Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2020 Summer Zone 3

For each sequence, write down the next two terms and find an expression for the nth term.

(a) 15, 11, 7, 3, -1, ...

Next two terms ....................... , .......................
nth term ............................................... [3]

(b) 1, 2, 4, 8, 16, ...

Next two terms ....................... , .......................
nth term ............................................... [3]

(c) 4, 10, 18, 28, 40, ...

Next two terms ....................... , .......................
nth term ............................................... [3]

10 students take a language examination.
The examination consists of two parts, a speaking test and a writing test.
Both tests are marked out of 100.
The marks for the students in each of the tests is shown in the table.

| Speaking mark \( (x) \) | 86 | 62 | 53 | 34 | 76 | 95 | 30 | 70 | 88 | 72 |
|-----------------------|----|----|----|----|----|----|----|----|----|----|
| Writing mark \( (y) \) | 73 | 48 | 44 | 12 | 62 | 66 | 26 | 44 | 90 | 75 |

(a) Complete the scatter diagram to show these results.
The first five points have been plotted for you.



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

(c) (i) Calculate the equation of the regression line in the form \( y = mx + c \).
\( y = \text{..............................................} \) [2]

(ii) Use this equation to estimate a mark in the writing test for a student who scored 48 in the speaking test. ..................................................... [1]

(a) Riaz invests $5000 at a rate of 2.5\% per year simple interest.
(i) Calculate the value of the investment at the end of 4 years.
$ \text{.................................} \quad [3]
(ii) Calculate the number of complete years it will take for the value of the investment to be $6500.
\text{................................................} \quad [2]

(b) Yasmin invests $5000 at a rate of 2\% per year compound interest.
(i) Calculate the value of Yasmin’s investment at the end of 4 years.
$ \text{.................................} \quad [3]
(ii) Calculate the number of complete years it will take for the value of Yasmin’s investment to first be worth more than $6500.
\text{................................................} \quad [4]

Given \( f(x) = x^3 - 4x^2 - 3x + 18 \)
(a) On the diagram, sketch the graph of \( y = f(x) \) for \(-3 \leq x \leq 5\). [2]
(b) Solve the equation \( f(x) = 10 \).
\( x = \text{..................} , \text{or} \ x = \text{..................} , \text{or} \ x = \text{..................}\) [3]
(c) Write down the coordinates of
(i) the local maximum,
\((\text{..................} , \text{..................})\) [2]
(ii) the local minimum.
\((\text{..................} , \text{..................})\) [1]
(d) \( f(x) = k \) has only 1 solution.
Find the ranges of values of \( k \).
\( \text{............................................}\) [2]

(a) (i) A reflection in the line $y = 3$ maps triangle $A$ onto triangle $B$.
Describe fully the single transformation that maps triangle $B$ onto triangle $A$.
................................................................................................................. .................................................................................................................. [1]
(ii) A translation using the vector $\begin{pmatrix} 5 \\ -4 \end{pmatrix}$ maps triangle $C$ onto triangle $D$.
Describe fully the single transformation that maps triangle $D$ onto triangle $C$.
................................................................................................................. .................................................................................................................. [2]
(iii) An enlargement, centre $(2, -1)$, scale factor $3$, maps triangle $G$ onto triangle $H$.
Describe fully the single transformation that maps triangle $H$ onto triangle $G$.
................................................................................................................. .................................................................................................................. [2]

(b) 
(i) Rotate triangle $A$ through $90^\circ$ anticlockwise, centre $(-1, 0)$.
Label the image $B$.
[2]
(ii) Enlarge triangle $A$ with scale factor $-\frac{1}{2}$, centre $(1, 3)$.
Label the image $C$.
[2]
(iii) Describe fully the single transformation that maps triangle $A$ onto triangle $D$.
................................................................................................................. .................................................................................................................. [3]

The cumulative frequency graph shows the heights, in centimetres, of 120 plants in location A.

(a) Use the graph to estimate
(i) the median, ....................................................... cm [1]
(ii) the interquartile range, ....................................................... cm [2]
(iii) the number of plants over 80 cm in height.
....................................................... [2]

(b) The table gives some information about 120 similar plants in location B.

[Table_1]
[Image of Table with columns: Minimum height (cm), Lower quartile (cm), Median (cm), Interquartile range (cm), Range (cm)]

(i) On the grid opposite, draw the cumulative frequency curve for the heights of the plants in location B. [3]

(ii) Use the curves to estimate how many more plants had heights of over 70 cm in location A than in location B. .............................................................. [2]

(iii) The heights of the plants in location A are more consistent than the heights of the plants in location B.
By comparing the shapes of the curves, explain how you know this is true.
...................................................................................................................................................[1]

The diagram shows a radio in the shape of a prism.
This diagram shows the base of the radio.

\( ABC \) is an equilateral triangle.
The circles have their centres at \( A, B \) and \( C \) and each has a radius of 5 cm.
\( DE, FG \) and \( HI \) are tangents to the circles.

(a) Show that \( AB = 8.66 \text{ cm} \), correct to 3 significant figures. [3]
(b) Calculate the area of the base of the radio. .......................................... \( \text{cm}^2 \) [4]
(c) The height of the radio is 12 cm. Calculate the volume of the radio. .......................................... \( \text{cm}^3 \) [1]

The number of people living in each house in a street of 100 houses is recorded. The results are shown in the table.
[Table_1]

(a) Find

(i) the range,
............................................................ [1]

(ii) the median,
............................................................ [1]

(iii) the mean.
............................................................ [2]

(b) Two of the houses are selected at random.
Find the probability that

(i) both had exactly one person living in them,
............................................................ [2]

(ii) one had exactly 2 people living in it and the other had exactly 3 people living in it,
............................................................ [3]

(iii) at least one house had fewer than 5 people living in it.
............................................................ [2]

A is the point (-2, 6), B is the point (3, 2) and C is the point (3, -4).
(a) Write down the equation of BC. ....................................................... [1]
(b) Find the coordinates of the point M, the mid-point of AC. (..................... , ......................) [1]
(c) The quadrilateral ABCD has rotational symmetry of order 2 about the point M.
Find the coordinates of the point D. (..................... , ......................) [2]
(d) Find the equation of the perpendicular bisector of AC. ....................................................... [4]

In this question, all lengths are in centimetres.



The areas of the two triangles are equal.

(a) Show that $8x^2 + 18x - 5 = 0$. [5]

(b) Solve $8x^2 + 18x - 5 = 0$. You must show all your working.

$x = \text{..................} \text{ or } x = \text{..................}$ [3]

(c) Find the area of each of the triangles.

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

The diagram shows the positions of three ports, $A$, $B$ and $C$.

(a) Calculate $BC$.

$BC = \text{........................................} \text{ km}$ [3]

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

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

(c) The bearing of $C$ from $A$ is $130^\circ$.
Find the bearing of $B$ from $C$.

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

(d) A ship leaves $B$ at 13 50 and sails in a straight line towards $C$.
Its constant speed is $37\text{ km/h}$.

Find the time when it is at its closest point to $A$.
Give your answer correct to the nearest minute.

................................................. [5]

Given functions: \( f(x) = 2x + 3 \) and \( g(x) = 5 - 3x \)
(a) Find \( f(4) \).
................................. [1]
(b) Solve \( f(x) - g(x) = 5 \).
\( x = .............................................. \) [2]
(c) Find \( g^{-1}(x) \).
\( g^{-1}(x) = ............................................ \) [2]
(d) Find and simplify \( f(g(x)) \).
............................................. [2]
(e) Simplify \( \frac{2}{f(x)} + \frac{3}{g(x)} \).
.............................................. [3]