Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2018 Winter Zone 1

(a) Solve the following equations.

(i) \( 12 - x = 4 \)
\( x = \text{.....................................................} \) [1]

(ii) \( 9x - 4 = 6x + 8 \)
\( x = \text{.....................................................} \) [2]

(iii) \( \frac{12}{x} + 5 = 9 \)
\( x = \text{.....................................................} \) [2]

(b) (i) Solve \( 6x^2 - 5x + 1 = 0 \).
\( x = \text{....................} \text{ or } x = \text{....................} \) [3]

(ii) Use your answer to \textit{part (b)(i)} to solve
\( 6\sin^2 x - 5\sin x + 1 = 0 \quad \text{for} \quad 0^{\circ} \leq x \leq 90^{\circ}. \)
\( x = \text{....................} \text{ or } x = \text{....................} \) [3]

The table shows the marks for 75 students in a test.

[Table_1]

(a) Write down the mode. ..................................................... [1]
(b) Find the range. ..................................................... [1]
(c) Find the median. ..................................................... [1]
(d) Find the inter-quartile range. ..................................................... [2]
(e) Calculate an estimate of the mean. ..................................................... [2]
(f) Give a reason why your answer to part (e) is an estimate. ..................................................... [1]
(g) Two of these students are chosen at random. Find the probability that the highest mark of these students is 2. ..................................................... [3]

Given the function $f(x) = 1 - 2 \sin(2x - 10)^{\circ}$:

(a) On the diagram sketch the graph of $y = f(x)$, for $-90 \leq x \leq 90$.

[3]

(b) Write down the co-ordinates of the x-intercepts.
$(\text{................} , \text{..............})$
[2]

(c) Write down the co-ordinates of the local maximum.
$(\text{................} , \text{..............})$
[1]

(d) The graph of $y = -\frac{x}{60}$ intersects the graph of $y = 1 - 2 \sin(2x - 10)^{\circ}$ three times.
Find the value of the x co-ordinate at each point of intersection.
$x = \text{..................} \text{ or } x = \text{.......................... or } x = \text{....................}$
[3]

ABCD is a rectangle.
The equation of the line $AB$ is $4x + 3y = 24$.
(a) Find the co-ordinates of
(i) point $A$, (............ , ............ ) [1]
(ii) point $B$, (............ , ............ ) [1]
(iii) the midpoint of $AB$. (............ , ............ ) [2]
(b) Rearrange the equation $4x + 3y = 24$ to make $y$ the subject.
$y = ..................................................$ [2]
(c) Find the equation of the line $BC$. Give your answer in the form $y = mx + c$.
$y = ..................................................$ [3]
(d) Find the co-ordinates of
(i) point $C$, (............ , ............ ) [1]
(ii) point $D$. (............ , ............ ) [3]

The number of fish in a lake decreases by 4\% each year. In January 2018 there are 30000 fish in the lake.
(a) Calculate the number of fish in the lake in

(i) January 2019, .......................................................... [2]

(ii) January 2029, .......................................................... [3]

(iii) January 2017. .......................................................... [3]

(b) Find the last year in which there were at least 50000 fish in the lake. .......................................................... [4]

(c) Philip runs a fishing business and he works 50 weeks every year. In 2018, he catches 800 kg of fish in each of these weeks. He sells all the fish he catches at a price of $\$3.50$ for each kilogram.
(i) Calculate the total amount he receives in 2018. $ .......................................................... [3]

(ii) For each of the 50 weeks, Philip’s business costs $2240 to run. Calculate his profit as a percentage of $2240. .......................................................... \% [3]

(d) In 2019, Philip’s business costs 8\% more to run than in 2018. The selling price of fish decreases by 10\%.
Find the amount of fish, in kilograms, Philip will need to catch each week to keep the percentage profit found in part (c)(ii) the same. .......................................................... kg [4]

(a) Reflect triangle $A$ in the line $x = -2$. Label the image $B$.
[2]
(b) Rotate triangle $A$ through $180^\circ$ about $(-2, -1)$. Label the image $C$.
[2]
(c) Describe fully the single transformation that maps triangle $C$ onto triangle $B$.
..............................................................................................
.............................................................................................. [2]
(d) Enlarge triangle $A$ with centre of enlargement $(1, 2)$ and scale factor $2$. Label the image $D$.
[2]

(a) Find an expression for the $n^{th}$ term for each of these sequences.
(i) 80, 77, 74, 71, ... ......................................................... [2]
(ii) 128, 64, 32, 16, ... ......................................................... [2]

(b) The $n^{th}$ term of a sequence is $n^2 - 1$.
Find the first four terms of this sequence.
............... , .............. , .............. , .............. [2]

(c) The $n^{th}$ term of a sequence is $|n - 3|$.
Find the first four terms of this sequence.
............... , .............. , .............. , .............. [2]

(d) The $n^{th}$ term of a sequence is $n^2 + n + 41$.
(i) Find the first three terms of this sequence.
............... , .............. , .............. [2]
(ii) Show that when $n = 41$ the number in this sequence is not prime. [1]

A, B, C \text{ and } D \text{ lie on a circle, centre } O.\newline ST \text{ is a tangent to the circle at } A.\newline ODT \text{ is a straight line that bisects angle } AOC.\newline\newline(a) \text{ Complete the statement.}\newline\newline \text{Angle } OAT = \text{ .......... because .................................................................} \newline............................................................................................................................ [2]\newline\newline(b) \quad DT = OC\newline\newline\text{Find angle } ABC.\newline\newline \text{Angle } ABC = .......................................................... [4]

f(x) = x^3 - 3x^2 - 4x + 1 \text{ for } -3 \leq x \leq 5.

(a) On the diagram, sketch the graph of \( y = f(x) \). [2]

(b) Write down the co-ordinates of the local minimum.
(\text{.................} , \text{.................}) [2]

(c) Find the range of values of \( k \) so that \( f(x) = k \) has only one solution.
\text{.................................................................} [2]

(d) \( g(x) = 3x^2 - 6x - 4 \text{ for } -3 \leq x \leq 5. \)
The graph of \( y = f(x) \) intersects the graph of \( y = g(x) \) twice.
Solve \( f(x) > g(x) \).
\text{..................................................} [2]

[Image: Coordinate Plane]

OAC is a triangle with $AB : BC = 1 : 2$ and $OD : DC = 1 : 2$.
The lines $OB$ and $AD$ intersect at $X$.
$$\overrightarrow{OA} = 6a \quad \text{and} \quad \overrightarrow{OC} = 6c.$$

(a) Find an expression, in terms of $a$ and/or $c$, for
 (i) $\overrightarrow{AC}$,
  $\overrightarrow{AC} = \text{.........................................................}$ [1]
 (ii) $\overrightarrow{BC}$,
  $\overrightarrow{BC} = \text{.........................................................}$ [1]
 (iii) $\overrightarrow{BD}$, giving your answer in its simplest form.
  $\overrightarrow{BD} = \text{.........................................................}$ [2]

(b) Use your answer to part (a)(iii) to explain why $OA$ and $BD$ are parallel.
 \text{..........................................................................................................................................................................} [1]

(c) Explain why triangle $OAX$ and triangle $BDX$ are similar.
 \text{..........................................................................................................................................................................}
 \text{..........................................................................................................................................................................} [2]

(d) Find an expression, in terms of $a$ and $c$, for
 (i) $\overrightarrow{AD}$,
  $\overrightarrow{AD} = \text{.........................................................}$ [2]
 (ii) $\overrightarrow{XD}$, giving your answer in its simplest form.
  $\overrightarrow{XD} = \text{.........................................................}$ [2]

(e) Find the ratio area $AXO :$ area $BXD$.
 ................ : ................ [2]

The area of triangle $ABC = 23.5 \text{ cm}^2$.

(a) Show that angle $BAC = 36.0^\circ$, correct to 1 decimal place.

[2]

(b) Use the cosine rule to find $BC$.

$BC = \text{........................................ cm}$ [3]

(c) All the angles in triangle $ABC$ are acute.
Use the sine rule to find the largest angle in the triangle $ABC$.

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