Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2024 Winter Zone 2

(a) The line $2x+3y = 12$ crosses the $x$-axis at $P$ and the $y$-axis at $Q$.
Find the coordinates of $P$ and the coordinates of $Q$.

$P ( ext{.....................} , ext{.....................} )$
$Q ( ext{.....................} , ext{.....................} )$ [2]

(b) (i) $A$ is the point $(-2, 5)$ and $B$ is the point $(4, -1)$.
Find the column vector of the translation from point $A$ to point $B$.

$$ \begin{pmatrix} \text{ } \\ \text{ } \end{pmatrix} $$ [1]

(ii) Find the coordinates of the mid-point of $AB$.

$( ext{.....................} , ext{.....................} )$ [2]

(c) Find the magnitude of $\begin{pmatrix} 7 \\ 9 \end{pmatrix}$.

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

(d) The line $L$ has gradient $3$ and passes through the point $(-2, 7)$.
Find the equation of line $L$.
Give your answer in the form $y = mx + c$.

$y = ext{..............................................}$ [2]

(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $-5$ and $3$. [2]
(b) On the diagram, draw the asymptote to the graph of $y = f(x)$. [1]
(c) Solve the equation $f(x) = -1$. ............................................................... [2]
(d) (i) Solve the equation $f(x) = x^2 - 1$. ............................................................... [2]
(ii) Solve the inequality $f(x) < x^2 - 1$. .............................................................. [2]
$f(x) = \frac{4}{(x+2)} - x$

(a) Write $1 \frac{1}{25}$ as a percentage. ............................................. \% [1]

(b) In a sale, all prices are reduced by 15\%. Lola buys a jacket which has an original price of \$64.
Calculate the sale price of this jacket. \$ ............................................... [2]

(c) Nina invests \$x at a rate of 1.8\% per year compound interest.
At the end of 3 years the value of this investment is \$3375.93, correct to the nearest cent.
Calculate the value of $x$.
$x = ...............................................$ [2]

(d) Olav buys a car for \$13000.
Each year the value of the car decreases by 12\% of its value in the previous year.
Calculate the number of complete years it takes for the value of Olav’s car to first become less than \$5000.
............................................... [4]

(a)

The diagram shows a solid metal shape made from a cylinder and two hemispheres.
The radius of the cylinder and of the hemispheres is 3 cm.
The length of the cylinder is 15 cm.
(i) Show that the total volume of the shape is 537 cm$^3$, correct to 3 significant figures. [3]
(ii) The shape is melted and all the metal is used to make 600 identical small cubes.
Calculate the side length of one of these cubes. Give your answer in millimetres. [3]
.......................................... mm

(b)

The diagram shows a sector $OAB$ with radius $r$ cm and centre $O$.
The sector angle is 120°.
The shaded segment has an area of 18.4 cm$^2$.
Calculate the length of the arc $AB$.
.......................................... cm

(a) $v^2 = u^2 - 2as$
(i) Find the value of $v$ when $u = 7$, $a = 1.5$ and $s = 10$.
$v = \text{...................................................}$ [2]
(ii) Rearrange the formula to write $u$ in terms of $v$, $a$ and $s$.
$u = \text{...................................................}$ [2]
(b) Complete the table for sequences $A$, $B$ and $C$.
[Table_1]
$$ \begin{array}{|c|c|c|c|c|c|c|} \hline \text{Sequence} & \text{1st term} & \text{2nd term} & \text{3rd term} & \text{4th term} & \text{5th term} & n\text{th term} \\ \hline A & 11 & 8 & 5 & 2 & & \\ \hline B & 3 & 8 & 15 & & & n^2 + 2n \\ \hline C & 4 & 1 & \frac{1}{4} & \frac{1}{16} & & \\ \hline \end{array} $$

(a) Describe fully the \textit{single} transformation that maps

(i) triangle $T$ onto triangle $P$

.................................................................................................................................
................................................................................................................................. [3]

(ii) triangle $T$ onto triangle $Q$.

.................................................................................................................................
................................................................................................................................. [3]

(b) Stretch triangle $T$ by factor 2 with invariant line $x = 5$. [2]

(a) $-3 < x \leq 1$
Show this inequality on the number line.


(b) (i) Solve the inequality $-7 \leq 2x + 1 < 5$.
...................................................... [2]
(ii) Write down the integers that satisfy the inequality $-7 \leq 2x + 1 < 5$.
.................................................................... [2]

(c) Solve.
$7(x - 3) - 3(2x + 1) = 1$
$x =$ .............................................. [3]

(d) $\frac{1}{y} + \frac{8}{y + 5} = 1$
(i) Show that $y^2 - 4y - 5 = 0$.
...................................................... [3]
(ii) Solve by factorisation.
$y^2 - 4y - 5 = 0$
$y = ............... \text{ or } y = ............... [3]$

The diagram shows four points, $A$, $B$, $C$ and $D$, on horizontal ground.
(a) Calculate $AD$.
\[ AD = \text{.............................} \text{ m} \] [2]
(b) Calculate angle $DAB$.
\[ \text{Angle } DAB = \text{.............................} \] [2]
(c) Calculate $CD$.
\[ CD = \text{.............................} \text{ m} \] [3]
(d) The point $P$ lies on $BC$ and is the nearest point to $D$.
Calculate $BP$.
\[ BP = \text{.............................} \text{ m} \] [3]
(e) $D$ is due north of $B$.
Calculate the bearing of $C$ from $A$.
\[ \text{.............................} \] [5]

f(x) = x^3 + 1 \quad g(x) = \tan x \quad h(x) = 3^x + 1
(a) Find f(-2).
................................................... [1]
(b) Find the exact value of g(120).
................................................... [1]
(c) On the diagram, sketch the graph of \( y = g(x) \) for values of \( x \) between \( 0^\circ \) and \( 180^\circ \). [2]
(d) Find \( x \) when \( h(x) = 82 \).
\( x = ................................................... \) [2]
(e) Find \( h^{-1}(x) \).
\( h^{-1}(x) = ................................................... \) [2]
(f) Simplify fully.
\[ f(x) - \frac{1}{f(x)} \]
Give your answer as a single fraction.
................................................... [3]

(a) Zola measures the height of each of 100 plants in her garden. The table shows her results.
[Table_1]

Height $(\text{h cm})$ | $0 < h \leq 10$ | $10 < h \leq 15$ | $15 < h \leq 20$ | $20 < h \leq 30$ | $30 < h \leq 60$
Frequency | 13 | 21 | 25 | 19 | 22

(i) Calculate an estimate of the mean. ........................................ cm [2]

(ii) One of the plants is chosen at random.
Find the probability that the plant has a height greater than 15 cm.
........................................ [1]

(iii) Two of the 100 plants are chosen at random without replacement.
Find the probability that one plant has a height of 15 cm or less
and one has a height greater than 30 cm. ........................................ [3]

(b) 50 students are asked if they like football $(F)$ and if they like swimming $(S)$.

3 do not like football and do not like swimming.
38 like football.
16 like swimming.
(i) Complete the Venn diagram.
[2]

(ii) Write down the number of students who like football and swimming.
........................................ [1]

(iii) One of the 50 students is chosen at random.
Find the probability that this student likes football or swimming but not both.
........................................ [1]

(iv) Two of the students who like swimming are chosen at random.
Find the probability that they both like football.
........................................ [2]

(a) $a$ is a positive integer.
Rationalise the denominator.
\[ \frac{3}{\sqrt{a} - 1} \]
\text{..................................................} \; [2]

(b) \((g + h\sqrt{3})(h - g\sqrt{3}) = p + q\sqrt{3}\)
Find $p$ and $q$ in terms of $g$ and $h$.

\[ p = \text{...............................................} \]
\[ q = \text{...............................................} \] \; [3]

(c) (i) $d$ is an integer.
Work out $3 \times 10^d + 3 \times 10^{d-2}$, giving your answer in standard form.
\text{..................................................} \; [2]

(ii) Find \( \sqrt[3]{8 \times 10^{2000}} \).
Give your answer in standard form.
\text{..................................................} \; [2]