2026 Cambridge IGCSE International Mathematics (0607) Practice Paper with Answers

Work out \(\frac{5}{6} - \frac{3}{8}\). Give your answer as a fraction in its simplest form.

Work out \(18 - 3 \times (4 - 7)^2\).

Work out \(0.3 \times (1.4 - 0.65)\).

Work out \(35\%\) of \(\$480\).

Find the lowest common multiple (LCM) of 24 and 36.

NOT TO SCALE

In the diagram, the line \(AB\) is parallel to the line \(CD\).
The line \(EF\) intersects \(AB\) at \(E\) and \(CD\) at \(F\).
\(G\) is a point on the line \(CD\) such that \(EF = EG\).
Angle \(AEF = 112^\circ\).

(a) Find the size of angle \(EFD\). Give a geometric reason for your answer. [2]
(b) Work out the size of angle \(FEG\). [1]

NOT TO SCALE

\(ABCDEF\) is a regular hexagon.
The line segment \(AB\) is extended to the point \(X\).

(a) Work out the size of one interior angle of the regular hexagon. [2]
(b) Find the size of angle \(CBX\). [1]

NOT TO SCALE

In the diagram, \(PT\) is a tangent to the circle, centre \(O\), at the point \(T\).
\(PO\) is a straight line that passes through the centre \(O\).
Angle \(TPO = 34^\circ\).

(a) Write down the size of angle \(PTO\). Give a geometric reason for your answer. [2]
(b) Work out the size of angle \(POT\). [1]

The coordinates of point \(A\) are \((1, -2)\) and the coordinates of point \(B\) are \((7, 10)\).

(a) Find the coordinates of the midpoint of the line segment \(AB\).

(b) Find the gradient of the line \(AB\).

(c) Write down the equation of the line parallel to \(AB\) that passes through the point \((0, 5)\).

A straight line \(L\) has the equation \(2y - 6x = -8\).

(a) Find the gradient of line \(L\).

(b) Write down the coordinates of the point where line \(L\) crosses the \(y\)-axis.

(c) Find the equation of the line parallel to \(L\) that passes through the point \((3, 1)\). Give your answer in the form \(y = mx + c\).

Solve the equation.

\[5(2x - 3) = 3x + 6\]

Expand and simplify.

\[(2x - 5)(x + 4)\]

Make \(w\) the subject of the formula.

\[P = 4w - 7\]

Factorise fully.

\[12a^2b - 18ab^2\]

The universal set \(\mathscr{U} = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}\).
\(A = \{\text{multiples of } 3\}\)
\(B = \{\text{factors of } 12\}\)

(a) List the elements of the set \(A \cap B\). [1]
(b) Find \(n((A \cup B)')\). [2]

A bag contains 7 red beads, 5 blue beads, and 8 green beads only.
A bead is chosen at random from the bag.

(a) Find the probability that the bead is blue. Give your answer as a fraction in its simplest form. [1]

(b) The bead is replaced in the bag. A bead is drawn at random and replaced 160 times.
Work out the expected number of times a red bead is drawn. [2]

Here is a list of the numbers of goals scored by a team in 8 matches:
\[2, \quad 3, \quad 5, \quad 6, \quad 6, \quad 7, \quad 10, \quad 17\]

(a) Write down the mode. [1]
(b) Find the range. [1]
(c) Calculate the mean. [1]

Rationalise the denominator and simplify fully.

\[\frac{5\sqrt{2} - 1}{3 + \sqrt{2}}\]

Give your answer in the form \(\frac{a\sqrt{2} + b}{c}\), where \(a\), \(b\) and \(c\) are integers.

Work out the exact value of

\[\frac{\sin 60^\circ}{\cos 45^\circ} + \sqrt{24}\]

Give your answer in the form \(k\sqrt{6}\), where \(k\) is a rational number in its simplest form.

Expand and simplify fully.

\[(3\sqrt{5} - 2\sqrt{3})^2 + \sqrt{240}\]

Give your answer in the form \(a + b\sqrt{15}\), where \(a\) and \(b\) are integers.

Work out the exact value of

\[\frac{4\cos^2 30^\circ - \tan 45^\circ}{2\sin 30^\circ \cos 60^\circ + \tan^2 60^\circ}\]

Give your answer as a fraction in its simplest form.

Rearrange the formula to make \(w\) the subject.

\[T = \sqrt{\frac{3w + 5k}{w - 2k}}\]

The line \(L\) is the perpendicular bisector of the line segment joining the points \(A(-3, 1)\) and \(B(5, 7)\).

Find the equation of line \(L\).
Give your answer in the form \(y = mx + c\).

The line \(L_1\) has the equation \(2x + 3y = 12\).
The line \(L_2\) is perpendicular to \(L_1\) and passes through the point \(P(1, -2)\).
Line \(L_2\) intersects the \(x\)-axis at the point \(Q\).

Find the exact length of the line segment \(PQ\).
Give your answer as a single fraction in simplest surd form.

NOT TO SCALE

\(A\), \(B\), \(C\) and \(D\) lie on the circumference of a circle, centre \(O\).
\(BD\) is a diameter of the circle.
Angle \(ADB = 38^\circ\) and angle \(CBD = 24^\circ\).

Find angle \(ADC\).

The interior angle of a regular polygon with \(n\) sides is \(7\) times its exterior angle.

(a) Find the value of \(n\). [2]
(b) Work out the sum of the interior angles of this polygon. [1]

NOT TO SCALE

\(A\), \(B\) and \(C\) lie on the circumference of a circle, centre \(O\).
The straight lines \(TA\) and \(TB\) are tangents to the circle.
Angle \(ATB = 44^\circ\).

(a) Work out angle \(AOB\). [1]
(b) Work out angle \(ACB\), where \(C\) lies on the major arc \(AB\). [1]
(c) Work out angle \(OAB\). [1]

The functions \(\mathrm{f}\) and \(\mathrm{g}\) are defined by
\[\mathrm{f}(x) = 3x - 4\]
\[\mathrm{g}(x) = \frac{6}{x - 2}, \quad x \ne 2.\]
(a) Find \(\mathrm{g}(\mathrm{f}(3))\).
(b) Find \(\mathrm{f}^{-1}(x)\).

Solve the equation
\[\log_{10}(5x + 15) - \log_{10}(x - 1) = 1.\]

The function \(\mathrm{h}\) is defined by \(\mathrm{h}(x) = 4 \times 8^{x - 1}\).
(a) Find the exact value of \(\mathrm{h}\left(\frac{5}{3}\right)\).
(b) Find the value of \(x\) when \(\mathrm{h}(x) = \frac{1}{16}\).

A bag contains 5 green discs and 4 yellow discs. Two discs are chosen at random without replacement. Find the probability that both discs are green. Give your answer as a fraction in its simplest form.

Elena changes €450 into Japanese yen (¥). The exchange rate is €1 = ¥162.40.
During her trip, she spends ¥52 000.

Work out the amount of Japanese yen Elena has left.

Work out the value of \(\frac{8.4 \times 10^7}{2.4 \times 10^{-3}}\).

Give your answer in standard form.

Hassan invests $3200 in a savings account at a rate of 3.5% per year simple interest.

Calculate the total interest earned at the end of 4 years.

The mass of one grain of mineral dust is \(4.8 \times 10^{-6}\text{ grams}\).

A sample contains \(3.5 \times 10^8\) of these grains.

Calculate the total mass of the sample in grams. Give your answer in standard form.

An end-of-year company bonus of $1440 is shared between Chloe and Dan in the ratio \(5 : 7\).

Work out how much Dan receives.

A laptop has an original price of $650.
In a special promotion, the price is reduced by 15%.

Calculate the sale price of the laptop.

The stem-and-leaf diagram shows the scores of 15 students in a mathematics quiz.

\(\begin{array}{r|lllll} 1 & 4 & 7 & 8 & & \\ 2 & 0 & 3 & 5 & 5 & 9 \\ 3 & 1 & 2 & 6 & 8 & \\ 4 & 0 & 4 & 5 & & \end{array}\)

Key: \(2 \mid 3\) represents a score of \(23\)

Find:
(a) the median score, [1]
(b) the interquartile range. [2]

The stem-and-leaf diagram shows the time, in seconds, taken by 11 athletes to complete a sprint drill.

\(\begin{array}{r|lllll} 1 & 5 & 8 & 9 & & \\ 2 & 1 & 4 & 4 & 7 & 8 \\ 3 & 2 & 5 & 6 & & \end{array}\)

Key: \(2 \mid 1\) represents \(2.1\text{ seconds}\)

(a) Write down the mode. [1]
(b) Find the interquartile range. [2]

The ages, in years, of 12 members of a community club are listed below.

\(28, \quad 35, \quad 41, \quad 24, \quad 39, \quad 52, \quad 45, \quad 31, \quad 35, \quad 48, \quad 29, \quad 50\)

(a) Complete the ordered stem-and-leaf diagram for this data.

\(\begin{array}{r|l} \text{Stem} & \text{Leaf} \\ \hline 2 & \\ 3 & \\ 4 & \\ 5 & \end{array}\)

Key: \(2 \mid 4\) represents \(24\text{ years}\) [2]

(b) Find the median age. [1]

A wire of length \(14.5\text{ m}\) is attached to the top of a vertical mast and fixed to a point on horizontal ground. The wire makes an angle of \(38^\circ\) with the ground. Calculate the height of the mast.

The size of each interior angle of a regular polygon is \(156^\circ\). Calculate the number of sides of this polygon.

Triangle \(P\) has vertices at \((1, 2)\), \((3, 2)\), and \((1, 5)\). Triangle \(Q\) has vertices at \((2, 4)\), \((6, 4)\), and \((2, 10)\). Describe fully the single transformation that maps triangle \(P\) onto triangle \(Q\).

A rectangular garden has a length of \(84\text{ m}\) and a width of \(35\text{ m}\). A straight path runs diagonally across the garden from one corner to the opposite corner. Calculate the length of the path.

\(AB\) is a diameter of a circle with centre \(O\). \(C\) is a point on the circumference such that angle \(BAC = 28^\circ\). The straight line \(TB\) is a tangent to the circle at point \(B\).
(a) Write down the size of angle \(ACB\).
(b) Find the size of angle \(ABC\).
(c) Find the size of angle \(CBT\).

\(f(x) = x^2 - 2x - 8\)

(a) Write down the coordinates of the \(y\)-intercept. [1]

(b) Write down the zeros of \(f(x)\). [1]

(c) Write down the coordinates of the local minimum. [1]

The function \(\mathrm{f}(x) = x^2 - 4x - 5\) is defined for all real numbers.

(a) Write down the coordinates of the point where the graph of \(y = \mathrm{f}(x)\) crosses the \(y\)-axis. [1]

(b) Find the coordinates of the local minimum point of the graph. [1]

(c) Write down the zeros of \(\mathrm{f}(x)\). [1]

Find the \(n\text{th}\) term of the sequence \(7,\; 12,\; 17,\; 22,\; 27,\; \dots\)

The \(n\text{th}\) term of a sequence is given by \(80 - 6n\).

Find the value of \(n\) when the term is \(-22\).

Here are the first four terms of a sequence:
\[43,\; 37,\; 31,\; 25,\; \dots\]

Find the \(n\text{th}\) term of this sequence.

The number of counters used to make each pattern forms an arithmetic sequence:

Pattern 1 has 5 counters.
Pattern 2 has 9 counters.
Pattern 3 has 13 counters.

Find the number of counters in Pattern 24.

\(f(x) = \frac{4x - 6}{x + 3}\), for \(x \neq -3\). (a) Write down the equations of the asymptotes of the graph of \(y = f(x)\). (b) Find the coordinates of the point where the graph of \(y = f(x)\) crosses the \(x\)-axis. (c) Solve the equation \(f(x) = -2\).

\(g(x) = \frac{2x^2 + 5}{x^2 - 4}\), for \(x \neq -2\) and \(x \neq 2\). (a) Write down the equations of the vertical asymptotes of the graph of \(y = g(x)\). (b) Write down the equation of the horizontal asymptote of the graph of \(y = g(x)\). (c) Write down the coordinates of the local maximum point. (d) Solve the equation \(g(x) = 3\).

\(f(x) = \frac{12}{x^2 - 2x - 8}\), for \(x \neq -2\) and \(x \neq 4\). (a) Write down the equations of the vertical asymptotes of the graph of \(y = f(x)\). (b) Write down the equation of the horizontal asymptote of the graph of \(y = f(x)\). (c) Write down the coordinates of the local maximum point. (d) Solve the equation \(f(x) = -2\).

The table shows the body mass (fit{x} ext{ grams}) and the resting metabolic rate (fit{y} ext{ units}) of a sample of six small rodents.

$$\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Body mass }(x) & 15 & 22 & 35 & 48 & 60 & 75 \\
\hline
\text{Metabolic rate }(y) & 1.8 & 2.3 & 3.1 & 4.0 & 4.6 & 5.5 \\
\hline
\end{array}$$

(a) Find the coordinates of the mean point $(\bar{x}, \bar{y})$. [1]
(b) Find the equation of the linear regression line for $y$ in terms of $x$. [2]
(c) Use your equation from part (b) to estimate the resting metabolic rate of a rodent with a body mass of $52\text{ g}$. [1]

A college survey recorded the preferences of 120 applicants regarding their field of study and whether they requested on-campus accommodation.

$$\begin{array}{|l|c|c|c|}
\hline
& \text{STEM} & \text{Humanities} & \text{Total} \\
\hline
\text{On-campus accommodation} & 42 & 28 & 70 \\
\hline
\text{Off-campus accommodation} & 18 & 32 & 50 \\
\hline
\text{Total} & 60 & 60 & 120 \\
\hline
\end{array}$$

(a) One applicant is chosen at random from those who prefer STEM. Find the probability that this applicant requested on-campus accommodation. [1]
(b) Two applicants are chosen at random without replacement from the 60 applicants who prefer Humanities. Find the probability that exactly one of the two applicants requested on-campus accommodation. [3]

A solid frustum of a cone is formed by removing a small cone from the top of a larger cone.
The bottom circular base of the frustum has radius \(8\text{ cm}\) and the top circular face has radius \(5\text{ cm}\).
The vertical height of the frustum is \(9\text{ cm}\).

Calculate the volume of the frustum.

A solid right circular cone has a base radius of \(12\text{ cm}\) and a vertical height of \(16\text{ cm}\).
A smaller cone with a base radius of \(6\text{ cm}\) is removed from the top of this cone to form a solid frustum.

Calculate the total surface area of the solid frustum.

A sailing yacht travels from a harbour \(A\) to a buoy \(B\) on a bearing of \(048^\circ\) for a distance of \(18.5\text{ km}\).
It then sails from buoy \(B\) to an island \(C\) on a bearing of \(132^\circ\) for a distance of \(24.0\text{ km}\).

NOT TO SCALE

Calculate the direct distance from \(A\) to \(C\).

Two observation stations \(P\) and \(Q\) are \(14.2\text{ km}\) apart on flat ground, with \(Q\) on a bearing of \(075^\circ\) from \(P\).
A fire tower \(R\) is on a bearing of \(020^\circ\) from \(P\) and on a bearing of \(310^\circ\) from \(Q\).

NOT TO SCALE

Calculate the distance \(PR\).

A helicopter flies from a base \(H\) to a checkpoint \(K\) on a bearing of \(110^\circ\) for \(35\text{ km}\).
It then flies from \(K\) to a landing pad \(L\) on a bearing of \(215^\circ\) for \(42\text{ km}\).

NOT TO SCALE

Calculate the bearing of \(H\) from \(L\).

A painting was bought for $15\,000 in the year 2018. Its value increases exponentially at a rate of 4.5% per year. (a) Calculate the value of the painting in the year 2026. Give your answer correct to the nearest dollar. [2] (b) Find the year in which the value of the painting first exceeds $28\,000. [2]

The mass, \(M\) grams, of a sample of a radioactive isotope decays according to the formula \(M = M_0 (0.88)^t\), where \(M_0\) is the initial mass in grams and \(t\) is the time in days. (a) Write down the percentage decrease in the mass of the sample each day. [1] (b) The initial mass of the sample is 450 g. Calculate the least number of whole days it takes for the mass to fall below 50 g. [2]

Expand and simplify fully.
\[ (2x - 3)(x + 4)(3x - 1) \]

Rearrange the formula to make \(w\) the subject.
\[ P = \frac{5w + 2}{3 - 4w} \]

Expand and simplify fully.
\[ (3x - 2)^2 - 2(x + 4)(3x - 5) \]

NOT TO SCALE

The points \(A\), \(B\), \(C\), and \(D\) lie on the circumference of a circle.

Angle \(ABC = (5x - 12)^\circ\),
Angle \(ADC = (3x + 24)^\circ\),
Angle \(DAB = (2x + 10)^\circ\).

Find the size of angle \(BCD\).

NOT TO SCALE

The points \(A\), \(B\), and \(C\) lie on the circumference of a circle with centre \(O\).

Angle \(OAB = (2x + 11)^\circ\) and angle \(ACB = (3x - 1)^\circ\).

Find the value of \(x\).

Square frames are constructed using small square tiles.
A frame of side length $n$ has shaded tiles along its outer border and an unshaded $(n-2) \times (n-2)$ square in the centre.
The table shows the number of shaded border tiles, $B$, for frames of different side lengths $n$.

$$\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Side length }(n) & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
\text{Number of border tiles }(B) & 8 & 12 & 16 & & & \\
\hline
\end{array}$$

Complete the table by finding the values of $B$ when $n = 6$, $n = 7$, and $n = 8$.

A sequence of honeycomb patterns is formed using equal-length rods.
- Pattern 1 consists of $1$ regular hexagon and uses $6$ rods.
- Pattern 2 adds a second hexagon sharing $1$ edge and uses $11$ rods.
- Pattern 3 adds a third hexagon in the chain sharing $1$ edge and uses $16$ rods.

$$\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{Pattern number }(n) & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
\text{Number of rods }(R) & 6 & 11 & 16 & & & \\
\hline
\end{array}$$

Complete the table by finding the number of rods, $R$, for $n = 4$, $n = 5$, and $n = 6$.

Straight cuts are made across a flat circular paper disc. No two cuts are parallel and no three cuts intersect at a single point.
The table shows the maximum number of non-overlapping regions, $R$, created by $c$ cuts.

$$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
\text{Number of cuts }(c) & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\text{Number of regions }(R) & 2 & 4 & 7 & 11 & & & \\
\hline
\end{array}$$

Complete the table for $c = 5$, $c = 6$, and $c = 7$.

A sequence of triangular dot patterns is formed.
- Pattern 1 has $1$ dot.
- Pattern 2 has $3$ dots.
- Pattern 3 has $6$ dots.
- Pattern 4 has $10$ dots.

$$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
\text{Pattern number }(n) & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\text{Total dots }(D) & 1 & 3 & 6 & 10 & & & \\
\hline
\end{array}$$

Complete the table by finding the value of $D$ when $n = 5$, $n = 6$, and $n = 7$.

L-shaped figures are constructed on a square grid using unit squares.
Pattern $k$ has two perpendicular arms of length $k$ joined at a single corner square.
The table shows the pattern number $k$, the total number of unit squares $S$, and the outer perimeter $P$.

$$\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Pattern number }(k) & 1 & 2 & 3 & 4 & 5 \\
\hline
\text{Number of squares }(S) & 1 & 3 & 5 & & \\
\hline
\text{Perimeter }(P) & 4 & 8 & 12 & & 20 \\
\hline
\end{array}$$

Complete the three missing entries in the table.

A sequence of fence designs is constructed using wooden slats. Design 1 uses 6 slats. Design 2 uses 11 slats. Design 3 uses 16 slats. Design 4 uses 21 slats. (a) State the difference between consecutive terms. (b) Find a formula for the number of slats, \(S\), needed for Design \(n\). Clearly communicate your reasoning.

A series of rectangular dot grids is investigated. Grid \(n\) has 3 rows and \(n + 2\) columns of dots. The total number of dots is \(T = 3(n + 2)\). The dots on the outer boundary of the grid are called border dots, \(B\). For Grid 1 (3 by 3), \(B = 8\). For Grid 2 (3 by 4), \(B = 10\). For Grid 3 (3 by 5), \(B = 12\). (a) Write down an expression for the number of border dots, \(B\), in terms of \(n\). (b) Using \(I = T - B\), communicate the algebraic steps to show that the number of interior dots, \(I\), is always equal to \(n\).

A sequence of total sums of consecutive even positive integers is given as follows: Row 1: 2 = 2. Row 2: 2 + 4 = 6. Row 3: 2 + 4 + 6 = 12. Row 4: 2 + 4 + 6 + 8 = 20. Let \(S_n\) be the sum for Row \(n\). (a) Write down the first two differences of the sequence 2, 6, 12, 20. (b) Formulate an expression for \(S_n\) in terms of \(n\), showing clearly how you established the formula.

The number of diagonals, \(D\), in a regular polygon with \(n\) sides is investigated for \(n \ge 3\): For a triangle (\(n = 3\)), \(D = 0\). For a quadrilateral (\(n = 4\)), \(D = 2\). For a pentagon (\(n = 5\)), \(D = 5\). For a hexagon (\(n = 6\)), \(D = 9\). By examining differences or the geometry of vertices, find the formula for \(D\) in terms of \(n\). Communicate your reasoning clearly.

This question is about investigating the number of factors of positive integers using their prime factorisations.

(a) Write 72 as a product of its prime factors in index form. [1]

(b) The number of factors of an integer written in the form \(N = 2^a \times 3^b\) is given by the rule:
$$\text{Number of factors} = (a + 1)(b + 1)$$
Show that this rule gives 12 factors for 72. [1]

(c) An integer \(M\) has only two distinct prime factors, 2 and 3, so that \(M = 2^a \times 3^b\) where \(a \ge 1\) and \(b \ge 1\).
\(M\) has exactly 10 factors.
Find the two possible values of \(M\). [3]

A student investigates the smallest positive integer multipliers needed to convert a given number into a square number or a cube number.

(a) Write 540 as a product of its prime factors in index form. [1]

(b) Find the smallest positive integer \(k\) such that \(540k\) is a square number. [2]

(c) Find the smallest positive integer \(m\) such that \(540m\) is a cube number. [2]

The prime factorisation of 168 is \(2^3 \times 3 \times 7\).

(a) Write down the total number of prime factors of 168, counting repeats. [1]

(b) Two positive integers \(A\) and \(B\) satisfy the relationship:
$$\text{HCF}(A, B) \times \text{LCM}(A, B) = A \times B$$
The highest common factor of 168 and an unknown integer \(W\) is 24. The lowest common multiple of 168 and \(W\) is 504.
Show that \(W = 72\). [2]

(c) Find the smallest integer \(V\) greater than 24 such that \(\text{HCF}(168, V) = 24\). [2]

This investigation looks at the number of unit squares crossed by the diagonal of an integer grid.

A straight diagonal line is drawn from \((0,0)\) to \((w, h)\) across a grid of \(w \times h\) unit squares, where \(w\) and \(h\) are positive integers.
Let \(N(w, h)\) be the number of \(1 \times 1\) unit squares whose interiors are crossed by the diagonal line.

(a) When \(w\) and \(h\) have no common factor greater than 1 (i.e. \(\gcd(w, h) = 1\)), the diagonal does not pass through any interior grid intersection.

Complete the table for cases where \(\gcd(w, h) = 1\):

$$\begin{array}{|c|c|c|}
\hline
w & h & N(w, h) \\
\hline
2 & 3 & 4 \\
3 & 4 & 6 \\
3 & 5 & \text{[i]} \\
4 & 7 & \text{[ii]} \\
\hline
w & h & \text{[iii]} \\
\hline
\end{array}$$

(b) When \(\gcd(w, h) = g > 1\), the diagonal crosses through \(g - 1\) interior grid intersection points.

Write down an expression for \(N(w, h)\) in terms of \(w\), \(h\), and \(\gcd(w, h)\).

(c) Use your expression from part (b) to find the number of unit squares crossed by the diagonal of a \(84 \times 126\) grid.

This investigation looks at the number of bounces a laser beam makes inside a rectangular grid table.

A laser beam is fired at a \(45^\circ\) angle from the bottom-left corner \((0,0)\) of a rectangle of width \(a\) and height \(b\), where \(a\) and \(b\) are integers with no common factors (i.e. \(\gcd(a, b) = 1\)). The beam reflects off the boundaries at \(45^\circ\) until it lands in one of the other three corners.

Let \(B(a, b)\) be the total number of reflections (bounces off the sides) before the beam hits a corner.

(a) Complete the table for coprime dimensions (\(\gcd(a, b) = 1\)):

$$\begin{array}{|c|c|c|}
\hline
a & b & B(a, b) \\
\hline
1 & 3 & 2 \\
2 & 3 & 3 \\
3 & 4 & 5 \\
3 & 5 & \text{[i]} \\
5 & 8 & \text{[ii]} \\
\hline
a & b & \text{[iii]} \\
\hline
\end{array}$$

(b) When \(a\) and \(b\) have a greatest common divisor \(\gcd(a, b) = g > 1\), the path scales.

Write down an expression for \(B(a, b)\) in terms of \(a\), \(b\), and \(\gcd(a, b)\).

(c) Calculate the total number of reflections \(B(60, 84)\).

This investigation looks at the distinct points visited by a cyclic hopper.

A circle has \(n\) equally spaced markers numbered \(0, 1, 2, \dots, n - 1\) in clockwise order. A hopper starts at marker \(0\) and jumps clockwise in steps of size \(s\), visiting markers \((k \times s) \pmod n\) for \(k = 0, 1, 2, \dots\) until it returns to \(0\).

Let \(V(n, s)\) be the number of distinct markers visited by the hopper.

(a) Complete the table for \(n = 24\):

$$\begin{array}{|c|c|c|c|}
\hline
n & s & \gcd(n, s) & V(n, s) \\
\hline
24 & 1 & 1 & 24 \\
24 & 4 & 4 & 6 \\
24 & 6 & 6 & 4 \\
24 & 9 & 3 & \text{[i]} \\
24 & 14 & \text{[ii]} & \text{[iii]} \\
\hline
\end{array}$$

(b) Write down an expression for \(V(n, s)\) in terms of \(n\) and \(\gcd(n, s)\).

(c) A circle has \(n = 360\) markers. The hopper jumps in steps of \(s = 48\).
Find the number of markers that are never visited by the hopper.

This investigation looks at integer coordinate points lying on straight line segments.

A line segment is drawn from the origin \((0,0)\) to the point \((x, y)\), where \(x\) and \(y\) are positive integers.
Let \(L(x, y)\) be the total number of integer coordinate points \((p, q)\) lying on this line segment, including both endpoints \((0,0)\) and \((x, y)\).

(a) Complete the table:

$$\begin{array}{|c|c|c|c|}
\hline
x & y & \gcd(x, y) & L(x, y) \\
\hline
2 & 3 & 1 & 2 \\
4 & 6 & 2 & 3 \\
6 & 9 & 3 & 4 \\
8 & 12 & \text{[i]} & \text{[ii]} \\
15 & 20 & \text{[iii]} & \text{[iv]} \\
\hline
\end{array}$$

(b) Write down a formula for \(L(x, y)\) in terms of \(x\) and \(y\).

(c) A line segment connects \((0,0)\) to \((144, 216)\).
Find the number of interior integer coordinate points on this segment (excluding both endpoints).

This investigation explores the synchronization of two flashing beacons.

Beacon A flashes every \(p\) seconds and Beacon B flashes every \(q\) seconds, where \(p\) and \(q\) are positive integers. Both beacons flash together at time \(t = 0\).
Let \(T(p, q)\) be the time in seconds between consecutive simultaneous flashes, and let \(S(p, q)\) be the total number of individual flashes (from either Beacon A or Beacon B or both) that occur in the half-open time interval \(0 < t \le T(p, q)\).

*(Note: When both beacons flash at the exact same second \(t\), it counts as 1 simultaneous event.)*

(a) Complete the table:

$$\begin{array}{|c|c|c|c|c|}
\hline
p & q & \gcd(p, q) & T(p, q) & S(p, q) \\
\hline
2 & 3 & 1 & 6 & 4 \\
3 & 5 & 1 & 15 & 7 \\
4 & 6 & 2 & 12 & \text{[i]} \\
6 & 10 & 2 & \text{[ii]} & \text{[iii]} \\
\hline
\end{array}$$

(b) Write down an expression for \(S(p, q)\) in terms of \(p\), \(q\), and \(\gcd(p, q)\).

(c) Two industrial warning lights flash every \(p = 36\) seconds and \(q = 84\) seconds respectively.
Find the value of \(S(36, 84)\).

A single strand of decorative copper wire is wound in a uniform cylindrical helix around a solid pillar of radius \(r = 3\text{ cm}\) and vertical height \(h = 8\pi\text{ cm}\).
The wire begins at the bottom base, completes exactly 1 full turn around the pillar, and ends at the top rim.

(a) By considering the unrolled curved surface of the cylinder as a rectangle, find the exact length of the wire, \(L_1\), in terms of \(\pi\).
(b) Calculate the angle of inclination, \(\alpha\), that the wire makes with the horizontal base. Give your answer correct to 1 decimal place.

A flexible cable is wound in a uniform helix of \(n\) identical complete turns around a cylindrical column of height \(h\) and radius \(r\).

(a) Show that the total length of the cable, \(L(n)\), is given by:
\[L(n) = \sqrt{h^2 + 4\pi^2 n^2 r^2}\]
(b) A support column has height \(h = 24\text{ cm}\) and radius \(r = \frac{7}{\pi}\text{ cm}\). Find the total length of cable required to make \(n = 5\) complete turns.

A flat protective ribbon of constant width \(w\text{ cm}\) is wound spirally around a pipe of radius \(R = 5\text{ cm}\) and length \(H = 60\text{ cm}\) in \(n = 4\) complete turns so that adjacent edges touch exactly with no overlap and no gaps.

(a) Find the pitch \(p\) (vertical rise per turn) of the helix.
(b) Using the relation \(w = p \cos \theta\), where \(\theta\) is the angle between the helical path and the horizontal circumference, calculate the required ribbon width \(w\). Give your answer correct to 3 significant figures.

An insulated sensor wire has a fixed maximum length of \(L = 150\text{ cm}\). It must be coiled in a uniform cylindrical helix around a pipe of outer radius \(r = 4\text{ cm}\).

The total vertical height \(h\) spanned along the pipe must be at least \(90\text{ cm}\).

Find the maximum integer number of complete turns, \(n\), that can be made.

A square-threaded screw operates mechanically as an inclined plane wrapped around a cylinder. The screw has mean radius \(r = 6\text{ mm}\) and pitch \(p = 4\text{ mm}\).

(a) The helix lead angle \(\psi\) satisfies \(\tan \psi = \frac{p}{2\pi r}\). Calculate \(\psi\) in degrees, correct to 2 decimal places.
(b) The mechanical efficiency \(\eta\) of the screw is given by the formula:
\[\eta = \frac{\tan \psi}{\tan(\psi + \phi)}\]
where \(\phi\) is the friction angle such that \(\tan \phi = 0.15\). Calculate the efficiency \(\eta\) as a percentage, correct to 1 decimal place.

A conical helical path runs along the curved surface of a right circular cone with base radius \(R = 12\text{ cm}\) and slant height \(s = 20\text{ cm}\).
When the curved surface of the cone is cut along a generator and unrolled flat, it forms a sector of a circle of radius \(s = 20\text{ cm}\) and central angle \(\theta\).

(a) Show that the sector angle \(\theta\) is \(216^\circ\).
(b) A path goes from a point \(A\) at the base of the seam to a point \(B\) halfway up the opposite generator (so \(OB = 10\text{ cm}\) where \(O\) is the vertex, and the angle between \(OA\) and \(OB\) on the unrolled sector is \(216^\circ\)). Find the shortest length of this path along the surface of the cone. Give your answer correct to 3 significant figures.