Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2019 Summer Zone 2

Louis and Maria share $50 in the ratio $11 : 14$.
(a) Show that Louis receives $22.
[1]

(b) Louis and Maria each spend $6 from their share of the $50.
Find the new ratio Louis' money : Maria’s money.
....................... : ....................... [2]

(c) Louis spends $\frac{17}{32}$ of his \text{remaining} money to buy a bus ticket.
Calculate the cost of the bus ticket.
$ ................................................... [1]

(d) In a sale, a bookshop reduces the price of each book by $10\%$.
Maria buys two of these books.
(i) The first book Maria buys has an original price of $\$6$.
Calculate how much Maria pays for this book.
$ ................................................... [2]

(ii) Maria pays $\$3.69$ for her second book.
Calculate the original price of this book.
$ ................................................... [3]

(a) On the diagram, sketch the graph of $y = \log\left(\frac{x+1}{x}\right)$ for $0 < x \leq 5$. [2]
(b) Write down the equations of the asymptotes to the graph of $y = \log\left(\frac{x+1}{x}\right)$.
...........................................................
........................................................... [2]
(c) Solve the equation $\log\left(\frac{x+1}{x}\right) = 0.5$.
$$x = \text{..................................................}$$ [1]
(d) On the same diagram, sketch the graph of $y = \frac{x}{2}$ for $0 < x \leq 5$. [1]
(e) Solve the equation $\log\left(\frac{x+1}{x}\right) = \frac{x}{2}$.
$$x = \text{...................................................}$$ [1]
(f) On your diagram, shade the region where $y \leq 0.5$, $y \geq \frac{x}{2}$ and $y \geq \log\left(\frac{x+1}{x}\right)$. [1]

Jono walks to school when the weather is fine. When the weather is not fine, Jono takes the bus. If Jono walks to school, the probability that he is late is 0.2. If Jono takes the bus, the probability that he is late is 0.05. On any day, the probability that the weather is fine is 0.7.

(a) Complete the tree diagram.



[3]

(b) (i) Find the probability that, on any day, Jono is late.

[3]

(ii) Jono attends school on 200 days.
Find the expected number of days that Jono is late.

[1]

The diagram shows a solid made from a cylinder and two hemispheres. The radius of the cylinder and each hemisphere is 3 cm. The total volume of the solid is $144\pi \text{ cm}^3$.

(a) The length of the cylinder is $l \text{ cm}$.

Find the value of $l$.


$$l = \text{..............................................}$$ [3]

(b) The solid is made of steel. $1 \text{ cm}^3$ of steel has a mass of $7.8 \text{ g}$.

Calculate the mass of the solid. Give your answer in kilograms. $$\text{................................................... kg}$$ [2]

(c) The solid is melted down and made into 20 cubes each of side length $2.8 \text{ cm}$.

Calculate the volume of steel not used for the cubes as a percentage of the $144\pi \text{ cm}^3$. $$\text{......................................................\%}$$ [3]

(d) A solid that is mathematically similar to the original solid has a volume of $18\pi \text{ cm}^3$.

Find the radius of the new cylinder. $$\text{.................................................... cm}$$ [3]

(a) Karl invests $200 at a rate of 1.5% per year simple interest.
Calculate the value of Karl’s investment at the end of 8 years.

$ \text{....................................................} \,[3]

(b) Lena invests $200 at a rate of 1.4% per year compound interest.
Calculate the value of Lena’s investment at the end of 8 years.

$ \text{....................................................} \,[3]

(c) The rates of interest remain the same as in part (a) and part (b).
Find how many extit{more} complete years it will take for the value of Lena’s investment to be greater than the value of Karl’s investment.

\text{....................................................} \,[2]

(a) Reflect shape $P$ in the $y$-axis. [1]
(b) Translate shape $P$ by the vector \( \begin{pmatrix} 6 \\ -3 \end{pmatrix} \). [2]
(c) Describe fully the \textit{single} transformation that maps shape $P$ onto shape $Q$.
..........................................................................................................................
.......................................................................................................................... [3]
(d) Stretch shape $P$ with stretch factor 2 and the $x$-axis invariant. [2]

A stone is thrown vertically upwards from ground level. Its height, $h$ metres above ground level, after $t$ seconds, is given by $h = 20t - 4.9t^2$.
(a) Find the height of the stone after 1 second. ..................................................... m [1]
(b) (i) On the diagram, sketch the graph of $h = 20t - 4.9t^2$ for $0 \leq t \leq 4.5$. [2]
(ii) Complete the statement. The maximum height reached by the stone is ..............................m when $t = .............................s.$ [2]
(iii) Find the length of time the stone is in the air before it hits the ground. ..................................................... s [1]
(iv) Find the length of time the stone is more than 18 m above ground level. ..................................................... s [3]

Find the $n^{th}$ term of each sequence.
(a) $7,\ 14,\ 21,\ 28,\ \ldots$ ............................................................ [1]
(b) $10,\ 7,\ 4,\ 1,\ \ldots$ ............................................................ [2]
(c) $8,\ 16,\ 32,\ 64,\ \ldots$ ............................................................ [2]
(d) $2,\ 6,\ 12,\ 20,\ \ldots$ ............................................................ [2]

240 students take part in a charity run.
The table shows information about the times, $t$ minutes, taken to complete the run.

[Table]
| Time ($t$ minutes) | $20 < t \leq 40$ | $40 < t \leq 50$ | $50 < t \leq 55$ | $55 < t \leq 75$ |
|-----------------|------------------|------------------|-------------------|-------------------|
| Number of students | 20 | 70 | 120 | 30 |

(a) Write down the time interval that contains the median.
.................... $< t \leq$ .................... [1]

(b) Calculate an estimate of the mean.
....................................... min [2]

(c) Complete the histogram to show the information in the table.
[4]

(d) (i) One of the 240 students is chosen at random.
Find the probability that this student took more than 55 minutes to complete the run.
............................................. [1]

(ii) Two students are chosen at random from the 240 students.
Calculate the probability that they both took more than 50 minutes.
............................................ [2]

(iii) Two students are chosen at random from the 240 students.
Complete the statement.
The probability that they both had times in the interval .................. $< t \leq$ ................. is $\frac{161}{1912}$.
[2]

(a) Amy buys 3 pencils and 1 ruler and pays 67 cents.
Ben buys 2 pencils and 3 rulers and pays 96 cents.

Find the cost of 1 pencil and the cost of 1 ruler. You must show all your working.

Pencil ........................................ cents
Ruler ........................................ cents [5]

(b) In this part, all measurements are in centimetres.



The area of the triangle is the same as the area of the rectangle.

(i) Show that $3x^2 - 10x - 48 = 0$.

(ii) Factorise $3x^2 - 10x - 48$. [2]

(iii) Find the area of the triangle. [2]

The diagram shows two fields on horizontal ground.
A is due south of D and C is due east of D.
(a) Calculate $DC$.
DC = ........................................ m [3]
(b) Calculate $AB$.
AB = ........................................ m [3]
(c) Calculate the total area of the fields.
........................................ $m^2$ [3]
(d) Calculate the bearing of A from B.
........................................ [4]

f(x) = 10 - x, g(x) = x^2 + 1, h(x) = \frac{1}{x}, j(x) = \log_{3}x.
(a) Find \ g(3). ..................................................... [1]
(b) Find \ f(h(2)). .................................................. [2]
(c) Find \ g(f(x)) \ in \ the \ form \ ax^2 + bx + c. .................................................. [3]
(d) For \ some \ functions, \ p^{-1}(x) = p(x).
Write \ down \ which \ two \ functions, \ f(x), \ g(x), \ h(x) \ or \ j(x), \ have \ this \ property. ...................... \ and \ ....................... [2]
(e) Write \ h(x) - \frac{1}{f(x)} \ as \ a \ single \ fraction \ in \ its \ simplest \ form. .................................................. [3]

(f) (i) \ Find \ j(243). ..................................................... [1]
(ii) \ Find \ x \ when \ j(x) = 1.5 .
\ x = ..................................................... [1]
(iii) \ Find \ j^{-1}(x).
j^{-1}(x)= ..................................................... [2]