Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2024 Winter Zone 3

The table shows the heights of 100 sunflower plants.

[Table_1]

(a) Calculate an estimate for the mean height of the sunflower plants. ........................................ cm [2]

(b) Complete the cumulative frequency table for the heights of the sunflower plants.

[Table_2]

(c) On the grid, draw a cumulative frequency curve to show this information.

[3]

(d) Use your cumulative frequency curve to estimate the number of sunflower plants that are more than 180 cm in height. ........................................ [2]

(a) Work out 24% of $15.50.
$\text{..................................................}$

(b) The price of a bookcase is $123.
This price is increased by 7%.
Calculate the new price.
$\text{..................................................}$

(c) An amount of money is shared between Ali, Kat, and Lena in the ratio $5 : 3 : 4$.
Lena’s share is $76.
Work out the total amount of money
$\text{..................................................}$

(d) A library has 32 800 books.
Each year the number of books in the library increases by 300.
Calculate the number of years it takes until there are 40 000 books in the library.
\text{..................................................}$

(e) A different library has 32 695 books at the end of 2024.
Each year the number of books increases by 0.6% of the number of books in the library at the end of the previous year.
(i) Calculate the number of books the library had at the end of 2023.
\text{..................................................}$
(ii) Calculate the number of complete years from 2024 that it takes for the number of books to first be greater than 40 000.
\text{..................................................}$

(a) $AC$ is a straight line. $B$ is the mid-point of $AC$. $A$ is the point $(-1, 11)$ and $B$ is the point $(3, 8)$.
(i) Find the length of $AB$. .......................................................... [3]

(ii) Find the coordinates of $C$. $( ext{........................} , ext{........................})$ [2]

(iii) Find the equation of the perpendicular bisector of $AC$. .......................................................... [4]

(b) $PQR$ is a straight line. $P$ is the point $(-6, -1)$ and $Q$ is the point $(-3, 1)$. $Q$ divides the line $PR$ in the ratio $PQ : QR = 1 : 2$.
Find the coordinates of $R$. $( ext{........................} , ext{........................})$ [2]

Given the functions $f(x) = x^3 - 4x + 2$ and $g(x) = \frac{3}{x} + x^2$:
(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $-3$ and $3$. [2]
(b) Find the solutions of $f(x) = 0$.
$x = \text{....................}, \, x = \text{....................}, \, x = \text{....................}$ [3]
(c) On the diagram, sketch the graph of $y = g(x)$ for values of $x$ between $-3$ and $3$. [3]
(d) Write down the equation of the asymptote of the graph of $y = g(x)$.
\[ \text{........................................................} \] [1]
(e) Solve $f(x) \leq g(x)$.
\[ \text{...................................................................................................................} \] [4]

(a) Describe fully the \textit{single} transformation that is the inverse of an enlargement with scale factor 3 and centre (2, 2).
........................................................................................................................................................................... [2]

(b)
(i) Draw the image of shape $A$ after a rotation of 90$^{\circ}$ anticlockwise with centre (0, 0). [2]

(ii) Draw the image of shape $A$ after a translation with vector $\begin{pmatrix} -5 \\ -4 \end{pmatrix}$ followed by an enlargement with scale factor $-2$ and centre (0, 0). [4]

(c)
Describe fully the \textit{single} transformation that maps shape $A$ onto shape $B$.
........................................................................................................................................................................... [3]

f(x) = 5x - 1, g(x) = x^2 + x, h(x) = (x - 1)^3
The domain for all three functions is x > 2.
(a) Find f(3). .................................................... [1]
(b) Find the range of f(x). .................................................... [1]
(c) Find g(f(4)). .................................................... [2]
(d) Find $h^{-1}(x)$.
h^{-1}(x) = .................................................... [2]
(e) Simplify fully.
$$\frac{10h(x)}{f(x) - 4}$$ .................................................... [3]

(a) Find the next term and the $n$th term for each of these sequences.

(i) 19 16 11 4

next term = ............................................
$n$th term = ............................................ [3]

(ii) 20 10 5 2.5

next term = ............................................
$n$th term = ............................................ [3]

(b) The $n$th term of a sequence is $2n^2 - 3n + 1$. The $k$th term is 465.
Work out the value of $k$.

$k = ............................................$ [3]

(a) The amount charged for electricity in one month is $E$. $E$ is the sum of a fixed charge $f$ and a cost of $d$ for each unit of electricity used.
Find a formula for the amount charged in one month when $u$ units of electricity are used.
.................................................. [2]

(b) Write as a single fraction in its simplest form.
$$\frac{x}{2} - \frac{2x}{3} + \frac{5x}{18}$$
.................................................. [2]

(c) Solve $7n - 9 > 21 + 2n$.
.................................................. [2]

(d) Solve the simultaneous equations. You must show all your working.
$$2x + 15y = -57$$
$$20x + 3y = 18$$
$x = $ ..................................................
$y = $ .................................................. [3]

(e) $y$ is proportional to the square of $(x - 3)$. $y = 5$ when $x = 7$.
Find the value of $y$ when $x = 27$.
$y = $ .................................................. [3]

Triangle $ABC$ is isosceles with $AB = BC$.
(a) Show that $AC = 13.5\text{ cm}$ correct to 3 significant figures. [3]
(b) Calculate the length $AB$. ............................................ cm [3]
(c) Find the area of $ABCD$. .......................................... $\text{cm}^2$ [3]

(a) !(https://via.placeholder.com/150)
A metal sphere has a volume of 9203 cm$^3$. The sphere is inside a cube and touches each face of the cube.
(i) Find the volume of the cube.
........................................ cm$^3$ [4]
(ii) The sphere is melted and poured into the cube.
Find the depth of the metal.
........................................ cm [2]

(b) !(https://via.placeholder.com/150)
$ABCDEFGH$ is a cuboid.
(i) Calculate the length $DF$.
........................................ cm [4]
(ii) Calculate the angle that the diagonal $DF$ makes with the base $EFGH$.
........................................ [2]

(a) One of these students is chosen at random.
Complete the sentence.
This student is most likely to belong in {students who like ......................................}. [1]

(b) Write down the number of students who like all three activities.
.................................................. [1]

(c) Find $n((S' \cup C) \cap W')$.
.................................................. [1]

(d) A student is chosen at random from the class.
Find the probability that this student likes both walking and swimming.
.................................................. [1]

(e) Two of the students who like swimming are chosen at random.
Find the probability that one of these students likes walking but not cycling and the other student only likes swimming.
.................................................. [3]

(f) Three of the 32 students are chosen at random.
Find the probability that one student likes exactly one of the activities and the other two students like exactly two of the activities.
.................................................. [3]