Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2019 Summer Zone 1

In a sale, a shop reduces all its prices by 15%.

(a) Calculate the sale price of a television originally costing $630.

$ \text{.................................} \enspace [2]

(b) The price of a fridge in the sale is $952.
Calculate the original price.

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

(c) After one week the shop reduces the price of the television in \textbf{part (a)} by a further 5% each week until it is sold.
Calculate the number of weeks from the start of the sale until the television reaches half the original price.

\text{.................................} \enspace [4]

(a) Describe fully the \textit{single} transformation that maps triangle $A$ onto triangle $B$.
............................................................................................................................................... [2]
(b) Translate triangle $A$ by the vector \( \begin{pmatrix} 6 \\ -3 \end{pmatrix} \). [2]
(c) Triangle $A$ can be mapped onto triangle $C$ by a rotation followed by an enlargement.
(i) Use trigonometry to calculate the angle of rotation.
.......................................................... [3]
(ii) The scale factor of the enlargement is $\sqrt{a}$ where $a$ is an integer.
Find the value of $a$.
$a = \text{.........................}$ [3]

The list shows the six factors of 45.

This is a method for finding how many factors a number has.

• Write the number as the product of its prime factors in index form.
• Add one to each of the powers and multiply these numbers together.

Rani planted some seeds in her garden. After two months she measured the heights, $h$ cm, of each of 120 plants. The results are shown in the table.

[Table_1]
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Height (} h \text{ cm)} & 0 < h \leq 10 & 10 < h \leq 20 & 20 < h \leq 25 & 25 < h \leq 30 & 30 < h \leq 35 & 35 < h \leq 40 & 40 < h \leq 50 \\ \hline \text{Frequency} & 0 & 16 & 28 & 32 & 24 & 14 & 6 \\ \hline \end{array} $$

(a) Calculate an estimate of the mean height.
$..............................................$ cm [2]

(b) Draw a cumulative frequency curve for this information.
[5]

(c) Use your cumulative frequency curve to estimate
(i) the median height,
$..............................................$ cm [1]
(ii) the interquartile range,
$..............................................$ cm [2]
(iii) the number of plants with a height of more than 37 cm.
$..............................................$ [2]

(d) (i) Complete this table of frequency densities for the 120 plants.
[Table_2]
$$ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \text{Height (} h \text{ cm)} & 0 < h \leq 10 & 10 < h \leq 20 & 20 < h \leq 25 & 25 < h \leq 30 & 30 < h \leq 35 & 35 < h \leq 40 & 40 < h \leq 50 \\ \hline \text{Frequency density} & 0 & 1.6 & & & & & \\ \hline \end{array} $$ [2]
(ii) Draw a histogram to show this information.
[3]

Jian asks 60 people what their favourite type of television programme is.
These are the results.

[Table_1]

(a) Jian draws a pie chart to show these results. Calculate the sector angle for Drama. .......................................................... [2]

(b) Jian chooses one of the 60 people at random. Write down the probability that the person says Factual. .......................................................... [1]

(c) Jian chooses two of the 60 people at random.
(i) Find the probability that one of them says Drama and the other says Game Show. .......................................................... [3]
(ii) Find the probability that at least one person says Sport. .......................................................... [3]

y is inversely proportional to \sqrt{x}. When $x = 9$, $y = 6$.
(a) (i) Find an equation connecting $x$ and $y$. .......................................................... [2]

(ii) Calculate $y$ when $x = 30$. .......................................................... [1]

(iii) Calculate $x$ when $y = 15$. .......................................................... [2]

(b) For the three variables $x$, $y$ and $z$, $z$ is also proportional to $(y + 5)$. When $x = 9$, $z = 33$. Find an equation connecting $x$ and $z$. .......................................................... [2]

The vectors a and b are shown on the grids.

(a) On the grid below, draw and label the following three vectors.
2b
2a + b
a - 2b

(b) Vectors p, q, and r are drawn on this grid.
Write each of the vectors in terms of a and/or b.

p = ................................................
q = ................................................
r = ................................................

ABCD is a quadrilateral.
(a) Show that $BD = 9.22 \text{ cm}$, correct to 3 significant figures. [3]
(b) Calculate angle $ABD$.
Angle $ABD = \text{....................................}$ [3]
(c) Calculate the total area of the quadrilateral $ABCD$.
$\text{............................................cm}^2$ [3]
(d) Calculate the length of the diagonal $AC$.
$AC = \text{............................................cm}$ [3]

In this question all lengths are in centimetres.



The diagram shows a picture frame with three pictures.
The frame and the pictures are rectangles.

Each picture measures 20 cm by 15 cm.
The width of the borders between each picture and between each picture and the frame are all $x$ cm.
The total area of the frame is 2208 cm².

(a) Show that $4x^2 + 85x - 654 = 0$. [3]

(b) Solve the equation $4x^2 + 85x - 654 = 0$.
You must show all your working.
$x = \text{...................}$ or $x = \text{...................}$ [3]

(c) Find the dimensions of the picture frame.
Length $\text{.........................}$ cm
Height $\text{.............................}$ cm [2]

(a) \( f(x) = 5 - 2x \)\hspace{0.5cm} \( g(x) = 3x + 2 \)
(i) Find \( f(-3) \). ................................................ [1]
(ii) Find \( f(g(4)) \). ........................................................ [2]
(iii) Solve \( \frac{f(x)}{g(x)} = 2 \).
\( x = .......................................................... \) [3]
(iv) Find \( f^{-1}(x) \).
\( f^{-1}(x) = .................................................... \) [2]
(v) Find and simplify \( g(f(x)) \). ...................................................... [2]
(vi) Write as a single fraction in its simplest form. \( \frac{3}{f(x)} + \frac{2}{g(x)} \) ................................................... [3]

(b) The function \( h(x) \) has an inverse function \( j(x) \).
Write down, in its simplest form, \( j(h(x)) \). .......................................... [1]

\( f(x) = \frac{(x+2)}{(x-1)(x-4)} \)

(a) On the diagram, sketch the graph of \( y = f(x) \) for values of \( x \) between \(-2\) and \(7\). [3]
(b) Write down the co-ordinates of the local maximum.
\(( \text{.........................} , \text{.........................} )\) [2]
(c) Write down the equation of each of the three asymptotes.
\( \text{..............................................} , \text{..............................................} , \text{..............................................} \) [3]
(d) \( g(x) = x - 5 \)
(i) Solve the equation \( f(x) = g(x) \).
\( x = \text{..........................} \) or \( x = \text{..........................} \) or \( x = \text{..........................} \) [3]
(ii) Solve the inequality \( f(x) > g(x) \).
\( \text{.............................................................................................................} \) [3]

Here is a sequence of patterns made using identical regular hexagons.

Pattern number | 1 | 2 | 3 | 4 | 5 | 6
Number of white hexagons | 1 | 1 | 13 | 13 | |
Number of grey hexagons | 0 | 6 | 6 | 24 | |
Total number of hexagons | 1 | 7 | 19 | 37 | 61 |
(a) Complete the table for Pattern 5 and Pattern 6. [5]
(b) The $n^{th}$ term of the sequence for the total number of hexagons is $3n^2 + pn + q$.
Find the value of $p$ and the value of $q$.
$p = \text{.............................}$
$q = \text{.............................}$ [2]