q$ is similar to quadrilateral $CDFE$.
The area of quadrilateral $Q$ is 158 cm$^2$.
Find the length of the shortest side of quadrilateral $Q$. ............................................ cm [2]
(c) Calculate angle $AFE$.
Angle $AFE =$ ............................................ [2]
A, D, B \text{ and } C \text{ lie on a circle, centre } O. \newline AP \text{ is a tangent to the circle at } A \text{ and } BP \text{ is a tangent to the circle at } B. \newline \text{Angle } AOB = 142° \text{ and angle } DAP = 42°. \newline (a) \text{ Find the value of} \newline (i) \text{ angle } ABD, \newline \text{Angle } ABD = \text{ .................................................. } [1] \newline (ii) \text{ angle } ACB, \newline \text{Angle } ACB = \text{ .................................................. } [1] \newline (iii) \text{ angle } ADB, \newline \text{Angle } ADB = \text{ .................................................. } [1] \newline (iv) \text{ angle } BAD, \newline \text{Angle } BAD = \text{ .................................................. } [1] \newline (v) \text{ angle } APB. \newline \text{Angle } APB = \text{ .................................................. } [1] \newline (b) \text{ The radius of the circle is 11 cm.} \newline \text{Find the area of triangle } ABD. \newline \text{...................................................... cm}^2 [5]
(a) Using a suitable sketch, solve $5^x = 10$.
$x = \text{....................................................}$
(b) Solve.
$$6x - 1 = \frac{5 + x}{2x + 3}$$
You must show all your working.
$x = \text{...................}$ or $x = \text{...................}$