Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2018 Summer Zone 2

(a) Work out.
$$\frac{\sqrt[3]{402}}{3.15^2}$$
.......................................................... [1]

(b) Write 130.47 correct to
(i) one decimal place,
.......................................................... [1]
(ii) one significant figure.
.......................................................... [1]

(c) Work out 23\% of $76.80.
$ .......................................................... [2]

(d) $4200 is shared in the ratio 3 : 4 : 6 : 8.
Find the difference between the largest share and the smallest share.
$ .......................................................... [3]

(e) Write down an irrational number less than 10.
.......................................................... [1]

(f) Work out $ 7.31 \times 10^2 + 1.56 \times 10^1 $.
Give your answer in standard form.
.......................................................... [2]

(a) On the diagram, sketch the graph of $y = f(x)$, for values of $x$ between $-5$ and $5$. [3]

f(x) = 1 - \frac{x}{(x^2 - 9)}
(b) Write down the equations of the three asymptotes.
........................, ........................, ........................ [3]
(c) The line $y = x$ intersects the curve $y = 1 - \frac{x}{(x^2 - 9)}$ three times.
Find the values of the $x$ co-ordinates of the points of intersection.
x = ..................... or x = ..................... or x = ..................... [3]

(a) $y$ varies directly as the square root of $x$.
$y = 32$ when $x = 16$.
(i) Find $y$ in terms of $x$.

$y = \text{................................................}$ [2]

(ii) Find the value of $y$ when $x = 4$.

$y = \text{................................................}$ [1]

(iii) Find $x$ in terms of $y$.

$x = \text{................................................}$ [2]

(b) $p$ varies inversely as $q + 2$.
$p = 3$ when $q = 2$.

Find the value of $p$ when $q = 4$.

$p = \text{................................................}$ [3]

(a) The mass, $x$ grams, of each of 100 oranges is found. The results are shown in the table.
[Table_1]
(i) Calculate an estimate of the mean mass of the oranges.
.................................................. g [2]
(ii) Two of these oranges are chosen at random.
Calculate the probability that they both have a mass of 140 g or less.
.................................................. [2]
(iii) The oranges with a mass of 140 g or less are removed. From the remaining oranges, two are chosen at random.
Calculate the probability that one orange has a mass of 250 g or less and the other has a mass of more than 250 g.
.................................................. [3]

(b) (i) Complete the frequency density column in this table.
[Table_2]
[2]
(ii) On the grid, draw a histogram to show this information.

[4]

In the diagram, $ABCD$ is a rectangle.
(a) Find $PS$.
$PS = \text{.........................}$ m [2]
(b) Find angle $BRS$.
Angle $BRS = \text{.............................}$ [2]
(c) Find the perimeter of $PQRS$.
$\text{.................................}$ m [3]
(d) Find the shaded area.
$\text{...............................}$ m$^2$ [3]
(e) Explain why triangle $ASP$ is similar to triangle $BSR$.
$\text{................................................ }$
$\text{................................................ }$ [2]

(a) Translate triangle A with vector \( \begin{pmatrix} -7 \\ -3 \end{pmatrix} \). Label the image B. [2]
(b) Rotate triangle A through 90^\circ anti-clockwise about \((-1, 2)\). Label the image C. [2]
(c) Describe fully the \textit{single} transformation that maps triangle C onto triangle B. ..............................................................................................................................................................................
........................................................................................................................... [3]
(d)Enlarge triangle A scale factor -2 with centre \((3, 1)\). Label the image D. [2]
(e) Describe fully the \textit{single} transformation that maps triangle D onto triangle A. ........................................................................................................................................................................ [2]

In this question, all lengths are measured in millimetres.

A small plastic cup, $A$, is shown in this diagram.

These plastic cups are stacked as shown in the diagram.

(a) Find the height of a stack of 8 of these cups.
....................................... mm [2]

(b) Find the number of these cups in a stack that has a total height of 105 mm.
....................................... [2]

(c) A similar cup, $B$, has base diameter 42 mm.
Find the height of this cup.
....................................... mm [2]

(d)
The formula for the volume of a similar cup is $V = \frac{\pi h (3r^2 + 3ar + a^2)}{3}$.

(i) For cup $A$, show that $a = 8$ mm. [2]

(ii) Find the volume of cup $A$.
....................................... mm$^3$ [2]

(iii) Find the volume of cup $B$.
....................................... mm$^3$ [3]

(iv) Rearrange $V = \frac{\pi h (3r^2 + 3ar + a^2)}{3}$ to make $h$ the subject.
$h = ......................................... $ [2]

A, B, C \text{ and } D \text{ lie on a circle, centre } O.\newline DE \bot \text{ is a tangent to the circle at } D.\newline AOCF \text{ and } BCE \text{ are straight lines.}
(a) Complete the statement.
Angle $ODE = 90^{\circ}$ because .........................................
..................................................... [1]
(b) Find the value of
(i) angle $AOD$,\newline $\quad$ Angle $AOD = \text{.........................} [2]$
(ii) angle $ODC$, \newline $\quad$ Angle $ODC = \text{.........................} [2]$
(iii) angle $ABC$, \newline $\quad$ Angle $ABC = \text{.........................} [1]$
(iv) angle $CFD$, \newline $\quad$ Angle $CFD = \text{.........................} [1]$
(v) angle $CAB$. \newline $\quad$ Angle $CAB = \text{.........................} [1]$

(a) Show that angle $BAC = 47.0^\circ$, correct to 1 decimal place.

[3]

(b) Use the sine rule to find angle $ABC$.

Angle $ABC = \text{.......................................................}$ [3]

(c) Find the area of triangle $ABC$.

\text{............................................. cm}^2$ [2]

(d) Find the length of the perpendicular from $B$ to $AC$.

\text{............................................. cm}$ [2]

Wasim sprays different amounts of fertiliser on some seedlings. He measures the amount, x millilitres, sprayed on each seedling. A week later he measures the height, y centimetres, of each seedling. His results are shown in the table.
[Table_1]
Amount of fertiliser (x ml) | 1 | 3 | 5 | 7 | 10 | 14 | 18 | 25 | 30 | 35 | 40
Height (y cm) | 15.1 | 15.6 | 16.5 | 16.6 | 17 | 19.8 | 21 | 25.1 | 28.8 | 28.6 | 29.1
(a) (i) Complete the scatter diagram. The first four points have been plotted for you.


(a) (ii) What type of correlation is shown by the scatter diagram? .......................................................

(b) (i) Find the mean amount of fertiliser, ................................................ ml

(b) (ii) Find the mean height. ................................................ cm

(c) (i) Find the equation of the regression line in the form $y = mx + c$. $y = ................................................$

(c) (ii) Use your answer to part (c)(i) to estimate the height of a seedling when the amount of fertiliser is 20 ml. ................................................ cm

(c) (iii) Write down the units of $m$ in the equation of the regression line, $y = mx + c$. ................................................

Given: $f(x) = 2x - 7$, $g(x) = \sqrt{x}$, $h(x) = \frac{1}{x}$, $x \neq 0$
(a) (i) Find $f(3)$.
    ................................................ [1]
(ii) Solve $f(x) = 1$.
    $x = ..................................................$ [2]
(b) Find $f^{-1}(x)$.
    $f^{-1}(x) = ..................................................$ [2]
(c) (i) Find $f(g(x))$ in terms of $x$.
    ................................................ [1]
(ii) Solve $f(g(x)) = 5$.
    $x = ..................................................$ [3]
(d) (i) Find $h(g(f(x)))$ in terms of $x$.
    ................................................ [2]
(ii) Find an inequality in terms of $x$ for which $h(g(f(x)))$ exists.
    ................................................ [2]