Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2023 Winter Zone 1

(a) Find $2.40$ as a percentage of $1.60$.
............................................... % [1]

(b) Calculate $7.2\%$ of $2.5$ g.
............................................... g [2]

(c) Amir invests $$400$ at a rate of $1.8\%$ per year compound interest.
Calculate the value of this investment at the end of $6$ years.
$$$ ............................................ [2]

(d) Each year the population of a small town increases by $4\%$ of its value in the previous year. The population is now $29640$.
(i) Calculate the population last year.
............................................... [2]
(ii) Calculate the number of complete years it will take for the population of $29640$ to be first greater than $40000$.
............................................... years [4]

On the diagram, sketch the graph of $y = f(x)$ where $f(x) = \frac{1}{\sin x^\circ}$ for $0 \le x < 360$.
(a) On the diagram, sketch the graph of $y = f(x)$. \[3\]
(b) Find the coordinates of the local minimum point. ( \text{.................... , ....................} ) \[1\]
(c) Write down the equations of the three asymptotes of the graph of $y = f(x)$. \text{........................ , ........................, ........................} \[2\]
(d) The equation $f(x) = k$ has no solutions.
Write down the range of values of $k$. \text{................................................} \[2\]
(e) By sketching another graph on the diagram, solve the equation $\frac{1}{\sin x^\circ} = 5 \sin \left(\frac{x}{2}\right)^\circ$ for $0 \le x < 360$. \text{..........................................................} \[3\]

Each of 200 students records their height, $h$ cm. The results are shown on the cumulative frequency curve.

(a) Use the cumulative frequency curve to find
(i) the median
......................................... cm [1]
(ii) the interquartile range
......................................... cm [2]
(iii) the number of students with a height greater than 150 cm.
......................................... [2]
(b) Use the cumulative frequency curve to complete the frequency table.
[Table_1]
| Height (h cm) | \(120 < h \leq 150\) | \(150 < h \leq 170\) | \(170 < h \leq 180\) | \(180 < h \leq 190\) |
|--------------|-----------------|-----------------|-----------------|-----------------|
| Frequency | | | | |
[2]
(c) Use the frequency table to calculate an estimate of the mean height.
......................................... cm [2]

(a) \( \mathbf{p} = \begin{pmatrix} 3 \\ -2 \end{pmatrix} \quad \mathbf{q} = \begin{pmatrix} -5 \\ 1 \end{pmatrix} \)

(i) Work out \( \mathbf{p} + 2\mathbf{q} \).

( \quad , \quad ) \quad [2]

(ii) \( A \) is the point \( (2, 6) \) and \( B \) is the image of point \( A \) after a translation by the vector \( \mathbf{p} \).
Find the coordinates of \( B \).

( ................. , ................. ) \quad [1]

(iii) Find the magnitude of \( \mathbf{q} \).

..................................................... \quad [2]

(b) Find the vector that translates the point \( (1, 5) \) to the point \( (-1, 7) \).

( \quad ) \quad [2]

(c)

(i) Describe fully the \textit{single} transformation that maps triangle \( T \) onto triangle \( A \).

.....................................................................................................
..................................................................................................... \quad [2]

(ii) Describe fully the \textit{single} transformation that maps triangle \( T \) onto triangle \( B \).

.....................................................................................................
.....................................................................................................
..................................................................................................... \quad [3]

(iii) Reflect triangle \( T \) in the \( y \)-axis. \quad [1]

(iv) Stretch triangle \( T \) with factor 3 and invariant line \( y = 3 \). \quad [2]

f(x) = 2x - 5, g(x) = x^2 + x + 3, h(x) = x^3, j(x) = 3^x
(a) The domain of f(x) is 0 ≤ x ≤ 10. Find the range of f(x).

(b) Solve.
(i) f(x) = -2
x = .................................................. [2]
(ii) g(x) = 3 - x
x = ................. or x = ................. [3]

(c) Find g(f(4)).

(d) Find h(2) - j(2).

(e) Find h^{-1}(x).
h^{-1}(x) = ..................................................

(f) Find j^{-1}(x).
j^{-1}(x) = ..................................................

(a) Jade and Kim share $160.
Jade receives $8 more than Kim.
Find the ratio Jade’s money : Kim’s money.
Give your answer in its simplest form.
...................... : ...................... [2]

(b) Each year the height of a bush increases by $x\%$ of its height at the start of the year.
It takes 6 years for the bush to grow from 1.2 m to 1.664 m.
Find the value of $x$.
$x =$ .................................................. [3]

(c) Work out, giving each answer in standard form.
(i) $(4.5 \times 10^{85}) \times (3 \times 10^{36})$
.................................................. [2]

(ii) $(2 \times 10^7) + (2 \times 10^{n-2})$
.................................................. [2]

(a) Marcus runs for 1 hour at $x$ km/h and then walks for 2 hours at $(x - 5)$ km/h.
He travels a total distance of 14 km.

Find his running speed.

................................... km/h [3]

(b) Nina runs 5 km at $y$ km/h and then walks 7 km at $(y - 7)$ km/h.
She takes a total of 2 hours.

(i) Show that $2y^2 - 26y + 35 = 0$.

[3]

(ii) Solve $2y^2 - 26y + 35 = 0$.

$y = ..................$ or $y = ..................$ [3]

(iii) Find Nina’s walking speed.

.............................. km/h [1]

The diagram shows the sector of a circle with radius 9 cm and sector angle 140°.
(a) Calculate the length of the arc $PQ$. ............................................... cm [2]
(b) Calculate the area of the sector. ............................................... cm$^2$ [2]
(c) The sector is the cross-section of a solid of length 20 cm.
Calculate the total surface area of the solid. ............................................... cm$^2$ [4]
(d) Another solid is mathematically similar to the solid in part (c).
The radius of the sector in this solid is 10 cm.
Calculate the total surface area of this solid. ............................................... cm$^2$ [2]

On any day the probability that Samira cycles to school is $\frac{5}{6}$.
When Samira cycles to school the probability that she arrives on time is $\frac{4}{5}$.
When Samira does not cycle to school the probability that she arrives on time is $\frac{2}{5}$.
(a) Find the number of days Samira is expected to cycle to school in a school term of 54 days.
................................................................. [1]
(b) Complete the tree diagram.

Cycles to school Arrives on time
        $\frac{4}{5}$
        Yes
 $\frac{5}{6}$
                                        Yes
                                        ..........
                                           No
    ..........
                                           Yes
                                           ..........
                                        No
        ..........
................................................................. [2]
(c) Calculate the probability that on any day Samira arrives at school on time.
................................................................. [3]
(d) In a school week of 5 days, find the probability that Samira cycles to school on exactly 1 day.
................................................................. [3]

(a) Simplify.
(i) \( \frac{k}{2p} \times \frac{t}{3} \) ............................................... [1]
(ii) \( \frac{u}{7} + \frac{2u}{21} \) ............................................... [2]
(b) Simplify.
\( \frac{x^2 - x - 42}{2x^2 - 98} \) ............................................... [4]
(c) Write as a single fraction in its simplest form.
\( \frac{g-1}{g+1} - \frac{2g}{5} + 4 \) ............................................... [3]

(a) Calculate the area of triangle $ABC$.

.................................... cm$^2$ [2]

(b) Calculate the shortest distance from $C$ to $AB$.
.................................... cm [3]

(c) Show that $BC = 7.41$ cm correct to 2 decimal places.
[3]

(d) In triangle $ABC$, $O$ is the centre of the circle that passes through $A$, $B$ and $C$.
Calculate the radius of this circle.
.................................... cm [4]