Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2016 Winter Zone 3

(a) Work out.
(i) $ \sqrt[3]{79507} $
.................................................... [1]

(ii) $ 3.6^2 + \frac{1}{0.63} $
.................................................... [1]

(b) $ p = 5.62 \times 10^5 \quad q = 6.83 \times 10^{-3} $
Work out, giving your answers in standard form.
(i) $ p^2 $
.................................................... [2]

(ii) $ \frac{p}{q} $
.................................................... [2]

Gennaro has $276480 in his Pension Fund.
(a) Gennaro has two options.
Option A Receive 25% of the $276480 now
plus 5.5% of the remaining 75% each year.
Option B Receive 5.5% of the whole $276480 each year.
(i) Show that the total amount Gennaro will have received at the end of 10 years, if he chooses option A, is $183168. [3]
(ii) After how many whole years will the total amount received using option B become more than the total amount received under option A? [4]

(b) The $276480 is 8% more than the amount the Pension Fund was worth one year ago.
Calculate how much it was worth one year ago.
$ \text{.........................} [3]

Describe fully the single transformation that maps
(a) triangle A onto triangle B,
.................................................................................................................................................................................
................................................................................................................................................................................. [2]
(b) triangle A onto triangle C,
.................................................................................................................................................................................
................................................................................................................................................................................. [3]
(c) triangle A onto triangle D.
.................................................................................................................................................................................
................................................................................................................................................................................. [3]

The diagram shows a solid, square-based pyramid $VABCD$. $O$ is the centre of the base $ABCD$ and $VO$ is perpendicular to the base.
$N$ is the midpoint of $AB$.
$AB = 6\, \text{cm}$ and $VO = 8\, \text{cm}$.
(a) Calculate
(i) the volume of the pyramid, ........................................ $\text{cm}^3$ [2]
(ii) the length of $VN$. ........................................ $\text{cm}$ [2]

(b) The similar pyramid $VPQRS$ is removed from the original pyramid to leave the solid below. The height of this solid is half the height of the pyramid $VABCD$.
(i) Find the volume of this solid. ........................................ $\text{cm}^3$ [3]
(ii) Find the total surface area of this solid. ........................................ $\text{cm}^2$ [5]

(a) On the diagram, sketch the graph of $y = f(x)$ for $-3 \leq x \leq 3$. [2]
(b) Solve the equation $f(x) = 2x + 3$.
$x = \text{..................... or } x = \text{..................... or } x = \text{.....................}$ [3]
(c) (i) Find the co-ordinates of the local maximum point and the local minimum point.
Maximum $(\text{.............. , ..............})$
Minimum $(\text{.............. , ..............})$ [3]
(ii) Find the range of values of $k$ for which $f(x) = k$ has only one solution.
\text{............................................} [1]
(d) Describe fully the symmetry of the graph of $y = f(x)$.
\text{........................................................................................................................................}
\text{........................................................................................................................................} [3]

The diagram shows the points $A(-1, -1)$, $B(1, 3)$ and $C(6, 3)$.
(a) The points $A$, $B$, $C$ and $D$ are the vertices of a parallelogram.
Write down the co-ordinates of the three possible positions of $D$.
$(................, ................)$
$(................, ................)$
$(................, ................)$ [3]
(b) $E$ is a point such that $C$ is the midpoint of the line $AE$.
Find the co-ordinates of the point $E$.
$(................, ................)$ [2]
(c) The line $L$ is perpendicular to the line $AC$ and goes through $A$.
Find the equation of the line $L$. ............................................................... [4]

A farmer measured the milk yield of each of his 120 cows over a one-year period. The results are shown in the frequency table.

[Table_1: Frequency Table]

(a) (i) Complete the cumulative frequency table. [1]
(ii) Complete the cumulative frequency curve. [3]
(iii) Use your graph to estimate the median.
.................................................. litres [1]
(iv) Use your graph to estimate the inter-quartile range.
.................................................. litres [2]
(v) The farmer sells the cows with a milk yield of less than 6200 litres.
Use your graph to estimate the number of cows he sells.
............................................................. [1]

(b) On the grid below, complete the histogram to represent the data in the first table. [4]

[Image_1: Cumulative Frequency Table and Curve Graph]
[Image_2: Histogram Grid]

A ship sails on the following course.

• 60 km on a bearing of 025\degree from $A$ to $B$
• 80 km on a bearing of 115\degree from $B$ to $C$
• 75 km on a bearing of 195\degree from $C$ to $D$

The diagram shows the course.

(a) Show that angle $ABC = 90\degree$.

(b) Calculate angle $BCA$.

(c) Calculate the distance $AC$.

(d) Calculate the distance $AD$.

(e) Calculate the bearing of $D$ from $A$.

Justine travels 760 km in her car.
(a) Justine’s average speed for the journey is 77 km/h.
Calculate the time Justine takes to complete the journey.
Give your answer in hours and minutes correct to the nearest minute.
......................... h .......................... min

(b) Justine travels 270 km on main roads and 490 km on autoroutes.
On main roads her car travels $x$ km on each litre of fuel.
On autoroutes her car travels $(x + 4)$ km on each litre of fuel.
(i) Write down an expression, in terms of $x$, for the fuel that Justine’s car uses on main roads on this journey.
........................................... litres
(ii) Altogether Justine’s car uses 62 litres of fuel for the whole journey.
Write down an equation in $x$ and show that it simplifies to $31x^2 - 256x - 540 = 0$.
(iii) Solve the equation $31x^2 - 256x - 540 = 0$ to find the distance Justine’s car travels on 1 litre of fuel on autoroutes.
.......................................... km

(a) (i) Factorise.
$$2x^2 - 3x + 1$$
.............................................. [2]

(ii) Show that $2x + 1 + \frac{3}{x - 2}$ can be written as $\frac{(2x - 1)(x - 1)}{(x - 2)}$.
[3]

(b)


(i) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $-3$ and $5$. [2]

(ii) On the same diagram, sketch the graph of $y = 2x + 1$. [2]

(iii) Write down the equations of the asymptotes to the graph of $y = f(x)$.
..............................................
.............................................. [2]

(iv) Solve $f(x) = 0$.
$x = ..................... \text{ or } x = .....................$ [2]

The 50 members of an activities group either go walking or cycling.
The table shows the choices of the males and females.

[Table_1]
| Walking | Cycling | Total |
|--------|--------|-------|
| Male | 16 | 29 |
| Female | | |
| Total | 22 | 50 |

(a) Complete the table. [2]

(b) Two of the 50 members are chosen at random.
Calculate the probability that they both go cycling.
................................................ [2]

(c) Two of those who go walking are chosen at random.
Calculate the probability that one is a male and the other is a female.
................................................ [3]

y is inversely proportional to the square root of x. When x = 25, y = 2.
(a) Find y in terms of x.

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

(b) Find the value of x when y = 3.

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

(c) $z = ax^n$
z is proportional to the cube of y.
When $x = 4$, $z = 500$.
Find the value of a and the value of n.

$a = \text{..................................................}$
$n = \text{..................................................}$ [3]