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

The number of matches in each of 140 matchboxes are counted. The table shows the results.
[Table_1]

(a) Write down the modal number of matches. ............................................................ [1]
(b) Write down the range. ............................................................ [1]
(c) Find the median. ............................................................ [1]
(d) Find the inter-quartile range. ............................................................ [2]
(e) Calculate the mean. Give your answer correct to one decimal place. ............................................................ [2]

Roberta starts from a point A and walks 1 km North to a point B. She then walks 2 km East to a point C, then walks 3 km South to a point D and finally walks 4 km West to a point E.



(a) Find the distance $AE$. .............................................km [3]

(b) Find the bearing of $E$ from $A$. ............................................. [2]

(c) Find the area $ABCDE$. .............................................km$^2$ [2]

Ten students at a school each recorded the number of hours they spent revising before an examination. The school compared the number of hours spent revising and the examination mark.

[Table_1]

Number of hours spent revising $(x)$ : 3, 4, 8, 9, 10, 12, 13.5, 17, 21, 24
Examination mark $(y)$ : 45, 36, 68, 55, 62, 66, 73, 81, 80, 94

(a) What type of correlation is there between the number of hours spent revising and the examination mark? .....................................................[1]

(b) Find
(i) the mean number of hours spent revising,
.....................................................[1]
(ii) the mean examination mark.
.....................................................[1]

(c) (i) Find the equation of the regression line for $y$ in terms of $x$.
$$ y = ext{.....................................................} $$ [2]
(ii) Estimate the examination mark for a student who spent 19 hours revising.
.....................................................[1]

A, B and C lie on a circle, centre O. The line QBP is a tangent to the circle at B. AC = BC = BP and angle QBA = 42°.
Find the value of
(a) angle OAB,
Angle OAB = ............................................. [1]
(b) angle AOB,
Angle AOB = ............................................. [2]
(c) angle BCA,
Angle BCA = ............................................. [1]
(d) angle CBP,
Angle CBP = ............................................. [2]
(e) angle CPB.
Angle CPB = ............................................. [2]

The age, $h$, of each of 120 passengers travelling on a train are shown in the table.
[Table_1]
(a) Calculate an estimate of the mean age of a passenger.
.....................................years [2]
(b) Complete the frequency density column in this table.
[Table_2]
[3]

Describe fully the single transformation that is the \textbf{inverse} of
(a) a reflection in the line $y = x$,
..................................................................................................................
.................................................................................................................. [2]
(b) a rotation of $90^{\circ}$ clockwise, centre $(2, 3)$,
..................................................................................................................
.................................................................................................................. [2]
(c) a translation with vector $\begin{pmatrix} 4 \\ -3 \end{pmatrix}$,
..................................................................................................................
.................................................................................................................. [2]
(d) an enlargement scale factor 3, centre $(0, 0)$.
..................................................................................................................
.................................................................................................................. [2]

Solve the simultaneous equations. You must show all your working.
$$ \begin{align*} 3x + 4y &= -8 \\ 5x - 6y &= -7 \end{align*} $$
\( x = \text{.........................} \)
\( y = \text{.................................} \) [4]

(a) $\cos x = \frac{1}{3}$ for $0^\circ < x < 90^\circ$.
Find the exact value of $\sin x$.
Give your answer as a surd.
$\sin x = \text{.........................................................}$ [3]

(b) [Image_1: Triangle ABC with AC = 11 cm, AB = 10 cm, BC = 9 cm]
(i) Show that $\cos B = \frac{1}{3}$. [2]
(ii) Using your answer to part (a), show that the exact value of the area of triangle $ABC$ is $30\sqrt{2}$ cm$^2$. [3]

A circle of radius 5 cm is inscribed inside a square. The square has one side on the base of an equilateral triangle, $ABC$. The other two vertices of the square touch the triangle as shown.

(a) Work out the shaded area.
........................................cm$^2$ [2]
(b) (i) Find the value of $x$.

$x$ = ........................................ [2]
(ii) Work out the length of a side of the equilateral triangle $ABC$.
........................................cm [2]
(iii) Calculate the area outside the square but inside triangle $ABC$.
........................................cm$^2$ [4]

(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $-6$ and $6$. [3]

(b) Write down the equations of the asymptotes of the graph of $y = f(x)$. .................... and ....................[2]

(c) Solve the equation $f(x) = -x$. ...................................................... [2]

(d) Solve the inequality $f(x) + x < 0$. ...................................................... [3]

(e) Describe fully the single transformation that maps (i) $y = 3 - \frac{6}{x}$ onto $y = 3 - \frac{6}{(x-2)}$, .............................................................................................................................................. ..............................................................................................................................................[2] (ii) $y = \frac{-6}{(x-2)}$ onto $y = 3- \frac{6}{(x-2)}$. .............................................................................................................................................. ..............................................................................................................................................[2]

Find the next term and the nth term in each of these sequences.
(a) 1, 8, 27, 64, 125, ...
Next term = ..................................................
nth term = ...................................................... [2]

(b) 3, 7, 13, 21, 31, ...
Next term = ..................................................
nth term = ...................................................... [4]

(c) -2, 1, 14, 43, 94, ...
Next term = ..................................................
nth term = ...................................................... [4]

A solid hemisphere has radius 6 cm.
(a) Find, in terms of \( \pi \),
(i) the volume of the hemisphere, .............................................. cm\(^3\) [2]
(ii) the total surface area of the hemisphere. .............................................. cm\(^2\) [2]

(b) Sixteen of these hemispheres, all with radius 6 cm, are made into one solid \textbf{sphere}.
(i) Find the radius of the sphere. .............................................. cm [3]
(ii) Find the ratio surface area of the sphere : total surface area of the 16 hemispheres. Give your answer in its simplest form. ............................. : .......................... [3]

(a) \( 3 \log p + 2 \log q - \log 6 = \log x \)
Find \( x \) in terms of \( p \) and \( q \).
\( x = \text{...........................................} \) [3]

(b) Solve the equations.
(i) \( 4^x = 6 \)
\( x = \text{...........................................} \) [3]
(ii) \((3x + 2)(2x - 3) = 1\)
You must show all your working.
\( x = \text{................................} \) or \( x = \text{................................} \) [5]

(a) On the diagram, sketch the graph of $y = f(x)$, for values of $x$ between $-2$ and $8$.

[4]
(b) Write down the $y$ co-ordinates of the local minimum points.
$y = \text{..................}$ and $y = \text{..................}$ [2]
(c) Write down the co-ordinates of the local maximum point.
$\text{(.................., ..................)}$ [2]
(d) Solve the equation $2^x - \frac{1}{3}x^3 = 2(1-x)$, for all real values of $x$.

$x = \text{..................}$ or $x = \text{..................}$ or $x = \text{..................}$ or $x = \text{..................}$ [4]