Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2019 Summer Zone 3

(a)
The diagram shows a cuboid.
The volume of this cuboid is 52.5 cm³.
Find the value of $h$.
$h = \text{.................................................}$ [2]

(b)
The diagram shows a pyramid.
The area of the base is 500 m².
The height of the pyramid is 27 m.
Find the volume of this pyramid.
............................................$m^3$ [2]

The table shows the marks of 10 students in a physics examination and a chemistry examination.

[Table_1]

There are 120 students at a school.
There are 30 students in each class.
The number of boys and the number of girls in each class is shown in the table.

[Table_1]

┌───────────┬───────────┬───────────┐
│ Class 1   │ Class 2   │ Class 3   │ Class 4   │
├───────────┼───────────┼───────────┼───────────┤
│ Boys      │ 16        │ 19        │ 12        │ 13        │
│ Girls     │ 14        │ 11        │ 18        │ 17        │
└───────────┴───────────┴───────────┴───────────┘

(a) On the diagram, sketch the graph of $y = f(x)$ where $$f(x) = \frac{1}{\cos x}$$ for values of $x$ between $-270$ and $270$. [3]

(b) Write down the range of $f(x)$. .................................................. [2]

(c) (i) On the same diagram, sketch the graph of $y = g(x)$ where $$g(x) = \frac{720 + x}{2x}$$ for values of $x$ between $-270$ and $270$. [2]

(ii) Find the values of the $x$ co-ordinates of the points of intersection of the two graphs.
$x = ext{................... or } x = ext{................... or } x = ext{...................}.$ [3]

(iii) Find the equation of each asymptote of the graph of $y = g(x)$.
............................................................... [2]

The Venn diagram shows the sets $A$, $B$ and $C$.



$U = \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13\}$
$A = \{\text{prime numbers}\}$
$B = \{\text{factors of 12}\}$
$C = \{\text{multiples of 3}\}$

(a) List the elements of set $A$. ............................................................... [1]

(b) Write all the elements of $U$ in the correct parts of the Venn diagram above. [3]

(c) List the elements of $(A \cup B)'$. ............................................................... [1]

(d) Find $n((B \cup C) \cap A')$. ............................................................... [1]

You may use this grid to help you answer this question.
The transformation P is a reflection in the line $y = x$.
The transformation Q is a rotation of $180^{\circ}$ about the origin.
The transformation R is a stretch, scale factor 2 with $x$-axis invariant.
The transformation S is a stretch, scale factor 2 with $y$-axis invariant.

(a) (i) Find the co-ordinates of the image of the point (5, 1) under the transformation P.
(..................... , .....................) [1]
(ii) Find the co-ordinates of the image of the point ($x$, $y$) under the transformation P followed by the transformation Q.
(..................... , .....................) [2]
(iii) Describe fully the single transformation equivalent to P followed by Q.
..........................................................................................
............................................................................................... [2]
(b) Describe fully the single transformation equivalent to R followed by S.
..........................................................................................
............................................................................................... [3]
(c) Describe fully the single transformation equivalent to the inverse of R.
..........................................................................................
............................................................................................... [2]

(a) Sergio invests $2000 at a rate of 3% per year compound interest.

(i) Find the value of his investment at the end of 5 years.

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

(ii) After how many complete years is the value of his investment greater than $4000?

................................................... \quad [3]

(b) Anna invests $2000 at a rate of 0.24% per month compound interest.

Find the value of her investment at the end of 5 years.

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

(c) Calculate the \textbf{monthly} compound interest rate that is equal to a compound interest rate of 3% per year.

...................................................% \quad [3]

(a) On the diagram, sketch the graph of $y = f(x)$, where
$$f(x) = |x^2 - 4|$$ for values of $x$ between $-3$ and $3$. [3]

(b) Write down the equation of the line of symmetry of the graph.
..................................................... [1]
(c) Write down the zeroes of $f(x)$.
..................... and .................. [1]
(d) (i) Find the value of $k$ when $y = k$ meets the curve $y = |x^2 - 4|$ three times.
$k =$ ..................................................... [1]
(ii) Find the range of values of $k$ when $y = k$ meets the curve $y = |x^2 - 4|$ four times.
..................................................... [2]

(a) Solve the following equations.
(i) \( \frac{135}{x} = 5 \)
\( x = \text{.............................................} \) [1]
(ii) \( 3x + 5 = 7x + 25 \)
\( x = \text{.............................................} \) [2]
(iii) \( 8x^2 = 11 - 2x \)
\( x = \text{.................} \text{ or } x = \text{.................} \) [4]

(b) Solve the following inequalities.
(i) \( 6 - 2x \geq 10 \)
\( \text{.............................................} \) [2]
(ii) \( \frac{1}{x-2} > 3 \)
\( \text{.............................................} \) [3]

(c) Solve the simultaneous equations. You must show all your working.
\( 3x + 5y = -3 \)
\( 5x - 2y = 26 \)
\( x = \text{.............................................} \)
\( y = \text{.............................................} \) [4]

(d) Solve the equation.
\( \log x + 4 \log 2 = \log 13 \)
\( x = \text{.............................................} \) [3]

The points $A (1, 2)$ and $B (7, 5)$ are shown on the diagram below.

(a) Write $\overrightarrow{AB}$ as a column vector. [1]
(b) Calculate the length of the line $AB$. [2]
(c) The point $C$ has co-ordinates $(10, k)$. $AB = BC$ and $k > 0$.
Show that $k = 11$. [3]
(d) Find the equation of the line that is perpendicular to $AC$ that passes through the midpoint of $AC$.
Give your answer in the form $y = mx + c$. [4]
(e) The points $A, B, C$ and $D$ form a rhombus.
Find the co-ordinates of $D$. [3]

The diagram shows four points $A$, $B$, $C$ and $D$ on horizontal ground.
There is a vertical flagpole, $FB$, held in place by straight wires $AF$, $CF$ and $DF$.
$BCD$ is a straight line, $AB = 5.5 \text{ m}$, $BC = 6.2 \text{ m}$ and angle $FAB = 41^\circ$.
(a) Show that $FB = 4.781 \text{ m}$, correct to 3 decimal places.

(b) Calculate angle $FCB$.
Angle $FCB = \text{.....................}$

(c) Angle $CDF = 18^\circ$.
Show that $CD = 8.514$, correct to 3 decimal places.

(d) Angle $ABC = 78^\circ$.
Find $AD$.
$AD = \text{......................... m}$

(e) Find the area of triangle $ABD$.
$\text{.......................................... m}^2$

(a) $y$ varies directly as the square root of $(x + 1)$.
$y = 8$ when $x = 24$.
(i) Find the value of $y$ when $x = 15$.
$y = \text{................................................} \: [3]$
(ii) Find the value of $x$ when $y = 16$.
$x = \text{................................................} \: [2]$

(b) Find the next term in each of the following sequences.
(i) 18, 13, 8, 3, $-2$, $\ldots$
\text{................................................} \: [1]
(ii) 3, 6, 11, 18, 27, $\ldots$
\text{................................................} \: [1]
(iii) $-1000$, 100, $-10$, 1, $\ldots$
\text{................................................} \: [1]
(iv) 0, 0, 0, 6, 24, 60, $\ldots$
\text{................................................} \: [2]