Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2021 Winter Zone 2

(a) Stella and Tomas share $200 in the ratio 11 : 14.
(i) Show that Stella receives $88.

[1]

(ii) Stella invests her $88 at a rate of 1.5\% per year simple interest.
Calculate the amount of interest Stella has at the end of 6 years.

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

(b) Urs buys some clothes in a sale.
(i) He buys a jacket for $22.
The original price of the jacket was $25.
Calculate the percentage reduction in the price of the jacket.

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

(ii) Urs buys a shirt for $13.50 .
This is the price after a reduction of 10\% of the original price.
Calculate the original price of the shirt.

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

f(x) = \sin x \text{ for } 0^\circ \leq x \leq 360^\circ
g(x) = (\sin x)^2 \text{ for } 0^\circ \leq x \leq 360^\circ

(a) On the diagram, sketch the graph of $y = f(x)$. [2]
(b) Write down the coordinates of the local minimum point on the graph of $y = f(x)$.
$\text{( ....................... , ....................... )}$ [1]
(c) Write down the period and amplitude of the graph of $y = f(x)$.
Period = $..................................................$
Amplitude = $..................................................$ [2]
(d) On the same diagram, sketch the graph of $y = g(x)$. [2]
(e) Write down the range of
(i) $f(x)$,
$..................................................$ [1]
(ii) $g(x)$.
$..................................................$ [1]
(f) On the diagram, shade the regions where $\sin x \geq (\sin x)^2$. [1]

(a) The number of members in a social media group increases exponentially at a rate of 5\% per month. At the start of the first month there are 882 members.
(i) Calculate the number of members at the end of 10 months. Give your answer correct to the nearest integer.
.................................................. [3]
(ii) Calculate the number of complete months from the start until the group has 2000 members.
.................................................. [4]
(b) The mass of a radioactive substance decreases exponentially at a rate of $r\%$ per month. At the end of 10 months, its mass has decreased from 500 g to 242 g.
Find the value of $r$.
$r = ..................................................$ [3]

The mass of each of 200 potatoes is measured. The cumulative frequency curve shows the results.
(a) (i) Write down the mass of the heaviest potato. ................................................. g [1]
(ii) Find the median. ................................................. g [1]
(iii) Find the interquartile range. ................................................. g [2]
(iv) Find the number of potatoes with a mass greater than 170 g. ................................................. [2]
(b) This frequency table also shows information about the masses of the 200 potatoes.
[Table_1]
Calculate an estimate of the mean mass. ................................................. g [2]

(a)
(i) Reflect shape $T$ in the $y$-axis. [1]
(ii) Translate shape $T$ by the vector \(\begin{pmatrix}-10\\5\end{pmatrix}\). [2]
(iii) Rotate shape $T$ through $90^\circ$ clockwise about the point $(2, 0)$. [2]
(iv) Enlarge shape $T$ with scale factor $-2$ and centre $(0, 0)$. [2]
(v) Describe fully the \textit{single} transformation that maps shape $T$ onto shape $P$. [3]

(b) $f(x) = x^2$
(i) The graph of $y = f(x)$ is mapped onto the graph of $y = g(x)$ by a translation with vector \(\begin{pmatrix}0\\2\end{pmatrix}\).
Find $g(x)$ in terms of $x$.
$g(x) = $ \text{..............................} [1]
(ii) The graph of $y = f(x)$ is mapped onto the graph of $y = h(x)$ by a stretch with factor $2$ and the $x$-axis invariant.
Find $h(x)$ in terms of $x$.
$h(x) = $ \text{..............................} [1]

(a) (i) Work out \( \begin{pmatrix} 3 \\ 5 \end{pmatrix} - 2 \begin{pmatrix} -1 \\ -2 \end{pmatrix} \).

(ii) \( A \) is the point \( (3, 5) \) and \( C \) is the point \( (4, 3) \).
Find the column vector that maps the point \( A \) onto the point \( C \).

(iii) \( D \) is the point \( (1, 3) \) and the vector from \( D \) to \( E \) is \( \begin{pmatrix} 3 \\ 2 \end{pmatrix} \).
Find the coordinates of \( E \).

(iv) Find the magnitude of the vector \( \begin{pmatrix} -3 \\ -4 \end{pmatrix} \).

(b) (i) \( P \) is the point \( (-1, 6) \) and \( Q \) is the point \( (3, 4) \).
Find the equation of the perpendicular bisector of the line \( PQ \).

(ii) Find the coordinates of the point where the perpendicular bisector in part(b)(i) crosses the \( x \)-axis.

(a) The cost of a newspaper is $p$.
The cost of a magazine is $m$.

The total cost of 3 newspapers and 5 magazines is $13.30.
The total cost of 1 newspaper and 7 magazines is $15.90.

Find the value of $p$ and the value of $m$.

$p = \text{..................................................}$
$m = \text{..................................................}$ [5]

(b) \[ \text{} \]
The area of the rectangle is equal to the area of the square.

Find the value of $x$.

$x = \text{.................................................}$ [7]

(a) $f(x) = 3x - 2$ \quad $g(x) = 5x - 1$ \quad $h(x) = \frac{1}{x + 1}$ , \; $x \neq -1$
(i) Find
(a) $f(3)$,
\[ \text{............................................... [1]} \]
(b) $h(f(3))$.
\[ \text{............................................... [1]} \]
(ii) Find $f(g(x))$ in its simplest form.
\[ \text{............................................... [2]} \]
(iii) Solve \; $f(x) = g(x)$.
$x = \text{...........................................}$ \; [2]
(iv) Find \;$g^{-1}(x)$.
$g^{-1}(x) = \text{...........................................}$ \; [2]
(v) Simplify \;$2h(x) + h(x + 1)$.
Give your answer as a single fraction, in terms of $x$, in its simplest form.
\[ \text{............................................... [4]} \]

(b) $j(x) = 5^x$
(i) Find the value of $x$ when \;$j(x) = \frac{1}{5 \sqrt{5}}$.
$x = \text{............................................... [1]}$
(ii) Find \;$j^{-1}(x)$.
$j^{-1}(x) = \text{............................................... [2]}$

(a) Complete the table for each sequence.

| Sequence | 1st term | 2nd term | 3rd term | 4th term | 5th term | nth term |
|----------|----------|----------|----------|----------|----------|----------|
| A | 7 | 5 | 3 | 1 | | |
| B | 16 | 25 | 36 | 49 | | |
| C | \( \frac{1}{2} \) | 1 | 2 | 4 | | |

(b) $ y \propto \frac{1}{\sqrt{x}} $ and $ z \propto y^{3} $.
When $ x = 36 $, $ y = 2 $ and $ z = 24 $.
Find $ z $ in terms of $ x $.

$ z = \text{......................................} $

Fast trains and slow trains travel from City A to City B.
40% of the trains from City A to City B are fast trains.

The probability that a fast train arrives in City B on time is 0.9.
The probability that a slow train arrives in City B on time is 0.95.

Manuela goes to the station in City A and takes the next train to City B.

(a) Complete the tree diagram.



(b) Find the probability that Manuela arrives in City B on time.

The diagram shows a solid triangular prism of length 20 cm.
The cross-section of the prism is triangle $BCP$ and three faces are rectangles.
$BC = 8$ cm, $CP = 5$ cm and angle $ADQ =$ angle $BCP = 100^{\circ}$.

(a) Calculate the total surface area of the prism. [7]
(b)
(i) On the diagram of the prism, draw two straight lines and mark angle $PAC$. [1]
(ii) Angle $APC = 73.45^{\circ}$.
Calculate angle $PAC$.
Angle $PAC = \text{................................................}$. [4]