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(a) These are the first four terms of a sequence.
27 \hspace{1cm} 20 \hspace{1cm} 13 \hspace{1cm} 6
(i) Write down the next two terms.
\text{......................., .....................} \hspace{1cm} [2]
(ii) Find the $n$th term.
\text{...............................................} \hspace{1cm} [2]
(b) These are the first four terms of another sequence.
8 \hspace{1cm} 16 \hspace{1cm} 32 \hspace{1cm} 64
(i) Write down the next two terms.
\text{......................., .....................} \hspace{1cm} [2]
(ii) Find the $n$th term.
\text{...............................................} \hspace{1cm} [2]
574
546
552
In a sale, a shop reduces all of its prices by 15%.
(a) Jake buys a jacket which had an original price of $65.
(i) Calculate how much Jake pays for the jacket.
$ ......................................................... [2]
(ii) After paying for the jacket, Jake has $24.75 left.
Work out $24.75 as a fraction of the total amount of money Jake had before he bought the jacket. Give your answer in its lowest terms.
......................................................... [2]
(b) In the sale, Amy pays $80.75 for a coat.
Calculate the original price of the coat.
$ ......................................................... [3]
(c) One day the shop reduces its sale prices by 10%.
Calculate the overall percentage reduction of the original prices.
....................................................... % [2]
565
581
539
(a) (i) Reflect shape $P$ in the line $y = 1$. Label the image $A$. [2]
(ii) Rotate shape $P$ through $90^\circ$ clockwise about $(-1, 1)$. Label the image $B$. [2]
(iii) Describe fully the \textbf{single} transformation that maps shape $A$ onto shape $B$.
................................................................................................................... ...................................................................................................................[2]
(b) Describe fully the \textbf{single} transformation that maps shape $P$ onto shape $Q$.
................................................................................................................... ...................................................................................................................[3]
(c) Stretch shape $P$ with the $x$-axis invariant and factor 2. [2]
554
568
592
(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ from $-3$ to $4$. [3]
(b) Find the two values of $x$ for which $f(x)$ does not exist.
..................... , ................... [2]
(c) When $k \neq 0$, write down the number of solutions to the equation $f(x) = k$.
............................................... [1]
(d) $g(x) = 2^{-x} + 1$
(i) On the diagram, sketch the graph of $y = g(x)$ for $-2 \leq x \leq 4$. [2]
(ii) Write down the equation of the asymptote to the graph of $y = g(x)$.
............................................... [1]
(e) Solve the equation $f(x) = g(x)$.
$x = $ .................... or $x = $ ..................... [2]
529
518
512
(a) Carlos owns a vintage car. Each year the value of the car increases by 4% of its value at the start of the year. At the start of 2012 the value of the car was \$17\,500.
Calculate the value of the car at the start of 2018. Give your answer correct to the nearest \$100.
\$ \text{.........................................................} [4]
(b) Alex invests \$200 at a rate of $r$% per year compound interest. After 12 years, Alex has a total amount of \$239.12.
Find the value of $r$.
$r = \text{.........................................................}$ [3]
564
501
576
(a) A factory tests the lifetime, $t$ hours, of each of 200 batteries.
The table shows the results.
[Table_1]
(i) Write down the modal interval.
................................................... [1]
(ii) Complete the \textit{cumulative} frequency curve.
(iii) Use your curve to find
(a) the median,
........................................ hours [1]
(b) the number of batteries with a lifetime greater than 65 hours.
............................................ [2]
(b) This table shows the lifetimes of the \textit{same} batteries but the time intervals are different.
[Table_2]
(i) Calculate an estimate of the mean.
...................................... hours [2]
(ii) Complete the table to show the frequency densities.
[Table_3]
(iii) Complete the histogram.
594
526
585
(a) Ali walks for 1 hour at $x$ km/h and then for 2 hours at $\left(x + \frac{1}{4}\right)$ km/h. He walks a total distance of 8 km.
Write an equation and solve it to find the value of $x$.
$x = \text{..........................................................} [3]$
(b) !
The volume of the cube is equal to the volume of the cuboid.
(i) Show that $x^3 - 8x^2 + 8x = 0$.
(ii) ! On the diagram, sketch the graph of $y = x^3 - 8x^2 + 8x$ for $0 \leq x \leq 7.5$. [2]
(iii) Find the volume of the cuboid. ............................................................... [2]
533
597
557
A fair 6-sided die is numbered 0, 1, 1, 2, 3, 3.
(a) The die is rolled and the number it shows is recorded.
Find the probability that the number is
(i) 3, ..................................................... [1]
(ii) not 3, ..................................................... [1]
(iii) an odd number. ..................................................... [1]
(b) The die is rolled twice.
Find the probability that
(i) both numbers are 0, ..................................................... [2]
(ii) one number is 2 and the other is 3. ..................................................... [3]
(c) The die is rolled three times and the three numbers shown are added.
Find the probability that the total is not 0. ..................................................... [2]
548
565
542
(a) (i) Find the equation of the line that passes through the points (1, 2) and (3, 12). Give your answer in the form $y = mx + c$.
$y = \text{..................................................}$ [3]
(ii) Find the equation of the line that passes through the point (0, 2) and is perpendicular to the line in part (a)(i).
.................................................. [2]
(b) (i) Solve the equation $3x^2 + 4x - 4 = 0$. You must show all your working.
$x = \text{....................... or } x = \text{.......................}$ [3]
(ii) Solve the inequality $3x^2 + 4x - 4 < 0$.
.................................................. [2]
(c) The graph of $y = ax^2 + bx + c$ has its vertex at the point (1, 5) and intersects the $y$-axis at (0, 1).
Find the values of $a, b$ and $c$.
$a = \text{..................................................}$
$b = \text{..................................................}$
$c = \text{..................................................}$ [3]
548
594
563
The diagram shows a school playing field, $ABCD$, which is on horizontal ground, with $D$ due East of $A$.
(a) Find the bearing of
(i) $C$ from $A$. ...................................................... [1]
(ii) $A$ from $C$. ...................................................... [2]
(b) Calculate the length of $CD$.
$CD = ......................................................$ m [3]
(c) Calculate angle $BAC$.
Angle $BAC = ......................................................$ [3]
(d) (i) Calculate the area of the school playing field.
...................................................... m² [4]
(ii) In the school office there is a plan of the school playing field. It is drawn to a scale of $1 : 500$.
Calculate the area of the school playing field on the plan. Give your answer in cm².
...................................................... cm² [3]
502
520
582
f(x) = 2x + 1, g(x) = x^2 + 1, h(x) = \log x
(a)
(i) Find the value of f(4.5).
.................................................. [1]
(ii) Find the value of h(f(4.5)).
.................................................. [1]
(b) Find $f^{-1}(x)$.
$f^{-1}(x) =$ .................................................. [2]
(c) Find g(f(x)) in the form $ax^2 + bx + c$.
.................................................. [3]
(d) $p(x) = x^2 - 1$
Find the single transformation that maps the graph of $y = g(x)$ onto the graph of $y = p(x)$.
................................................................................................................................. [2]
(e) Solve the equation $h^{-1}(x) = 1000$.
$x =$ .................................................. [1]
526
575
508
