Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2017 Summer Zone 2

In 2016, Carla’s salary was $23,970 per year.

(a) From her salary she pays tax at a rate of 20%. She is paid monthly in equal amounts.
Calculate the amount Carla receives each month after tax has been paid.

$ ..................................................... [3]

(b) Carla’s salary of $23,970 was 2% more than her salary in 2015.

(i) Calculate her yearly salary in 2015.

$ ..................................................... [3]

(ii) From 2016, Carla’s employer agrees to pay her an increase of 3% each year.
Calculate the year in which her salary is first greater than $30,000.

..................................................... [3]

(a) (i) Reflection in the line $y = x$ maps triangle $A$ onto triangle $B$.
Describe fully the single transformation that maps triangle $B$ onto triangle $A$.
$\text{..................}$ [1]

(ii) Enlargement, with centre $(2, 1)$ and scale factor $4$, maps triangle $C$ onto triangle $D$.
Describe fully the single transformation that maps triangle $D$ onto triangle $C$.
$\text{..................}$ [2]
$\text{..................}$ [2]

(iii) Translation by the vector $\begin{pmatrix} -3 \\ 5 \end{pmatrix}$ maps triangle $F$ onto triangle $F'$.
Describe fully the single transformation that maps triangle $F'$ onto triangle $E$.
$\text{..................}$ [2]
$\text{..................}$ [2]

(b)
(i) Rotate triangle $P$ through $90^\circ$ anticlockwise about $(0, 0)$.
Label the image $Q$.
$\text{..................}$ [2]

(ii) Stretch triangle $P$ with stretch factor $2$ and the $y$-axis invariant.
Label the image $R$.
$\text{..................}$ [2]

Two judges each give a mark out of ten for each dancer in a competition. Their marks for ten dancers are shown in the table.

| Mark from judge A (x) | 4.0 | 4.6 | 5.2 | 6.2 | 8.8 | 6.8 | 7.0 | 7.4 | 8.0 | 8.6 |
|----------------------|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|
| Mark from judge B (y) | 3.8 | 4.0 | 4.4 | 5.0 | 7.6 | 5.2 | 5.6 | 6.8 | 6.6 | 7.0 |

(a) Complete the scatter diagram. The first four points have been plotted for you.


(b) What type of correlation is shown on your scatter diagram?
.................................................. [1]

(c) (i) Find the equation of the regression line, in the form $y = mx + c$.
$y = ..................................................$ [2]

(ii) Judge A gives another dancer a mark of 6.4.
Use your equation to estimate the mark judge B gives this dancer.
.................................................. [1]

Given $\mathbf{p} = \begin{pmatrix} -3 \\ 2 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 4 \\ 3 \end{pmatrix}$

(a) Find
(i) the column vector $\frac{1}{2} \mathbf{p}$, [1]

(ii) the column vector $\mathbf{q} - 2\mathbf{p}$, [2]

(iii) $|\mathbf{p}|$, leaving your answer in surd form. [2]

(b) $\overrightarrow{AB} = \mathbf{p} + \mathbf{q}$
Mark and label point $B$ on the grid.
[2]

(a) Nitini changes 119050 Indian rupees (INR) to Singapore dollars (SGD). The exchange rate is 1 SGD = 47.62 INR. Find how many Singapore dollars he receives. ......................................... SGD [2]

(b) The flight from New Delhi to Singapore takes 5 hours and 45 minutes. The distance of the flight is 4150km.
(i) The time in New Delhi when the flight leaves is 2155. The time in Singapore is 2\frac{1}{2} hours ahead of the time in New Delhi. Find the time in Singapore when the flight arrives. ......................................... [2]
(ii) Find the average speed of the aircraft. ......................................... km/h [3]
(iii) On the return flight the average speed is 750 km/h. Find the time of this flight in hours and minutes. ................ h .................... min [3]

(a) (i) $x$ is proportional to $v$.
Write down an expression for $x$ in terms of $v$ and a constant $c$.
$$x = ext{.........................................}$$ [1]

(ii) $y$ is proportional to $v^2$.
Write down an expression for $y$ in terms of $v$ and a constant $k$.
$$y = ext{.........................................}$$ [1]

(iii) $d = x + y$
Write down an expression for $d$ in terms of $v$, $c$ and $k$.
$$d = ext{.........................................}$$ [1]

(b) The table shows two values of $v$ and the corresponding values of $d$.

A ship sails 65 km on a bearing of 310° from $A$ to $B$. It then changes course and sails 40 km on a bearing of 250° from $B$ to $C$. The ship then returns to $A$.
(a) On the diagram, sketch the path of the ship from $A$. On your diagram show the bearings and distances.
[3]
(b) Find angle $ABC$.
..................................................
[1]
(c) Calculate $AC$ and show that it rounds to 91.8 km, correct to the nearest tenth of a kilometre.
[3]
(d) Find the bearing of $C$ from $A$.
..........................................................
[4]

(a) Sketch the graph of $y = f(x)$ for $-400^\circ \leq x \leq 600^\circ$.
(b) Find the $x$ co-ordinates of the local maximum points of $f(x)$ for $-400^\circ \leq x \leq 600^\circ$.
$x = \text{.....................}$ or $x = \text{.....................}$ or $x = \text{.....................}$.
(c) The point $(30, \sqrt{3})$ is on the graph. The point $(a, \sqrt{3})$ is also on the graph where $600^\circ < a < 900^\circ$.
Find the two possible values of $a$.
$a = \text{.....................}$ or $a = \text{.....................}$.
(d) $g(x) = 3 - \frac{x}{100}$
Solve the inequality $g(x) > f(x)$.
$$f(x) = 3^{\sin x}$$

In the diagram $AC = x$ cm, $AB = (x + 2)$ cm and angle $A = 60^\circ$.
(a) (i) Find an expression, in terms of $x$, for the area of triangle $ABC$. Give your answer in surd form. ................................................ cm² [2]
(ii) The area of triangle $ABC = 18 \sqrt{3}$ cm². Show that $x^2 + 2x - 72 = 0$. [2]
(b) (i) Solve the equation $x^2 + 2x - 72 = 0$. $x = \text{....................... or } x = \text{.......................}$ [2]
(ii) Find the shortest distance between the line $AB$ and the point $C$. ................................................ cm [2]

A is the point (2, 2), B is the point (11, 4) and C is the point (14, 8).

(a) Find the equation, in the form $y = mx + c$, of

(i) the line AC,

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

(ii) the line through B that is perpendicular to AC.

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

(b) Show that the point (10, 6) is on both the lines you found in part (a). [2]

(c) AC is the perpendicular bisector of BD. Find the co-ordinates of D. [1]

(d) Find the exact area of the quadrilateral ABCD. [4]

A farmer sorts the grapefruit he grows into sizes, according to their diameter. The diameters, $d$ cm, of 170 grapefruit are shown in the table.

[Table_1]

(a) Calculate an estimate of the mean diameter of the grapefruit.
.................................................. cm [2]

(b) On the grid, draw a histogram to represent this information. Complete the scale on the frequency density axis.
[4]

(c) Two of the 170 grapefruit are chosen at random. Calculate the probability that
(i) they are both Very Large,
.................................................. [2]
(ii) one is Small and the other is Medium.
.................................................. [3]

f(x) = 4x + 2\quad g(x) = 5 - 2x\quad h(x) = x^2 - 3
(a) Find g(-3).
[1]
(b) Find f(h(2)).
[2]
(c) Find x when f(x) = -10.
x = ................................................ [2]
(d) Write down the range of h(x).
[1]
(e) Find $f^{-1}(x)$.
$f^{-1}(x)$ = ................................................ [2]
(f) k(x) = 10 - 4x
Describe fully the single transformation that maps the graph of $y = g(x)$ onto the graph of $y = k(x)$.
.....................................................................................................................
..................................................................................................................... [3]
(g) The graph of $y = h(x)$ is translated by the vector \begin{pmatrix} 2 \\ 0 \end{pmatrix}.
Find the equation of the graph of the image.
Write your answer in the form $y = ax^2 + bx + c$.
y = ................................................ [3]