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

A shopkeeper buys 60 identical notebooks for a total cost of $150.
She sells each notebook for $3.20.

Work out her total profit.

Marco invests $800 at a rate of \(3.5\%\) per year simple interest.

Calculate the total interest earned after 4 years.

The normal price of a jacket is $75.
In a sale, the price is reduced by \(15\%\).

Calculate the sale price of the jacket.

An adult cinema ticket costs $12.50 and a child ticket costs $7.80.
A family buys 2 adult tickets and 3 child tickets.
They pay with a $100 note.

Work out the change they receive.

The line \(L\) passes through the points \((0, -5)\) and \((4, 7)\).

(a) Work out the gradient of line \(L\). [1]

(b) Write down the equation of line \(L\) in the form \(y = mx + c\). [2]

In a diagram, straight line \(AB\) is parallel to straight line \(CD\).
Triangle \(PQR\) has vertex \(P\) on line \(AB\), and vertices \(Q\) and \(R\) on line \(CD\).
\(PQ = PR\) and angle \(APQ = 52^\circ\).

(a) Write down the size of angle \(PQR\). [1]

(b) Work out the size of angle \(QPR\). [2]

The point \(M(3, 7)\) is the midpoint of the line segment joining \(A(-1, 2)\) and \(B(k, 12)\).

(a) Find the value of \(k\). [2]

(b) Work out the gradient of the line segment \(AB\). [1]

A circle has centre \(O\). The points \(A\), \(B\) and \(C\) lie on the circumference of the circle, and \(AC\) is a diameter.
Angle \(BAC = 38^\circ\).

(a) Write down the size of angle \(ABC\) and give a geometrical reason for your answer. [2]

(b) Find the size of angle \(ACB\). [1]

Triangle \(T\) has vertices at \((1, 2)\), \((4, 2)\) and \((1, 6)\).

(a) Write down the coordinates of the image of the point \((4, 2)\) after a reflection in the line \(y = 0\). [1]

(b) Triangle \(T\) is enlarged by a scale factor of 3 with centre of enlargement \((0, 0)\) to give triangle \(U\).
Write down the coordinates of the vertices of triangle \(U\). [2]

Here are the first four terms of a sequence.

\[7, \quad 13, \quad 19, \quad 25, \quad \dots\]

(a) Write down the next term of this sequence. [1]

(b) Find an expression for the \(n\)th term of this sequence. [2]

(c) Work out the 20th term of this sequence. [1]

(a) The point \(A(2, 4)\) is rotated \(90^\circ\) clockwise about the origin \((0, 0)\) to give point \(B\).
Find the coordinates of \(B\). [2]

(b) The point \(C(-1, 3)\) is reflected in the line \(x = 2\) to give point \(D\).
Find the coordinates of \(D\). [2]

The first four terms of a sequence are

\[42, \quad 35, \quad 28, \quad 21, \quad \dots\]

(a) Write down the term-to-term rule for this sequence. [1]

(b) Find an expression for the \(n\)th term of this sequence. [2]

(c) Find the first negative term in this sequence. [1]

(a) A shape is enlarged with centre \((0, 0)\) and scale factor 3.
The point \(P(4, 1)\) is on the original shape.
Find the coordinates of the image of \(P\). [1]

(b) Triangle \(T\) has vertices at \((1, 1)\), \((3, 1)\) and \((1, 4)\).
Triangle \(U\) has vertices at \((-1, -1)\), \((-3, -1)\) and \((-1, -4)\).

Describe fully the single transformation that maps triangle \(T\) onto triangle \(U\). [3]

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

(a) List the elements of set \(A\). [1]
(b) Write down the elements of \(A \cap B\). [1]
(c) Find \(\text{n}(A \cup B)\). [1]
(d) A number is chosen at random from the universal set \(\mathcal{U}\).
Find the probability that this number is not in set \(A\). [1]

A bag contains counters that are red, blue, green, or yellow.
A counter is taken at random from the bag.
The table shows the probabilities of choosing a red, blue, or green counter.

$$\begin{array}{|l|c|c|c|c|} \hline \text{Colour} & \text{Red} & \text{Blue} & \text{Green} & \text{Yellow} \\ \hline \text{Probability} & 0.25 & 0.4 & 0.15 & \\ \hline \end{array}$$

(a) Complete the table to find the probability of choosing a yellow counter. [2]
(b) Work out the probability of choosing a counter that is red or green. [1]
(c) There are 120 counters in the bag altogether.
Work out the number of blue counters in the bag. [2]

Rationalise the denominator and simplify fully.

\[\frac{12}{3 - \sqrt{3}}\]

\(\mathbf{a} = \begin{pmatrix} 5 \\ -1 \end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix} -2 \\ 3 \end{pmatrix}\)

(a) Work out \(3\mathbf{a} - 2\mathbf{b}\).

(b) Find \(|\mathbf{a} + \mathbf{b}|\), giving your answer in exact surd form.

Rationalise the denominator and simplify fully.

\[\frac{7\sqrt{2} + 2}{\sqrt{2} + 1}\]

\(\mathbf{p} = \begin{pmatrix} w \\ 7 \end{pmatrix}\) and \(\mathbf{q} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}\).

Given that \(|\mathbf{p} - \mathbf{q}| = \sqrt{106}\) and \(w > 0\), find the value of \(w\).

Simplify fully.

\[\frac{2x^2 - 7x + 3}{2x^2 + x - 1} \div \frac{x^2 - 5x + 6}{x^2 + 3x + 2}\]

The coordinates of point \(A\) are \((-2, 7)\) and the coordinates of point \(B\) are \((6, -1)\).

Find the equation of the perpendicular bisector of the line segment \(AB\).
Give your answer in the form \(y = mx + c\).

Solve the equation.

\[\frac{6}{x - 1} - \frac{1}{x - 2} = 2\]

Line \(L_1\) passes through the points \((-3, -5)\) and \((5, 7)\).
Line \(L_2\) is parallel to line \(L_1\) and passes through the point \((4, 1)\).
Line \(L_3\) is perpendicular to line \(L_1\) and passes through the point \((0, 8)\).

Find the coordinates of the point of intersection of line \(L_2\) and line \(L_3\).

The points \(P(1, k)\), \(Q(k, 4)\) and \(R(5, 10)\) lie on the same straight line.

Find the two possible values of \(k\).

Solve the equation.

$$\log_3(x + 5) - \log_3(2x - 1) = 2$$

The function $\mathrm{f}(x)$ is defined by
$$\mathrm{f}(x) = \frac{2x + 3}{x - 4}, \quad x \neq 4.$$

Find $\mathrm{f}^{-1}(x)$.

$\mathrm{f}(x) = x^2 - 3$ and $\mathrm{g}(x) = 2x + 1$.

Find the values of $x$ for which $\mathrm{f}(\mathrm{g}(x)) = 13$.

Solve the equation.

$$2\log_3(x) - \log_3(x + 6) = 1$$

Work out the exact value of
$$\frac{2\sin 30^\circ + \sqrt{3}\tan 60^\circ}{\cos^2 45^\circ}$$
Give your answer as an integer.

$TA$ and $TB$ are tangents to a circle, centre $O$, touching the circle at $A$ and $B$ respectively.
$C$ is a point on the major arc $AB$.
Angle $ATB = 44^\circ$.

(a) Find angle $AOB$.
(b) Find angle $ACB$.

Solve the equation
$$2\cos x + \sqrt{3} = 0$$
for $0^\circ \leqslant x \leqslant 360^\circ$.

A rectangle has length \((2x + 5)\text{ cm}\) and width \((x - 2)\text{ cm}\).
A square has sides of length \((x + 1)\text{ cm}\).
The area of the rectangle is \(19\text{ cm}^2\) greater than the area of the square.

(a) Show that \(x^2 - x - 30 = 0\). [3]

(b) (i) Factorise \(x^2 - x - 30\). [1]

(ii) Find the value of \(x\). [1]

(iii) Work out the perimeter of the rectangle. [1]

A coach travels a distance of \(120\text{ km}\) at an average speed of \(x\text{ km/h}\).
On the return journey, the coach travels at an average speed of \((x + 10)\text{ km/h}\).
The return journey takes \(1\) hour less than the outward journey.

(a) Write down an expression, in terms of \(x\), for the time taken, in hours, for:
(i) the outward journey, [1]
(ii) the return journey. [1]

(b) (i) Form an equation in \(x\) and show that it simplifies to \(x^2 + 10x - 1200 = 0\). [2]

(ii) Factorise \(x^2 + 10x - 1200\). [1]

(iii) Work out the average speed of the coach on the return journey. [1]

A bakery produces loaves of bread in three varieties: white, wholemeal, and rye in the ratio \(5 : 3 : 2\).

On Tuesday, the bakery produces a total of \(760\) loaves of bread.

Calculate the number of wholemeal loaves produced.

Eleni invests \(\$3200\) in a savings account that pays simple interest at a rate of \(4.5\%\) per year.

Calculate the total amount in the account at the end of \(3\) years.

Find the lowest common multiple (LCM) of \(36\) and \(84\).

In a survey of \(450\) students, \(\frac{2}{5}\) travel to school by bus, \(32\%\) walk to school, and the remaining students travel by car.

Work out the number of students who travel by car.

A shopkeeper buys a box of \(24\) headphones for a total cost of \(\$480\).
She sells each pair of headphones for \(\$26.50\).

Calculate her percentage profit on the total cost.

A box contains 7 blue pens and 3 black pens.
A pen is chosen at random from the box, its colour is noted, and it is replaced.
A second pen is then chosen at random from the box.

(a) Write down the probability that the first pen chosen is blue. [1]

(b) Work out the probability that both pens chosen are blue. [2]

NOT TO SCALE

Triangle \(ABC\) is right-angled at \(B\).
\(AB = 6.8\text{ cm}\) and \(BC = 9.5\text{ cm}\).

(a) Calculate the length \(AC\).
(b) Calculate angle \(BAC\).

A straight ladder of length \(5.2\text{ m}\) rests against a vertical wall on horizontal ground.
The bottom of the ladder is \(1.8\text{ m}\) from the base of the wall.

(a) Calculate the angle the ladder makes with the ground.
(b) Calculate the height the ladder reaches up the wall.

(a) A solid metal cylinder has a radius of \(4.5\text{ cm}\) and a height of \(12\text{ cm}\).
Calculate the volume of the cylinder.

(b) A solid metal sphere has a radius of \(3.2\text{ cm}\).
(i) Calculate the volume of the sphere.
(ii) The cylinder and the sphere are melted down and recast together into a single block of metal.
Calculate the total volume of metal in the block.

The function \(f(x) = 3 + 2x - x^2\) is defined for \(-2 \le x \le 4\).

(a) Sketch the graph of \(y = f(x)\) for \(-2 \le x \le 4\). [2]
(b) Write down the coordinates of the maximum point of the graph. [1]
(c) Find the zeros of the function \(f(x)\). [1]

The function \(f(x) = x^3 - 3x - 2\) is defined for \(-2.5 \le x \le 2.5\).

(a) Sketch the graph of \(y = f(x)\) for \(-2.5 \le x \le 2.5\). [2]
(b) Write down the coordinates of the local maximum point. [1]
(c) Find the roots of the equation \(x^3 - 3x - 2 = 0\). [1]

(a) Sketch the graph of \(y = 4 - x^2\) for \(-3 \le x \le 3\). [2]

(b) Use your graphic display calculator to solve the equation \(4 - x^2 = 2x - 1\). [2]

Simplify fully.
$$\frac{2x^2 + 5x - 12}{4x^2 - 9}$$

Solve the equation.
$$3x^2 - 8x - 5 = 0$$
Show all your working and give your answers correct to 2 decimal places.

Write as a single fraction in its simplest form.
$$\frac{5}{x + 2} - \frac{3}{2x - 1}$$

Solve the equation.
$$2(x - 3)^2 = 5x + 7$$
Show all your working and give your answers correct to 3 significant figures.

$$\mathrm{f}(x) = \frac{x - 3}{x^2 - 4} \quad \text{for } x \ne -2, x \ne 2$$

(a) Write down the equations of the three asymptotes of the graph of $y = \mathrm{f}(x)$. [3]

(b) Use your graphic display calculator to find the coordinates of the local maximum point on the graph of $y = \mathrm{f}(x)$ for $x > 2$. Give your answers correct to 3 significant figures. [2]

$$\mathrm{f}(x) = \frac{3x - 6}{x + 2} \quad \text{for } x \ne -2$$

(a) Write down the equation of
(i) the vertical asymptote of the graph of $y = \mathrm{f}(x)$, [1]
(ii) the horizontal asymptote of the graph of $y = \mathrm{f}(x)$. [1]

(b) Find the coordinates of the points where the graph of $y = \mathrm{f}(x)$ crosses the coordinate axes. [2]

(c) Write down the range of $\mathrm{f}(x)$ for the domain $x > -2$. [1]

A forest area decreases exponentially at a rate of \(3.5\%\) per year due to deforestation.
After 8 years, the area of the forest is \(14\,800\text{ hectares}\).

Calculate the original area of the forest 8 years ago.
Give your answer correct to 3 significant figures.

The 3rd term of a geometric sequence is \(72\) and the 7th term is \(1152\).
All the terms in the sequence are positive.

(a) Find the common ratio of the sequence. [2]

(b) Find the first term of the sequence. [1]

(c) Calculate the 10th term of the sequence. [1]

The value of a piece of industrial machinery depreciates exponentially at a constant rate of \(r\%\) per year.
Two years after purchase, the value of the machine is \(\$32\,000\).
Four years after purchase, the value of the machine is \(\$20\,480\).

(a) Show that \(r = 20\). [2]

(b) Calculate the original purchase price of the machine. [2]

The diagram shows a plot of land formed by a quadrilateral \(ABCD\) and an adjoining circular sector \(CDE\) with centre \(D\).

NOT TO SCALE

In triangle \(ABC\), \(AB = 14\text{ m}\), \(BC = 18\text{ m}\), and angle \(ABC = 64^\circ\).
In triangle \(ACD\), angle \(DAC = 40^\circ\) and angle \(ADC = 75^\circ\).
The sector \(CDE\) has radius \(DC\) and angle \(CDE = 55^\circ\).

(a) Calculate the length \(AC\). [3]

(b) Calculate the area of the sector \(CDE\). [3]

The diagram shows a quadrilateral \(KLMN\).

NOT TO SCALE

In triangle \(KLM\), \(KL = 8.5\text{ cm}\), \(LM = 10.4\text{ cm}\), and angle \(KLM = 82^\circ\).
In triangle \(KMN\), \(KN = 14.0\text{ cm}\), angle \(KNM = 58^\circ\), and angle \(KMN\) is acute.

(a) Calculate the length \(KM\). [3]

(b) A circular sector with centre \(M\) and radius \(6.5\text{ cm}\) subtends the angle \(KMN\).
Calculate the area of this sector. [3]

The diagram shows a circular sector \(OAB\) attached to a triangle \(ABC\).

NOT TO SCALE

\(O\) is the centre of the circle with radius \(OA = OB = 7.5\text{ cm}\) and angle \(AOB = 84^\circ\).
In triangle \(ABC\), angle \(ABC = 125^\circ\) and angle \(BAC = 32^\circ\).

(a) Calculate the length of the chord \(AB\). [2]

(b) Calculate the length \(AC\). [2]

(c) Calculate the area of the minor sector \(OAB\). [2]

The table shows the cumulative frequency distribution of the times, \(t\) minutes, taken by 120 runners to complete a 10\text{ km} race.

\[ \begin{array}{|c|c|} \hline \text{Time } (t \text{ minutes}) & \text{Cumulative frequency} \\ \hline t \le 40 & 8 \\ t \le 45 & 26 \\ t \le 50 & 64 \\ t \le 55 & 98 \\ t \le 60 & 114 \\ t \le 65 & 120 \\ \hline \end{array} \]

(a) Find the number of runners who took more than 50 minutes. [1]

(b) Two runners are chosen at random from those who took 50 minutes or less.

Calculate the probability that both runners took 45 minutes or less. [2]

(c) A runner is chosen at random from the 120 runners.

Given that this runner took more than 40 minutes, calculate the probability that this runner took at most 55 minutes. [2]

A bag contains 7 blue discs and 5 yellow discs.

(a) Nadia picks two discs at random from the bag without replacement.

Calculate the probability that the two discs are of different colours. [2]

(b) If the two discs picked are of the same colour, the game ends.
If the two discs picked are of different colours, Nadia picks a third disc at random from the remaining discs.

Calculate the probability that Nadia picks exactly two blue discs in total. [2]

\(VABCD\) is a pyramid with a horizontal rectangular base \(ABCD\).
\(AB = 16\text{ cm}\) and \(BC = 12\text{ cm}\).
The vertex \(V\) is vertically above the centre of the base, \(O\).
The length of each sloping edge \(VA = VB = VC = VD = 18\text{ cm}\).

Calculate the angle between the edge \(VA\) and the base \(ABCD\).

Simplify fully.

\[\frac{2x^2 + 5x - 12}{4x^2 - 9}\]

Two players, Amy and Ben, play a game with a single counter placed on a strip of numbered squares.
The counter starts on square 0. On each turn, a player must move the counter forward by either 1, 2, or 3 squares.
The player who lands the counter exactly on square \(n\) wins the game. Amy moves first.

A square is a Winning position (W) if the player whose turn it is can guarantee a win with optimal play.
Otherwise, it is a Losing position (L).

Amy begins to complete a strategy table for target square \(n\):

$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\text{Target } n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
\text{Result for Amy (Player 1)} & \text{W} & \text{W} & \text{W} & \text{L} & \text{W} & \text{W} & \text{W} & \text{L} & \text{W} \\
\hline
\end{array}$$

(a) Write down the next target number \(n\) that is a losing position (L) for Amy.

(b) Write down an algebraic expression, in terms of the positive integer \(k\), for the \(k\)th target number \(n\) that is a losing position for Amy.

(c) Amy plays a game with target \(n = 35\). State who wins the game if both players play optimally.

In a subtraction game, a pile contains \(N\) counters. Two players take turns removing either 1, 3, or 4 counters from the pile. The player who takes the last counter wins. Player 1 moves first.

The strategy table shows whether the starting number of counters \(N\) is a winning (W) or losing (L) position for Player 1:

$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
N & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 \\
\hline
\text{Outcome} & \text{W} & \text{L} & \text{W} & \text{W} & \text{W} & \text{W} & \text{L} & \text{W} & \text{L} & \text{W} & \text{W} & \text{W} & \text{W} & \text{L} \\
\hline
\end{array}$$

(a) State the repeating length (period) of the winning and losing pattern.

(b) Work out whether \(N = 50\) is a winning (W) or losing (L) position for Player 1.

(c) On Player 1's first turn with \(N = 8\) counters, write down the number of counters Player 1 must remove to ensure a win.

A token is placed at the bottom-left corner \((1, 1)\) of an \(n \times n\) grid. Two players take turns moving the token either one square to the right or one square up. The player who moves the token to the top-right corner \((n, n)\) wins the game. Player 1 moves first.

Every game from \((1, 1)\) to \((n, n)\) takes the same total number of moves, \(T\).

$$\begin{array}{|c|c|c|c|c|}
\hline
\text{Grid size } (n \times n) & 2 \times 2 & 3 \times 3 & 4 \times 4 & 5 \times 5 \\
\hline
\text{Total number of moves } (T) & 2 & 4 & 6 & 8 \\
\hline
\text{Winner under optimal play} & \text{Player 2} & \text{Player 2} & \text{Player 2} & \text{Player 2} \\
\hline
\end{array}$$

(a) Write down an expression for \(T\) in terms of \(n\).

(b) Work out the total number of moves \(T\) for a \(12 \times 12\) grid.

(c) Explain why Player 2 always wins on an \(n \times n\) grid for any integer \(n \ge 2\).

Two players play a scoring game over several rounds on an \(n\)-sided polygon board.
In each round, Player A scores \(3n - 2\) points and Player B scores \(n + 6\) points.

Let \(d\) be the difference in points scored between Player A and Player B in one round, where \(d = \text{Score}_A - \text{Score}_B\).

$$\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Number of sides } (n) & 3 & 4 & 5 & 6 & 7 \\
\hline
\text{Score}_A & 7 & 10 & 13 & 16 & 19 \\
\hline
\text{Score}_B & 9 & 10 & 11 & 12 & 13 \\
\hline
\text{Difference } d & -2 & 0 & 2 & 4 & 6 \\
\hline
\end{array}$$

(a) Find an expression for \(d\) in terms of \(n\) in its simplest form.

(b) Find the value of \(n\) for which Player A scores 18 points more than Player B in a single round.

(c) When \(n = 8\), work out the total lead (cumulative point difference) Player A has over Player B after 15 rounds.

Two players play a game with two piles of beads, Pile X and Pile Y. On a player's turn, the player may take any positive number of beads from either Pile X or Pile Y, but not from both.
The player who takes the very last bead from the table wins.

A strategy table investigates positions \((x, y)\), where \(x\) is the number of beads in Pile X and \(y\) is the number of beads in Pile Y.

$$\begin{array}{|c|c|l|}
\hline
\text{Position } (x, y) & \text{Status for Player 1} & \text{Winning Strategy / Reason} \\
\hline
(1, 1) & \text{Losing (L)} & \text{Player 1 leaves }(1, 0)\text{ or }(0, 1)\text{, then Player 2 wins} \\
\hline
(2, 2) & \text{Losing (L)} & \text{Any move leaves unequal piles; Player 2 can make them equal again} \\
\hline
(3, 3) & \text{Losing (L)} & \text{Player 2 copies Player 1 to maintain equal piles} \\
\hline
(m, m) & \text{Losing (L)} & \text{Symmetric copy strategy guarantees Player 2 wins} \\
\hline
\end{array}$$

(a) State whether the starting position \((9, 9)\) is a winning (W) or losing (L) position for Player 1.

(b) The game starts at position \((14, 6)\). Describe the exact move Player 1 should make on turn 1 to guarantee a win.

(c) If the game starts at position \((25, k)\), write down the value of \(k\) that makes this a losing position for Player 1.

A sequence of grid fences is made using identical metal rods of length \(1\text{ m}\).

Diagram 1 uses \(6\) rods to form a single \(1\times 1\) square frame with two internal diagonals.
Diagram 2 uses \(11\) rods to form two adjacent squares with diagonals.
Diagram 3 uses \(16\) rods to form three adjacent squares with diagonals.

(a) Write down the number of rods needed for Diagram 4.
(b) Find an expression, in terms of \(n\), for the number of rods, \(R\), needed to make Diagram \(n\).

Pathways are constructed using regular hexagonal paving slabs in a single straight line.

- A path with \(1\) hexagon has \(6\) exposed outer edges.
- A path with \(2\) hexagons joined side-by-side has \(10\) exposed outer edges.
- A path with \(3\) hexagons has \(14\) exposed outer edges.

(a) Complete the statement:
Each time \(1\) hexagon is added to the path, the number of exposed outer edges increases by ______.

(b) Find an expression, in terms of \(n\), for the number of exposed outer edges, \(E\), for a path of \(n\) hexagons.

A designer creates a sequence of tiled decorative borders. Each border is formed by a row of \(n\) black square tiles surrounded entirely by white square tiles of the same size.

- Pattern 1 has \(1\) black tile and \(8\) white tiles.
- Pattern 2 has \(2\) black tiles and \(10\) white tiles.
- Pattern 3 has \(3\) black tiles and \(12\) white tiles.

(a) Write down an expression for the number of white tiles, \(W\), in Pattern \(n\).
(b) Work out the number of black tiles in a pattern that contains \(86\) white tiles.

Circular counters are arranged in concentric rings around a single centre counter to form star formations.

- Stage 1 consists of \(1\) centre counter and \(1\) ring of \(7\) counters (total \(8\) counters).
- Stage 2 has the centre counter and \(2\) rings, with the second ring containing \(12\) counters (total \(20\) counters).
- Stage 3 has the centre counter and \(3\) rings, with the third ring containing \(17\) counters (total \(37\) counters).

(a) Write down the number of counters in the outer ring alone for Stage \(1\), Stage \(2\), and Stage \(3\).
(b) Find an expression, in terms of \(k\), for the number of counters in the \(k\text{th}\) ring alone.

L-shaped tetromino blocks are joined at their ends to form a continuing staircase ribbon.

The perimeter, \(P\text{ cm}\), of each shape is recorded in the table below.

$$\begin{array}{|l|c|c|c|c|c|}
\hline
\text{Number of blocks }(n) & 1 & 2 & 3 & 4 \\
\hline
\text{Perimeter }(P\text{ cm}) & 10 & 16 & 22 & 28 \\
\hline
\end{array}$$

(a) Find an expression for \(P\) in terms of \(n\).
(b) Is it possible to have a ribbon shape with a perimeter of exactly \(140\text{ cm}\)? Give a clear mathematical reason for your answer.

A three-digit number has three identical non-zero digits, so it can be written as \(ddd = 100d + 10d + d\), where \(d\) is a digit from 1 to 9.

(a) Show that any such three-digit number \(ddd\) is always divisible by 37. [1]

(b) The sum of the digits of the number is \(S = d + d + d = 3d\).
Use algebra to justify that \(ddd \div S\) is always equal to the same integer value for every choice of \(d\), and state this integer. [2]

A four-digit palindromic number is written in the form \(abba\), where \(a\) and \(b\) are digits and \(a \ne 0\).
Its place value expansion is \(abba = 1000a + 100b + 10b + a\).

(a) Simplify the expression for \(abba\) in terms of \(a\) and \(b\). [1]

(b) By factorising out 11 from your expression, justify why every four-digit palindromic number must be divisible by 11, and write down the quotient in terms of \(a\) and \(b\). [2]

A two-digit number is \(N = 10t + u\), where \(t\) and \(u\) are digits such that \(t > u \ge 1\).
The number formed by reversing its digits is \(R = 10u + t\).

(a) Find an expression for \(N - R\) in its simplest factorised form. [1]

(b) Explain why \(N - R\) must always be divisible by 9, and find the maximum possible value of \(N - R\). [2]

Consider the sum of five consecutive positive integers:
\[S = n + (n + 1) + (n + 2) + (n + 3) + (n + 4)\]
where \(n\) is a positive integer.

(a) Simplify the expression for \(S\) in terms of \(n\). [1]

(b) (i) Justify why the sum \(S\) is always divisible by 5. [1]
(ii) Explain why \(S\) can never be a prime number for any positive integer \(n\). [1]

Section A: Investigation – Powers and Remainders

This investigation looks at the remainders when powers of 3 are divided by 7.

Let \(R_n\) be the remainder when \(3^n\) is divided by 7, where \(n\) is a positive integer.

(a) Complete the table for \(R_n\).

$$\begin{array}{|c|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
3^n & 3 & 9 & 27 & 81 & 243 & 729 \\
\hline
R_n & 3 & 2 & & & 5 & \\
\hline
\end{array}$$

[2]

(b) Write down the length of the repeating cycle (period) of the sequence \(R_n\).

[1]

(c) Using your pattern, work out the value of \(R_{2026}\).
Show your working.

[2]

Section A: Investigation – Digital Roots of Square Numbers

The digital root \(D(n)\) of a positive integer \(n\) is the single digit obtained by repeatedly adding all the digits of \(n\).
For example, \(D(49) = D(4 + 9) = D(13) = 1 + 3 = 4\).

(a) Complete the table for the digital roots of the first 9 square numbers.

$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline
n^2 & 1 & 4 & 9 & 16 & 25 & 36 & 49 & 64 & 81 \\
\hline
D(n^2) & 1 & 4 & 9 & 7 & & 9 & 4 & & 9 \\
\hline
\end{array}$$

[2]

(b) Write down the complete list of all possible values of \(D(n^2)\) for any positive integer \(n\).

[1]

(c) A student claims that \(123\,456\,788\) is a square number.
Find the digital root of \(123\,456\,788\) and use it to explain why the student is incorrect.

[2]

Section A: Investigation – Remainders of the Fibonacci Sequence

The Fibonacci sequence is defined by:
$$F_1 = 1, \quad F_2 = 1, \quad \text{and} \quad F_n = F_{n-1} + F_{n-2} \quad \text{for } n \ge 3.$$

Let \(r_n\) be the remainder when \(F_n\) is divided by 4.

(a) Complete the table for \(F_n\) and \(r_n\).

$$\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
F_n & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 \\
\hline
r_n & 1 & 1 & 2 & 3 & 1 & & & 1 \\
\hline
\end{array}$$

[2]

(b) State the length of the repeating cycle of \(r_n\).

[1]

(c) Find the remainder when \(F_{302}\) is divided by 4.
Show your working.

[2]

Section A: Investigation – Stepping on a Modular Ring

A counter moves around a circle with 8 positions labelled \(0, 1, 2, 3, 4, 5, 6, 7\) in a clockwise direction.
The counter starts at position 0.
At step \(n\), the total distance moved from the start is given by the triangular number:
$$T_n = \frac{n(n+1)}{2}$$
The landing position \(P_n\) after step \(n\) is the remainder when \(T_n\) is divided by 8.

(a) Complete the table for \(T_n\) and \(P_n\).

$$\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline
T_n & 1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 \\
\hline
P_n & 1 & 3 & 6 & 2 & 7 & & 4 & \\
\hline
\end{array}$$

[2]

(b) The sequence of positions \(P_n\) repeats every 16 steps.
Write down the landing position \(P_{803}\).

[1]

(c) Explain why the counter will never land on position 0 during any step \(n\).

[2]

Section A: Investigation – Repeating Decimal Patterns

The decimal expansion of \(\frac{1}{13}\) is \(0.\dot{0}7692\dot{3}\), which has a repeating block of 6 digits.
The digits can be found by looking at the remainders \(r_k\) when \(10^k\) is divided by 13.

(a) Complete the table of remainders \(r_k\) when \(10^k\) is divided by 13.

$$\begin{array}{|c|c|c|c|c|c|c|}
\hline
k & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
10^k & 10 & 100 & 1000 & 10\,000 & 100\,000 & 1\,000\,000 \\
\hline
r_k & 10 & 9 & & 3 & & 1 \\
\hline
\end{array}$$

[2]

(b) The remainder sequence repeats every 6 terms.
Find the remainder when \(10^{500}\) is divided by 13.

[1]

(c) The decimal expansion of \(\frac{4}{13}\) is \(0.\dot{3}0769\dot{2}\).
Find the 2025th digit after the decimal point in the expansion of \(\frac{4}{13}\).
Show your working.

[2]

At the start of each year, Maya deposits \(\$500\) into an investment plan that earns a compound interest rate of \(4\%\) per year.
Interest is calculated and added to the account at the end of each year, just before the next deposit is made.

Let \(A_n\) represent the total amount in the account at the end of year \(n\).

(a) Show that \(A_2 = 1060.80\).
(b) Write an expression for \(A_3\) as a sum of three terms and work out its value to the nearest cent.

A saver deposits an amount \(\$P\) at the start of each year into a fund paying compound interest at an annual rate of \(r\%\).
Let \(R = 1 + \frac{r}{100}\).

The total value in the fund at the end of \(n\) years is given by
\[ A_n = PR + PR^2 + PR^3 + \dots + PR^n \]

(a) Show algebraically that
\[ A_n = \frac{PR(R^n - 1)}{R - 1} \]

(b) Hence, find the total value in the fund after 15 years when \(P = 1200\) and \(r = 5\). Give your answer correct to the nearest dollar.

Lucas wants to save a total of \(\$50\,000\) by the end of 10 years.
He deposits a fixed amount \(\$P\) at the start of each year into an account paying a constant compound interest rate of \(6\%\) per year.

The future value after \(n\) years is modelled by
\[ A_n = \frac{PR(R^n - 1)}{R - 1}, \quad \text{where } R = 1.06 \]

(a) Find the minimum annual deposit \(P\), rounded up to the nearest whole dollar, required to reach at least \(\$50\,000\) at the end of 10 years.
(b) Using your rounded value of \(P\), calculate the total interest earned over the 10-year period.

A retiree places \(\$200\,000\) into an investment fund that yields \(4.5\%\) compound interest per year.
At the end of each year, after interest has been added, the retiree withdraws \(\$18\,000\).

The balance \(W_n\) in the fund at the end of year \(n\), immediately after the withdrawal, is modelled by
\[ W_n = 200\,000(1.045)^n - 18\,000\left( \frac{1.045^n - 1}{0.045} \right) \]

(a) Show that this formula simplifies to \(W_n = 400\,000 - 200\,000(1.045)^n\).
(b) Use logarithms or a graphic display calculator to determine the maximum number of full annual withdrawals of \(\$18\,000\) that can be made before the balance drops below \(\$18\,000\).

Section B: Modelling

A student investigates the solubility of a mineral compound in water. The table shows the mass, \(y\) grams, of the mineral dissolved after stirring continuously for \(x\) minutes.

$$\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Time, } x\text{ (minutes)} & 2 & 4 & 6 & 8 & 10 \\
\hline
\text{Mass, } y\text{ (grams)} & 14.2 & 22.8 & 30.5 & 39.1 & 47.4 \\
\hline
\end{array}$$

(a) Find the equation of the linear regression model for \(y\) in terms of \(x\). Give coefficients correct to 3 significant figures.

(b) Sketch the graph of your linear regression model for \(0 \le x \le 12\), clearly labelling the \(y\)-intercept.

(c) Use the linear regression model to estimate the mass of dissolved mineral when \(x = 7.5\) minutes.

Section B: Modelling

An engineer monitors the remaining energy, \(E\) watt-hours (Wh), in an electric scooter battery after travelling a distance of \(d\) kilometres.

$$\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Distance, } d\text{ (km)} & 5 & 10 & 15 & 20 & 25 \\
\hline
\text{Remaining energy, } E\text{ (Wh)} & 362 & 325 & 291 & 252 & 218 \\
\hline
\end{array}$$

(a) Calculate the equation of the linear regression line of \(E\) on \(d\). Give numerical values correct to 3 significant figures.

(b) Sketch the linear regression model on axes for \(0 \le d \le 60\), showing clearly the coordinates of the intercepts on both axes.

(c) Use your model to calculate the total distance travelled when the battery is fully depleted (\(E = 0\)).