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A shopkeeper buys a box of 40 pens for $12. She sells each pen for 45 cents. Calculate the percentage profit she makes on the box of pens.

A savings account pays simple interest at a rate of 4% per year. Karl invests $350 in this account. Calculate the total amount in the account at the end of 3 years.

Given that \(f(x) = 7 - 4x\), find the value of \(x\) when \(f(x) = -5\).

Given that \(h(x) = 2x^2 - 3\), find the value of \(h(-4)\).

Factorise fully: \(6x^2y - 9xy^2\).

Expand and simplify: \((x - 4)(x + 7)\).

Find the \(n\)-th term of the sequence: \(5, 11, 17, 23, 29, \dots\)

A sequence has \(n\)-th term given by \(T_n = 3n^2 - 2\). Find the 5th term of this sequence.

Arthur changes $300 into Euros (€) when the exchange rate is €1 = $1.20. How many Euros does Arthur receive?

A shop owner buys a box of 50 notebooks for $40. She sells each notebook for $1.00. Calculate her percentage profit.

Given that \( f(x) = 3x - 5 \), find the value of \( x \) when \( f(x) = 13 \).

Given that \( g(x) = \frac{12}{x+1} \), find \( g(3) \).

Factorise completely: \( 6ax - 9ay \)

Expand and simplify: \( (2x - 3)(x + 5) \)

Find the \( n \)-th term of the sequence: \( 5, 11, 17, 23, 29, \dots \)

The \( n \)-th term of a sequence is \( n^2 + 3 \). Find the 8th term of this sequence.

The first four terms of a sequence are \(4, 11, 18, 25, \dots\). Find an expression, in terms of \(n\), for the \(n\)-th term of this sequence.

A tourist changes €500 into Dollars ($) when the exchange rate is €1 = $1.24. She spends $380 of this money while on holiday. She then changes all of her remaining dollars back into Euros when the exchange rate is €1 = $1.20. Calculate the amount of Euros she receives back.

The function \(f(x)\) is defined as \(f(x) = \frac{2x + 5}{3x - 4}\) for \(x \neq \frac{4}{3}\). Find an expression for \(f^{-1}(x)\).

Simplify fully the algebraic fraction: \(\frac{2x^2 - 5x - 3}{4x^2 - 1}\)

Find the \(n\)-th term of the sequence: \(3, 8, 15, 24, 35, \dots\)

A shopkeeper sells a camera for $276, making a profit of 15% on the cost price. Find the cost price of the camera.

Solve the equation: \(3^{2x - 1} = 27^{x - 2}\)

Rearrange the formula to make \(t\) the subject: \(w = \frac{3t + 2}{5 - t}\)

The \(n\)-th term of a sequence is given by \(T_n = a \cdot 2^{n-1} + b\). Given that \(T_1 = 7\) and \(T_2 = 11\), calculate the value of \(T_4\).

An investment of \(\$400\) earns compound interest at a rate of \(3\%\) per year. Calculate the total interest earned at the end of 2 years.

A book is sold in London for \(\pounds15\) and in New York for \(\$24\). The exchange rate is \(\pounds1 = \$1.45\). Calculate the difference in price of the book in dollars. Give your answer in dollars.

Given that \(f(x) = \frac{3x - 5}{2x + 1}\) for \(x \neq -\frac{1}{2}\), find \(f^{-1}(x)\).

The graph of \(y = f(x)\) is transformed to the graph of \(y = g(x)\) by a translation of vector \(\begin{pmatrix} 2 \\ 0 \end{pmatrix}\) followed by a stretch parallel to the \(y\)-axis with scale factor 4. Find an expression for \(g(x)\) in terms of \(f(x)\).

Factorise completely: \(6x^2 - 5xy - 6y^2\)

Rearrange the formula to make \(x\) the subject: \(t = \sqrt{\frac{x+3}{x-2}}\)

Find the \(n\)-th term of the sequence: \(4, 14, 30, 52, \dots\)

Find the \(n\)-th term of the sequence: \(\frac{2}{3}, \frac{5}{9}, \frac{8}{27}, \frac{11}{81}, \dots\)

A bank exchange rate is $1 = 0.85 Euros. A traveler exchanges $600 into Euros. They spend 340 Euros and exchange the remaining Euros back to dollars at a rate of 1 Euro = $1.15. Calculate the final amount in dollars they have.

Given that \(f(x) = \frac{2x + 3}{x - 1}\) for \(x \neq 1\), find an expression for \(f^{-1}(x)\).

Factorise completely: \(12a^2b - 3b^3\).

The \(n\)-th term of a sequence is given by \(T_n = an^2 + bn - 3\). Given that the second term \(T_2 = 13\) and the third term \(T_3 = 30\), find the value of \(a\) and the value of \(b\).

The graph of \(y = f(x)\) is mapped onto the graph of \(y = g(x)\) by a stretch, parallel to the \(y\)-axis, with scale factor 3, followed by a translation of \(\begin{pmatrix} 0 \\ -5 \end{pmatrix}\). Write an expression for \(g(x)\) in terms of \(f(x)\).

Elena invests \(\$4500\) at a rate of \(3.2\%\) per year compound interest. Calculate the total value of her investment at the end of 5 years. Give your answer to the nearest cent.

A shopkeeper buys a box of 50 calculators for \(\$400\). He sells 40 of them at \(\$11\) each, and the remaining 10 at \(\$6\) each. Calculate his percentage profit on the cost price.

Given the function \(f(x) = \frac{3x + 8}{2}\), find the value of \(x\) when \(f(x) = 13\).

The graph of the quadratic function \(y = x^2 - 4x - 12\) crosses the \(x\)-axis at two points. Find the positive \(x\)-coordinate where the graph crosses the \(x\)-axis.

Expand and simplify the expression: \(4(2x - 3) - 3(x - 5)\)

Solve the simultaneous equations to find the value of \(x\): \(3x + 2y = 12\) and \(4x - y = 5\)

Here are the first five terms of an arithmetic sequence: \(4, 11, 18, 25, 32\). Find an expression for the \(n\)-th term of this sequence.

The \(n\)-th term of a sequence is given by \(T_n = 2n^2 - 5\). Find the 12th term of this sequence.

Amelia changes €550 into Japanese Yen (¥). The bank charges a 2% commission fee, which is subtracted from the €550 before the conversion. The exchange rate is €1 = ¥162.40. Calculate how many Japanese Yen Amelia receives, giving your answer to the nearest Yen.

Liam invests $4500 at a rate of 3.5% per year compound interest. Calculate the total value of his investment at the end of 4 years. Give your answer to the nearest cent.

An arithmetic sequence has first term 7 and common difference 4. Find the 50th term of this sequence.

Find the \(n\)-th term of the sequence: 5, 12, 21, 32, ...

Simplify fully: \(4(2x - 3) - 3(x - 5)\)

Factorise fully: \(12a^2b - 18ab^2\)

Given the function \(f(x) = 3x^2 - 5x + 2\), find the positive value of \(x\) for which \(f(x) = 10\). Give your answer as a simplified fraction.

Find the inverse function \(g^{-1}(x)\) of \(g(x) = \frac{4x - 3}{5}\).

Lia invests \(\$4500\) at a rate of \(2.3\%\) per year simple interest. Calculate the total value of her investment after 6 years.

Here are the first five terms of a sequence:

\[8, \quad 13, \quad 18, \quad 23, \quad 28\]

Find an expression for the \(n\)-th term of this sequence.

Aravind invests $5000 USD in a savings account with an interest rate of 3.2% per year, compounded quarterly. After 5 years, he withdraws the entire amount and converts it to Euros (EUR). The exchange rate is 1 USD = 0.92 EUR, and the bank charges a 1.5% commission on the converted amount in EUR. Calculate the final amount of Euros Aravind receives. Give your answer to 2 decimal places.

The functions \(f(x)\) and \(g(x)\) are defined as follows:
\(f(x) = \frac{2x - 3}{x + 4}\) for \(x \neq -4\)
\(g(x) = 3x + 1\)

Find the value of \(x\) for which \(f^{-1}(x) = g(2)\).

Solve the equation:
\(\frac{3}{2x - 1} + \frac{4}{x + 3} = 1\)

The first five terms of a sequence are:
5, 14, 29, 50, 77, ...

Find an expression, in terms of \(n\), for the \(n\)-th term of this sequence.

A machine is purchased for $12,000 and depreciates at a rate of 8% per year compounded annually. At the same time, $8,000 is invested in a savings account earning 4.5% simple interest per year. Calculate the positive difference between the value of the savings account and the value of the machine after 6 years. Give your answer to 2 decimal places.

A rational function is defined by \(f(x) = \frac{a}{x - 2} + b\). The graph of \(y = f(x)\) passes through the coordinates \((3, 7)\) and \((5, 4)\). Find the value of \(a\) and the value of \(b\).

Simplify completely:
\(\frac{2x^2 - 5x - 3}{4x^2 - 1} \times \frac{2x - 1}{x^2 - 9}\)

The 1st, 3rd, and 11th terms of an arithmetic progression are the first three terms of a geometric progression. The first term of the arithmetic progression is 4. Given that the common difference of the arithmetic progression is non-zero, find the common difference, \(d\), of the arithmetic progression and the common ratio, \(r\), of the geometric progression.

An investment of $8000 is placed in a bank account that pays an interest rate of 4.5% per year, compounded quarterly. Find the number of years, to the nearest tenth of a year, for the value of the investment to reach $11,500.

Consider the functions \(f(x) = \frac{3x - 5}{2x + 1}\) for \(x \neq -0.5\) and \(g(x) = x^2 - 3\). Find the values of \(x\) for which \(f^{-1}(x) = g(2)\). Give your answer as a fraction in its simplest form.

Solve the equation:

\(\frac{4}{x - 3} - \frac{3}{x + 2} = 2\)

Show all your working and give your answers correct to 2 decimal places.

The first four terms of a sequence are \(7, 18, 35, 58, \dots\). Find an expression, in terms of \(n\), for the \(n\)-th term of this sequence.

A shopkeeper reduces the price of a television by 12%. Later, she increases this reduced price by 15%. The final price of the television is $556.60. Calculate the original price of the television.

The graph of a quadratic function \(y = ax^2 + bx + c\) passes through the points \((-1, -2)\), \((1, 4)\), and \((2, 13)\). Find the equation of the quadratic function in the form \(y = ax^2 + bx + c\).

Write as a single fraction in its simplest form:

\(\frac{2x - 3}{x^2 - 4} - \frac{3}{x + 2}\)

The \(n\)-th term of a geometric sequence is given by \(T_n = p \cdot q^{n-1}\). Given that the second term of this sequence is 6 and the fifth term is 162, calculate the sum of the first 6 terms of this sequence.

An amount of $6500 is invested in an account that pays compound interest at a rate of \(r\%\) per year. After 8 years, the total value of the investment is $8200. Find the value of \(r\), giving your answer correct to 2 decimal places.

The \(n\)-th term of a sequence is given by \(T_n = an^2 + bn + 7\). Given that the 3rd term of the sequence is 16 and the 5th term of the sequence is 42, find the value of the 12th term.

A triangular pyramid of coins is built with \( n \) rows, where Row 1 is at the top. Row 1 has 1 coin, Row 2 has 2 coins, Row 3 has 3 coins, and in general, Row \( r \) has \( r \) coins. The value of each coin in Row \( r \) is \( 5r \) cents. Let \( T(r) \) be the total value, in cents, of all the coins in Row \( r \). Find the value of \( T(4) \), the total value of the coins in Row 4.

Using the context from Question 1, write down an expression in terms of \( r \) for the total value \( T(r) \), in cents, of the coins in Row \( r \).

Let \( S(n) \) be the total value, in cents, of all the coins in a pyramid of height \( n \). Find \( S(4) \), the total value of a pyramid with 4 rows.

The sum of the first \( n \) square numbers is given by the formula: \( 1^2 + 2^2 + 3^2 + \dots + n^2 = \frac{n(n+1)(2n+1)}{6} \). Using this and your result from Question 2, find a formula for \( S(n) \), the total value in cents of a pyramid of height \( n \), in terms of \( n \).

Use the formula from Question 4 to find the total value, in dollars ($), of a pyramid of height 10.

Suppose instead, a different pyramid is built where Row \( r \) has \( r \) coins, and each coin in Row \( r \) has a value of \( 2r - 1 \) cents. Let \( U(r) \) be the total value, in cents, of the coins in Row \( r \). Write down an expression for \( U(r) \) in terms of \( r \), and expand the brackets.

Let \( V(n) \) be the total value, in cents, of this new pyramid of height \( n \). Using the formula for the sum of the first \( n \) integers, \( 1 + 2 + \dots + n = \frac{n(n+1)}{2} \), and the sum of squares formula, the total value \( V(n) \) can be shown to simplify to: \( V(n) = \frac{n(n+1)(4n-1)}{a} \). Find the integer value of \( a \).

An investigation of dot patterns arranged in concentric equilateral triangles. Let \(n\) represent the side length of the outermost triangle. The total number of dots in the pattern is denoted by \(S_n\). The number of dots on the outer boundary is denoted by \(B_n\). The number of dots inside the boundary is denoted by \(I_n\). For \(n = 1\): \(S_1 = 3, B_1 = 3, I_1 = 0\). For \(n = 2\): \(S_2 = 6, B_2 = 6, I_2 = 0\). For \(n = 3\): \(S_3 = 10, B_3 = 9, I_3 = 1\). (a) State the values of \(S_4\), \(B_4\), and \(I_4\). (b) Find an expression for \(B_n\) in terms of \(n\). (c) The formula for the total number of dots is \(S_n = \frac{(n+1)(n+2)}{2}\). Use this and your answer to part (b) to show that \(I_n = \frac{n^2 - 3n + 2}{2}\). (d) Find the value of \(n\) for which the number of inside dots is exactly 45.

This investigation looks at the behavior of infinite nested square roots of the form \(x_{k+1} = \sqrt{c + x_k}\). Let \(c = 6\) and the initial term \(x_1 = \sqrt{6}\). (a) Calculate the values of \(x_1\) and \(x_2\) correct to 3 decimal places. (b) Assuming the sequence of terms approaches a positive limit \(L\) as \(k\) increases, then \(L = \sqrt{6 + L}\). (i) Show that \(L^2 - L - 6 = 0\). (ii) Solve this equation to find the value of \(L\), explaining why only one solution is valid. (c) For a general constant \(c > 0\), the limit \(L\) satisfies \(L = \sqrt{c + L}\). (i) Write down the quadratic equation in \(L\) for this general case. (ii) Show that the positive solution to this equation is \(L = \frac{1 + \sqrt{1 + 4c}}{2}\). (d) Find the positive integer value of \(c\) for which the limit \(L = 7\).

A logistics company operates delivery drones. The total average monthly cost of maintaining and operating a drone battery over \(t\) months is modelled by: \(T(t) = 15 + 3.2t + \frac{320}{t}\), for \(t \ge 1\). Here, \(15 + 3.2t\) represents the monthly operating cost, and \(\frac{320}{t}\) represents the average monthly purchase cost of the battery. (a) Complete the following table of values for \(T(t)\): At \(t = 5\), \(T(t) = \dots\); At \(t = 16\), \(T(t) = \dots\). (b) Use your graphic display calculator (GDC) to find the coordinates of the local minimum of \(T(t)\) for \(t \ge 1\). (c) State the optimal number of months to use a battery before replacing it, and the corresponding minimum average monthly cost. (d) If the purchase price of a new battery rises to $500, write down the new model for the total average monthly cost \(T_{\text{new}}(t)\), and find the new optimal replacement interval in months.

The expansion of a steel bridge span under varying temperatures is modeled by the quadratic function: \(L(T) = a T^2 + b T + c\) where \(L(T)\) is the length of the span in meters at temperature \(T\) in degrees Celsius (\(^\circ\text{C}\)). The following measurements were recorded: At \(T = 0^\circ\text{C}\), \(L(0) = 150\) meters. At \(T = 10^\circ\text{C}\), \(L(10) = 150.018\) meters. At \(T = 30^\circ\text{C}\), \(L(30) = 150.066\) meters. (a) State the value of \(c\). (b) Form a system of simultaneous equations and show that \(b = 0.0016\) and \(a = 0.00002\). (c) Calculate the length of the bridge span when the temperature is \(50^\circ\text{C}\). (d) Find the temperature, to 1 decimal place, at which the bridge span reaches a length of \(150.25\) meters.

In this investigation, we look at the difference between the squares of consecutive terms in an arithmetic progression. Let the terms of an arithmetic progression be \(a_n\). The sequence of differences is defined as \(D_n = a_{n+1}^2 - a_n^2\). First, consider the sequence of consecutive odd numbers: \(1, 3, 5, 7, 9, \dots\) (a) (i) Write down the 6th term in this sequence and its square. (ii) The first three differences are \(D_1 = 3^2 - 1^2 = 8\), \(D_2 = 5^2 - 3^2 = 16\), \(D_3 = 7^2 - 5^2 = 24\). State the values of \(D_4\) and \(D_5\). (iii) Find an expression for \(D_n\) in terms of \(n\). Next, consider the arithmetic progression \(a_n = 3n - 1\). (b) (i) Write down the first four terms of this progression and their squares. (ii) Find the first three terms of its difference sequence \(D_n = a_{n+1}^2 - a_n^2\). Finally, consider a general arithmetic progression \(a_n = d(n - 1) + a\), where \(a\) is the first term and \(d\) is the common difference. (c) Show algebraically that the difference of squares is given by \(D_n = d(2a + d(2n - 1))\).