Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2017 Winter Zone 3

f(x) = 3x - 2 \hspace{0.3cm} g(x) = \frac{1}{x^2}, \hspace{0.3cm} x \neq 0 \hspace{0.3cm} h(x) = (x+2)^2
(a) Find
(i) \hspace{0.2cm} f(4), ...................................................................... [1]
(ii) \hspace{0.2cm} gf(4). ...................................................................... [1]
(b) \hspace{0.2cm} Find \hspace{0.2cm} g(g(5)). ...................................................................... [2]
(c) \hspace{0.2cm} Solve \hspace{0.2cm} f(h(x)) = 10.
x = ....................... or \hspace{0.2cm} x = ....................... [3]
(d) \hspace{0.2cm} Find \hspace{0.2cm} g(h(f(x))) \hspace{0.2cm} in \hspace{0.2cm} terms \hspace{0.2cm} of \hspace{0.2cm} x.
...................................................................... [2]

Alan, Brendan and Cieran work as gardeners.
(a) The total amount of money they earn is shared in the ratio of the time each person works. One day Alan works for 2 hours 40 minutes, Brendan works for 5.5 hours and Cieran works for 200 minutes. They earn, in total, $379.50 .
By changing all the times into minutes, find the amount of money each person earns.

Alan $ ..................................................
Brendan $ ..................................................
Cieran $ ..................................................

(b) (i) Alan needs to buy some gardening tools. In shop A, the price of the tools is $70.20 . In shop B, the price of the tools is 5\% less than in shop A.
Find the price of the tools in shop B.
$ .................................................. [2]

(ii) The price of $70.20 is 8\% higher than it was last year.
Find the price last year.
$ .................................................. [3]

(c) (i) Brendan invests $450 for 5 years at a rate of 3.5\% per year simple interest.
Show that the total value of this investment after 5 years is $528.75 .

[2]

(ii) Cieran invests $450 for 5 years at a rate of x \% compound interest. The value of Cieran's investment after 5 years is $530.60 .
Find the value of x.
x = .................................................. [3]

(a) (i) Complete the scatter diagram. The first eight points have been plotted for you.

[2]
(ii) What type of correlation is shown by the scatter diagram?
........................................................ [1]

(b) (i) Find the mean number of hours of sunshine.
......................................................... hours [1]
(ii) Find the mean rainfall.
......................................................... cm [1]

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

$y = .........................................................$ [2]
(ii) Estimate the rainfall when the number of hours of sunshine is 7.7.
......................................................... cm [1]

The masses of 120 peaches are recorded in the table.

[Table_1]

\begin{tabular}{|c|c|}\hline Mass (m grams) & Frequency \\ \hline 0 < m \leq 120 & 12 \\ \hline 120 < m \leq 150 & 27 \\ \hline 150 < m \leq 180 & 33 \\ \hline 180 < m \leq 210 & 15 \\ \hline 210 < m \leq 250 & 28 \\ \hline 250 < m \leq 300 & 5 \\ \hline \end{tabular}

(a) Calculate an estimate of the mean mass of a peach.
Give your answer correct to the nearest gram.
........................................................ g [3]

(b) Two peaches are chosen at random.
Find the probability that they both have a mass of more than 210g.
Give your answer as a fraction in its simplest form.
........................................................ [3]

(c) (i) Complete the frequency density column in this table.

[Table_2]

\begin{tabular}{|c|c|c|}\hline Mass (m grams) & Frequency & Frequency density \\ \hline 0 < m \leq 120 & 12 & \\ \hline 120 < m \leq 150 & 27 & \\ \hline 150 < m \leq 180 & 33 & \\ \hline 180 < m \leq 210 & 15 & \\ \hline 210 < m \leq 250 & 28 & \\ \hline 250 < m \leq 300 & 5 & 0.1 \\ \hline \end{tabular}
[2]

(ii) On the grid, draw an accurate histogram to show this information.



[4]

ABC is a triangle and BCFD is a parallelogram.

AD = \frac{1}{3}AB \text{ and } AE = \frac{1}{3}AC.
\overrightarrow{AB} = 6\mathbf{p} \text{ and } \overrightarrow{AC} = 6\mathbf{q}.

(a) Find an expression, in terms of \mathbf{p} and/or \mathbf{q}, for

(i) \overrightarrow{BC},
.................................................. [1]

(ii) \overrightarrow{DE},
.................................................. [2]

(iii) \overrightarrow{FC},
.................................................. [1]

(iv) \overrightarrow{BE}.
.................................................. [2]

(b) The area of triangle ADE is 24 units^2.

(i) Find the area of triangle ABC.
.................................................. \text{units}^2 [2]

(ii) Find the area of triangle EFC.
.................................................. \text{units}^2 [3]

The diagram shows a pyramid with a square base $ABCD$ of side $\sqrt{72}$ cm. The diagonals of the base, $AC$ and $BD$, meet at $P$. The vertex, $V$, is vertically above $P$ and $VP = 8$ cm.
(a) Find the volume of the pyramid. Give the units of your answer.

(b) Find the length $AC$.

(c) Find the length $DV$.

(d) Find angle $VDP$.

(e) $X$ is the midpoint of the side $CD$.
(i) Find the length $VX$.

(e) $X$ is the midpoint of the side $CD$.
(ii) Find angle $VXP$.

(f) The pyramid is cut parallel to $ABCD$ to form a smaller pyramid $VEFGH$. The volume of $VEFGH$ is $24$ cm$^3$. Find the vertical height of this pyramid.

The diagram shows a hollow metal hemisphere.
The outside diameter of the hemisphere is 30 cm and the inside diameter is 10 cm.
(a) Find the volume of metal used to make the hemisphere. .......................................................... cm³ [3]
(b) Find the total surface area of the hemisphere. .......................................................... cm² [5]

You may use the grid to help you in answering this question.
The transformation P is a rotation through 90° anti-clockwise about the origin.
The transformation Q is a reflection in the line $y = -x$.
(a) Find the image of the point (5, 1) under the transformation P. (................., .................) [2]
(b) Find the image of the point (5, 1) under the transformation Q. (................., .................) [2]
(c) Describe fully the single transformation equivalent to the transformation P followed by the transformation Q. ........................................................................................................................................................................ ................................................................................................................................................................ [2]

(a) (i) On the diagram, sketch the graph of $y = f(x)$. [2] (ii) Write down the co-ordinates of the points where the graph crosses the axes.
(................ , ................) or (................ , ................) [2] (iii) Solve $f(x) = 1$.
$x =$ ......................................................... [1]
$$f(x) = 10 + x - x^2 \text{ for } 0 \leq x \leq 4$$

(b) $g(x) = x^2 - 10\log x$ (i) On the same diagram, sketch the graph of $y = g(x)$, for $0 < x \leq 4$. [2] (ii) Write down the co-ordinates of the minimum point of $g(x)$.
(................ , ................) [2] (iii) Solve the equation.
$f(x) = g(x)$
........................................................ [2] (iv) Solve the equation.
$f(x - 1) = g(x - 1)$
........................................................ [2]

(a) Solve the equation $4x^2 = 12 - 3x$.
Give your answers correct to 2 decimal places. You must show all your working.
x = ....................... or x = ....................... [4]
(b) Solve the inequality $4x^2 > 12 - 3x$.
.................................................. [2]
(c) Solve the inequality $4x^2 + 5 \le 12 - 3x$.
.................................................. [4]

(a) Solve the simultaneous equations. You must show all your working.
$$3x - 2y = 11$$
$$4x - 5y = 10$$
$$x = \text{...................................................}$$
$$y = \text{...................................................} \ [4]$$

(b) Use your answers to part (a) to solve the simultaneous equations.
$$3a - 2b = 22$$
$$4a - 5b = 20$$
$$a = \text{...................................................}$$
$$b = \text{...................................................} \ [2]$$

(c) (i) Use your answers to part (a) to find the exact answers to these simultaneous equations.
$$3 \times 10^p - 2 \times 10^q = 11$$
$$4 \times 10^p - 5 \times 10^q = 10$$
$$p = \text{...................................................}$$
$$q = \text{...................................................} \ [3]$$
(ii) Find the value of $p + q$.
$$\text{...........................................................} \ [1]$$