Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2024 Winter Zone 1

(a) Solve the equations.
(i) \( 6x + 5 = -19 \)
\( x = \text{............................} \) \ [2\]

(ii) \( 8x - 13 = 11 - 4x \)
\( x = \text{............................} \) \ [2\]

(iii) \( \frac{8}{2x - 3} = -5 \)
\( x = \text{............................} \) \ [3\]

(b) Solve the equation \( 6x^2 - 2x - 1 = 0 \).
Give your answers correct to 2 decimal places.
You must show all your working.
\( x = \text{.............. or } x = \text{..............} \) \ [3\]

(a) Younous wants to calculate $\frac{78.8}{2.46^2} + \frac{153 + 9.81^2}{\sqrt{9.47}}$.

(i) He finds an estimate for the answer by rounding each number correct to 1 significant figure.
Find this estimate.
You must show all your working.
.......................................................... [2]

(ii) Explain why his answer to part (i) is greater than the actual answer.
.............................................................................................................................. [1]

(iii) Work out.
$$\frac{78.8}{2.46^2} + \frac{153 + 9.81^2}{\sqrt{9.47}}$$
.......................................................... [1]

(b) Work out $3\frac{1}{4} \times \frac{8}{39}$.
.......................................................... [1]

(c) (i) Write 506 grams in kilograms.
.......................................................... kg [1]

(ii) Write 2000 m$^2$ in km$^2$.
.......................................................... km$^2$ [1]

(d) An athlete runs 20 km in 100 minutes and then walks at 8 km/h for 50 minutes.
Find the athlete’s average speed in km/h.
.......................................................... km/h [3]

Paulo compares the fuel consumption of his car and the average speed of his car for ten journeys.
The results are shown in the table.

[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? $\text{..............................}$ [1]

(b) Find the mean fuel consumption. $\text{..................................... km/l}$ [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 fuel consumption when the average speed is $80$ km/h.
$\text{..................................... km/l}$ [1]

(iii) Paulo drives his next journey at an average speed of $30$ km/h. Give a reason why the regression line is unlikely to give a reliable estimate of the fuel consumption for this journey.
$\text{.....................................................................................}$ [1]

(a) Alan, Beth and Imran share an amount of money in the ratio $3x : 2x : (x + 1)$ where $x$ is an integer.

(i) Find the amount Beth receives when $x = 4$ and they share $400 in total.
$\text{S ....................................................}$ [3]

(ii) Find the amount that Alan receives when Beth receives $66.
$\text{S ....................................................}$ [2]

(iii) Find the value of $x$ when Alan receives 2.5 times the amount Imran receives.
$x = \text{ ....................................................}$ [2]

(b) In a sale, a shop reduces the price of all furniture by 12%.

(i) Find the sale price of a chair that has an original price of $90.
$\text{S ....................................................}$ [2]

(ii) Find the original price of a table that has a sale price of $440.
$\text{S ....................................................}$ [2]

(c) Kurt invests $X$ in a bank which pays simple interest at a rate of 4% each year. The total amount of money that Kurt has in the bank at the end of 6 years is $930.

Show that $X = 750$.

(d) Ivana invests $750 in a bank which pays compound interest at a rate of $y\%$ each year. The total amount of money that Ivana has in the bank at the end of 6 years is $921.94$.

Find the value of $y$.
$y = \text{ ....................................................}$

The area of triangle $ABC$ is $262\ \text{cm}^2$.
(a) Show that $AC = 22.0\ \text{cm}$, correct to 1 decimal place. [2]
(b) Find $BC$.
$$BC = \text{........................................ cm}$$ [3]
(c) Use the sine rule to find angle $ABC$.
Angle $ABC = \text{........................................}$ [3]
(d) Find the length of the perpendicular line from $A$ to the line $BC$.
$$\text{........................................ cm}$$ [2]

(a)
(i) Draw the image of triangle \( A \) after a reflection in the line \( y = -x \). Label the image \( B \). [2]
(ii) Draw the image of triangle \( B \) after a reflection in the \( y \)-axis. Label the image \( C \). [1]
(iii) Describe fully the single transformation that maps triangle \( C \) onto triangle \( A \).
....................................................................................................................................... [3]

(b) The transformation \( P \) is a translation with vector \( \begin{pmatrix} -1 \\ 3 \end{pmatrix} \).
The transformation \( Q \) is a stretch, factor 3 with invariant line \( y = 2 \).
(i) Describe the transformation that is the inverse of \( P \).
....................................................................................................................................... [2]
(ii) Describe the transformation that is the inverse of \( Q \).
....................................................................................................................................... [2]

(a) There are 49 students in a year group.

Each student studies at least one of the sciences, biology ($B$), chemistry ($C$) and physics ($P$).

$x$ students study all 3 sciences.
$y$ students study chemistry only.
12 students study physics only.

6 students study biology and chemistry but not physics.
11 students study biology and physics but not chemistry.
2 students study physics and chemistry but not biology.

25 students study only one science.

(i) Show this information on the Venn diagram.



[2]

(ii) Find the number of students who study all 3 sciences.

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

(iii) The number of students that study biology is two times the number of students that study chemistry.

Find the number of students who study

(a) chemistry only ............................................................ [2]

(b) biology only. ............................................................. [1]

(b) A bag contains 7 red balls and 3 blue balls.
In an experiment, three balls are chosen at random without replacement.

Find the probability that at least two of the balls chosen are red.

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

Find the next term and the nth term in each of the following sequences.

(a) \( 16, \ 9, \ 2, \ -5, \ -12, \ \ldots \)
next term = ........................................
\( nth \ term = \text{..............................................} \) \([3]\)

(b) \( 2, \ 8, \ 18, \ 32, \ 50, \ \ldots \)
next term = ........................................
\( nth \ term = \text{..............................................} \) \([3]\)

(c) \( 1, \ -3, \ 5, \ -7, \ 9, \ \ldots \)
next term = ........................................
\( nth \ term = \text{..............................................} \) \([3]\)

(d) \( 6, \ 9, \ 10, \ 9, \ 6, \ \ldots \)
next term = ........................................
\( nth \ term = \text{..............................................} \) \([3]\)

Given $f(x) = \frac{1}{(2x - 3)(2x + 1)}$:
(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $-2$ and $4$. [3]
(b) Write down the equations of the asymptotes parallel to the $y$-axis.
\[ \text{........................................................ ...} \] [2]
(c) Write down the coordinates of the local maximum.
\( ( \text{.....................} , \text{.....................} ) \) [2]
(d) The line $y = x - 2$ intersects the curve $y = f(x)$ three times.
Find the $x$-coordinate of each point of intersection.
\( x = \text{.................. or } x = \text{.................. or } x = \text{..................} \) [3]
(e) Solve the inequality $f(x) \geq x - 2$.
\[ \text{..................................................................................................................} \] [3]

In this question all lengths are in centimetres.

A solid cone has radius $r$ and vertical height $h$.
A solid hemisphere also has radius $r$.
The curved surface area of the cone is the same as the curved surface area of the hemisphere.

(a) Show that $h = r\sqrt{3}$. [4]

(b) The cone is placed directly on top of the hemisphere.

Show that the volume of this solid is $\frac{1}{3}\pi r^3 (2 + \sqrt{3})$. [2]

(c) A larger solid is mathematically similar to the solid in part (b).

The larger solid has volume $243\pi r^3 (2 + \sqrt{3})$.

(i) Find, in terms of $r$, the radius of the hemisphere of the larger solid.
................................................ [2]

(ii) The surface area of the larger solid is $5000\text{ cm}^2$.
Find the volume of this solid.
................................................. cm$^3$ [4]

[Image: A solid cone with radius r and height h next to a solid hemisphere with radius r.]

(a) Solve the equation.

$$2 \log 5 - 5 \log 2 = 3 \log 4 - 2 \log x$$

Give your answer in the form $\frac{a \sqrt{b}}{c}$, where $a, b$ and $c$ are integers.

$x = \text{.....................}$ [4]

(b) Make $x$ the subject of the formula.

$$y = \sqrt{\frac{x}{2x+1}}$$

$x = \text{.....................}$ [4]