Solving Equations with Technology: exam review Exam review

Three equations are \(2x + y = 3z + 11\), \(x - 2y + z = -2\) and \(4x + 5 = y - 2z + 13\). [5]

1. (a) Write the equations in the form \(ax + by + cz = d\). [2]
2. (b) Use your calculator to find the values of \(x\), \(y\) and \(z\). [3]

Use your calculator in this question. Give answers to 3 significant figures. [6]

1. (a) Show that \(2x + \dfrac{5}{x} = 9\) can be written as \(2x^2 - 9x + 5 = 0\). [2]
2. (b) Hence find the possible values of \(x\). [2]
3. (c) Find the real solution of \(3x^3 + 2x + 11 = 0\). [2]

Solve each equation. Give real solutions to 3 significant figures. [6]

1. (a) Solve \(9x^2 - 12x + 4 = 0\). [2]
2. (b) Show that \(x^2 - 4x + 9 = 0\) has no real solutions. [2]
3. (c) Find the real solutions of \(x^4 - 5x^2 + x + 2 = 0\). [2]

The graph shows \(y = f(x)\), where \(f(x) = x^3 - 3x^2 - 9x\). Use your calculator. [6]

1. (a) Find the \(x\)-intercepts of the graph. [2]
2. (b) Solve \(f(x) = −10\). [2]
3. (c) The line \(y = c\) meets the curve at exactly two points. Find the possible values of \(c\). [2]

Consider the equations \(3x - 2y = 5\) and \(6x - 4y = k\), where \(k\) is a constant. [4]

1. (a) Find the value of \(k\) for which the system has infinitely many solutions. [2]
2. (b) Explain, using the graphs of the two equations, why the system has no solution when \(k = 12\). [2]

Three numbers \(x\), \(y\) and \(z\) satisfy \(x \le y \le z\). Their mean is 8, their median is 7 and their range is 9. [5]

1. (a) Write down three equations in \(x\), \(y\) and \(z\). [3]
2. (b) Find the three numbers. [2]

A curve has equation \(y = ax^2 + bx\). It passes through the point \((2,\ 6)\) and its gradient at that point is 7. [7]

1. (a) Write down two equations in \(a\) and \(b\). [2]
2. (b) Find the values of \(a\) and \(b\). [2]
3. (c) The line \(y = 3x + 1\) meets the curve at the points P and Q. Use your calculator to find the distance PQ. [3]

A company invests $24 000 in three funds for one year. Fund P pays 2% per year, fund Q pays 3.5% per year and fund R pays 5% per year. There is $4 000 more in fund Q than in fund P. The total interest earned over the year is $870. Let \(p\), \(q\) and \(r\) be the amounts, in dollars, in funds P, Q and R. [8]

1. (a) Write down an equation for the total amount invested. [1]
2. (b) Write down an equation for the total interest earned. [2]
3. (c) Write down an equation using the fact that fund Q has $4 000 more than fund P. [1]
4. (d) Find the amount invested in fund R. [2]
5. (e) Find the percentage of the total interest that comes from fund R. Give your answer to 3 significant figures. [2]

A craft stall makes candles, soaps and bath bombs. Each candle uses 300 g of wax and 10 ml of fragrance and sells for $15. Each soap uses 120 g of wax and 6 ml of fragrance and sells for $6. Each bath bomb uses 60 g of wax and 8 ml of fragrance and sells for $9. In one week the stall uses 10.8 kg of wax and 590 ml of fragrance, and everything made is sold for a total of $720. Let \(c\), \(s\) and \(b\) be the numbers of candles, soaps and bath bombs made. [7]

1. (a) Write down three equations in \(c\), \(s\) and \(b\). [3]
2. (b) Find the number of bath bombs made. [2]
3. (c) Find the mean price at which an item is sold. Give your answer in dollars to 2 decimal places. [2]

A rectangular sheet of metal measures 36 cm by 20 cm. A square of side \(x\) cm is cut from each corner and the sides are folded up to make an open tray. [8]

1. (a) Show that the volume of the tray is \(V = 4x^3 - 112x^2 + 720x\) cm³. [2]
2. (b) Explain why \(x\) must be less than 10. [1]
3. (c) The volume of the tray is 1000 cm³. Find the possible values of \(x\). [3]
4. (d) Find the length and the width of the tray when \(x\) takes the smaller possible value. [2]

A charity records the money, in dollars, collected on days 1, 2, 3, … of a campaign. The amounts form an arithmetic sequence. On day 5 it collects $23. In the first 10 days it collects $250 in total. In a second campaign the amounts collected on days 1, 2 and 3 form a geometric sequence with first term 8. The total collected over these three days is $38. [10]

1. (a) Find the amount collected on day 1 and the common difference. [3]
2. (b) The total amount collected in the first \(n\) days is $900. Find \(n\). [3]
3. (c) In the second campaign, find the possible values of the common ratio \(r\). [3]
4. (d) State which value of \(r\) is possible for amounts of money collected, giving a reason. [1]

A school hall is hired for a concert. The weekly income from \(x\) tickets sold is \(R = 80x - 0.5x^2\) dollars and the weekly cost is \(C = 1500 + 12x\) dollars. [8]

1. (a) Find the break-even quantities. [2]
2. (b) Find the smallest and the largest whole number of tickets that give a profit. [2]
3. (c) Find the greatest profit and the number of tickets that gives it. [2]
4. (d) The organisers want a profit of at least $600. Find the range of whole numbers of tickets that achieves this. [2]

A ball is thrown upwards from a platform. Its height above the ground after \(t\) seconds is \(h = −4.9t^2 + 14t + 2\) metres, for \(t \ge 0\). [7]

1. (a) Write down the height of the platform. [1]
2. (b) Find the greatest height of the ball and the time at which it is reached. [2]
3. (c) Find the time at which the ball hits the ground. [2]
4. (d) Find the length of time for which the ball is higher than 6 m above the ground. [2]