Indices, surds and quadratic functions – Year 12 A level maths worksheet with answers to download as PDF

Work out each value without a calculator, then simplify the algebraic expression.

1. \(27^{2/3}\) [2 marks]
2. \(16^{-3/4}\) [2 marks]
3. \(\left(\dfrac{8}{27}\right)^{-2/3}\) [2 marks]
4. \(\dfrac{(2x^3)^2\times3x^{-5}}{6x^{-1}}\) [3 marks]

Write each answer in the form \(a+b\sqrt c\) or \(a\sqrt c\) where \(a\), \(b\), \(c\) are integers.

1. \(\sqrt{75}+\sqrt{12}-\sqrt{27}\) [3 marks]
2. \((3+\sqrt5)(2-\sqrt5)\) [2 marks]
3. \(\dfrac{6}{\sqrt3}\) [2 marks]
4. \(\dfrac{4}{3-\sqrt5}\) [3 marks]
5. \(\dfrac{\sqrt2+1}{\sqrt2-1}\) [3 marks]

Write each function in completed-square form, state the coordinates of the turning point, and say whether it is a maximum or minimum.

1. \(f(x)=x^2-8x+19\) [3 marks]
2. \(g(x)=-2x^2+12x-7\) [4 marks]
3. Solve \(g(x)=0\), giving exact answers. [3 marks]

Solve each equation. Give exact answers where they are not integers or fractions.

1. \(2x^2-5x-12=0\) [3 marks]
2. \(x^2+6x-2=0\) [3 marks]
3. \(3x^2-2x-4=0\) [3 marks]

Use the discriminant \(b^2-4ac\) to answer each part.

1. The equation \(x^2+kx+9=0\) has equal roots. Find the possible values of \(k\). [3 marks]
2. The equation \(2x^2-4x+k=0\) has no real roots. Find the range of values of \(k\). [3 marks]
3. The equation \(kx^2+8x+2=0\) has two real roots. Find the range of values of \(k\). [4 marks]

Use a substitution to solve each equation.

1. \(x^4-13x^2+36=0\) [3 marks]
2. \(x-5\sqrt x+6=0\) [3 marks]
3. \(2^{2x}-6\times2^x+8=0\) [4 marks]

A rectangular patio has length \((5+\sqrt2)\) m and width \((5-\sqrt2)\) m. Paving slabs cost £18.90 per square metre and edging costs £14.50 per metre.

1. Show that the area of the patio is exactly 23 m\(^2\). [2 marks]
2. Find the exact perimeter. [2 marks]
3. Show that the length of a diagonal is \(3\sqrt6\) m. [3 marks]
4. Work out the total cost of the paving and edging. [3 marks]

A football is kicked from the ground. Its height \(h\) metres after \(t\) seconds is modelled by \(h=1.5+20t-5t^2\), where 1.5 m is the height at which it is struck.

1. Write \(h\) in the form \(a-b(t-c)^2\). [3 marks]
2. State the maximum height and the time at which it occurs. [2 marks]
3. The ball hits the ground when \(h=0\). Find this time to 2 decimal places. [3 marks]
4. For how long is the ball higher than 15 m? Give your answer to 2 decimal places. [3 marks]

A school sells tickets for its summer show at £\(x\) each. Past sales suggest that \(200-10x\) tickets are sold. Each ticket costs the school £3 to produce (programme and refreshments).

1. Show that the profit is \(P=-10x^2+230x-600\) pounds. [2 marks]
2. Find the ticket price that gives the greatest profit and state this profit. [4 marks]
3. Find the prices at which the school breaks even. [2 marks]
4. The school wants a profit of at least £500. Find the range of prices, to the nearest penny. [4 marks]

A sailing club makes a triangular sail in the shape of an isosceles triangle with base \(4\sqrt3\) m and perpendicular height 6 m. The cloth costs £12.40 per square metre.

1. Show that the area of the sail is \(12\sqrt3\) m\(^2\). [2 marks]
2. Find the exact length of each sloping side. Write your answer in the form \(a\sqrt b\). [3 marks]
3. Work out the cost of the cloth to the nearest pound. [3 marks]
4. The club wants a similar sail with area \(48\sqrt3\) m\(^2\). Show that all lengths must be doubled. [2 marks]

A farmer has 60 m of fencing to make a rectangular sheep pen against a straight wall. The wall forms one side, so only three sides need fencing. The width of the pen (the sides at right angles to the wall) is \(x\) metres.

1. Show that the area of the pen is \(A=60x-2x^2\) m\(^2\). [2 marks]
2. Complete the square to find the maximum area and the dimensions that give it. [4 marks]
3. The farmer needs an area of at least 400 m\(^2\). Find the range of possible widths. [3 marks]
4. Explain why the farmer cannot enclose 500 m\(^2\) using this fencing. [2 marks]

Kepler found that the orbital period \(T\) (in Earth years) of a planet is related to its distance \(r\) from the Sun (in astronomical units, AU) by \(T=r^{3/2}\).

1. Find \(T\) for Mars and Jupiter, to 2 decimal places. [3 marks]
2. A comet has period 64 years. Find its mean distance from the Sun. [3 marks]
3. Saturn\'s period is 29.5 years. Use the model to find \(r\) and compare with the table. [3 marks]
4. Show that \(T^2=r^3\). [1 mark]

On a map (units: hundreds of metres) a canal bank follows the curve \(y=x^2-4x+7\). A straight cycle path is the line \(y=2x+c\), where \(c\) depends on where it is built.

1. Show that the \(x\)-coordinates of the points where the path meets the canal satisfy \(x^2-6x+7-c=0\). [2 marks]
2. Find the value of \(c\) for which the path just touches the canal bank. [3 marks]
3. Find the coordinates of the point of contact. [2 marks]
4. For what values of \(c\) does the path cross the canal at two points? [2 marks]

A seaside kiosk opens for the season. Its weekly profit, in hundreds of pounds, is modelled by \(P=t-7\sqrt t+10\), where \(t\) is the number of weeks since opening (\(t>0\)).

1. Show that \(P=0\) when \(t=4\) and when \(t=25\). [3 marks]
2. For which weeks is the kiosk making a loss? [2 marks]
3. Find the greatest weekly loss, in pounds. [4 marks]

A farmer has 150 m of fencing to enclose a rectangular field with an area as large as possible. Let one side be \(x\) m. The second side is therefore \((75-x)\) m. Here are some values of the area \(A=x(75-x)\).

1. Check two of the table values by calculation. [2 marks]
2. Write \(A\) in completed-square form and find the maximum area. [3 marks]
3. The farmer wants an area of exactly 1200 m\(^2\). Find both possible values of \(x\), to 1 decimal place. [4 marks]
4. Use the discriminant to show that an area of 1450 m\(^2\) is impossible. [3 marks]

Triangle \(ABC\) has a right angle at \(B\). \(AB=(\sqrt5+1)\) cm and \(BC=(\sqrt5-1)\) cm.

1. Show that the area of the triangle is exactly 2 cm\(^2\). [2 marks]
2. Show that \(AC=2\sqrt3\) cm. [3 marks]
3. Show that \(\tan A=\dfrac{3-\sqrt5}{2}\). [4 marks]
4. Find the exact perimeter of the triangle. [2 marks]

The quadratic equation \(x^2-6x+k=0\) has roots \(\alpha\) and \(\beta\).

1. Find the range of \(k\) for which the equation has two distinct real roots. [2 marks]
2. Given that one root is twice the other, find \(k\). [4 marks]
3. Given that the roots differ by 4, find \(k\) and the roots. [4 marks]
4. Show that when \(0\lt k<9\) both roots are positive. [2 marks]

Solve each equation, showing your method.

1. \(8^{x-1}=4^{x+2}\) [4 marks]
2. \(9^x-4\times3^{x+1}+27=0\) [5 marks]
3. \(x^{3/2}=125\) [2 marks]
4. \(\dfrac{2^{x+3}}{4^{x-1}}=16\) [4 marks]

A footbridge has a parabolic arch. Measured in metres from the left foot of the arch, the height of the arch is \(h\) at horizontal distance \(x\). The arch is 20 m wide at ground level and 8 m high at its highest point, which is in the middle.

1. Show that \(h=-0.08x^2+1.6x\) fits the table. [3 marks]
2. A narrowboat is 6 m wide and 6.5 m tall. It passes under the centre of the arch. Show that it fits. [3 marks]
3. Find the width of the arch at a height of 7 m. [3 marks]
4. Find the widest 7 m high boat. How does it compare with part (c)? [2 marks]

Consider the equation \(x^2+kx+(k-2)=0\) where \(k\) is a real constant.

1. Prove that the equation has two distinct real roots for all values of \(k\). [4 marks]
2. Let the roots be \(\alpha\) and \(\beta\). Show that \(\alpha^2+\beta^2=k^2-2k+4\) and find its minimum value. [5 marks]
3. For what values of \(k\) are both roots negative? [3 marks]