Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2024 Summer Zone 1

(a) Solve the equations.
(i) $3x - 2 = -14$
$x = \text{..............................................}$ [2]
(ii) $7x + 11 = 26 - 3x$
$x = \text{..............................................}$ [2]

(b) Solve the simultaneous equations.
You must show all your working.
$$\begin{align*} 5x + 3y &= -15 \\ 3x + 5y &= -17 \end{align*}$$
$x = \text{..............................................}$
$y = \text{..............................................}$ [4]

(c) Solve the inequality.
$$|2x + 1| > 9$$
$\text{..............................................}$ [4]

(a) The heights, $x$ cm, of 100 plants are shown in the table.
[Table: "Height (x cm) | 0 < x \leq 20 | 20 < x \leq 35 | 35 < x \leq 40 | 40 < x \leq 60 | 60 < x \leq 80\nFrequency | 7 | 13 | 20 | 32 | 28"]
(i) Calculate an estimate of the mean height of the plants.
................................................... cm [2]
(ii) (a) Complete the cumulative frequency table for the plants.
[Table: "Height (x cm) | x \leq 20 | x \leq 35 | x \leq 40 | x \leq 60 | x \leq 80\nCumulative frequency | 7 | | | | 100"] [1]
(b) On the grid, draw the cumulative frequency curve.
[Image: Cumulative frequency graph grid] [3]
(c) Use your cumulative frequency curve to find an estimate for the interquartile range.
................................................... cm [2]
(b) The heights, $h$ cm, of 50 different plants are shown in the table, where $k$ is an integer.
[Table: "Height (h cm) | Frequency\n0 < h \leq 20 | 25\n20 < h \leq k | 15\nk < h \leq 80 | 10"]
An estimate of the mean height of these plants is 27 cm.
Find the value of $k$.
$k = ..........................................................$ [3]

(a) Translate triangle $A$ with vector $\begin{pmatrix} 2 \\ -6 \end{pmatrix}$. Label the image $B$. [2]
(b) Describe fully the \textbf{single} transformation that maps triangle $B$ onto triangle $A$.
...........................................................................................................................
........................................................................................................................... [2]
(c) Rotate triangle $A$ through $90^\circ$ clockwise about $(0, 0)$. Label the image $C$. [2]
(d) Reflect triangle $A$ in the line $y = x$. Label the image $D$. [2]
(e) Describe fully the \textbf{single} transformation that maps triangle $C$ onto triangle $B$.
...........................................................................................................................
........................................................................................................................... [3]

(a) The price of a coat is $84.
The price is reduced by 12%.
Find the new price of the coat.

$ \text{................................................} \ [2]

(b) The price of a table is reduced by 25%.
The price is now $960.
Find the original price of the table.

$ \text{................................................} \ [2]

(c) Samir invests $600 in a bank that pays compound interest at a rate of 5.1% each year.

(i) Find the value of Samir’s investment after 4 complete years.

$ \text{................................................} \ [2]

(ii) Find the number of complete years for the value of Samir’s investment to be first worth more than $1000.

\text{................................................} \ [4]

(d) Amir and Bob work together and share their earnings in the ratio \ 3 : 5.

(i) Find the amount Bob receives when their earnings are $120.

$ \text{................................................} \ [2]

(ii) They decide to change the ratio for all further earnings.
Amir’s share of the earnings is increased by 20% of his original share.
Bob’s share of the earnings is decreased by 20% of his original share.
Show that the ratio of their earnings is now \ 9 : 10.

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

A, B, C \text{ and } D \text{ lie on a circle, centre } O.
AP \text{ is a tangent to the circle at } A.
OP \text{ is perpendicular to } AC \text{ and } AD \text{ is parallel to } BC.
\text{Angle } ABC = 61^\circ \text{ and angle } PAD = 22^\circ.

(a) \text{ Write down the mathematical name of the cyclic quadrilateral } ABCD.
\text{ ......................................................... [1]}

(b) \text{ Complete the statement.}
\text{Angle } OAP = 90^\circ \text{ because ..............................................................}
............................................................................................................................
\text{ ............................................................................................................................ [1]}

(c) \text{ Find}
(i) \text{ angle } ADC
\quad \text{Angle } ADC = \text{ .............................................................. [1]}
(ii) \text{ angle } ACD
\quad \text{Angle } ACD = \text{ .............................................................. [1]}
(iii) \text{ angle } ACB
\quad \text{Angle } ACB = \text{ .............................................................. [2]}
(iv) \text{ angle } OCA.
\quad \text{Angle } OCA = \text{ .............................................................. [2]}

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

\( f(x) = 2 - \lvert 1 - 0.5x^2 \rvert \)
(b) The graph cuts the $x$-axis at points $A$ and $B$.
Work out the length $AB$.
$$AB = \text{..................................................}$$ [2]
(c) Solve $f(x) = 0.5$.
\text{.................................................................} [2]
(d) Write down the coordinates of the minimum point of the graph.
$$ (\text{............................. , .............................}) $$ [1]
(e) The equation $f(x) = k$ has two solutions.
Find the range of values of $k$.
\text{.................................................................} [2]

(a) Spinner A and spinner B are each fair 5-sided spinners. Spinner A is numbered 1, 2, 2, 3, 4. Spinner B is numbered 1, 2, 3, 4, 4. The two spinners are each spun once and the number on each spinner is recorded. Find the probability that (i) the number on spinner A is 6 .................................................. [1]
(ii) the number on spinner B is not 4 .................................................. [1]
(iii) the number on spinner A is the same as the number on spinner B ................................................. [3]
(iv) the sum of the two numbers is 6. .................................................. [3]

(b) (i) On the Venn diagram, shade $A \cup B$. [1]
(ii) Describe the shaded region using set notation. .................................................. [1]
(iii) The Venn diagram below shows the number of elements in each subset. Find $n((A \cap B) \cap C')$. .................................................. [1]

The diagram shows a shape $ABDC$ formed from triangle $ABC$ and a sector of a circle $BCD$, centre $B$.

(a) Show that $BC = 9.0 \text{ m}$, correct to 1 decimal place. [3]

(b) Use the sine rule to find angle $BCA$.

Angle $BCA = \text{.............................................}$ [3]

(c) Find the area of triangle $ABC$.

\text{.............................................} \text{ m}^2 [2]

(d) Find the area of the shaded region.

\text{.............................................} \text{ m}^2 [3]

(e) Find the perimeter of the shape $ABDC$.

\text{.............................................} \text{ m} [2]

The diagram shows a solid cone with base radius 40 cm and slant height 60 cm.
(a) Find the volume of the cone.
........................................ cm³ [3]
(b) Show that the total surface area of the cone is $4000\pi \text{ cm}^2$. [2]
(c) A mathematically similar cone has a surface area of $1000\pi \text{ cm}^2$.
Show that the radius of this cone is 20 cm. [2]
(d) A cone with radius 20 cm is removed from the top of the cone with radius 40 cm to leave a solid.
Calculate the surface area of the remaining solid.
........................................ cm² [3]

y varies inversely as the square root of (x + 1).
y = 18 when x = 3.
(a) (i) Find the value of y when x = 8.
y = ................................................ [3]
(ii) Find the value of x when y = 1.5.
x = ................................................ [2]
(b) w varies directly as the square root of (x + 1).
w = 18 when x = 3.
Find the value of \( \sqrt{wy} \).
\( \sqrt{wy} = ................................................ \) [3]

f(x) = 3x - 1 \quad g(x) = 5 - 2x \quad h(x) = \frac{1}{2x - 3}, \quad x \neq 1.5
(a) Find \ f(4). ............................................\ [1]
(b) Solve \ \ f(x) = -7. ............................................\ [2]
(c) Find \ g^{-1}(x). \quad g^{-1}(x) = ............................................\ [2]
(d) Solve \ g(x) = 7h(f(x)).
You must show all your working. \quad x = ............................................\ [6]