AS June 2022 Paper 1 Q15

AQACurrent spec6 marksHyperbolic Functions Using Roots of Polynomials

15 The two values of \(\theta\) that satisfy the equation

\[\sinh^2\theta - \sinh\theta - 2 = 0\]

are \(\theta_1\) and \(\theta_2\)

(a) Hamzah is asked to find the value of \(\theta_1 + \theta_2\)

He writes his answer as follows:

The quadratic coefficients are \(a = 1\), \(b = -1\), \(c = -2\)

The sum of the roots is \(-\dfrac{b}{a}\)

So \(\theta_1 + \theta_2 = -\dfrac{-1}{1} = 1\)

Explain Hamzah’s error. [1 mark]

(b) Find the correct value of \(\theta_1 + \theta_2\)

Give your answer as a single logarithm. [5 marks]

Mark scheme (a)

Scheme	Marks	AO
Assesses the validity of Hamzah’s work by explaining his error. eg the roots of the quadratic are not the solutions of the equation. eg the roots of the equation are not \(\theta\)	E1	2.3
	(1)

Typical solution

The roots of the quadratic are \(\sinh\theta_1\) and \(\sinh\theta_2\)

So \(\sinh\theta_1 + \sinh\theta_2 = 1\)

Mark scheme (b)

Scheme	Marks	AO
Finds the roots of the quadratic. PI by two of 1.44 or \(-0.88\) or 0.56 or better.	M1	1.1a
Selects a method to find a value of \(\theta_1\) or \(\theta_2\) PI by 1.44 or \(-0.88\) or 0.56 or better Condone an incorrect base.	M1	3.1a
Obtains the correct exact values of \(\theta_1\) and \(\theta_2\) in log form. PI by correct sum in log form May be unsimplified.	A1	1.1b
Correctly changes the sum of two log expressions into one log. Condone an incorrect base.	M1	1.1a
Obtains the correct sum as a single log ACF	A1	3.2a
	(5)
	(6 marks)

Typical solution

\[(\sinh\theta - 2)(\sinh\theta + 1) = 0\]\[\sinh\theta = 2 \quad \text{or} \quad \sinh\theta = -1\]\[\theta_1 = \ln\left(2 + \sqrt{5}\right) \quad \text{and}\]\[\theta_2 = \ln\left(-1 + \sqrt{2}\right)\]\[\begin{aligned}\theta_1 + \theta_2 &= \ln\left(2 + \sqrt{5}\right) + \ln\left(-1 + \sqrt{2}\right) \\ &= \ln\left(\left(2 + \sqrt{5}\right)\left(-1 + \sqrt{2}\right)\right)\end{aligned}\]