Moderate -0.3 This is a straightforward logarithm equation requiring application of the power law (2ln(x+1) = ln(x+1)²), followed by exponentiating both sides and solving a simple quadratic. It's slightly easier than average as it's a standard textbook exercise with clear steps and no conceptual surprises, though it does require careful algebraic manipulation.
Use law of the logarithm of a power or quotient and remove logarithms
M1
Obtain a 3-term quadratic equation \(x^2 - x - 3 = 0\), or equivalent
A1
Solve 3-term quadratic obtaining 1 or 2 roots
M1
Obtain answer 2.30 only
A1
[4]
OR1:
Answer
Marks
Use an appropriate iterative formula, e.g. \(x_{n+1} = \exp\left(\frac{1}{2}\ln(3x_n + 4)\right) - 1\) correctly at least once
M1
Obtain answer 2.30
A1
Show sufficient iterations to at least 3 d.p. to justify 2.30 to 2 d.p., or show there is a sign change in the interval (2.295, 2.305)
A1
Show there is no other root
A1
OR2:
Answer
Marks
Guidance
Use calculated values to obtain at least one interval containing the root
M1
Obtain answer 2.30
A1
Show sufficient calculations to justify 2.30 to 3 s.f., e.g. show it lies in (2.295, 2.305)
A1
Show there is no other root
A1
[4]
| Use law of the logarithm of a power or quotient and remove logarithms | M1 | |
| Obtain a 3-term quadratic equation $x^2 - x - 3 = 0$, or equivalent | A1 | |
| Solve 3-term quadratic obtaining 1 or 2 roots | M1 | |
| Obtain answer 2.30 only | A1 | |
| | | [4] |
**OR1:**
| Use an appropriate iterative formula, e.g. $x_{n+1} = \exp\left(\frac{1}{2}\ln(3x_n + 4)\right) - 1$ correctly at least once | M1 | |
| Obtain answer 2.30 | A1 | |
| Show sufficient iterations to at least 3 d.p. to justify 2.30 to 2 d.p., or show there is a sign change in the interval (2.295, 2.305) | A1 | |
| Show there is no other root | A1 | |
**OR2:**
| Use calculated values to obtain at least one interval containing the root | M1 | |
| Obtain answer 2.30 | A1 | |
| Show sufficient calculations to justify 2.30 to 3 s.f., e.g. show it lies in (2.295, 2.305) | A1 | |
| Show there is no other root | A1 | [4] |
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