Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2021 Winter Zone 1

Amir, Bibi and Caitlyn are each given $1500 to invest.

(a) Amir invests his $1500 in an account which pays compound interest. The interest rate is 3% per year for 5 years, after which it is 2% per year.
Find the value of Amir's investment at the end of 11 years.

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

(b) Bibi invests her $1500 in an account which pays $r\%$ per year simple interest. At the end of 11 years, the investment is worth $1962.
Calculate the value of $r$.

$r = ............................................. [3]

(c) Caitlyn invests her $1500 in an account which pays $t\%$ per year compound interest. At the end of 11 years, the investment is worth $1968.13 .
Calculate the value of $t$.

$t = ............................................. [3]

(a) In part (a) enlargements and stretches have scale factors greater than 1.

(i) A transformation maps triangle \( A \) onto triangle \( B \). Triangle \( A \) is congruent to triangle \( B \).

Tick all the possible transformations it could be.

[Table_1]

Transformation | Tick (✔)
----------------|----------
Rotation |
Reflection |
Translation |
Enlargement |
Stretch |

[1]

(ii) A transformation maps triangle \( C \) onto triangle \( D \). The angles of triangle \( C \) are the same as the corresponding angles of triangle \( D \).

Tick all the possible transformations it could be.

[Table_2]

Transformation | Tick (✔)
----------------|----------
Rotation |
Reflection |
Translation |
Enlargement |
Stretch |

[1]

(iii) A transformation maps triangle \( E \) onto triangle \( F \). Triangle \( F \) has a larger area than triangle \( E \).

Tick all the possible transformations it could be.

[Table_3]

Transformation | Tick (✔)
----------------|----------
Rotation |
Reflection |
Translation |
Enlargement |
Stretch |

[1]

(b)

(i) Describe fully the \textit{single} transformation that maps triangle \( P \) onto triangle \( Q \).

\text{...................................................................................................}
\text{...................................................................................................}

[3]

(ii) Stretch triangle \( P \) with the \( x \)-axis invariant and scale factor 2.

[2]

The table shows the number of days, $d$, since planting and the heights, $h$ cm, of some plants.

[Table_1]

(a) Complete the scatter diagram.
The first five points have been plotted for you.

[2]

(b) What type of correlation is shown in the scatter diagram?

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

(c) Find the equation of the regression line for $h$ in terms of $d$.

$h = \text{.............................}$ [2]

(d) Use your regression line to estimate the height of a plant that was planted 28 days ago.

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

(e) A plant was planted 140 days ago.
Explain why you should not use the equation of the regression line to estimate the height of this plant.

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

The table shows a set of data.

[Table_1]

(a) When $x$ represents the number of emails Essa receives each day, find

(i) the median,
................................................ [1]

(ii) the range,
................................................ [1]

(iii) the upper quartile,
................................................ [1]

(iv) the mean.
................................................ [2]

(b) When $x$ represents the height of a seedling, correct to the nearest centimetre, explain why you cannot work out the range of the heights.
.................................................................................................................................................. .................................................................................................................................................. [1]

(a) On the diagram, sketch the graph of $y = f(x)$.

f(x) = \frac{2x^2 + 3}{(x+1)(2-x)} \text{ for } -7 \leq x \leq 7
[3 marks]
(b) Write down the equation of each asymptote parallel to the $y$-axis.
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
[2 marks]
(c) Write down the coordinates of the local minimum.
( \_\_\_\_\_\_\_\_\_\_\_\_ , \_\_\_\_\_\_\_\_\_\_\_\_)
[2 marks]
(d) Find the range of values of $x$ for which the gradient of $f(x)$ is negative.
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
[3 marks]
(e) Solve $f(x) = -x$.
x = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_
[1 mark]

The masses of 300 apples are shown in the table.
[Table_1]
Mass (m grams): $0 < m \leq 25$, $25 < m \leq 50$, $50 < m \leq 75$, $75 < m \leq 100$, $100 < m \leq 125$, $125 < m \leq 150$
Frequency: 4, 26, 60, 88, 106, 16

(a) Draw a cumulative frequency curve to show these results.

(b) Use your curve to find the interquartile range.
......................................... [2]
(c) Apples with a mass below 80 g are used to make drinks. Find the percentage of the 300 apples that are used to make drinks.
......................................... % [2]

(a) The nth term of a sequence is $\frac{n(n+1)(2n+1)}{6}$.
Find the first three terms of this sequence.
..............., ..............., ...............

(b) For each of the following sequences:
• find the next two terms
• find an expression for the nth term.
(i) 11 8 5 2
Next two terms ...................... , ......................
nth term ............................................. [3]
(ii) -2 -2 0 4 10 18
Next two terms ...................... , ......................
nth term ............................................. [3]
(iii) 3 5 9 17 33
Next two terms ...................... , ......................
nth term ............................................. [3]

The diagram shows a right-angled triangular prism.
$ABCD, ADFE$ and $BCFE$ are rectangles.
$AD = 11 \text{ cm}, DC = 6 \text{ cm}$ and the height $CF = 4 \text{ cm}$.

(a) Calculate the volume of the prism.
.................................... $\text{cm}^3 \ [2]$
(b) Calculate the total surface area of the prism.
.................................... $\text{cm}^2 \ [4]$
(c) Calculate the length $AF$.
.................................... $AF = .................................... \text{cm} \ [3]$
(d) Calculate angle $FAC$.
.................................... $\text{Angle} \ FAC = .................................... \ [2]$
(e) The volume of a mathematically similar prism is $445.5 \text{ cm}^3$.
Calculate the total surface area of this similar prism.
.................................... $\text{cm}^2 \ [3]$

The equation of the circle is $x^2 + y^2 = 16$.
The equation of the straight line is $y = 3x + 1$.
The line crosses the circle at the points $A$ and $B$.

(a) Use substitution to show that the $x$-coordinates of the points $A$ and $B$ satisfy the equation $10x^2 + 6x - 15 = 0$. [3]

(b) Solve the equation $10x^2 + 6x - 15 = 0$ to find the coordinates of the points $A$ and $B$. Show your working and give your answers correct to 2 decimal places. [4]

$A ( ext{.....................} , ext{.....................} )$
$B ( ext{.....................} , ext{.....................} )$

f(x) = 3x - 2\qquad g(x) = (x - 3)^2
(a) Find $f(g(1))$.
.................................................. [2]
(b) Solve $g(x) = 25$.
$x = \text{..................... or } x = \text{.....................}$ [2]
(c) Find $f^{-1}(4)$.
.................................................. [2]
(d) Write down $f(f^{-1}(x))$.
.................................................. [1]

A, B and C are three ports.
(a) Show that angle $ABC = 107.2^\circ$ correct to 1 decimal place. [3]
(b) The bearing of B from A is $305^\circ$.
(i) Using the sine rule, show that angle $BAC = 44.4^\circ$ correct to 1 decimal place. [3]
(ii) Find the bearing of C from A. [1]
(c) A ship leaves A at 2250 and sails at a constant speed of 24 km/h towards C.
Calculate the time, correct to the nearest minute, when the ship is nearest to B. [5]

(a) (i) For each Venn diagram, shade the given set.

U A \cup B U A \cap B' [2]

(ii) Use set language to describe the shaded set.

U A B ................................................... [1]

(b) 40 people are asked which of 3 television programmes, $P$, $Q$ and $R$, they watch.
The results are shown in the Venn diagram.


(i) Two of the 40 people are chosen at random.
Find the probability that they both watch exactly 2 of the 3 programmes.
................................................... [2]

(ii) Two of the people who watch programme $P$ are chosen at random.
Find the probability that one of them watches both other programmes and one watches just one of the other programmes.
................................................... [3]

(iii) Three of the 40 people are chosen at random.
Find the probability that two of them watch only programme $Q$ and one of them watches only programme $R$.
................................................... [3]

(a) Rearrange $y = \frac{ax+b}{ex+f}$ to make $x$ the subject.
$x = \text{..................................................}$ [4]

(b) $f(x) = 3\sin(2x)^{\circ}$
(i) Write down the amplitude and the period of $f(x)$.

Amplitude = \text{..................................................}
Period = \text{..................................................} [2]

(ii) The graph of $y = f(x)$ is stretched with the $x$-axis invariant and scale factor 3 to give the graph of $y = g(x)$.
Find $g(x)$.

$g(x) = \text{..................................................}$ [1]

(iii) The graph of $y = f(x)$ is translated through $\begin{pmatrix}-90\\0\end{pmatrix}$ to give the graph of $y = h(x)$.
Find $h(x)$, giving your answer in its simplest form.

$h(x) = \text{..................................................}$ [2]