Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2016 Summer Zone 2

(a) Translate shape $T$ by the vector $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$. [2]
(b) Reflect shape $T$ in the line $y = -x$. [2]
(c) Rotate shape $T$ by 90° anticlockwise about $(-2, 1)$. [3]
(d) Describe fully the \textbf{single} transformation that maps:
(i) shape $T$ onto shape $U$. ...................................................................................................................... ...................................................................................................................... [3]
(ii) shape $T$ onto shape $V$. ...................................................................................................................... ...................................................................................................................... [3]

Nikhil and Padma share $630 in the ratio 5 : 4.
(a) Show that Nikhil receives $350 and that Padma receives $280.

(b) (i) In a sale, prices are reduced by 18\%.
Padma buys a jacket for $98.40 in this sale.
Calculate the original price of the jacket.
$ \text{.....................................................} \ [3]
(ii) Padma decides that she does not like the jacket and sells it for $30.
Calculate the percentage loss made by Padma.
\text{.....................................................} \% \ [3]
(iii) Calculate how much of the $280 Padma now has.
$ \text{.....................................................} \ [1]
(iv) Padma invests $150 at a rate of 2\% per year compound interest.
Calculate the total value of this investment after 10 years.
Give your answer correct to the nearest dollar.
$ \text{.....................................................} \ [4]

(c) On January 1st 2016, Nikhil invested all of his $350 at a rate of 0.15\% per \text{month} compound interest.
Find in which month and in which year Nikhil's investment will first have a total value of at least $500.
\text{month} \text{.........................} \ \text{year} \text{........................} \ [5]

(a) The cumulative frequency curve shows information about the average speeds of 200 cars on the same journey.

(i) Find the median.
...................................................... km/h [1]
(ii) Find the inter-quartile range.
...................................................... km/h [2]
(iii) Find the number of cars with an average speed of more than 70 km/h.
.......................................................... [2]
(b) A bus completes a journey in 2 h 24 min at an average speed of 50 km/h.
A car completes the same journey in 1 h 45 min.
Calculate the average speed of the car.
...................................................... km/h [3]

(a) The cost of a drink of water is $w$ cents.
The cost of a drink of juice is $(w + 30)$ cents.
The total cost of 6 drinks of water and 5 drinks of juice is $4.14$.
Find the value of $w$.

$w =$ ext{...............................................} [3]

(b)

The total area of the square and the rectangle is $10$ cm$^2$.
Find the perimeter of the square.
Give your answer correct to 2 decimal places.

ext{............................................... cm} [5]

A, B, C and D lie on the circle. The chords AC and BD intersect at X.
(a) Show that triangles ADX and BCX are similar. Give a reason for each statement that you make. [2]
(b) $AX = 5 \text{ cm}$, $DX = 2 \text{ cm}$ and $CX = 3 \text{ cm}$.
Calculate $BX$.
$$BX = \text{...........................................} \text{ cm}$$ [2]
(c) $AD = 4.61 \text{ cm}$.
Calculate angle $AXD$.
$$\text{Angle } AXD = \text{.........................................................}$$ [3]

f(x) = \sin(x^2) \text{ where } x^2 \text{ is in degrees.}
(a) On the diagram, sketch the graph of y = f(x) for 0 \leq x \leq 20. [2]
(b) One solution of the equation f(x) = 0, for 0 \leq x \leq 20 is x = 0.
Find the other two solutions.
$x = \text{......................... or } x = \text{.........................}$ [2]
(c) Find the co-ordinates of the local maximum point.
$\text{(.......................... , .......................... )}$ [2]
(d) There is a local minimum point at (0, 0).
Find the co-ordinates of the other local minimum point when 0 \leq x \leq 20.
$\text{(.......................... , .......................... )}$ [2]
(e) Write down the range of f(x).
$\text{..........................................................................}$ [1]
(f) By sketching another graph on the diagram, solve this equation.
$\sin(x^2) = \frac{x^2}{20} - 1$
$x = \text{.............................................................}$ [2]

(a)

The diagram shows a plastic solid made by joining a hemisphere to a cone. The radius of the hemisphere is 5 cm and the height of the cone is 12 cm.

(i) Calculate the volume of the solid.

.......................................................... cm$^3$ [3]

(ii) One cubic centimetre of the plastic has a mass of 0.95g.

Calculate the mass of the solid. Give your answer in kilograms.

.......................................................... kg [2]

(iii) Find the number of these solids that can be made from 1 tonne of plastic.

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

(iv) Calculate the total surface area of the solid.

.......................................................... cm$^2$ [4]

(b)

A solid cone has radius $r$ cm and height $3r$ cm. The total surface area of the cone is 377cm$^2$.

Find the value of $r$.

$r =$ .......................................................... [5]

(a) The equations of the asymptotes to the graph are $x = a$, $x = b$, $x = c$ and $y = d$.
Find the values of $a$, $b$, $c$ and $d$.
$a = \text{..............................................................}$
$b = \text{..............................................................}$
$c = \text{..............................................................}$
$d = \text{..............................................................}$

(b) $f(x) = k$ has only one solution, where $k$ is an integer and $k \neq 0$.
Find the value of $k$.
$k = \text{..............................................................}$

(c) Find the integer value of $x$ such that $f(x) < 0$.
$x = \text{..............................................................}$

(d) $g(x) = x^2 - p$
On the diagram, sketch a possible graph of $y = g(x)$ so that $f(x) = g(x)$ has 5 solutions.

The Venn diagram shows the following information.
U = \{\text{students in a music group}\} \quad P = \{\text{students who play the piano}\} \quad G = \{\text{students who play the guitar}\}
\( n(P \cup G)' = 2 \quad n(P \cap G') = 7 \quad n(G \cap P') = 3 \).

(a) \( n(U) = 23 \)
Find \( n(P \cap G) \).
....................................................................... [1]

(b) A student is chosen at random from the music group.
Find the probability that this student plays the piano but does not play the guitar.
....................................................................... [1]

(c) Two students who play the guitar are chosen at random.
Find the probability that they both also play the piano.
....................................................................... [3]

(d) On the Venn diagram, shade the region \( P \cup G' \).
[1]

Given the functions:
f(x) = x^2 - x - 30
g(x) = x^2 - 36
h(x) = 2x + 7

(a) Find h(f(7)). .............................................................. [2]

(b) Find $h^{-1}(x)$.

$h^{-1}(x) =$ .............................................................. [2]

(c) Find g(h(x)) in its simplest factorised form.
.............................................................. [3]

(d) Simplify \frac{f(x)}{g(x)}.
.............................................................. [4]

In the diagram, $ADC$ is a straight line.

(a) Calculate $AB$.
$$AB = \text{....................................................} \text{ cm}$$ [2]

(b) Calculate angle $DRC$.
$$\text{Angle } DBC' = \text{............................................................}$$ [5]

(c) Calculate the area of triangle $ABC$.
$$\text{........................................................} \text{ cm}^2$$ [2]

(a) Find the \textit{n}th term of the sequence.

1, \ 8, \ 27, \ 64, \ 125, \ \ldots \hspace{250pt}[1]

(b) (i) Find the next term in the sequence.

2, \ 12, \ 36, \ 80, \ 150, \ 252, \ \ldots \hspace{220pt}[2]

(ii) Find the \textit{n}th term of the sequence.

2, \ 12, \ 36, \ 80, \ 150, \ 252, \ \ldots \hspace{220pt}[2]