Solving Equations with Technology: exam review Exam review: second paper

Three equations are \(3x + 2y = 4z - 4\), \(x - y + 2z = -3\) and \(2x + 9 = 3y - z - 3\). [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 \(3x + \dfrac{4}{x} = 9\) can be written as \(3x^2 - 9x + 4 = 0\). [2]
2. (b) Hence find the possible values of \(x\). [2]
3. (c) Find the real solution of \(2x^3 + x + 8 = 0\). [2]

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

1. (a) Solve \(16x^2 - 24x + 9 = 0\). [2]
2. (b) Show that \(x^2 + 6x + 13 = 0\) has no real solutions. [2]
3. (c) Find the real solutions of \(x^4 - 6x^2 + x + 3 = 0\). [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 \(5x + 2y = 4\) and \(15x + 6y = k\), 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 \(k = 15\). [2]

Three numbers \(x\), \(y\) and \(z\) satisfy \(x \le y \le z\). Their mean is 10, their median is 9 and their range is 11. [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 \((3,\ 9)\) and its gradient at that point is 9. [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 \(y = 2x + 3\) meets the curve at the points P and Q. Use your calculator to find the distance PQ. [3]

A gardener mixes three fertilisers to make 300 kg of mixture. Fertiliser A contains 4% nitrogen, fertiliser B contains 6% nitrogen and fertiliser C contains 10% nitrogen. There is 20 kg less of fertiliser B than of fertiliser A. The mixture contains 18.8 kg of nitrogen. Let \(a\), \(b\) and \(c\) be the masses, in kg, of fertilisers A, B and C. [8]

1. (a) Write down an equation for the total mass of the mixture. [1]
2. (b) Write down an equation for the mass of nitrogen in the mixture. [2]
3. (c) Write down an equation using the fact that there is 20 kg less of fertiliser B than of fertiliser A. [1]
4. (d) Find the mass of fertiliser C in the mixture. [2]
5. (e) Find the percentage of the nitrogen that comes from fertiliser C. Give your answer to 3 significant figures. [2]

A school kitchen makes pies, quiches and tarts. Each pie uses 200 g of flour and 80 g of butter and sells for $4.50. Each quiche uses 150 g of flour and 90 g of butter and sells for $5.00. Each tart uses 100 g of flour and 60 g of butter and sells for $3.50. One day the kitchen uses 14.5 kg of flour and 7.1 kg of butter, and everything made is sold for a total of $400. Let \(p\), \(q\) and \(t\) be the numbers of pies, quiches and tarts made. [7]

1. (a) Write down three equations in \(p\), \(q\) and \(t\). [3]
2. (b) Find the number of quiches 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 piece of fabric measures 60 cm by 40 cm. A square of side \(x\) cm is cut from each corner and the sides are sewn up to make an open storage bag. [8]

1. (a) Show that the volume of the bag is \(V = 4x^3 - 200x^2 + 2400x\) cm³. [2]
2. (b) Explain why \(x\) must be less than 20. [1]
3. (c) The volume of the bag is 7000 cm³. Find the possible values of \(x\). [3]
4. (d) Find the length and the width of the bag when \(x\) takes the smaller possible value. [2]

A farmer records the mass of apples, in kg, picked on days 1, 2, 3, … of a harvest. The masses form an arithmetic sequence. On day 6 she picks 29 kg. In the first 8 days she picks 172 kg in total. In a second orchard the masses picked on days 1, 2 and 3 form a geometric sequence with first term 6. The total picked over these three days is 42 kg. [10]

1. (a) Find the mass picked on day 1 and the common difference. [3]
2. (b) The total mass picked in the first \(n\) days is 664 kg. Find \(n\). [3]
3. (c) In the second orchard, find the possible values of the common ratio \(r\). [3]
4. (d) State which value of \(r\) is possible for masses, giving a reason. [1]

A theatre is hired for a show. The income from \(x\) tickets sold is \(R = 95x - 0.4x^2\) dollars and the cost is \(C = 2000 + 15x\) 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 $1500. Find the range of whole numbers of tickets that achieves this. [2]

A stone is thrown upwards from a cliff top. Its height above the sea after \(t\) seconds is \(h = −4.9t^2 + 12t + 1.5\) metres, for \(t \ge 0\). [7]

1. (a) Write down the height of the cliff top. [1]
2. (b) Find the greatest height of the stone and the time at which it is reached. [2]
3. (c) Find the time at which the stone hits the sea. [2]
4. (d) Find the length of time for which the stone is higher than 5 m above the sea. [2]