Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2017 Summer Zone 3

(a) Translate triangle $T$ by the vector $\begin{pmatrix} -2 \\ 7 \end{pmatrix}$. [2]

(b) (i) Reflect triangle $T$ in the $x$-axis. Label the image $P$. [1]
(ii) Reflect triangle $T$ in the line $x = -1$. Label the image $Q$. [1]
(iii) Describe fully the \textbf{single} transformation that maps triangle $P$ onto triangle $Q$.
..............................................................................................................................
.............................................................................................................................. [3]

(c) Describe fully the \textbf{single} transformation that maps triangle $T$ onto triangle $U$.
..............................................................................................................................
.............................................................................................................................. [3]

(a)
In the diagram, $ABC$ is a triangle and $AB$ is parallel to $DE$.
Angle $BCA = 68^{\circ}$ and $DE = DC$.
(i) Find angle $BAC$.
Angle $BAC = \text{....................................................} \; [2]$
(ii) scalene equilateral isosceles right-angled
Choose one word from the list to complete the statement.
Triangle $ABC$ is \text{...........................................} \; [1]

(b) Calculate the interior angle of a regular 20-sided polygon.
\text{....................................................} \; [3]

(c)
In the diagram, angle $A = \text{angle} \; P$ and angle $B = \text{angle} \; Q$.
(i) Explain why angle $C = \text{angle} \; R$.
\text{.............................................................................................................................} \; [1]
(ii) $AB = 8 \text{ cm}, \; AC = 5 \text{ cm}, \; BC = 9 \text{ cm}$ and $PR = 3 \text{ cm}$.
(a) Complete the statement.
Triangle $ABC$ is \text{...........................................} to triangle $PQR$ \; [1]
(b) Calculate $QR$.
$QR = \text{...........................................} \text{ cm} \; [2]$

(a) 12 students take part in a quiz. The table shows the number of correct answers given by each student.
[Table_1]

(a) Marie has $260.50 and Luk has $208.40.
(i) Find, in its simplest form, the ratio Marie’s money : Luk’s money.
Marie’s money : Luk’s money = ..................... : ..................... [2]
(ii) Marie spends 16% of her money to buy a new coat.
Calculate the cost of the coat.
$ .......................................................... [2]
(iii) In a sale, the prices of all books are reduced by 10\%.
Luk buys a book for $11.25.
Calculate the original price of the book.
$ .......................................................... [3]
(iv) Marie invests $200 at a rate of 2\% per year simple interest.
Calculate the total value of this investment at the end of 25 years.
$ .......................................................... [3]
(v) Luk invests $190 at a rate of 2\% per year compound interest.
Calculate the value of this investment at the end of 25 years.
$ .......................................................... [3]

(b) Fredrik invests $120 at a rate of 5.7\% per year compound interest.
Calculate the number of complete years it will take until the value of this investment is first greater than $300.
.......................................................... [3]

The diagram shows a solid sphere of radius 4 cm inside a hollow cone of radius 8 cm and height 16 cm. The sphere touches the interior of the cone.
(a) Calculate the volume of the cone that is not occupied by the sphere. [3 marks]
(b) Calculate the curved surface area of the cone. [3 marks]
(c) The centre, $O$, of the sphere is directly above the vertex, $V$, of the cone.
Calculate the length $OV$. [4 marks]

f(x) = x - 5\log x
(a) On the diagram, sketch the graph of $y = f(x)$ for $0 < x \leq 10$. [2]
(b) Find the co-ordinates of the local minimum point.
( ..................... , ..................... ) [2]
(c) Find the range of $f(x)$ for the domain $1 \leq x \leq 5$.
.............................................................. [2]
(d) Solve the equation $f(x) = 2$.
$x = ..................$ or $x = ..................$ [2]
(e) Solve the inequality $f(x) < 2$.
.............................................................. [1]
(f) (i) Find $f(0.001)$, $f(0.000\,01)$ and $f(0.000\,001)$.
$f(0.001) = ..........................$, $f(0.000\,01) = ..........................$, $f(0.000\,001) = ..........................$ [1]
(ii) Complete the statement.
The y-axis is ............................................................. to the graph of $y = f(x)$. [1]

(a) Calculate $AB$.

$AB = \text{.........................................} \text{ cm} \ [3]$

(b) Calculate angle $BCD$.
Angle $BCD = \text{...........................................} \ [3]$

f(x) = x^2 + 1 \
g(x) = 3 + 2x \
h(x) = \frac{1}{x+1}, \; x \neq -1 \
(a) Find f(-3). ................................................. [1]
(b) Find the value of g(h(1)). ................................................. [2]
(c) Simplify f(g(x)) + f(x). ................................................. [3]
(d) Find \( h^{-1}(x) \). .................................................
h^{-1}(x) = ................................................. [3]
(e) Solve.
(i) \( g(x) = 1 \)
x = ................................................. [2]
(ii) \( g^{-1}(x) = 1 \)
x = ................................................. [1]

In a survey, 40 students are asked if they like football, $F$, and if they like baseball, $B$. 22 like football, 19 like baseball and 6 do not like either football or baseball.

(a) Complete the Venn diagram to show this information.


[2]

(b) How many of these students
(i) like both football and baseball, ....................................................... [1]
(ii) either like football or do not like baseball? ....................................................... [1]

(c) Find $n(F \cap B^\prime)$. ....................................................... [1]

(d) Two of these students are chosen at random.
Find the probability that they both like football. ....................................................... [2]

(e) (i) One of the 19 students who like baseball is chosen at random.
Find the probability that this student also likes football. ....................................................... [1]
(ii) Two of the 19 students who like baseball are chosen at random.
Find the probability that one likes football and one does not like football. ....................................................... [3]

(f) Another $n$ students take part in the survey.
They all like both baseball and football.
A student is then chosen at random from the $(40 + n)$ students.
The probability that a student likes both football and baseball is $\frac{5}{16}$.
Find the value of $n$.

$n =$ ....................................................... [3]

(g) On the Venn diagram, shade the region $F^\prime \cup B^\prime$. [1]

(a) The time, $t$ hours, taken by each of 200 cars to complete a journey of 200 km is recorded. The results are shown in the table.

[Table_1]

(i) Calculate an estimate of the mean.
\[ \text{..............................} \hspace{10pt} \text{h [2]} \]

(ii) On the grid, draw the histogram to show the information in the table.

[3]

(b) One car completes the 200 km journey at an average speed of $x$ km/h.
Another car completes the 200 km journey at an average speed of $(x + 10)$ km/h.
The difference between the times taken by the two cars is 20 minutes.

(i) Show that $x^2 + 10x - 6000 = 0$.

[4]

(ii) Find the time taken for the slower journey.
Give your answer in hours and minutes correct to the nearest minute.
\[ \text{................... h ..................... min [4]} \]

In the diagram, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OB} = \mathbf{b}$. $M$ is the midpoint of $AB$ and $N$ is the midpoint of $AM$.

(a) Find each of these vectors in terms of $\mathbf{a}$ and $\mathbf{b}$. Give each vector in its simplest form.

(i) $\overrightarrow{AB}$

$\overrightarrow{AB} = \text{..........................}$ [1]

(ii) $\overrightarrow{AN}$

$\overrightarrow{AN} = \text{..........................}$ [1]

(iii) $\overrightarrow{ON}$

$\overrightarrow{ON} = \text{..........................}$ [2]

(b) $O$ is the point $(0, 0)$. $\overrightarrow{OA} = \begin{bmatrix} 8 \\ 0 \end{bmatrix}$ and $\overrightarrow{OB} = \begin{bmatrix} 2 \\ 6 \end{bmatrix}$.
Find the co-ordinates of $N$.

$( \text{....................}, \text{....................} )$ [3]