Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2020 Summer Zone 1

(a)
(i) Reflect shape $T$ in the $y$-axis. [1]

(ii) Translate shape $T$ by the vector $\begin{pmatrix} -5 \\ 3 \end{pmatrix}$. [2]

(iii) Enlarge shape $T$ by scale factor 2, centre (2, 0). [2]

(b) Describe fully the single transformation that maps shape $T$ onto
(i) shape $P$,
...............................................................
............................................................... [3]

(ii) shape $Q$.
...............................................................
............................................................... [3]

(a) These are Tom's ten homework marks.
8 7 10 8 9 5 8 10 6 8
Find
(i) the range,
..................................................... [1]
(ii) the mean,
..................................................... [1]
(iii) the median,
..................................................... [1]
(iv) the upper quartile.
..................................................... [1]

(b) The mass, $m$ kg, of each of 120 parcels is recorded.
The cumulative frequency curve shows the results.

(i) Find the median.
..................................... kg [1]
(ii) Find the lower quartile.
..................................... kg [1]
(iii) Find the interquartile range.
..................................... kg [1]
(iv) Find the number of parcels with a mass of more than 3 kg.
...................................................... [2]
(v) (a) Use the cumulative frequency curve to complete the frequency table.
[Table_1]
$$\begin{array}{|c|c|c|c|c|c|}\hline \text{Mass (m kg)} & 0 < m \leq 1 & 1 < m \leq 1.5 & 1.5 < m \leq 2 & 2 < m \leq 3 & 3 < m \leq 4 \\ \hline \text{Frequency} & 30 & 30 & & & \\ \hline\end{array}$$
[3]
(b) Use the frequency table to calculate an estimate of the mean.
........................................................ kg [2]

ABCD is a parallelogram.
A is the point (3, 1), B is the point (10, 2) and D is the point (2, 3).

(a) Find the coordinates of C.

(..................... , .....................) [2]

(b) Calculate the length of $AB$.
Give your answer as a surd in its simplest form.

$AB = ................................................$ [3]

(c) The diagonals of the parallelogram meet at $X$.
Find the coordinates of $X$.

(..................... , .....................) [2]

(d) The straight line $BA$ is extended to meet the $y$-axis at $P$ and the $x$-axis at $Q$.
Find the coordinates of $P$ and the coordinates of $Q$.

$P(..................... , .....................)$
$Q(..................... , .....................)$ [5]

Find the $n^{th}$ term of each sequence.
(a) $16,\ 25,\ 36,\ 49,\ 64,\ \ldots$ ........................................................... [2]
(b) $3,\ 10,\ 29,\ 66,\ 127,\ \ldots$ ........................................................... [2]
(c) $64,\ 32,\ 16,\ 8,\ 4,\ \ldots$ ........................................................... [2]

(a) Expand the brackets and simplify.
(i) $5(2-p) - 3(3+2p)$ ............................................................. [2]
(ii) $(7g-2h)(3g+11h)$ ......................................................... [3]

(b) Factorise completely.
(i) $2x^2y^3 - 4x^3y^2$ .......................................................... [2]
(ii) $49t^2 - 9u^2$ ................................................................. [2]
(iii) $6d^2 + d - 2$ ............................................................... [2]

(a)

(i) On the diagram, sketch the graph of $y = |\log x|$ for $0 < x \leq 5$. [2]
(ii) Solve the equations.
(a) $|\log x| = 0.2$
$x = \text{.....................} \text{ or } x = \text{.....................}$ [2]
(b) $|\log x| = 1 - \frac{x}{4}$
$x = \text{.....................} \text{ or } x = \text{.....................}$ [4]

(b)

(i) On the diagram, sketch the graph of $y = \log |x|$ for values of $x$ between $-5$ and $5$. [2]
(ii) Solve the equation $\log |x| = 0.2$.
$x = \text{.....................} \text{ or } x = \text{.....................}$ [2]
(c) Write down the range of values of $x$ for which the graph of $y = |\log x|$ is the same as the graph of $y = \log |x|$.
............................................... [1]

(a) Louis invests $500 at a rate of 2.5\% per year simple interest. Calculate the total amount of interest at the end of 8 years.
$ \text{.................................} \ [2]

(b) Martha invests $500 at a rate of 2.4\% per year compound interest. Calculate the total amount of interest at the end of 8 years.
$ \text{.................................} \ [4]

(c) Naomi invests an amount of money at a rate of 2.1\% per year compound interest. Find the number of complete years it takes for the value of Naomi’s investment to double.
\text{.................................} \ [4]

(d) Oscar invests an amount of money at a rate of \(r\%\) per year compound interest. At the end of 31 years the value of Oscar’s investment is 2.5 times greater than the original amount of money. Find the value of \(r\).
\(r= \text{.................................} \) [3]

(a) When the weather is fine, the probability that Sara goes to the park is 0.9. When the weather is not fine, the probability that Sara goes to the park is 0.2.
On any day, the probability that the weather is fine is 0.7.

(i) Complete the tree diagram.


[3]

(ii) Find the probability that, on any day, Sara goes to the park.

.......................................................... [3]

(b) 30 students are asked if they like Mathematics ($M$) and if they like English ($E$). The Venn diagram shows the number of students in each subset.



(i) Find $n(M \cup E')$.

...................................................... [1]

(ii) Two students are chosen at random.
Find the probability that they both like Mathematics but not English.

...................................................... [3]

Given the function $f(x) = x^3 - 6x^2 + 8x$ for $-0.5 \leq x \leq 4.5$.
(a) On the diagram, sketch the graph of $y = f(x)$. [2]
(b) Solve the inequality $f(x) < 0$. .................................................. [3]
(c) Find the positive value of $k$ when $f(x) = k$ has two different solutions.
\[ k = \text{..................................................} \] [2]

f(x) = 2x + 3 \hspace{15px} g(x) = 5^x

\begin{align*} & (a) \text{ Find } f(g(3)). \quad \text{..................................................... [2]} \\ & (b) \text{ Find } f^{-1}(x). \quad f^{-1}(x) = \text{..................................................... [2]} \\ & (c) \text{ Find } x \text{ when } g(x) = \frac{1}{25\sqrt{5}}. \quad x = \text{..................................................... [2]} \\ & (d) \text{ Find } g^{-1}(x). \quad g^{-1}(x) = \text{..................................................... [2]} \end{align*}

(a)

Calculate the shortest distance from $B$ to $AC$.

(b)

The diagram shows a pyramid on a rectangular base $PQRS$.
The diagonals of the base meet at $M$ and $V$ is vertically above $M$.

$PQ = 8\, \text{cm}$, $QR = 6\, \text{cm}$ and $VM = h\, \text{cm}$.
The volume of the pyramid is $112\, \text{cm}^3$.

(i) Show that $h = 7$. [2]

(ii) Calculate the length of $VR$. [3]

$VR = \text{............................ cm}$ [3]

(iii) $K$ is the mid-point of $PS$ and $L$ is the mid-point of $QR$.
Calculate angle $KVL$.

Angle $KVL = \text{...............................}$ [3]