Simultaneous equations, inequalities and polynomials – Year 12 A level maths worksheet with answers to download as PDF

Solve each pair of simultaneous equations. Show all your working.

1. \(4x-3y=15\) and \(2x+5y=1\) [3 marks]
2. \(y=2x-1\) and \(x^2+y^2=10\) [5 marks]
3. \(xy=12\) and \(x+y=7\) [4 marks]

Solve each inequality and write the solution using inequality notation.

1. \(5-3x\ge20\) [2 marks]
2. \(2(x+4)<3x-1\) [2 marks]
3. \(x^2-x-12>0\) [3 marks]
4. \(3x^2\le12x\) [3 marks]

Use the factor theorem to factorise each cubic completely.

1. \(f(x)=x^3-7x+6\) [4 marks]
2. \(g(x)=2x^3+x^2-13x+6\) [5 marks]

\(f(x)=x^3+2x^2-5x+1\).

1. Divide \(f(x)\) by \((x-2)\), giving the quotient and the remainder. [3 marks]
2. Check the remainder using the remainder theorem. [1 mark]
3. Find the remainder when \(f(x)\) is divided by \((x+1)\). [2 marks]

The polynomial \(p(x)=x^3+ax^2+bx-6\) has \((x-2)\) as a factor. When \(p(x)\) is divided by \((x+1)\) the remainder is \(-12\).

1. Write down two equations in \(a\) and \(b\). [3 marks]
2. Solve them to find \(a\) and \(b\). [3 marks]
3. Show that \(p(x)=0\) has only one real root. [3 marks]

Find the set of values of \(x\) that satisfy both inequalities in each case.

1. \(2x+1>-5\) and \(x^2\le16\) [3 marks]
2. \(x^2-5x+4<0\) and \(3x-2>1\) [4 marks]

Three gyms advertise their prices. Gym A charges £25 to join and then £18 per month. Gym B has no joining fee and charges £23 per month. Gym C charges £60 to join and then £14 per month.

1. After how many months do gyms A and B cost the same? [3 marks]
2. Find the total cost of each gym for one year (12 months). [3 marks]
3. After how many whole months does gym C become cheaper than gym A? [3 marks]
4. Mia plans to stay for 6 months. Which gym is cheapest? [2 marks]

A rectangular garden has perimeter 34 m and area 60 m\(^2\). Let the length be \(l\) m and the width \(w\) m.

1. Write down two equations in \(l\) and \(w\). [2 marks]
2. Solve them to find the dimensions of the garden. [4 marks]
3. A straight path joins opposite corners. Find its length. [2 marks]
4. Fencing costs £22 per metre. Work out the cost of fencing all four sides. [2 marks]

A bakery makes \(x\) hundred loaves a day. Its weekly profit model is \(P=-0.5x^2+12x-50\), where \(P\) is in hundreds of pounds.

1. Show that the bakery makes a profit when \(x^2-24x+100<0\). [2 marks]
2. Solve this inequality, giving the boundaries to 2 decimal places. [4 marks]
3. Find the number of loaves that gives the greatest profit and state that profit in pounds. [3 marks]
4. The bakery can bake between 600 and 2000 loaves. Between which production levels does it make a profit? [2 marks]

A school hires a coach for £420 and pays £8 per student for entry. The school budget is at most £1 000. The trip only runs with at least 40 students and the cost per student must not be more than £14. Let \(n\) be the number of students.

1. Show that the budget condition gives \(n\le72\). [2 marks]
2. Show that the cost-per-student condition gives \(n\ge70\). [3 marks]
3. List the possible numbers of students. [1 mark]
4. For each possible value, work out the cost per student to 2 decimal places. [3 marks]

A water jet leaves a nozzle at the origin and follows the path \(y=4x-x^2\), where \(x\) is the horizontal distance and \(y\) the height, both in metres. A hillside rises along the straight line \(y=0.5x\).

1. Find the coordinates of the point where the water hits the hillside. [4 marks]
2. Find the greatest height of the jet above the horizontal. [2 marks]
3. The vertical gap between the jet and the hillside at distance \(x\) is \(g=3.5x-x^2\). Find the greatest gap. [3 marks]
4. Find the horizontal distance at which the jet is 3 m above the hillside. [3 marks]

A rectangular sheet of card is 20 cm by 14 cm. A square of side \(x\) cm is cut from each corner and the sides are folded up to make an open box.

1. Show that the volume is \(V=4x^3-68x^2+280x\). [3 marks]
2. Find \(V\) when \(x=2\). [1 mark]
3. The volume is 288 cm\(^3\). Show that \(x=4\) is a solution and find the other valid solution to 2 decimal places. [6 marks]
4. State the range of values that \(x\) can take. [1 mark]

At a cinema, 2 adult tickets and 3 child tickets cost £47.50. 3 adult tickets and 2 child tickets cost £52.50.

1. Form two simultaneous equations. [2 marks]
2. Solve them to find the price of an adult ticket and a child ticket. [4 marks]
3. A family has 2 adults and 4 children. Find the cost. [2 marks]
4. The cinema offers a family ticket for £50. Does this save money? [2 marks]

A simple model for the stopping distance \(d\) metres of a car travelling at \(v\) miles per hour is \(d=0.02v^2+0.6v\).

1. Check the entries for 30 and 50 mph. [2 marks]
2. A school has a stopping-distance limit of 60 m. Show that \(v\) must satisfy \(v^2+30v-3000\le0\). [2 marks]
3. Find the maximum speed to 1 decimal place. [4 marks]
4. Explain why the negative root is not used. [1 mark]

A theatre sells adult tickets and child tickets for its pantomime. The takings on two nights are shown. Prices were the same on both nights.

1. Let an adult ticket cost £\(a\) and a child ticket £\(c\). Write two equations. [2 marks]
2. Solve to find \(a\) and \(c\). [4 marks]
3. The hall seats 400. At least 150 seats must be sold to children. Find the greatest possible takings from one full night. [4 marks]
4. Each performance costs £2 900 plus £2.50 per ticket sold. How many tickets must be sold (all adult) to make a profit? [3 marks]

The curve \(C\) has equation \(y=x^2-4x+1\) and the line \(L\) has equation \(y=x+k\).

1. Show that the \(x\)-coordinates of the intersections satisfy \(x^2-5x+1-k=0\). [2 marks]
2. Find the range of \(k\) for which \(L\) meets \(C\) at two distinct points. [3 marks]
3. When \(k=-5\) find the coordinates of the two intersections and the distance between them. [4 marks]
4. Find the value of \(k\) for which \(L\) is a tangent to \(C\) and the point of contact. [4 marks]

\(f(x)=2x^3-3x^2-11x+6\).

1. Show that \((x-3)\) is a factor and factorise \(f(x)\) completely. [4 marks]
2. Sketch the graph of \(y=f(x)\), showing where it crosses the axes. [3 marks]
3. Solve \(f(x)>0\). [3 marks]
4. Find the remainder when \(f(x)\) is divided by \((x+1)\). [2 marks]

A rectangular allotment has length \((x+4)\) m and width \((x-1)\) m. The area must be less than 50 m\(^2\) and the perimeter must be at least 20 m.

1. Check the table entries for \(x=4\). [2 marks]
2. Show that the area condition gives \(x^2+3x-54<0\) and solve it. [4 marks]
3. Show that the perimeter condition gives \(x\ge3.5\). [2 marks]
4. Find the range of \(x\) and list the whole-number dimensions that satisfy both conditions. [3 marks]

Part of a roller coaster track is modelled by \(y=x^3-6x^2+9x\) for \(0\le x\le4\), where \(y\) is the height in tens of metres.

1. Show that \(y=x(x-3)^2\). What does this tell you about the point \((3,0)\)? [3 marks]
2. The ride has a horizontal safety beam at height \(y=4\). Show that the beam meets the track where \((x-1)^2(x-4)=0\). [4 marks]
3. Explain what the repeated root at \(x=1\) means for the track. [2 marks]
4. For which values of \(x\) in the range is the track above ground level? [2 marks]

\(p(x)=x^4+ax^3-7x^2+bx+6\). \((x-2)\) and \((x+3)\) are both factors of \(p(x)\).

1. Find \(a\) and \(b\). [5 marks]
2. Factorise \(p(x)\) completely. [4 marks]
3. Solve \(p(x)<0\). [3 marks]
4. Find the remainder when \(p(x)\) is divided by \((x-3)\). [2 marks]