Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2020 Summer Zone 2

A class of 40 students complete a science test. The table shows the marks of the 40 students.
[Table_1]
(a) Write down the mode. ..................................................... [1]
(b) Work out the range. ..................................................... [1]
(c) Find the median. ..................................................... [1]
(d) Find the interquartile range. ..................................................... [2]
(e) Calculate the mean. ..................................................... [2]
(f) Two of the students are chosen at random. Find the probability that the difference in their marks is 8. ..................................................... [3]

(a)
(i) Describe fully the single transformation that maps triangle $A$ onto triangle $B$.
..................................................................................................................................................
................................................................................................................................................ [2]
(ii) Describe fully the single transformation that maps triangle $A$ onto triangle $C$.
..................................................................................................................................................
................................................................................................................................................ [3]

(b) You may use the grid to help you in answering this question.

The transformation $P$ is a rotation of $90^\circ$ clockwise about the origin.
The transformation $Q$ is a reflection in the line $y = -x$.
(i) Find the image of the point $(5, -2)$ under the transformation $P$.
( .................... , .................... ) [1]
(ii) Find the image of the point $(5, -2)$ under the transformation $Q$.
( .................... , .................... ) [1]
(iii) Describe fully the single transformation equivalent to $P$ followed by $Q$.
..................................................................................................................................................
................................................................................................................................................ [2]
(iv) Describe fully the single transformation equivalent to $Q$ followed by $P$.
..................................................................................................................................................
................................................................................................................................................ [2]

Petra is a singer. She wants to estimate how much to spend on advertising. The table shows the amount spent on advertising, $x$, and the number of tickets sold, $y$, for 10 performances.

[Table_1]

(a) (i) Complete the scatter diagram. The first six points have been plotted for you.



(ii) What type of correlation is shown by the scatter diagram? ............................................................. [1]

(b) Find the mean amount of money spent on advertising. $ \text{...................................................} $ [1]

(c) (i) Find the equation of the regression line for $y$ in terms of $x$.
$ y = \text{................................................} $ [2]

(ii) Use your regression line to estimate the number of tickets sold when Petra spends $130 on advertising.
............................................................. [1]

(iii) Explain why Petra should not rely on this regression line to estimate the number of tickets she will sell if she spends $500 on advertising.
.......................................................................................................................................................... .......................................................................................................................................................... [1]

A piece of metal is in the shape of a cuboid. The cuboid has length 18 cm, width 12 cm and height 12 cm. A cylinder is removed from the cuboid. The cylinder has length 18 cm and radius 4 cm.
(a) (i) Find the volume of the metal remaining after the cylinder has been removed. .......................................... cm^3 [3]
(ii) Write your answer to part (i) in standard form. .......................................... cm^3 [1]
(b) Find the total surface area of the metal remaining after the cylinder has been removed. .......................................... cm^2 [4]
(c) The cylinder removed is melted and formed into 16 identical spheres.
(i) Calculate the volume of one sphere. .......................................... cm^3 [1]
(ii) Calculate the radius of one sphere. .......................................... cm [2]

Fifty students, 25 boys and 25 girls, were asked which sport they prefer. The results are shown in the table.
[Table_1]

(a) A student is selected at random.
Calculate the probability that the student chosen is

(i) a girl who prefers swimming,
..................................................... [1]

(ii) a boy who does not prefer football,
..................................................... [1]

(iii) a student who prefers athletics.
..................................................... [1]

(b) Two of the girls are chosen at random.
Calculate the probability they both prefer tennis.
..................................................... [2]

(c) Two of the students who prefer athletics are chosen at random.
Calculate the probability that one is a boy and one is a girl.
..................................................... [3]

(d) Three of the 50 students are chosen at random.
Calculate the probability that one is a boy and two are girls and they all prefer swimming.
..................................................... [4]

Herman bought a motorbike on 1 January 2014. By 1 January 2015 the value of the motorbike had reduced by 16%. By 1 January 2016 the value of the motorbike had reduced by 12% of the value on 1 January 2015. The value of the motorbike on 1 January 2016 was $7392.

(a) Find how much Herman paid for the motorbike.
$ \text{.............................................................} \hspace{5mm} [3]

(b) From 2016, the value of the motorbike reduced by 8% each year.
Calculate the number of complete years it will take for the value of the motorbike to decrease from $7392 to $5000.
\text{.............................................................} \hspace{5mm} [4]

(a) $f(x) = 2 + \frac{1}{x + 2}$
(i) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $-6$ and $2$. [2]
(ii) Write down the coordinates of the points where the graph crosses the axes.
$(............. , .............)$ and $(............. , .............)$ [2]
(iii) Write down the equations of the asymptotes of the graph.
............................. , ............................. [2]
(b) $g(x) = (x + 4)^2$
On the diagram, sketch the graph of $y = g(x)$ for $-6 \leq x \leq -1$. [2]
(c) Solve the equation.
$f(x) = g(x)$
............................................................................................. [3]
(d) Solve the inequality.
$f(x) \geq g(x)$
............................................................................................. [2]

The diagram shows four points $A$, $B$, $C$ and $D$ on horizontal ground.
$B$ is due North of $C$ and $C$ is due East of $A$.
(a) Find the bearing of
 (i) $D$ from $A$, .......................................................... [1]
 (ii) $A$ from $D$. .......................................................... [1]
(b) Calculate angle $ABC$.
Angle $ABC = .........................................................$ [2]
(c) Calculate the area of quadrilateral $ABCD$. ...................................................... km$^2$ [3]
(d) Calculate $CD$.
$CD = .........................................$ km [3]
(e) Angle $ACD$ is acute.
Find the bearing of $D$ from $C$.
......................................................... [4]

For the functions \(f(x) = 4 - 3x\), \(g(x) = \frac{1}{x-1}\), \(x \neq 1\), and \(h(x) = x^2\):

(a) Find
(i) \(f(2)\).
................................................... [1]

(ii) \(f(g(4))\).
................................................... [2]

(b) Find \(g(g(-1))\).
................................................... [2]

(c) Solve \(h(f(x)) = 9\).

\(x = \text{.....................} \text{ or } x = \text{..................} \) [3]

(d) Find \((f(x))^2 - 1\) in terms of \(x\). Give your answer in the form \(k(ax + b)(cx + d)\) where \(a, b, c, d\) and \(k\) are integers.
................................................... [3]

The diagram shows a vertical pole $CD$. $ABC$ is a straight line on level ground.
Find $DC$.


$DC = \text{..........................}$ m [6]

(a) Solve the equations.

(i) \( 5 + 2x = 1 \)

\( x = \text{.......................................} \) [2]

(ii) \( 6 - \frac{10}{x} = 1 \)

\( x = \text{.......................................} \) [2]

(iii) \( 3(1 - 2x) = 2 - 4(x - 7) \)

\( x = \text{.......................................} \) [3]

(b) (i) Solve \( 6x^2 = 7 - 3x \).

Give your answers correct to 3 decimal places.
You must show all your working.

\( x = \text{..................... or } x = \text{.....................} \) [4]

(ii) Solve \( 6y^4 = 7 - 3y^2 \).

Give your answers correct to 3 decimal places.

\( y = \text{..................... or } y = \text{.....................} \) [2]

(c) Solve \( 2 \log x + \log 5 = 1 \).

\( x = \text{.......................................} \) [4]