Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2019 Winter Zone 2

(a) Reflect triangle $A$ in the $x$-axis. Label the image $B$. [1]
(b) Translate triangle $A$ by the vector $\begin{pmatrix} 0 \\ -3 \end{pmatrix}$. Label the image $C$. [1]
(c) Describe fully the single transformation that maps triangle $B$ onto triangle $C$.
.................................................................................................................................
................................................................................................................................. [2]
(d) Rotate triangle $A$ through $90^\circ$ anti-clockwise, about the origin. Label the image $D$. [2]
(e) Describe fully the single transformation that maps triangle $B$ onto triangle $D$.
.................................................................................................................................
................................................................................................................................. [2]

(a) Find \( f(6) \). ............................................... [1]

(b) Solve \( f(x) = -2 \).
\( x = \) ............................................... [2]

(c) Find \( h(g(x)) \). ............................................... [1]

(d) Solve \( h(g(x)) = h(x) + 2 \).
\( x = \) ............................................... [4]

(e) Find \( f^{-1}(x) \).
\( f^{-1}(x) = \) ............................................... [3]

(f) (i) On the diagram, sketch the graph of \( y = f(x) \) and the graph of \( y = h(x) \) for values of \( x \) between \(-3\) and \(3\). [3]
(ii) Write down the equation of the line of symmetry of \( y = h(x) \). ............................................... [1]
(iii) Solve \( f(x) > h(x) \). ............................................... [2]

Alana and Bill share some money in the ratio $5 : 4$. Alana's share is $160$.

(a) Show that Bill’s share is $128$.

[1]

(b) Alana spends $x$. The ratio of Alana's money : Bill's money is now $4 : 5$.
Find the value of $x$.

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

(c) A shop has a sale. Bill buys a jacket in the sale for $32$.
(i) Write $32 as a percentage of $128$.

$\text{.............................................}\%$ [1]

(ii) The original price of the jacket was reduced by $20\%$ to $32$.
Work out the original price.

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

(a) Solve the following equations.
(i) $2x - 3 = -11$
$x =$ .................................................. [2]

(ii) $\frac{36}{x} = -4$
$x =$ .................................................. [2]

(iii) $6x + 13 = 17 - 2x$
$x =$ .................................................. [2]

(b) Solve the simultaneous equations. You must show all your working.
$5x + 3y = -19$
$3x + 5y = -21$

$x =$ ..................................................
$y =$ .................................................. [4]

A, B, C and D lie on a circle, centre O.
AX is a tangent to the circle at A and BX is a tangent to the circle at B.
Angle OAB = 20\degree and angle DAX = 25\degree.

(a) Find the value of

(i) angle AOB,

\text{Angle } AOB = \text{.........................................} \hspace{0.5cm} [2]

(ii) angle ACB,

\text{Angle } ACB = \text{..........................................} \hspace{0.5cm} [1]

(iii) angle ADB,

\text{Angle } ADB = \text{..........................................} \hspace{0.5cm} [1]

(iv) angle BAD,

\text{Angle } BAD = \text{.........................................} \hspace{0.5cm} [1]

(v) angle DBA,

\text{Angle } DBA = \text{.........................................} \hspace{0.5cm} [1]

(vi) angle AXB.

\text{Angle } AXB = \text{.........................................} \hspace{0.5cm} [1]

(b) What type of quadrilateral is ACBD?

\text{.................................................} \hspace{0.5cm} [1]

[Image of Circle and Tangents]

Spinner A is numbered 1, 2, 3, 4.
Spinner B is numbered 1, 2, 3, 4, 5, 6.
Each spinner is equally likely to land on any of its numbers.
The two spinners are each spun once and the number that each spinner lands on is recorded.

Find the probability that
(a) the number on spinner A is greater than 4,
................................................ [1]
(b) the number on spinner B is not a 3,
................................................ [1]
(c) the number on spinner A is the same as the number on spinner B,
................................................ [2]
(d) one number is odd and one number is even,
................................................ [3]
(e) the sum of the numbers is 6.
................................................ [2]

(a) (i) Factorise $2x^2 - 11x - 6$.
[Image_1: .................................................... [2]]

(ii) Using your answer to part (i), solve $2x^2 - 11x - 6 < 0$.
[Image_1: .......................................................... [2]]

(b) Solve the equation $3x^2 - x - 5 = 0$.

Give your answers correct to 2 decimal places.
You must show all your working.

$x = .................$ or $x = ............$ [3]

There are 100 students in a year group. Each student studies at least one of the languages, French \((F)\), Italian \((I)\) and Spanish \((S)\).

x students study all 3 languages.
y students study French only.
18 students study Italian only.
4 students study French and Italian but not Spanish.
12 students study French and Spanish but not Italian.
2 students study Italian and Spanish but not French.
74 students study only one language.

(a) Show this information on the Venn diagram.



(b) Twice as many students study French as Italian.
Find the number of students who study
(i) all 3 subjects,

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

(ii) French only,

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

(iii) Spanish only.

\(\text{.............................................}\) \([1]\)

The diagram shows three solids, a prism, a sphere and a cone.
The radius of the sphere is equal to the base radius of the cone.
The volume of each solid is the same.

(a) Show that the volume of the prism is 7392 cm³. [3]

(b) A similar prism has a volume of 924 cm³.
The length of the original prism is 24 cm.
Find the length of this similar prism. [3]

(c) Find the value of $r$.
$r = \text{........................................... cm}$ [2]

(d) Find the value of $h$.
$h = \text{........................................... cm}$ [2]

(e) When exact values of $h$ and $r$ are used, $h = 4r$.
Find, in terms of $r$, an exact expression for the curved surface area of the cone.
Give your answer in its simplest form. [3]

The mass of each of 80 apples is shown in the table.

[Table_1]

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

(b) Find the interval which contains the upper quartile.
.................. $< m \leq$ .................. [1]

(c) Two of these apples are chosen at random.

Find the probability that they both have a mass of 120 g or less. Give your answer as a fraction in its simplest form.
................................................. [3]

(d) (i) Complete the frequency density column in this table.

[Table_2]
[2]

(ii) On the grid, draw a histogram to show this information.


[3]

The diagram shows a quadrilateral $ABCD$.
$AB = 8\, \text{cm}, AD = 9\, \text{cm}, CD = 6.5\, \text{cm}$ and angle $BAD = 64^\circ$.
(a) Calculate $BD$ and show that your answer rounds to $9.05\, \text{cm}$, correct to $2$ decimal places. [2]
(b) The area of the quadrilateral $ABCD$ is $57.3\, \text{cm}^{2}$.
(i) Calculate angle $BDC$ and show that your answer rounds to $58^\circ$, correct to the nearest degree. [4]
(ii) Calculate angle $BCD$.
angle $BCD = ....................................................$ [5]

(a) On the diagram, sketch the graph of $y = f(x)$, where $$f(x) = \frac{1}{x(x-1)(x+1)}$$ for values of $x$ between $-3$ and $3$.
[4]

(b) Write down the equations of the asymptotes.
..................... , ..................... , ..................... , .....................
[3]

(c) Write down the co-ordinates of the local maximum.
(................ , ................ )
[2]

(d) The line $y = 2x + 1$ intersects the curve $y = f(x)$ twice.
Find the value of the $x$ co-ordinate of each point of intersection.
$x = ................$ or $x = ................$
[2]