Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2019 Winter Zone 3

(a) Aisha invests $12000 at a compound interest rate of 3.5% per year. Calculate the value of her investment at the end of 4 years.

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

(b) 2 years ago, Byron invested $P at a compound interest rate of 3% per year. The value of his investment is now $10078.55. Calculate the value of $P$.

$P = \text{................................................}$ [3]

(c) 5 years ago Cheng invested $Q at a simple interest rate of 4% per year. The value of his investment is now $20400. Calculate the value of $Q$.

$Q = \text{................................................}$ [3]

The table shows the number of goals scored in 100 matches.

[Table_1]

Find

(a) the mode, ................................................................. [1]

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

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

(d) the inter-quartile range, ................................................................. [2]

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

f(x) = 2x^3 - 5x^2 + 3 \text{ for } -1.5 \leq x \leq 3 \newline (a) \text{ On the diagram, sketch the graph of } y = f(x). \hspace{1cm} [2] \newline (b) \text{ Find the zeros of } f(x). \newline \hspace{1cm} ........................................................ \hspace{1cm} [3] \newline (c) \text{ Find the co-ordinates of the local maximum.} \newline \hspace{1cm} (............. , .............) \hspace{1cm} [1] \newline (d) \text{ Find the co-ordinates of the local minimum.} \newline \hspace{1cm} (............. , .............) \hspace{1cm} [2] \newline (e) \text{ The equation } 2x^3 - 5x^2 + 3 = k \text{ has three solutions.} \newline \text{Find the range of values of } k. \newline \hspace{1cm} ........................................................ \hspace{1cm} [2]

The table shows the mathematics mark and the physics mark for each of 10 students in an examination.

[Table_1]

Mathematics mark $(m)$ 14   28   38   41   60   66   76   82   90   98
Physics mark $(p)$ 8   28   66   43   67   56   51   74   85   88

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



[2]

(b) Write down the type of correlation shown by the scatter diagram.
................................................... [1]

(c) Find the equation of the regression line.
Write the answer in the form $p = am + b$.

$p = ext{................................................}$ [2]

(d) A student was absent for the physics examination but gained 56 marks in the mathematics examination.
Use your answer to part (c) to estimate a physics mark for this student.

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

(e) The school decided that the physics examination was too difficult and added 5 marks to each of the physics marks.
Write down the new equation of the regression line.

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

(a) Reflect triangle $A$ in the line $y = 1$.
[2]

(b) Rotate triangle $B$ through $90^\circ$ clockwise about $(1, 0)$.
[3]

(c) Describe fully the single transformation that maps triangle $A$ onto triangle $B$.
..........................................................................................................................................
..........................................................................................................................................
[3]

(d) Describe fully the single transformation that maps triangle $B$ onto triangle $C$.
..........................................................................................................................................
..........................................................................................................................................
[3]

(a) $P$ is the point $(3, 5)$ and $Q$ is the point $(7, -2)$. $Q$ is the midpoint of $PR$.
Find the co-ordinates of the point $R$.
[Image Placeholder]
$(\text{....................} , \text{....................})$ [2]

(b) [Image Placeholder] \( \overrightarrow{OA} = \mathbf{a} \) and \( \overrightarrow{OB} = \mathbf{b} \).
$C$ divides $AB$ in the ratio $4:3$.
Find these vectors, in terms of \( \mathbf{a} \) and \( \mathbf{b} \), in their simplest form.
(i) \( \overrightarrow{AB} \)
\( \overrightarrow{AB} = \text{..................................................} \) [1]
(ii) \( \overrightarrow{OC} \)
\( \overrightarrow{OC} = \text{..................................................} \) [3]

The diagram shows a child's toy made of a cone joined to a hemisphere. The cone and the hemisphere each have a radius of 4 cm. The perpendicular height of the cone is 6 cm.

(a) (i) Find the volume of the hemisphere. ....................................cm^3 [2]
(ii) Find the volume of the cone. ....................................cm^3 [2]
(iii) Each cubic centimetre of the hemisphere has a mass of 7.85 g.
Each cubic centimetre of the cone has a mass of 0.65 g.
Find the total mass of the toy. .....................................g [2]

(b) Find the total surface area of the toy. ....................................cm^2 [5]

(c) The height of the cone on a similar toy is 9 cm. Find the total surface area of this toy. ....................................cm^2 [2]

A dance club has 90 members.
Here is some information about types of dancing members like.

50 like Ballroom ($B$)
37 like Latin ($L$)
47 like Modern ($M$)
18 like Ballroom and Latin
15 like Ballroom and Modern
22 like Latin and Modern
8 like Ballroom, Latin and Modern

(a) Complete the Venn diagram.


[2]
(b) Write down the number of members who do not like any of these three types of dancing.
.................................................. [1]
(c) Two of the 90 members are chosen at random.
Find the probability that they both like Ballroom and Latin but not Modern.
........................................................ [2]
(d) Two of the members who like Ballroom are chosen.
Find the probability that one of these members likes Latin but not Modern and the other likes Modern but not Latin.
............................................................ [3]

A, B \text{ and } C \text{ are points on the circle, centre } O.
ON \text{ is perpendicular to } BC.
AB = 14 \text{ cm}, \; AC = 12 \text{ cm and angle } BAC = 58^\circ.
(a) \text{ Show that } BC = 12.73 \text{ cm, correct to 2 decimal places.} [3]
(b) \text{ Explain why angle } BON = 58^\circ. [1]
(c) \text{ Calculate } OB, \text{ the radius of the circle.} \; OB = \text{.................. cm} [1]
(d) \text{ Calculate the area of the shaded segment.} \; \text{.................. cm}^2 [3]

All lengths in this question are in metres and all areas are in square metres.

The length of this rectangle is $(2x + 3)$ and the area is 840.
(a) Write down an expression, in terms of $x$, for the width of the rectangle.
.................................................. [1]
(b) The perimeter of the rectangle is 118.
Show that $2x^2 - 53x + 336 = 0$. [3]
(c) Solve the equation $2x^2 - 53x + 336 = 0$.
Show all your working.
$x = .........................$ or $.........................$ [3]
(d) Find the length and the width of the rectangle.
Length = ................................. m
Width = .................................. m [2]

(a) Simplify.
(i) $\frac{a^5 \times a^4}{a^3}$ .............................................................. [2]
(ii) $\log_5(5^x)$ .............................................................. [1]
(iii) $\log_9(3^x)$ .............................................................. [1]
(b) Solve.
\[3 \log 10 - 2 \log 5 = \log x \]
$x =$ .............................................................. [2]

$$f(x) = \frac{3x+2}{(x+2)(x-3)}$$
(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $-8$ and $8$. [3]
(b) Write down the equations of the asymptotes.
......................... , ......................... , ......................... [3]
(c) $g(x) = x-2$
(i) On the diagram, sketch the graph of $y = g(x)$ for $-6 \le x \le 8$. [1]
(ii) Solve $f(x) = g(x)$.
$$x = ......................... \text{ or } x = ......................... \text{ or } x = ......................... $$ [3]
(iii) Solve $f(x) > g(x)$.
....................................................................................................................... [3]

f(x) = 2x + 5 g(x) = 1 - 2x

(a) Find \( g(-4) \). .................................................... [1]

(b) Find \( f^{-1}(-7) \). .................................................... [2]

(c) Find \( g(f(3)) \). .................................................... [2]

(d) Find and simplify \( f(g(x)) \). .................................................... [2]

(e) Find and simplify \( g^{-1}(x) \).

\( g^{-1}(x) = .................................................... \) [2]

(f) Write as a single fraction, simplifying your answer.

\( 2 + \frac{3}{f(x)} \) .................................................... [2]