Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2015 Winter Zone 2

(a) By writing each number correct to 1 significant figure, find an estimate for $$ \frac{\sqrt[3]{987}}{5.13} + \frac{(16.3 + 1.91^2)}{\sqrt{9.12}}. $$
You must show your working.

Answer(a) .......................................................... [2]

(b) Explain why your answer to part (a) is greater than the actual answer.

Answer(b) ........................................................................................................................................... ....................................................................................................................................................................... [2]

(c) Work out.
$$ \frac{\sqrt[3]{987}}{5.13} + \frac{(16.3 + 1.91^2)}{\sqrt{9.12}}. $$

Answer(c) .......................................................... [1]

(a) Solve the equations.
(i) $4 \log 3 - 3 \log 4 = \log x - 5 \log 2$ [3]
Answer(a)(i) $x = \text{...........................................................}$
(ii) $4 \sin x + 3 = 1$ for $0^\circ \leq x \leq 360^\circ$ [3]
Answer(a)(ii) $\text{............................................................}$
(b) Make $x$ the subject of the formula.
$a = \sqrt{\frac{x - 1}{x}}$ [3]
Answer(b) $x = \text{...........................................................}$

The table gives the marks of 10 students in a geography exam and a history exam.

[Table_1]

Geography mark (x): 12, 23, 36, 41, 57, 62, 78, 81, 89, 93
History mark (y): 32, 43, 41, 51, 52, 60, 68, 65, 76, 80

(a) Find

(i) the mean geography mark,

Answer(a)(i) ...............................................................[1]

(ii) the mean history mark.

Answer(a)(ii) ...............................................................[1]

(b) (i) Find the equation of the regression line for $y$ in terms of $x$.

Answer(b)(i) $y = $ .......................................................[2]

(ii) Estimate the history mark when the geography mark is 51.

Answer(b)(ii) ...............................................................[1]

The transformation P is a reflection in the $x$-axis.
The transformation Q is a rotation of $90^\circ$ clockwise about the origin.

(a) Write down the transformation that is

(i) the inverse of P,

Answer(a)(i) ..................................................................................................................................................
...................................................................................................................................................................... [1]

(ii) the inverse of Q.

Answer(a)(ii) ...............................................................................................................................................
....................................................................................................................................................................... [2]

(b) Describe fully the \textit{single} transformation equivalent to P followed by Q.

Answer(b) ................................................................................................................................................
....................................................................................................................................................................... [2]

Find the next term and the $n^{th}$ term in each of the following sequences.
(a) 27, 20, 13, 6, -1, ...
Answer(a) next term = ...................................................
$n^{th}$ term = ....................................................... [3]
(b) 1024, 512, 256, 128, 64, ...
Answer(b) next term = ...................................................
$n^{th}$ term = ................................................... [3]

The marks, $x$, of 800 students in a mathematics exam are given in the table.

[Table_1]
| Mark $(x)$ | Frequency |
| --------- | --------- |
| $0 < x \leq 20$ | 62 |
| $20 < x \leq 30$ | 84 |
| $30 < x \leq 40$ | 140 |
| $40 < x \leq 50$ | 160 |
| $50 < x \leq 60$ | 142 |
| $60 < x \leq 80$ | 112 |
| $80 < x \leq 100$ | 100 |

(a) Calculate an estimate of the mean mark.
Answer(a) ................................................................. [2]
(b) Complete the cumulative frequency table.

[Table_2]
| Mark $(x)$ | Cumulative frequency |
| --------- | ------------------ |
| $0 < x \leq 20$ | 62 |
| $0 < x \leq 30$ |
| $0 < x \leq 40$ |
| $0 < x \leq 50$ |
| $0 < x \leq 60$ |
| $0 < x \leq 80$ |
| $0 < x \leq 100$ | 800 | [1]

(c) On the grid below, draw a cumulative frequency curve.
[Graph_1] [3]
(d) Use your graph in part (c) to find estimates for
(i) the median mark,
Answer(d)(i) ................................................................. [1]
(ii) the interquartile range,
Answer(d)(ii) ................................................................. [2]
(iii) the minimum mark for a candidate to obtain a grade A, given that 15% of students gain a grade A.
Answer(d)(iii) ................................................................. [3]

(a) (i) On the diagram, sketch the graph of $y = f(x)$, for values of $x$ between $x = -4$ and $x = 6$. [2]
(ii) Write down the equations of the asymptotes.
Answer(a)(ii) ..........................., ........................... [2]
(iii) Write down the co-ordinates of the points where the graph crosses the axes.
Answer(a)(iii) (.......... , ..........), (.......... , ..........) [2]
(b) Solve the inequality.
$$x < \frac{(6x + 11)}{(2x - 3)}$$
Answer(b)................................................................................................. [4]

Freddo lives in Manchester. He drives to Cambridge for a meeting.
The distance from Manchester to Cambridge is 300km.

(a) Freddo leaves Manchester at 07 05 and arrives in Cambridge at 10 50.
Calculate his average speed.

Answer(a) ..........................................................km/h [3]

(b) After the meeting Freddo drives back to Manchester.
His average speed for this journey is 5\% more than his average speed driving to Cambridge.
He leaves Cambridge at 17 45.
Find the time Freddo arrives in Manchester.

Answer(b) .............................................................. [3]

(c) Freddo's car uses fuel at the rate of 8.1 km per litre.
Fuel costs £1.45 per litre.
Find the total cost of fuel for Freddo's journey from Manchester to Cambridge and back to Manchester.

Answer(c) £ ................................................................. [2]

(a) A coat costs $100.
The price is increased by 10\% and then decreased by 10\%.
Find the new price of the coat.

Answer (a) $ \text{...............................................................} [2]

(b) A chair costs $1000.
The price is increased by 20\% and then decreased by 20\%.
Find the new price of the chair.

Answer (b) $ \text{..............................................................} [2]

(c) A car costs $10\ 000.
The price is increased by $x\%$ and then decreased by $x\%$.
Find an expression, in terms of $x$, for the new price of the car.
Give your answer in its simplest form.

Answer (c) $ \text{...............................................................} [3]

A bag contains 3 red balls and 5 blue balls. In an experiment, three balls are chosen at random without replacement.
(a) Find the probability that the three balls chosen are
(i) all red,

Answer(a)(i) .......... [2]
(ii) two red and one blue,

Answer(a)(ii) .......... [3]
(iii) at least one of each colour.

Answer(a)(iii) .......... [3]
(b) This experiment is to be carried out 1680 times. Find the expected frequency of 3 red balls being chosen.

Answer(b) .......... [2]

A is the point (2, 6) and C is the point (5, 4).
The equation of the line $AB$ is $y + 4x = 14$.
The equation of the line $BC$ is $y = x - 1$.
(a) $B$ is the point where the lines $AB$ and $BC$ intersect.
Find the co-ordinates of the point $B$.
Answer(a) ( \text{.....................} , \text{.....................} ) [3]
(b) $M$ is the midpoint of $AC$.
Find the co-ordinates of $M$.
Answer(b) ( \text{.....................} , \text{.....................} ) [2]
(c) Find the equation of the line $BM$.
Answer(c) .............................................................. [3]
(d) The point $D$ lies on the line $BM$.
The co-ordinates of $D$ are $(k, k + 9)$.
Find the value of $k$.
Answer(d) $k =$ ................................................... [2]

In the diagram, $ABC$ is a straight line and $BFED$ is a rectangle.
(a) Find $BC$.
Answer(a) ................................................................. cm [3]
(b) Show that angle $DBC = 34.7^\circ$, correct to 3 significant figures. [3]
(c) Find the perimeter of the quadrilateral $ACDE$.
Answer(c) ................................................................. cm [4]
(d) Find the area of the quadrilateral $ACDE$.
Answer(d) ................................................................. cm${}^2$ [3]

(a) (i) On the diagram, sketch the graph of $y = f(x)$, for $0 \leq x \leq 5$.
(ii) Write down the $x$ co-ordinate of the point where the graph crosses the $x$-axis.
(iii) Write down the range of $f(x)$.
Answer(a)(ii) ..................................................[1]
Answer(a)(iii) ..................................................[1]
(b) Solve the equation. $$ \frac{100}{2^x} - 10 = 20 $$ Answer(b) $x =$ ..................................................[1]
(c) Describe fully the single transformation that maps the graph of $y = \frac{10}{2^x}$ onto the graph of $y = \frac{100}{2^x} - 10$.
Answer(c) ..........................................................................................................[2]

A fraction \( P \) has denominator \( x \).
The numerator of the fraction is 3 less than the denominator.

(a) Write down fraction \( P \) in terms of \( x \).
Answer(a) ............................................................ [1]

(b) The numerator and the denominator of fraction \( P \) are each increased by 3 to give fraction \( Q \).
Write down fraction \( Q \) in terms of \( x \).
Answer(b) ............................................................ [1]

(c) \( Q - P = \frac{9}{40} \)
(i) Write down an equation in \( x \) and show that it simplifies to \( x^2 + 3x - 40 = 0 \). [3]
(ii) Solve the equation \( x^2 + 3x - 40 = 0 \).
Answer(c)(ii) \( x = \) .................... or \( x = \) .................... [2]
(iii) Write down the original fraction, \( P \).
Answer(c)(iii) ............................................................ [1]

Solve the inequalities.
(a) \( \frac{5}{2x - 1} < 3 \)
Answer(a) ......................................................... [3]
(b) \( \log(2^x) > 10 \)
Answer(b) ......................................................... [2]