Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2015 Winter Zone 3

The diagram shows triangle $\triangle ACD$ with a perpendicular from $D$ to $AC$. Given that $AB = 17 \text{ cm}$, $BD = 15 \text{ cm}$, and $\angle BDC = 41^\circ$.

(a) Calculate the length of $BD$. Answer(a) \text{..................... cm} \ [2]
(b) Calculate the area of triangle $\triangle ACD$. Answer(b) \text{..................... cm}^2 \ [2]
(c) Use the cosine rule to find the length of $AD$. Answer(c) \text{..................... cm} \ [3]

(a) Jay buys a bicycle for $220. He later sells it for $160. Calculate his percentage loss.
Answer(a) .............................................................. % [3]

(b) A television has a sale price of $216 after a reduction of 10\%. Calculate the original price of the television.
Answer(b) $ ........................................................... [3]

(c) The population of a village is 2180. The population decreases by 3\% each year.
(i) Calculate the population in 20 years time.
Answer(c)(i) ............................................................... [3]
(ii) Calculate the number of whole years it takes for the population to decrease from 2180 to less than 1000.
Answer(c)(ii) .............................................................. [2]

(a) The speeds, v km/h, of 120 cars passing under a bridge are measured. The table shows the results.

[Table_1]

(i) Write down the interval that contains the lower quartile.

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

(ii) Calculate an estimate of the mean.

Answer(a)(ii) .......................................................... km/h [2]

(iii) Complete the table of frequency densities.

[Table_2]

[3]

(b) The table below shows the monthly rainfall and the average midday temperatures of a city.

[Table_3]

Find the equation of the line of regression, giving t in terms of r.

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

(a) (i) Shade in one more square so that the diagram has one line of symmetry.

[1]
(ii) Shade in two more squares so that the diagram has rotational symmetry of order 2 and no lines of symmetry.

[1]
(b) Triangle $ABC$ and triangle $PQR$ are mathematically similar. $AB : PQ = 3 : 2$.
(i) $CB = 10.5 \text{ cm}$. Calculate the length of $RQ$.
Answer(b)(i) .......................................................... cm [2]
(ii) The area of triangle $ABC$ is $45 \text{ cm}^2$. Calculate the area of triangle $PQR$.
Answer(b)(ii) .......................................................... cm$^2$ [2]

(a) (i) Describe fully the \textit{single} transformation that maps triangle $T$ onto triangle $U$.
Answer(a)(i) \text{...........................}\text{...........................}\text{...........................} [3]
(ii) Describe fully the inverse of the transformation in \text{part(a)(i)}.
Answer(a)(ii) \text{...........................}\text{...........................}\text{...........................} [2]
(b) (i) Draw the image of triangle $T$ under a reflection in the line $y=x$. [2]
(ii) Draw the image of triangle $T$ under a rotation of $90^\circ$ anti-clockwise about the point $(6, 8)$. [2]
(c) Describe fully the \textit{single} transformation equivalent to a rotation $90^\circ$ clockwise about $(0,0)$ followed by a reflection in the line $y=-x$. You may use the grid to help you.
Answer(c) \text{...........................}\text{...........................}\text{...........................} [3]

The diagram shows a solid cone inside a cylinder.
The base radius of the cone and the radius of the cylinder are both 10 cm.
The height of both the cone and the cylinder is 30 cm.

(a) Find the volume of the cylinder not occupied by the cone.
Answer(a) ..................cm^3 [3]
(b) Water is poured into the cylinder until it reaches a depth of 15 cm.

(i) Calculate the volume of the part of the cone that is below the water level and show that it rounds to 2749 cm^3, correct to the nearest cubic centimetre.
[4]
(ii) Calculate the amount of water that has been poured into the cylinder.
Give your answer in litres.
Answer(b)(ii) .................. litres [3]

(a) Kim walks 10 km at 4 km/h and then a further 6 km at 3 km/h.

Calculate Kim’s average speed.

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

(b) Chung runs at \( x \) km/h for 45 minutes and then at \( (x - 2) \) km/h for 30 minutes.

Find an expression, in terms of \( x \), for Chung’s average speed in km/h. Give your answer in its simplest form.

Answer(b) ........................................................ km/h [4]

(a) (i) Solve the inequality.

$2(x−3) < 5(x+3)$

Answer(a)(i) .......................................................... [3]

(ii) Show your answer to part(a)(i) on the number line.




[1]

(b) Solve the equation.

$(x+3)^2 + (x+1)^2 = 25$

Give your answers correct to 2 decimal places.

Answer(b) $x = ..........................$ or $x = ..........................$ [6]

(c) Solve the equations.

(i) $\log x = 5 - x$

Answer(c)(i) $x = ..........................................................$ [3]

(ii) $\log x = |5 - x|$

Answer(c)(ii) $x = ..........................$ or $x = ..........................$ [2]

(d) Simplify, giving your answer as a single fraction.

$$\frac{x}{x-1} - \frac{2}{x+1}$$

Answer(d) .......................................................... [3]

(a)
In the quadrilateral $ABCD$, $DA = AB$ and $DA$ is parallel to $CB$.
Angle $DAB = 124^{\circ}$ and angle $BDC = 25^{\circ}$.
Calculate angle $BCD$.

(b) Nine of the angles of a 10-sided polygon are each $142^{\circ}$.
Calculate the other angle.

(c)
$A$, $B$, $C$ and $D$ lie on the circle, centre $O$.
$BD$ is a diameter and $EDF$ is a tangent at $D$.
$AC$ and $BD$ intersect at $X$.
Angle $BCA = 25^{\circ}$ and angle $BDC = 20^{\circ}$.
Calculate
(i) angle $ADE$,
(ii) angle $DAC$,
(iii) angle $AXD$.

In this question, the weather is only considered to be either wet or dry.
When the weather is dry the probability that Sara will go walking is $\frac{3}{5}$.
When the weather is wet the probability that Sara will go walking is $\frac{1}{10}$.
The probability of a dry day is $\frac{2}{3}$.
(a) Complete the tree diagram.
[Image_1: Tree Diagram]
(b) Show that the probability that Sara goes walking is $\frac{13}{30}$.
(c) The probability that Sara does not go walking when the weather is wet is $\frac{9}{30}$.
Complete this tree diagram.
[Image_2: Tree Diagram]

f(x) = x^2 - 16
g(x) = \frac{2}{x + 1}, x \neq -1
h(x) = 2^x

(a) Find h(3).

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

(b) Find the range of g(x) for the domain \{0, 1\}.

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

(c) f(x - 2) can be written as (x + a)(x + b).
Find the value of a and the value of b.

Answer(c) a = ........................................................
b = .......................................................... [4]

(d) Find the inverse of
(i) g(x),

Answer(d)(i) ...................................................... [3]

(ii) h(x).

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

(e) Describe fully the single transformation that maps the graph of y = f(x) onto the graph of y = 2x^2 - 32.
........................................................................................................................... [2]

(a) On the diagram, sketch the graphs of $y = \frac{12}{(x + 2)}$ and $y = 2^x - 5$ for values of $x$ between $x = -6$ and $x = 4$. [4]
(b) Write down the equation of each asymptote of the graph of
(i) $y = \frac{12}{x + 2}$,
Answer(b)(i) ............................................................. ............................................................. [2]
(ii) $y = 2^x - 5$.
Answer(b)(ii) ............................................................. [1]
(c) Solve the inequality.
$2^x - 5 > \frac{12}{x + 2}$ for $x > 0$.
Answer(c) ............................................................. [2]