Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2022 Summer Zone 3

(a) Anneka invests $2500 in an account paying compound interest at a rate of 1.6% per year.
Find the amount in the account at the end of 3 years.

$ .................................................. [2]

(b) Bashir invests $2500 in an account paying simple interest at a rate of $r\%$ per year.
At the end of 5 years the amount in the account is $2718.75$ .
Calculate the value of $r$.

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

(c) Chanda invests $2500 in an account paying compound interest at a rate of 1.55% per year.
Find the number of complete years until Chanda’s investment is first worth more than $4000.

.................................................. [4]

The heights, $h$ cm, of 100 seedlings are shown in the table.

[Table_1]

(a) Calculate an estimate for the mean.
......................................... cm [2]

(b) Write down the modal group.
................. $< h \leq$ ................. [1]

(c) (i) Draw a cumulative frequency curve for the heights of the seedlings.
[Graph_1] [4]
(ii) Use your curve to estimate the median.
......................................... cm [1]
(iii) Use your curve to estimate the interquartile range.
......................................... cm [2]
(iv) Find an estimate of the percentage of the seedlings that were more than 8 cm in height.
......................................... % [2]

The diagram shows triangles A, B, C and D and the line with equation $x+y=9$.
(a) Enlarge triangle A with centre (4, 3) and scale factor 3. [2]
(b) Describe fully the single transformation that maps triangle A onto
(i) triangle B,
......................................................................................................................................
...................................................................................................................................... [2]
(ii) triangle C.
......................................................................................................................................
...................................................................................................................................... [3]
(c) Triangle A can be mapped onto triangle D by a rotation of $90^\circ$ clockwise about a point on the line $x+y=9$ followed by a reflection.
Find one possible centre of rotation and the equation of the corresponding mirror line.
Centre ( ........................ , ........................ )
Equation of mirror line .............................................. [2]

(a) Solve $4x - 3 = 7$.
\[ x = \text{..............................................} \quad [2] \]

(b) $y = \frac{3x + 1}{z}$
Find the value of $y$ when $x = 4.3$ and $z = -2$.
\[ y = \text{..............................................} \quad [2] \]

(c) Solve the simultaneous equations.
You must show all your working.
\[\begin{align}4x - 3y &= 14\\3x + 5y &= 25\end{align}\]
\[ x = \text{..............................................} \quad \]
\[ y = \text{..............................................} \quad [4] \]

(d) Simplify $\frac{2x^2 + 4x}{5y^2} \div \frac{x^2 - 4}{10y}$.
\[ \text{..............................................} \quad [4] \]

f(x) = x^3 - 5x + 3 \text{ for } -3 \leq x \leq 3

(a) On the diagram, sketch the graph of $y = f(x)$. [2]

(b) Find the coordinates of the local maximum.
(\,\,\,\,\,\,\,\,\, , \;\,\,\,\,\,\,\,\,\, ) [2]

(c) Describe fully the symmetry of the graph of $y = f(x)$.
..........................................................................................................................
.......................................................................................................................... [3]

(d) Find the zeros of the graph of $y = f(x)$.
.................................................................................. [3]

(e) $g(x) = x^2 - 2x + 2 \text{ for } -3 \leq x \leq 3$
 (i) On the same diagram, sketch the graph of $y = g(x)$. [2]
 (ii) Use your graphs to solve $x^3 - x^2 - 3x + 1 = 0$.
....................................................................... [3]

VABC is a pyramid with a triangular base.
All the edges have length 12 cm.
O is vertically below V.
D is the mid-point of AC and $BO = \frac{2}{3} BD$.
(a) Show that $BO = 6.928 \text{ cm}$, correct to 3 decimal places.
(b) Calculate the volume of the pyramid.

(a) Shade the region indicated below each of these Venn diagrams.
[Image_1: Venn diagram showing \( (A \cup B)' \) and \( (P \cap Q') \cup (P' \cap Q) \)]
(b) Bag A contains 4 white balls and 3 black balls. Bag B contains 4 white balls and 5 black balls.
A ball is taken at random from bag A.
If the ball is white, it is replaced in Bag A. If the ball is black, it is put in bag B.
A ball is then taken at random from bag B.
Find the probability that
(i) the ball taken from bag A is white, ................................................ [1]
(ii) both balls are black, ................................................ [2]
(iii) the balls are different colours. ................................................ [3]

The diagram shows the route of a ship between three ports, A, B and C. The bearing of B from A is 055\(^\circ\) and the bearing of C from B is 120\(^\circ\). BC = 65 \text{ km}.
The ship takes 7 hours to sail from A to B. It sails at a speed of 20 \text{ km/h}.

(a) Find the distance AB. 

(b) Show that angle ABC = 115\(^\circ\).

(c) (i) Calculate the distance CA. 

(ii) Calculate the bearing of A from C. 

(d) The ship takes 3.6 hours to sail from B to C. It then sails from C to A at a speed of 21.5.  Find the average speed for the complete journey from A to B to C and back to A.

The functions are defined as $f(x) = 2 - 3x$, $g(x) = (x + 1)^2$, $h(x) = \log x$.
(a) Find.
(i) $f(-4)$ ........................................ [1]
(ii) $f(g(3))$ ........................................ [2]
(iii) $f^{-1}(4)$ ........................................ [2]
(iv) $h^{-1}(2)$ ........................................ [2]
(b) Solve $(f(x))^{-1} = 5$.
$x = $ ........................................ [3]
(c) Find $g(f(x))$. Write your answer in the form $ax^2 + bx + c$.
........................................ [3]
(d) $y = h(f(x))$
Find $x$ in terms of $y$.
$x = $ ........................................ [3]

(a)

A, B, C, D \text{ and } E \text{ are points on the circle centre } O.\newline FBG \text{ is a tangent to the circle at } B.\newline \text{Angle } ABF = 62^\circ \text{ and angle } BED = 54^\circ.\newline \text{Find}

(i) \text{ angle } AEB,\newline \hspace{1cm} \text{Angle } AEB = \text{...................................................} \hspace{1cm} [1]

(ii) \text{ angle } BAD,\newline \hspace{1cm} \text{Angle } BAD = \text{...................................................} \hspace{1cm} [1]

(iii) \text{ angle } EAD,\newline \hspace{1cm} \text{Angle } EAD = \text{...................................................} \hspace{1cm} [1]

(iv) \text{ angle } BCD,\newline \hspace{1cm} \text{Angle } BCD = \text{...................................................} \hspace{1cm} [1]

(v) \text{ angle } FBD.\newline \hspace{1cm} \text{Angle } FBD = \text{...................................................} \hspace{1cm} [1]

(b)

PA \text{ and } PB \text{ are tangents to the circle centre } O.\newline \text{The radius of the circle is } 6\text{ cm and angle } AOB = 120^\circ.\newline \text{The shaded area } = (a\sqrt{3} - b\pi) \text{ cm}^2.\newline \text{Find the value of } a \text{ and the value of } b.

\hspace{1cm} a = \text{...................................................} \newline \hspace{1cm} b = \text{...................................................} \hspace{1cm} [5]

A tank has a capacity of 400 litres.

Water from Tap A flows at $x$ litres per minute.
Water from Tap B flows at 2 litres per minute less than the water from tap A.

(a) Write down an expression in terms of $x$ for the time, in minutes, for tap A to fill the tank.

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

(b) Tap B takes 10 minutes longer to fill the tank than tap A.

Write down an equation in terms of $x$ and show that it simplifies to

$$x^2 - 2x - 80 = 0.$$

[4]

(c) Solve $x^2 - 2x - 80 = 0$ and find the time it takes to fill the tank when both taps are turned on.
Give your answer in minutes and seconds, correct to the nearest second.

.......... minutes .......... seconds [4]