Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2022 Winter Zone 3

(a) Rotate triangle $T$ through $90^\circ$ clockwise about the point $(9, 6)$. [2]
(b) Enlarge triangle $T$ with scale factor $\frac{1}{2}$, centre $(0, 0)$. [2]
(c) Describe fully the \textit{single} transformation that maps triangle $T$ onto triangle $P$.
...................................................................................................................
............................................................................................................. [2]
(d) Describe fully the \textit{single} transformation that maps triangle $T$ onto triangle $Q$.
...................................................................................................................
............................................................................................................. [3]

Consider the function $f(x) = \frac{1}{x} - \frac{1}{x^2}$.

(a) On the diagram, sketch the graph of $y = f(x)$ for values of $x$ between $-5$ and $5$. [2]

(b) Find $f(-2)$.
............................................... [1]

(c) Solve the equation $f(x) = 0$.
$x = $ ............................................... [1]

(d) Find the maximum value of $f(x)$.
............................................... [1]

(e) Write down the equation of each asymptote.
................................................................. [2]

(f)
(i) Solve the equation $\frac{1}{x} - \frac{1}{x^2} = x^2 - 2$.
............................................................................. [3]

(ii) The equation $\frac{1}{x} - \frac{1}{x^2} = x^2 - 2$ can be rearranged to the form $x^4 + ax^2 + bx + c = 0$.
Find the values of $a$, $b$ and $c$.

$a = $ ...............................................
$b = $ ...............................................
$c = $ ............................................... [2]

(a) Amira buys a magazine that costs $n and a book that costs $(2n + 5).
She pays with a $20 note and receives $1.62 change.

Find the cost of a magazine.

$ .............................................

(b) The cost of a bar of chocolate is $x and the cost of a bag of sweets is $y.

Bruce buys 2 bars of chocolate and 1 bag of sweets for a total of $3.60.
Charlie buys 3 bars of chocolate and 2 bags of sweets for a total of $6.05.

Find the total cost of 1 bar of chocolate and 3 bags of sweets.
You must show all your working.

$ .............................................

Complete the table for the 5th term and the nth term of each sequence.

[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} & \text{nth term} \\ \hline A & 3 & 5 & 7 & 9 & & \\ \hline B & 1 & 8 & 27 & 64 & & \\ \hline C & \frac{1}{4} & \frac{1}{2} & 1 & 2 & & \\ \hline D & 0 & 2 & 6 & 12 & & \\ \hline \end{array}[11]

(a) Kris and Laila share $200 in the ratio 2 : 3.
(i) Show that Kris receives $80. [1]

(ii) Kris spends 30.8\% of his $80 on a book.
Calculate the cost of the book.
$ \text{................................................} [2]

(iii) Laila invests her $120 at a rate of 1.16\% per year simple interest.
Calculate the total amount Laila has at the end of 5 years.
$ \text{................................................} [3]

(b) On 1 January 2020, Sangita invests an amount of money at a rate of 2\% per year compound interest.
On 1 January 2023 the value of the investment is $5306.04.
(i) Calculate the amount Sangita invested on 1 January 2020.
$ \text{................................................} [2]

(ii) Calculate the value of the investment on 1 January 2025.
$ \text{................................................} [2]

(c) Tomas invests an amount of money at a rate of 1.4\% per year compound interest.
Find the number of complete years it takes for the value of his investment to increase by 50\%.
\text{................................................} [4]

(a) \ \mathbf{p} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}, \ \mathbf{r} = \begin{pmatrix} -1 \\ 7 \end{pmatrix}
(i) \ Find \ 2\mathbf{p}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\ \ ) [1]
(ii) \ Find \ \frac{1}{4}\mathbf{p} - \mathbf{r}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\ \ ) [2]
(iii) \ Find \ the \ magnitude \ of \ \mathbf{p}. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\ \ ) [2]

(b) \ K \ is \ the \ point \ (3, \ 4).
(i) \ The \ vector \ from \ K \ to \ L \ is \ \begin{pmatrix} -1 \\ 1 \end{pmatrix}.
Find \ the \ coordinates \ of \ L. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\ .........., \ ..........) [1]
(ii) \ The \ vector \ from \ J \ to \ K \ is \ \begin{pmatrix} 5 \\ -2 \end{pmatrix}.
Find \ the \ coordinates \ of \ J. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\ .........., \ ..........) [1]

(c) \ A \ is \ the \ point \ (-1, \ 3) \ and \ B \ is \ the \ point \ (5, \ 7).
The \ perpendicular \ bisector \ of \ the \ line \ AB \ meets \ the \ x-axis \ at \ C.
Find \ the \ coordinates \ of \ C. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\ .........., \ ..........) [7]

(a) The time, $t$ hours, spent watching television in one week by each of 100 students is shown in the table.

[Table]
| Time, $t$ hours | $0 < t \leq 10$ | $10 < t \leq 20$ | $20 < t \leq 25$ | $25 < t \leq 30$ | $30 < t \leq 60$ |
|-----------------|------------------|------------------|------------------|------------------|------------------|
| Frequency | 3 | 11 | 42 | 40 | 4 |

(i) A pie chart is drawn to show the results.
Calculate the sector angle for the number of students who spend more than 30 hours watching television.
.................................................... [2]

(ii) Calculate an estimate of the mean.
............................................... h [2]

(b) A shopkeeper records the midday temperature, $t$ °C, and the number of ice creams, $n$, sold each day in one week.
The table shows the results.

[Table]
| Midday temperature, $t$ °C | 20 | 24 | 20 | 17 | 18 | 20 | 25 |
|----------------------------|----|----|----|----|----|----|----|
| Number of ice creams, $n$ | 103 | 106 | 95 | 91 | 93 | 98 | 114 |

(i) Write down the type of correlation shown in the table.
..................................................... [1]

(ii) Find the equation of the regression line, giving $n$ in terms of $t$.
$$ n = ..................................................... $$ [2]

(iii) Use your answer to part(b)(ii) to find the number of ice creams expected to be sold when the midday temperature is 22 °C.
..................................................... [1]

(iv) During this week, the shopkeeper sells 700 ice creams.
She estimates that she will sell a total of 9800 ice creams during the next 14 weeks.
Give a reason why this may not be a good estimate.
............................................................................................................................. [1]

(c) When the weather is fine, the probability that Lance goes cycling is $\frac{7}{9}$.
When the weather is not fine, the probability that Lance goes cycling is $\frac{1}{5}$.
The probability that the weather is fine is $\frac{3}{4}$.

(i) Complete the tree diagram.

[Tree Diagram]
Weather  Cycles
   ❱⠅ Yes
  ⠅ Fine❱ $\frac{7}{9}$
  ⠅ $\frac{3}{4}$❱ .......... No
  ⠅ Not fine❱ .......... Yes
  ⠅ .......... No
[2]

(ii) Find the probability that Lance goes cycling.
..................................................... [3]

The diagram shows a pentagon $ABCDE$ and diagonals $BD$ and $BE$.
(a) (i) Calculate angle $BCD$.
Angle $BCD = \text{..................................................}$ [1]
(ii) Calculate $BC$.
$BC = \text{.......................................... cm}$ [3]
(b) Calculate angle $EBD$.
Angle $EBD = \text{..............................................}$ [3]
(c) Calculate the area of the pentagon $ABCDE$.
$\text{........................................ cm}^2$ [4]
(d) Calculate the shortest distance from $C$ to $AE$.
$\text{.......................................... cm}$ [4]

(a) $f(x) = 2x + 3$, $g(x) = x^2 + 1$, $h(x) = 2 \sin(2x)$
(i) Find $f(-2)$.
....................................................... [1]
(ii) Find $f^{-1}(x)$.
$f^{-1}(x) = .......................................................$ [2]
(iii) Find $x$ when $g(x) = 2 f(x)$.
$x = .............$ or $x = .............$ [3]
(iv) Find $g(f(x))$, giving your answer in the form $ax^2 + bx + c$.
....................................................... [3]
(v) Find the amplitude and period of $h(x)$.
Amplitude = .............................
Period = ............................. [2]
(vi) Solve the equation $h(x) = \sqrt{3}$ for $0^\circ \leq x \leq 180^\circ$.
....................................................... [2]
(b) $j(x) = \log_a x$, $x > 0$
(i) Find the value of $j(\sqrt[3]{a})$.
....................................................... [1]
(ii) Find $j^{-1}(x)$.
$j^{-1}(x) = .......................................................$ [2]

(a) A machine lays a pipe of length 2.5 km in 18 hours. The machine always works at the same rate.
Calculate the time it takes to lay a pipe of length 4 km.
..................................... hours [2]
(b) $t$ varies inversely as the square root of $x$. $x$ varies directly as the square of $y$.

When $x = 4$, $t = 3$.
When $y = 4$, $x = 81$.

$ty = h$
Find the value of $h$.

$h = ........................................ [5]$