Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2017 Summer Zone 1

(a) Find the next term and the nth term in each of the following sequences.
(i) 4, 8, 12, 16, 20, ...
next term = ....................................................
nth term = .................................................... [2]
(ii) -1, -3, -5, -7, -9, ...
next term = ....................................................
nth term = .................................................... [3]
(iii) 3, 12, 27, 48, 75, ...
next term = ....................................................
nth term = .................................................... [3]
(iv) 1, 8, 27, 64, 125, ...
next term = ....................................................
nth term = .................................................... [2]
(b) Use your answers to part (a), to find the next term and the nth term in the following sequence.
7, 25, 61, 121, 211, ...
next term = ....................................................
nth term = .................................................... [3]

(a) The heights, $x$ cm, of some plants are shown in the table.

[Table_1]

Calculate an estimate of the mean height of the plants.
............................................. cm [2]

(b) (i) Complete the cumulative frequency table for the plants.

[Table_2]

(ii) On the grid below, draw the cumulative frequency curve.


[3]

(c) Use your graph in part (b)(ii) to find estimates for

(i) the median height,
................................................ cm [1]

(ii) the interquartile range,
................................................ cm [2]

(iii) the range of heights of plants that are between the 45th and the 55th percentile.
.................................................. cm [3]

In the diagram, BCD is a straight line.
(a) Find AC.
$AC = \text{.......................................} \text{m}$ [3]
(b) Find BC.
$BC = \text{.......................................} \text{m}$ [3]
(c) Find CD.
$CD = \text{.......................................} \text{m}$ [3]
(d) Find the area of triangle ACD.
$\text{.......................................} \text{m}^2$ [2]

Triangle A is given in the coordinate grid.

(a) Translate triangle $A$ with vector $\begin{bmatrix} 0 \\ -4 \end{bmatrix}$. Label the image $B$. [2]
(b) Rotate triangle $A$ through $90^{\circ}$ anticlockwise about $(0, 0)$. Label the image $C$. [2]
(c) Describe fully the \textit{single} transformation that maps triangle $C$ onto triangle $A$. [2]
............................................................................................................................
............................................................................................................................
(d) Reflect triangle $A$ in the line $y = -x$. Label the image $D$. [3]
(e) Describe fully the \textit{single} transformation that maps triangle $C$ onto triangle $D$. [2]
............................................................................................................................
............................................................................................................................

A, B, C and D lie on a circle, centre O.
AP and BP are tangents to the circle.
Angle $APB = 46^{\circ}$.

(a) Complete the statement.

Angle $OAP = 90^{\circ}$ because ................

(b) Find the value of

(i) angle $AOB$

(ii) angle $OAB$

(iii) angle $ACB$

(iv) angle $ADB$

(c) $OB$ bisects angle $ABC$.
Find angle $OAC$.

y varies inversely as the square of x.
y = 32 when x = 2.
(a) Find the value of y when x = 4.
y = ............................................. [3]
(b) Find the value of x when y = 512.
x = ............................................. [2]
(c) Find x in terms of y.
x = ............................................. [3]

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

[4]
(b) Solve $f(x) = 7$.
.....................................................................................
[2]
(c) The equation $|9 - x^2| = k$ has two solutions.
Find the range of values of $k$.
....................................................................................
[2]

The Venn diagram shows the sets $M$, $E$ and $T$.

$U = \{ \text{students at a school} \}$
$M = \{ \text{students who study mathematics} \}$
$E = \{ \text{students who study English} \}$
$T = \{ \text{students who study technology} \}$
$n(M \cap E \cap T) = 8$
$n(M \cup E \cup T)' = 4$
$n(M \cap E) = 12$, $n(M \cap T) = 14$ and $n(E \cap T) = 20$
$n(M) = 25$, $n(E) = 30$, $n(T) = 35$ and $n(U) = 56$
(a) Complete the Venn diagram.

(b) Find
(i) $n(M \cap (E' \cup T'))$. ...........................................................

(b) (ii) $n(M \cap T')$. ...........................................................

(c) One of these students is chosen at random.
Find the probability that this student studies English and mathematics but not technology. ...........................................................

(d) Two of the 56 students are chosen at random.
Find the probability that they both study technology. ...........................................................

(e) A student who studies mathematics is chosen at random.
Find the probability that this student also studies technology but not English. ...........................................................

(f) Two students who study English are chosen at random.
Find the probability that they both study mathematics but not technology. ...........................................................

The diagram shows triangle $ABC$.
(a) Use the cosine rule to find angle $ABC$.

Angle $ABC = \text{..................................................}$ [3]
(b) Use the sine rule to find angle $BAC$.
Angle $BAC = \text{..................................................}$ [3]

$$f(x) = 2 \sin x + \cos x \quad \text{for} \quad 0^\circ \leq x \leq 360^\circ$$
$$g(x) = 2 - \log x \quad \text{for} \quad 0^\circ \leq x \leq 360^\circ$$
(a) On the diagram, sketch the graph of $y = f(x)$. [3]
(b) On the same diagram, sketch the graph of $y = g(x)$. [2]
(c) Solve the equation.
$$2 \sin x + \cos x = 2 - \log x$$ .......................................................... [3]

Vito lives in Sicily.
Table A shows the distances, in km, between different towns.
Table B shows the average speed, in km/h, that Vito drives his car between towns.

Table A (distances, in km)
[Table_1]

Table B (average speeds, in km/h)
[Table_2]

(a) (i) Write down the distance from Agrigento to Messina.
.............................................km [1]
(ii) Find the time taken for Vito to drive from Agrigento to Messina.
........................................hours [2]

(b) On another day, Vito drives from Agrigento to Trapani. He arrives at Trapani at 1042.
At what time did he leave Agrigento?
.................................................. [3]

(c) One day Vito drives from Catania to Palermo. Vito’s car uses fuel at the rate of 12.5 km/litre.
The cost of fuel is 1.432 euros per litre.
Find the cost of this journey.
......................................euros [3]

(d) The time for Vito to drive from Catania to Trapani is $1\frac{1}{2}$ hours longer than the time for Vito to drive from Palermo to Trapani.
(i) Show that $x^2 - 75x + 1400 = 0$.
[5]
(ii) Find the two possible average speeds that Vito drives from Catania to Trapani.
......................km/h, ......................km/h [3]