Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2020 Winter Zone 1

Ten students at a school each study chemistry and physics.
Their marks in an examination in each subject are recorded.
[Table_1]
\[\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline \text{Chemistry mark } (x) & 27 & 36 & 48 & 52 & 53 & 62 & 75 & 80 & 86 & 93 \\ \hline \text{Physics mark } (y) & 45 & 68 & 36 & 55 & 62 & 73 & 66 & 81 & 94 & 80 \\ \hline \end{array}\]
(a) What type of correlation is there between the chemistry mark and the physics mark?
.................................................... [1]
(b) Find
(i) the mean chemistry mark,
.................................................... [1]
(ii) the mean physics mark.
.................................................... [1]
(c) (i) Find the equation of the regression line for $y$ in terms of $x$.
\( y = .................................................... \) [2]
(ii) Another student scored 40 in the chemistry examination but was absent for the physics examination.
Estimate a physics mark for this student.
.................................................... [1]

(a) Write the number 25.0467
(i) correct to 1 decimal place,
................................................... [1]
(ii) correct to 3 significant figures,
................................................... [1]
(iii) correct to the nearest 10,
................................................... [1]
(iv) correct to the nearest 0.001,
................................................... [1]
(v) in standard form.
................................................... [1]

(b) Change
(i) 20 cm into metres,
................................................... m [1]
(ii) 20m$^2$ into square centimetres,
................................................. cm$^2$ [1]
(iii) 18 km/h into metres per second.
................................................... m/s [2]

(a) Solve the simultaneous equations.
You must show all your working.

$2x + 5y = -12$
$7x - 3y = -1$

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

(b) Solve $(4x - 1)(2x + 3) = -5$.
You must show all your working.

$x = \text{................. or } x = \text{..................}$ [5]

(a)
(i) Describe fully the single transformation that maps triangle $A$ onto triangle $B$.
...............................................................................................................................
............................................................................................................................... [3]
(ii) Describe fully the single transformation that maps triangle $A$ onto triangle $C$.
...............................................................................................................................
............................................................................................................................... [3]
(iii) On the grid, draw the stretch of triangle $A$, scale factor 2, $y$-axis invariant.
............................................................................................................................... [2]
(b) Describe fully the single transformation that is the inverse of
(i) a reflection in $y = 2$,
...............................................................................................................................
............................................................................................................................... [1]
(ii) a translation with vector $\begin{pmatrix} -5 \\ 2 \end{pmatrix}$.
...............................................................................................................................
............................................................................................................................... [2]

A, B, C \text{ and } D \text{ lie on a circle, centre } O. \text{ } AD = CD \text{ and } XBT \text{ is a tangent to the circle at } B. \text{ } TCD \text{ is a straight line. } \text{ Angle } XBA = 47^\circ \text{ and angle } TBC = 65^\circ.
\text{Find the value of}
\text{(a) angle } OBX,\quad \text{Angle } OBX = \text{..........................................} \;[1]
\text{(b) angle } AOB,\quad \text{Angle } AOB = \text{..........................................} \;[2]
\text{(c) angle } CAO,\quad \text{Angle } CAO = \text{..........................................} \;[2]
\text{(d) angle } CDA,\quad \text{Angle } CDA = \text{..........................................} \;[2]
\text{(e) angle } DAC,\quad \text{Angle } DAC = \text{..........................................} \;[2]
\text{(f) angle } CTB.\quad \text{Angle } CTB = \text{..........................................} \;[2]

Find the next term and the nth term in each of these sequences.

(a) 125, 64, 27, 8, 1, ...

Next term ..........................................
nth term ................................................. [3]

(b) 6, 12, 20, 30, 42, ...

Next term ..........................................
nth term ................................................. [4]

(a) On the diagram, sketch the graph of $y = f(x)$, for values of $x$ between $-1.5$ and $1.5$. [3]

(b) Write down the equation of the asymptote of the graph. ................................... [1]

(c) Solve the equation $f(x) = 2$ for values of $x$ between $-1.5$ and $0$.
$x = \text{..................}$ or $x = \text{..................}$ [2]

(d) Solve the inequality $f(x) + x^2 \leq 2$ for values of $x$ between $-1.5$ and $1.5$. ................................... [3]

$f(x) = \left|x^3 - \frac{1}{x}\right|$

AB C DEF \text{ is a triangular prism.}\
AB C D \text{ is a rectangle.}
\text{Find}
\text{(a) } AC,\quad AC = \text{.....................cm } [2]
\text{(b) } ED,\quad ED = \text{.....................cm } [2]
\text{(c) } \text{angle } EAD,\quad \text{Angle } EAD = \text{..................... } [2]
\text{(d) } \text{angle } FAC.\quad \text{Angle } FAC = \text{..................... } [2]

The diagram shows a solid made from a cuboid and a solid hemisphere. The cuboid measures 12 cm by 16 cm by 24 cm.
The hemisphere has radius 5 cm.

(a) Find
(i) the volume of the solid, ......................................... $\text{cm}^3$ [3]
(ii) the volume of a similar solid where the radius of the hemisphere is 3 cm.
......................................... $\text{cm}^3$ [2]
(b) Find
(i) the total surface area of the original solid, ......................................... $\text{cm}^2$ [3]
(ii) the total surface area of a similar solid where the radius of the hemisphere is 6 cm.
......................................... $\text{cm}^2$ [2]

ABC \text{ and } ADC \text{ are triangles.}
AD = 8.1 \text{ cm and } CD = 9.6 \text{ cm.}
\text{Angle } ABC = 46^\circ, \text{ angle } ADC = 78^\circ \text{ and angle } BAD = 15^\circ.
(a) \text{ Find } AC.
\[ AC = \text{...................................... cm} \] [3]
(b) \text{ Show that angle } DAC = 57^\circ, \text{ correct to the nearest degree.} [3]
(c) \text{ Find } BC.
\[ BC = \text{...................................... cm} \] [3]
(d) \text{ Find the area of quadrilateral } ABCD.
\[ \text{....................................... cm}^2 \] [4]

A bag contains 4 red balls, 5 black balls and 3 white balls only.
(a) In an experiment, one ball is chosen at random.
(i) Find the probability that the ball chosen is not black. [1]
(ii) This experiment is carried out 1440 times. Find the expected number of times the ball chosen is not black. [1]

(b) In a different experiment, one ball is chosen at random, the colour is noted, and the ball is replaced in the bag. Another ball is then chosen at random and the colour is noted.
Find the probability that the balls chosen are
(i) both white, [2]
(ii) both the same colour, [3]
(iii) different colours. [1]

(c) In another experiment, three balls are chosen at random without replacement.
(i) Find the probability that the first ball is not black, the second ball is black and the third ball is white. [3]
(ii) Find the probability that exactly two of the balls are red. [4]

Solve the equations.
(a) \( 6 - \frac{2}{x} = -2 \)
\( x = \text{.................................} \) [3]
(b) \( 3 + 2(4x + 5) = 1 - 2(x + 8) \)
\( x = \text{.................................} \) [3]
(c) \( 3 \log x + 2 \log 3 = 2 \log 6 + \log 2 \)
\( x = \text{.................................} \) [3]
(d) \( 2^x = 10 \)
\( x = \text{.................................} \) [3]