Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2016 Winter Zone 1

Toby takes a journey from Johannesburg to Zurich.

(a) He changes 2500 rand into Swiss francs (CHF).
1 Swiss franc = 12.43 rand.

Calculate the amount Toby receives in Swiss francs.
Give your answer correct to the nearest Swiss franc.

............................................... CHF [2]

(b) Toby leaves Johannesburg at 19 30 and arrives in Zurich at 06 10 the next morning.
Local time in Zurich is the same as local time in Johannesburg.
The distance from Johannesburg to Zurich is 8350 km.

(i) Calculate the average speed of the journey.

................................................ km/h [3]

(ii) After arriving at 06 10, Toby takes a further 1 hour 55 minutes to reach his office.

Work out the time he arrives at his office.

................................................ [1]

(iii) Later, Toby takes a taxi from his office to a hotel.
The taxi fare is made up of a fixed charge of 20 CHF plus 2.40 CHF per kilometre.
Toby paid 36.80 CHF altogether.

Work out the distance of Toby’s taxi journey.

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

(a) $ \mathbf{u} = \begin{pmatrix} -3 \\ -2 \end{pmatrix} $ and $ \mathbf{v} = \begin{pmatrix} -5 \\ -3 \end{pmatrix} $

(i) Find $ \mathbf{u} + \mathbf{v} $.\hspace{8cm}[1]

(ii) Draw the image of triangle $ T $ under the translation by the vector $ \mathbf{u} + \mathbf{v} $. \hspace{8cm}[2]

(iii) Calculate $ |\mathbf{u} + \mathbf{v}| $.



(b) Describe fully the single transformation that maps

(i) triangle $ T $ onto triangle $ P $,\hspace{8cm}[2]

..........................................................................................................................
..........................................................................................................................

(ii) triangle $ T $ onto triangle $ Q $,\hspace{8cm}[3]

..........................................................................................................................
..........................................................................................................................

(iii) triangle $ T $ onto triangle $ R $.\hspace{8cm}[3]

..........................................................................................................................
..........................................................................................................................

f(x) = 2^{\sin x}
(a) On the diagram, sketch the graph of $y = f(x)$ for $-360^\circ \leq x \leq 360^\circ$. [3]
(b) Find the range of $f(x)$. [2]
(c) Find the value of $f(x)$ when
(i) $x = 3780^\circ$, [1]
(ii) $x = 4050^\circ$. [1]
(d) (i) Find the four values of $x$ from $-360^\circ$ to $1080^\circ$ for which $f(x) = 0.5$. [2]
(ii) The values in the answer to part (d)(i) form the first four terms of a sequence. Find the $n^{th}$ term of this sequence. [2]
(e) $g(x) = \frac{x(360-x)}{16200}$
(i) On the diagram, sketch the graph of $y = g(x)$ for $0^\circ \leq x \leq 360^\circ$. [2]
(ii) Solve the equation $f(x) = g(x)$. [2]

The diagrams show a solid hemisphere and a solid cone.
Both the hemisphere and the base of the cone have radius 9cm.
The volumes of the two shapes are equal.
(a) Show that the perpendicular height of the cone is 18cm. [2]
(b) (i) Calculate the total surface area of the hemisphere. [2]
(ii) Calculate the curved surface area of the cone. [3]
(c) The hemisphere is made from metal.
The metal is melted down and made into spheres of radius 2cm.
Calculate the number of spheres that are made. [3]

(a) Show that $n(P \cap Q) = 8$. [2]
(b) An element is chosen at random from $U$.
Find the probability that the element is a member of
(i) $P \cup Q$. ...................................................... [1]
(ii) $P \cup Q'$. ...................................................... [1]
(c) An element is chosen at random from $P$.
Find the probability that this element is also a member of $Q$. ...................................................... [1]
(d) The probability of a single event is $\frac{2}{3}$.
Describe this event in terms of $P$ and $Q$. ................................................................................ [1]

A is the point (0, 6) and B is the point (4, 0).
(a) Find the equation of the perpendicular bisector of $AB$. [5]
(b) The line $y = 2x + 3$ cuts the $y$-axis at $C$.
The perpendicular bisector of $AB$ cuts the $y$-axis at $D$.
Find the length $CD$.
$$CD = \text{............................}$$ [2]

The diagram shows a triangular prism with a horizontal base $ABCD$. $X$ is a point on the line $AQ$. $AB = 20 \text{cm}$, $BC = 10 \text{cm}$, $CQ = 9 \text{cm}$ and angle $BCQ = 90^\circ$.
(a) Calculate angle $QBC$.
Angle $QBC' = \text{...............................}$ [2]
(b) Calculate angle $BAQ$ and show that it rounds to $33.9^\circ$, correct to $1$ decimal place. [3]
(c) $AX = 22 \text{cm}$.
Calculate the length of $BX$.
$BX = \text{............................... cm}$ [3]

(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between -4 and 4. [2]

(b) Find the zeros of $f(x)$.
......................................................... [2]

(c) Solve the inequality $f(x) < 0$.
......................................................... [3]

(d) The asymptotes of the graph are $x = a$ and $y = x + b$, where $a$ and $b$ are integers.
Find the value of $a$ and the value of $b$.
$a = .........................................................
b = .........................................................$ [2]

(e) $g(x) = x + \frac{1}{x}$
Describe fully the single transformation that maps the graph of $y = f(x)$ onto the graph of $y = g(x)$.
............................................................................................................................. [2]

In one day a delivery company delivers 93 parcels. The histogram shows information about the masses, m kg, of these parcels.



(a) Complete the frequency table.

[Table_1]

(b) Calculate an estimate of the mean mass.
.......................................................... kg [2]

(c) Two parcels are chosen at random.
Find the probability that they both have a mass greater than 1 kg.
Give your answer as a decimal, correct to 3 significant figures.
.......................................................... [2]

(a) Solve.
$$7x + 2 = 11$$

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

(b) Write as a single fraction, in its simplest form.
$$\frac{x+1}{2} + \frac{x-1}{3}$$

........................................................... [2]

(c) Simplify the following.
(i) $$\frac{8x^4y^2}{4x^3y^4}$$

........................................................... [2]
(ii) $$\frac{x^2-9}{x^2-2x-3}$$

........................................................... [4]

f(x) = 3x + 1 \hspace{1cm} g(x) = \log x
(a) Find the value of g(f(33)). ............................................... [2]
(b) Find the value of x when g(x) = f(-1).
\hspace{4cm} x = ............................................... [2]
(c) Find
(i) \hspace{0.5cm} f^{-1}(x),
\hspace{2cm} f^{-1}(x) = ............................................... [2]
(ii) \hspace{0.5cm} g^{-1}(x).
\hspace{2cm} g^{-1}(x) = ............................................... [2]

(a) In 2015, Ahmed had a monthly salary of $1375.
In 2016, his monthly salary is $1540.
(i) Calculate the percentage increase in Ahmed’s monthly salary.
................................................ % [3]
(ii) Work out $1375 as a percentage of $1540.
................................................ % [1]
(iii) In 2015, Ahmed’s monthly salary of $1375 was 10% more than his monthly salary in 2014.
Calculate Ahmed’s monthly salary in 2014.
$ ................................................ [3]

(b) Samia invested $500 in each of two Schemes.
Scheme A    3% per year simple interest.
Scheme B    2.5% per year compound interest.
(i) Calculate the difference between the value of Scheme A and the value of Scheme B after 5 years. Show all your working.
$ ................................................ [5]
(ii) Find the number of complete years it will take for the value of Scheme B to be greater than the value of Scheme A.
................................................ [4]