Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2022 Summer Zone 2

(a) Describe fully the \textit{single} transformation that maps triangle \( A \) onto triangle \( B \).
......................................................................................................
...................................................................................................... [2]

(b) Rotate triangle \( B \) through \( 90^{\circ} \) clockwise with centre of rotation \((1, 0)\). Draw this triangle and label it \( C \). [2]

(c) Describe fully the \textit{single} transformation that maps triangle \( C \) onto triangle \( A \).
......................................................................................................
...................................................................................................... [2]

The number of hours, $x$, spent revising and the mark scored, $y$, in an examination for each of 10 students are shown in the table.
[Table_1]

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


(a) (ii) Write down the type of correlation shown by the scatter diagram.
..........................................................

(b) Find the mean mark.
..........................................................

(c) (i) Find the equation of the regression line for $y$ in terms of $x$. Give your answer in the form $y = mx + c$.
$y = ..........................................................$

(c) (ii) The value for $m$ represents a connection between time and mark. Write down the units of $m$.
..........................................................

(d) Use your answer to part (c)(i) to estimate
(i) the mark scored for a student who revised for 10 hours,
..........................................................
(ii) the number of hours spent revising for a student to score a mark of 36.
..........................................................

A, B, C, and D lie on a circle centre O. PQA is a tangent to the circle. QBC and PBOD are straight lines.

Angle $BQA = x^\circ$ and angle $ODA = (x - 30)^\circ$.
Find, in terms of x, expressions for each of the following angles. Give each answer in its simplest form.
(a) angle $BOA$

Angle $BOA = \text{.............................................}$ [1]
(b) angle $QBO$

Angle $QBO = \text{.............................................}$ [3]
(c) angle $CDB$

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

(a) On the diagram, sketch the graph of $y = f(x)$, where
$f(x) = 4 - |2x|$ for values of $x$ between $-4$ and $4$. [3]
(b) Write down the $x$-coordinates of the points where the graph meets the $x$-axis.
$x = \text{................}$ and $x = \text{................}$ [1]
(c) On the diagram, sketch the graph of $y = g(x)$, where
$g(x) = 0.25x^2$ for values of $x$ between $-4$ and $4$. [2]
(d) Write down the equation of the line of symmetry of the graph of $y = g(x)$.
$\text{.........................................................}$ [1]
(e) Find the value of the $x$-coordinate of each point of intersection of the two graphs.
$x = \text{................}$ and $x = \text{................}$ [2]
(f) On your diagram shade the region defined by
$f(x) \geq g(x)$. [1]

(a) Alenia, Bob and Cara share some money in the ratio $5 : 3 : 4$. Alenia's share is $1240$.
(i) Show that Bob's share is $744$.

[1]

(ii) Cara spends $x$ from her share. The ratio of Bob's money : Cara's money is now $4 : 3$. Find the value of $x$. $x =$ .................................................. [3]

(b) A shop has a sale and all prices are reduced by $20\%$.
(i) Bob buys a coat. The original price of the coat was $92$. Work out the sale price of the coat. $\$ \text{..................................................}$ [2]

(ii) Cara buys a jacket in the sale for $132$. Work out the original price of the jacket. $\$ \text{.................................................}$ [2]

(c) On 1 January 2022 Alenia buys a scooter for $1240$. On 1 January 2023 the value of the scooter is reduced by $18\%$. On 1 January 2024 the value of the scooter is reduced by $12\%$ of its 1 January 2023 value.
(i) Calculate the value of the scooter on 1 January 2024. $..................................................$ [3]

(ii) After 1 January 2024, the value of the scooter is reduced by $12\%$ each year. Find the year in which the value of the scooter on 1 January will first be below $310$. .................................................. [4]

The lifetimes, $x$ hours, of 80 electric light bulbs are shown in the table.

[Table_1]

(a) Calculate an estimate of the mean lifetime.

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

(b) Complete the cumulative frequency table.

[Table_2]

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



(d) Use your graph in part (c) to find an estimate for
(i) the median lifetime,
..................................................... h [1]
(ii) the interquartile range.
..................................................... h [2]

(e) Find the percentage of bulbs that have a lifetime of more than 900 hours.
..................................................... % [2]

The diagram shows four points $A$, $B$, $C$ and $D$.
$B$ is due north of $C$ and $C$ is due east of $A$.
$AC = 420\,m$, $AD = 750\,m$, $BC = 535\,m$ and angle $CAD = 28^{\circ}$.

(a) Find the bearing of

(i) $D$ from $A$,

................................... [1]

(ii) $A$ from $D$.

................................... [1]

(b) Calculate $AB$.

$AB$ = ................................... $m$ [2]

(c) Calculate $CD$.

$CD$ = ................................... $m$ [3]

(d) Calculate the area of quadrilateral $ABCD$.

......................................... $m^2$ [3]

(e) Angle $ACD$ is obtuse.
Find the bearing of $D$ from $C$.

................................... [4]

The Venn diagram shows the sets $A$, $B$ and $C$.



$U = \{\text{integers from 10 to 20, including 10 and 20}\}$
$A = \{\text{prime numbers}\}$
$B = \{\text{multiples of 3}\}$
$C = \{\text{multiples of 4}\}$

(a) List the elements of set $A$.
.................................................. [1]

(b) Write all the elements of $U$ in the correct parts of the Venn diagram. [2]

(c) List the elements of $(A \cup B)'$.
................................................................................... [1]

(d) Find $n((A \cup B) \cap C')$.
................................................. [1]

Find the next term and the $n\text{th}$ term in each of the following sequences.
(a) 100,\ 91,\ 82,\ 73,\ 64,\ \ldots
Next term = \text{..............................................}
$n\text{th}$ term = \text{..........................................}\ [3]
(b) 64,\ -32,\ 16,\ -8,\ 4,\ \ldots
Next term = \text{..............................................}
$n\text{th}$ term = \text{..........................................}\ [3]
(c) -1,\ 8,\ 21,\ 38,\ 59,\ \ldots
Next term = \text{..............................................}
$n\text{th}$ term = \text{..........................................}\ [3]

(a) $P = \frac{(x+3)y}{5}$
Work out the value of $P$ when $x = -18$ and $y = 28$.
$P = \text{..............................................}$

(b) Simplify fully.
$\frac{5y}{2x} \times \frac{4x}{3}$
$\text{..............................................}$

(c) Factorise fully.
(i) $15ab - 25bc$
$\text{..............................................}$
(ii) $6x^2y^5 - 16x^3y^3$
$\text{..............................................}$
(iii) $6cd - 3 - 9d + 2c$
$\text{..............................................}$

(d) Make $x$ the subject of the formula.
$3ax = 1 - \frac{2x}{a+2}$
$x = \text{..............................................}$

(e) Solve the inequality.
$\frac{3-x}{2+x} > 1$
$\text{..............................................}$

(a) A pyramid has a square base with sides of length 9 cm and vertical height $h$ cm.
Find an expression, in terms of $h$, for the volume of the pyramid.
........................................ cm³

(b) \[\begin{array}{c} \text{} \end{array}\]
$ADE$ is an isosceles triangle.
$BC$ is parallel to $DE$, $BC = a$ cm and $DE = 9$ cm.
The vertical height of triangle $ADE$ is $h$ cm and the vertical height of triangle $ABC$ is 10 cm.
Show that $a = \frac{90}{h}$

(c) A square-based pyramid with base of side 9 cm and vertical height $h$ cm contains some water.
When the pyramid is placed on level ground the surface of the water is 10 cm below the vertex of the pyramid (see Diagram 1).
When the pyramid stands vertically on its vertex, the surface of the water is 1 cm below the base of the pyramid (see Diagram 2).
\[ \begin{array}{cc} \text{} & \text{} \end{array} \]
(i) Use Diagram 1 to find an expression, in terms of $a$ and $h$, for the volume of the water.
........................................ cm³
(ii) Use Diagram 2 to find an expression, in terms of $b$ and $h$, for the volume of the water.
........................................ cm³
(iii) Show that $h^3 - 1000 = (h-1)^3$.
(iv) The equation $h^3 - 1000 = (h-1)^3$ simplifies to $h^2 - h - 333 = 0$.
Use a graphical method to find the value of $h$.
$h = ..................................................$

A bag contains 7 red balls, 4 blue balls and 1 green ball.
In an experiment, three balls are chosen at random without replacement.
(a) Find the probability that the three balls chosen are
(i) all green, .......................................... [1]
(ii) all red, .......................................... [2]
(iii) two red and one blue. .......................................... [3]
(b) This experiment is to be carried out 2640 times.
Use your answer from part (a)(ii) to find the expected frequency of 3 red balls being chosen.
.......................................... [1]