Binomial expansion for rational n and series modelling – Year 13 A level maths worksheet with answers to download as PDF

Expand each expression in ascending powers of \(x\) up to and including the term in \(x^3\), and state the values of \(x\) for which each expansion is valid.

1. \((1+x)^{-2}\) [3 marks]
2. \((1-x)^{-3}\) [3 marks]
3. \((1+2x)^{-1}\) [3 marks]

Find the first four terms in the expansion of each expression in ascending powers of \(x\), and state the range of validity.

1. \((1+x)^{1/3}\) [3 marks]
2. \((1-x)^{-1/2}\) [3 marks]
3. \(\sqrt{1+4x}\) [3 marks]

Expand each expression up to and including the term in \(x^2\). State the range of values of \(x\) for which the expansion is valid.

1. \((4+x)^{-1/2}\) [4 marks]
2. \((2+3x)^{-2}\) [4 marks]
3. \(\dfrac1{3-2x}\) [3 marks]

Find the coefficient of \(x^3\) in the expansion of each expression.

1. \((1-3x)^{-1/3}\) [3 marks]
2. \((2-x)(1+x)^{-1}\) [4 marks]
3. \(\dfrac{1+x}{\sqrt{1-x}}\) [4 marks]

Use the expansion of \((1+x)^{1/2}\) with the first three terms to estimate each value, giving your answer to 6 decimal places. Compare with your calculator.

1. \(\sqrt{1.04}\) [3 marks]
2. \(\sqrt{0.98}\) [3 marks]
3. \(\sqrt{50}\), by writing \(50=49\left(1+\tfrac1{49}\right)\) [4 marks]

A student wants to use \((1+2x)^{-1/2}\) to estimate \(\dfrac1{\sqrt3}\).

1. State the range of \(x\) for which the expansion of \((1+2x)^{-1/2}\) is valid. [1 mark]
2. The student substitutes \(x=1\). Explain why this is not allowed. [1 mark]
3. Find the expansion up to \(x^2\) and use \(x=0.01\) to estimate \(\dfrac1{\sqrt{1.02}}\) to 5 decimal places. [4 marks]

The period of a pendulum of length \(L\) is \(T=2\pi\sqrt{\dfrac Lg}\). A grandfather clock in a hallway in Leeds is adjusted so that its pendulum has period exactly 2 seconds. In summer the brass rod expands and the length increases by 3%.

1. Show that the new period is \(T_0(1.03)^{1/2}\), where \(T_0\) is the original period. [1 mark]
2. Use the first three terms of the binomial expansion to find \((1.03)^{1/2}\) to 5 decimal places. [3 marks]
3. The clock still counts one tick per swing. Find the number of seconds the clock loses in one day, giving your answer to the nearest second. [4 marks]

In special relativity a particle of rest mass \(m_0\) moving at speed \(v\) has kinetic energy \(E=m_0c^2(\gamma-1)\) where \(\gamma=\left(1-\dfrac{v^2}{c^2}\right)^{-1/2}\) and \(c\) is the speed of light.

1. Expand \(\gamma\) up to and including the term in \(\dfrac{v^4}{c^4}\). [3 marks]
2. Hence show that for small \(v\), \(E\approx\tfrac12m_0v^2\). [2 marks]
3. A proton travels at \(v=0.1c\). Use your expansion to estimate \(\gamma\) to 7 decimal places. [2 marks]

A bubble-blowing toy makes spherical bubbles. The volume of a sphere is \(V=\dfrac43\pi r^3\) and its surface area is \(4\pi r^2\).

1. The volume of a bubble increases by 5%. Show that the radius is multiplied by \((1.05)^{1/3}\). [2 marks]
2. Use the binomial expansion to find \((1.05)^{1/3}\) to 4 decimal places, using three terms. [3 marks]
3. Find the percentage increase in surface area, to 1 decimal place. [3 marks]

The intensity of light from a lamp at distance \(d\) is \(I=\dfrac k{d^2}\). A photographer in a studio moves a lamp 5% further from a model.

1. Show that the intensity is multiplied by \((1.05)^{-2}\). [1 mark]
2. Expand \((1+x)^{-2}\) up to \(x^3\) and use \(x=0.05\) to estimate \((1.05)^{-2}\) to 4 decimal places. [3 marks]
3. By what percentage does the intensity fall? [2 marks]

At height \(h\) above the Earth’s surface the gravitational acceleration is \(g=g_0\left(1+\dfrac hR\right)^{-2}\) where \(R=6371\) km is the radius of the Earth and \(g_0=9.81\) m s\(^{-2}\).

1. Use the first two terms of a binomial expansion to approximate \(g\) at the cruising height of an aeroplane, \(h=10\) km. [3 marks]
2. Find the percentage decrease in \(g\). [2 marks]
3. Explain why the expansion is valid for the heights in this question. [1 mark]

A school raffle promises a prize of £5000 to be paid in two years’ time. The school bank account pays 3% compound interest per year. The present value is \(5000(1.03)^{-2}\).

1. Use the binomial expansion of \((1+x)^{-2}\) up to \(x^2\) to estimate the present value. [3 marks]
2. Find the exact present value to the nearest penny. [1 mark]
3. Find the error in the estimate, in pounds. [1 mark]

The power output of a wind turbine is proportional to the cube of the wind speed, \(P=kv^3\). Engineers want to raise the power by 8%.

1. Show that the wind speed must rise by a factor of \((1.08)^{1/3}\). [1 mark]
2. Use the first three terms of the binomial expansion to estimate this factor to 4 decimal places. [3 marks]
3. Hence estimate the percentage rise in speed to 1 decimal place. [1 mark]

A builder lays a square paving slab with sides 5 m. The length of the diagonal is \(\sqrt{50}\) m but the builder does not have a calculator.

1. Write \(\sqrt{50}\) as \(7\sqrt{1+\dfrac1{49}}\). [1 mark]
2. Use the binomial expansion with two terms to estimate the diagonal. [2 marks]
3. Use three terms, and find the percentage error of the two-term estimate. [3 marks]

In the binomial expansion of \((1+ax)^n\), where \(a\) and \(n\) are constants and \(n\) is not a positive integer, the coefficients of \(x\) and \(x^2\) are 6 and 24 respectively.

1. Show that \(na=6\) and \((n-1)a=8\). [3 marks]
2. Find the values of \(a\) and \(n\). [2 marks]
3. Find the coefficient of \(x^3\). [2 marks]
4. State the range of \(x\) for which the expansion is valid. [1 mark]
5. Check your answer by substituting \(x=0.01\) into both sides. [2 marks]

\(\(f(x)=\dfrac{4-5x}{(1+x)(2-x)}\)\)

1. Find constants \(A\) and \(B\) such that \(f(x)=\dfrac A{1+x}+\dfrac B{2-x}\). [3 marks]
2. Hence show that \(f(x)=2-\dfrac72x+\dfrac{11}4x^2-\dfrac{25}8x^3+\cdots\). [4 marks]
3. State the range of values of \(x\) for which this expansion is valid. [2 marks]
4. Use \(x=0.1\) to evaluate \(f(0.1)\) exactly and by the series, and find the percentage error. [3 marks]

A student investigates how accurately the expansion of \(\sqrt{1+x}\) approximates its true value. She uses one, two and three terms after the constant 1, giving approximations \(A_1\), \(A_2\) and \(A_3\).

1. Verify the entry for \(A_3\) when \(x=0.5\). [2 marks]
2. Describe how the accuracy changes as terms are added when \(x=0.1\) and when \(x=0.5\). [2 marks]
3. Explain why the approximations for \(x=2\) are poor. [3 marks]
4. Find the percentage error in \(A_2\) for \(x=0.1\). [2 marks]

The mass of a particle moving at speed \(v\) is \(m=m_0\left(1-\dfrac{v^2}{c^2}\right)^{-1/2}\). A research team at a university accelerator measures the kinetic energy \(E=(m-m_0)c^2\) of electrons and compares it with the classical value \(\tfrac12m_0v^2\).

1. Show that \(E\approx\tfrac12m_0v^2+\tfrac38\dfrac{m_0v^4}{c^2}\). [3 marks]
2. For Run B, find \(\dfrac{E-\frac12m_0v^2}{E}\) as a percentage using the full expression, to 2 significant figures. [4 marks]
3. For which run is the classical formula least accurate? Explain, using a calculation for Run C. [3 marks]

\(\(g(x)=\dfrac{1+x}{\sqrt{1-2x}}\)\)

1. Expand \(g(x)\) in ascending powers of \(x\) up to and including \(x^3\). [5 marks]
2. State the range of \(x\) for which the expansion is valid. [1 mark]
3. Use \(x=0.01\) to estimate \(\dfrac{1.01}{\sqrt{0.98}}\) to 6 decimal places. [2 marks]
4. The equation \(g(x)=1.05\) has a small root. Use your expansion up to \(x^2\) to find an approximate value for it. [3 marks]

Consider the expansion of \((1-4x)^{-1/2}\).

1. Show that \((1-4x)^{-1/2}=1+2x+6x^2+20x^3+70x^4+\cdots\). [4 marks]
2. The coefficient of \(x^n\) is \(\dbinom{2n}n\). Verify this for \(n=1,2,3,4\). [2 marks]
3. Hence write down the first four terms of \(\dfrac1{\sqrt{1-4x}}\) when \(x=0.01\) and estimate its value to 5 decimal places. [3 marks]
4. Explain why the series cannot be used to find \(\dfrac1{\sqrt{1-4x}}\) when \(x=\tfrac14\). [1 mark]