Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2020 Winter Zone 2

Asif buys a one-year old car. He pays $19975 which is 15% less than its price when it was new.
(a) Calculate the price when it was new.

$ ................................................
(b) Option 1: Pay 10% of the $19975 and then pay $345 per month for 5 years.
Option 2: Borrow $19975 and pay this back at the end of 5 years at a rate of 2.5% per year compound interest.
Asif can pay for the car using Option 1 or Option 2.
(i) Using Option 1, find how much Asif would pay in total for the car.
$ ................................................
(ii) By how much is Option 2 cheaper than Option 1?
$ ................................................

(a) Describe fully the \textit{single} transformation that maps triangle \( A \) onto triangle \( B \).
......................................................................................................................
...................................................................................................................... [2]
(b) Reflect triangle \( A \) in the line \( y = -x \). Label the image \( C \). [2]
(c) Rotate triangle \( A \) through \( 90^\circ \) clockwise about centre \( (1, -1) \). Label the image \( D \). [2]
(d) Describe fully the \textit{single} transformation that maps triangle \( C \) onto triangle \( D \).
......................................................................................................................
......................................................................................................................
[2]
(e) Describe fully the \textit{single} transformation that maps triangle \( A \) onto triangle \( E \).
......................................................................................................................
......................................................................................................................
[3]

The table shows the engine capacity, $x$ litres, and the fuel consumption, $y$ kilometres per litre, for each of nine cars.

[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 your scatter diagram? ................................................... [1]

(c) Find the equation of the regression line for $y$ in terms of $x$.

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

(d) Use your answer to part (c) to estimate the fuel consumption for a car with engine capacity 2.8 litres.

................................................ km/l [1]

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 minimum point.
\quad ( \text{.......................} , \text{.......................} ) [2]
(c) Describe fully the symmetry of the diagram.
\quad \text{...............................................................................................................................}
\quad \text{...............................................................................................................................} [3]
(d) $g(x) = 2x - 1$
\quad (i) Solve $f(x) = g(x)$ for $-3 \leq x \leq 3$.
\quad \text{.................... , ...................., ....................} [3]
\quad (ii) Use your answers to \text{part(i)} to solve $f(x) > g(x)$.
\quad \text{...............................................................} [2]

Naomi flies non-stop from London, England, to Perth, Australia. The flight takes 16 hours 45 minutes. The distance is 14498 km.
(a) Find the average speed of the plane in km/h.
.......................................... km/h [2]

(b) The plane leaves London at 13 15. The time in Perth is 8 hours ahead of the time in London.
Find the time in Perth when the plane lands.
................................................ [3]

(c) The cost, in pounds (£), of the flight is £827.75. The exchange rate is 1 Australian dollar = £0.55.
Calculate the cost of the flight in Australian dollars.
................................... Australian dollars [2]

The diagram shows a field $ABCD$ with a straight path from $A$ to $C$. The bearing of $B$ from $A$ is $075^\circ$ and angle $ADC = 90^\circ$.
(a) Show that angle $BAC = 31.6^\circ$, correct to 1 decimal place. [3]
(b) Find the bearing of $D$ from $A$. [3]
(c) Find the shortest distance from $B$ to $AC$. ......................................... m [2]
(d) Find the total area of the field $ABCD$. ......................................... $\text{m}^2$ [3]

A is the point (3, 2) and B is the point (9, 5).
(a) Find the length $AB$.
$AB = \text{....................................................}$ [3]

(b) Find the equation of the line $AB$.
Give your answer in the form $y = mx + c$.
$y = \text{....................................................}$ [3]

(c) $C$ is the point (8, 2).
Find the equation of the line perpendicular to $AB$ which passes through $C$.
Give your answer in the form $y = mx + c$.
$y = \text{....................................................}$ [3]

(d) Find the coordinates of the point where the line in part (c) intersects $AB$.
$(\text{..................} , \text{..................})$ [2]

(e) $D$ is the reflection of $C$ in the line $AB$.
(i) Find the coordinates of $D$.
$(\text{..................} , \text{..................})$ [2]
(ii) What is the special name of quadrilateral $ACBD$?
$\text{....................................................}$ [1]

(f) Find the area of the quadrilateral $ACBD$.
$\text{....................................................}$ [3]

Bag $A$ contains 5 black balls and 2 white balls. Bag $B$ contains 4 black balls and 5 white balls.
(a) Gustav picks one ball at random from bag $A$ and replaces it. Write down the probability that the ball Gustav picks is black. [1]
(b) Sharia picks one ball at random from bag $A$, notes its colour, and places it in bag $B$. She then picks a ball at random from bag $B$. Find the probability that
(i) both balls are white, [2]
(ii) one ball is black and the other ball is white. [3]
(c) The balls are returned to their original bags. Jean picks a ball at random from bag $A$. He then replaces the ball. He continues to do this until he gets a white ball. Find the probability that the first time he gets a white ball is on the 5th pick. [2]
(d) The balls are returned to their original bags. Leanne picks a ball at random from bag $B$. She continues to do this without replacement until she gets a white ball. The probability that she picks the first white ball on her $n$th attempt is $\frac{5}{126}$. Find the value of $n$. [3]

The cumulative frequency curve shows the marks of 120 students in an examination.

(a) Use the graph to estimate
(i) the median,
...................................................... [1]
(ii) the interquartile range.
...................................................... [2]
(b) The top 15% of the students gained a grade A in the examination.
Estimate the minimum mark for a grade A.
...................................................... [3]

y is inversely proportional to the square root of x.
When $x = 25$, $y = 4$.

(a) Find y in terms of x.

y = \text{........................................} [2]

(b) Find y when $x = 0.25$.

y = \text{........................................} [1]

(c) Find x when $y = 5$.

x = \text{........................................} [2]

(d) z is proportional to $y + 2$.
When $x = 4$, $z = 84$.

Find z in terms of x.

z = \text{........................................} [3]

Given $f(x) = 5 - 3x$ and $g(x) = 2x + 7$:

(a) Solve $f(x) = g(x)$. [2]

(b) Find and simplify $g(f(x))$. [2]

(c) (i) Find $f(x^2) + g(x^2)$, simplifying your answer. [2]

(ii) Find $(f(x) + g(x))^2$, giving your answer in the form $ax^2 + bx + c$. [3]

(d) Find $f^{-1}(x)$. [2]

(e) Write as a single fraction in its simplest form:
$$\frac{2}{f(x)} - \frac{3}{g(x)}$$ [3]

(a) The vector $\mathbf{a} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}$ and the vector $\mathbf{b} = \begin{pmatrix} 2 \\ -1 \end{pmatrix}$.
On the grid, draw and label these vectors.
(i) $2\mathbf{a}$ [1]
(ii) $-\mathbf{b}$ [1]
(iii) $\mathbf{a} - 2\mathbf{b}$ [2]


(b) $p\begin{pmatrix} 2 \\ 3 \end{pmatrix} + q\begin{pmatrix} -1 \\ 4 \end{pmatrix} = \begin{pmatrix} 10 \\ -7 \end{pmatrix}$
By solving a pair of simultaneous equations, find the value of $p$ and the value of $q$.
Show all your working.
$p = \text{.....................}$
$q = \text{.....................}$ [4]