Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2022 Winter Zone 1

(a) Describe fully the \textbf{single} transformation that maps triangle $A$ onto triangle $B$.
...................................................................................................................................................
................................................................................................................................................... [3]
(b) Translate triangle $A$ by vector \( \begin{pmatrix} 3 \\ 6 \end{pmatrix} \). Label the image $C$. [2]
(c) Rotate triangle $B$ through $90^\circ$ clockwise about $(3, 6)$. Label the image $D$. [2]
(d) Reflect triangle $B$ in the line $y = 1$. Label the image $E$. [2]

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


(a) (ii) What type of correlation is shown by the scatter diagram? ....................................................

(b) Find the mean price of one barrel of oil. $ ...............................................

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

(d) Use your answer to part (c) to estimate
(i) the price of one barrel of oil when the number of barrels produced is 90 million, $ ............................................... $
(ii) the price of one barrel of oil when the number of barrels produced is 120 million, $ ............................................... $

(e) Which of your two answers to part (d) is likely to be more reliable? Give a reason for your answer. Part ...................... because .................................................................................................... .......................................................................................................................................

(a) The total amount of money they earn is shared in the ratio of the time each person works. During one week Alana works for 16 hours 40 minutes, Bev works for 30 hours and Cara works for 1200 minutes.
They earn a total of $1680.80 .
Change all the times into minutes and find the amount of money each person earns.
Alana $ ...............................................
Bev $ ...............................................
Cara $ ...............................................

(b) (i) Alana pays a weekly rent of $255 for her apartment. The price of this weekly rent is 2\% higher than one year ago.
Find the price of her weekly rent one year ago.
$ .......................................

(b) (ii) Alana can pay one full year's rent in advance. One year is 52 weeks. She will receive a discount of 3\% on each weekly rent of $255.
Calculate the cost of paying the full year’s rent in advance.
$ .......................................

(c) One week Bev earns $x.
She spends \frac{1}{4} of these earnings on rent and \frac{2}{9} on food.
She spends \frac{1}{3} \text{ of the remaining money} on clothes and saves the rest. She saves $152.
Find the value of $x.
$x = ...............................................

(d) Cara invests $500 for 5 years at a rate of $y\%$ per year simple interest. The value of Cara’s investment at the end of 5 years is $530.75 .
Find the value of $y$.
$y = ...............................................

A, B, C and D lie on a circle, centre O.
AP and BP are tangents to the circle.
AC and BD intersect at X.
Angle APB = 52°.

(a) Complete the statement.

Angle OAP = 90° because ...............

(b) Find

(i) angle AOB

(ii) angle OAB

(iii) angle ACB

(c) ABCD is a trapezium with AB parallel to DC.

(i) Write down a triangle that is similar to triangle ABX.

(ii) The length CD = 4 cm and the length AB = 12 cm.
Find the ratio area CDX : area ABX.

(a) On the diagram, sketch the graph of $y = f(x)$, where $f(x) = \frac{1}{(x-1)(x-2)(x-3)}$ for values of $x$ between $-1$ and $5$. [3]
(b) Write down the $y$-coordinate of the point where the curve meets the $y$-axis.
$y =$ .................................................. [1]
(c) Write down the equations of all the asymptotes to the graph of $y = f(x)$.
........................................................................................................................... [3]
(d) On the diagram, sketch the graph of $y = g(x)$, where $g(x) = x-1$, for values of $x$ between $-1$ and $5$. [1]
(e) Find the $x$-coordinate of each point of intersection of the two graphs.
$x = ..................$ or $x = ..................$. [2]
(f) Solve the inequality $f(x) > g(x)$. [3]

y \text{ varies inversely as } (2x - 1)^2.
y = 4 \text{ when } x = 3.
(a) \text{ Find the value of } y \text{ when } x = 2.5.
y = \text{...............................................} \hspace{2cm} [3]
(b) \text{ Find the values of } x \text{ when } y = 16.
x = \text{.................. or } x = \text{..................} \hspace{2cm} [4]

(a) Solve the simultaneous equations.
You must show all your working.
$$4x + 3y = -21$$
$$6x - 2y = 1$$

\( x = \text{.............................................} \)
\( y = \text{.............................................} \) [4]

(b) \( f(x) = 5x - 2 \), \( g(x) = \frac {1}{2x - 1} \), \( x \neq 0.5 \), \( h(x) = (x - 1)^3 \)
(i) Find \( f(3) \).
\( \text{.............................................} \) [1]

(ii) Find \( h(f(2)) \).
\( \text{.............................................} \) [2]

(iii) Solve \( f(h(x)) = -7 \).
\( \text{.............................................} \) [2]
\( x = \text{............................................} \) [3]

(iv) Find \( g(g(x)) \) in terms of \( x \).
Give your answer in its simplest form.
\( \text{.............................................} \) [3]

The cumulative frequency table shows the masses, in grams, of 1200 potatoes.

[Table_1]

$\begin{array}{|c|c|} \hline \text{Mass }(x \text{ grams}) & \text{Cumulative frequency} \\ \hline x \leq 150 & 22 \\ \hline x \leq 180 & 160 \\ \hline x \leq 200 & 480 \\ \hline x \leq 250 & 860 \\ \hline x \leq 300 & 1120 \\ \hline x \leq 400 & 1200 \\ \hline \end{array}$

(a) On the grid below, draw a cumulative frequency curve. [3]



(b) Use your curve to estimate

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

(ii) the interquartile range. .................................................. g [2]

(c) Find the percentage of potatoes that have a mass of at least 280 grams. .................................................. % [2]

(d) Complete the table to show the masses of the 1200 potatoes.

[Table_2]

$\begin{array}{|c|c|} \hline \text{Mass }(x \text{ grams}) & \text{Frequency} \\ \hline 100 < x \leq 150 & 22 \\ \hline 150 < x \leq 180 & \\ \hline 180 < x \leq 200 & \\ \hline 200 < x \leq 250 & \\ \hline 250 < x \leq 300 & \\ \hline 300 < x \leq 400 & 80 \\ \hline \end{array}$ [1]

(e) Calculate an estimate of the mean mass of a potato. .................................................. g [2]

The diagram shows a square of side $2a$ cm inside a square of side $(2a + 2x)$ cm.
(a) (i) Find an expression, in terms of $a$ and $x$, for the area of the shaded region.
Give your answer in the form $px^2+qax$, where $p$ and $q$ are integers.
................................................ [2]
(ii) Calculate the area of the shaded region when $a = 6$ and $x = 1$.
............................................ cm$^2$ [1]
(b) Find an expression, in terms of $a$ and $x$, for the total perimeter of the shaded region.
Give your answer in its simplest form.
................................................ [2]
(c) The numerical value of the shaded area is equal to the numerical value of the perimeter of the shaded region.
Find $x$ when $a = 10$.
You must show all your working.
$x =$ ................................................ [4]

A bag contains 5 red balls, 3 blue balls and 2 green balls.

(a) Rosa chooses a ball at random from the bag, notes its colour and replaces it. She then chooses a ball at random from the bag a second time, notes its colour and replaces it.
Find the probability that the two balls chosen are
(i) both green,
............................................... [2]
(ii) the same colour.
............................................... [2]

(b) Savio chooses a ball at random from the bag and does not replace it. He then chooses another ball from the bag.
Find the probability that the two balls chosen are different colours.
............................................... [3]

The diagram shows a triangular prism $ABCDEF$.
$X$ is a point on $FE$.
$AB = 8\,m$, $AD = 15\,m$, $AF = 10\,m$, $EC = 6\,m$ and $FX = 5\,m$.
Angle $ABF = 90^\circ$ and angle $DCE = 90^\circ$.
(a) Calculate angle $CDE$.

Angle $CDE = \text{..............................}$ [2]
(b) Calculate $AC$.

$AC = \text{..........................................} \ m$ [2]
(c) Calculate angle $CXA$.

Angle $CXA = \text{...............................}$ [5]
(d) Calculate the area of triangle $CXA$.

$\text{....................................} \ m^2$ [2]

(a) $a^b = 1$ where $a > 0$
(i) When $b = 13$, write down the value of $a$.
$a = \text{................................................} \; [1]$
(ii) When $a = 17$, write down the value of $b$.
$b = \text{.................................................} \; [1]$

(b) Write down the solution to each equation.
(i) $3^{x-5} = 1$
$x = \text{............................................} \; [1]$
(ii) $(x - 5)^3 = 1$
$x = \text{.............................................} \; [1]$

(c) Use part (a) to find all the solutions to the following equations.
(i) $(4x - 1)^{(x-3)} = 1$
$\text{.................................................} \; [3]$
(ii) $(x^2 - 4x + 4)^{(x^2 - 9x + 20)} = 1$
$\text{.................................................} \; [4]$