June 2025 Paper 3 Q6
6 The first four terms in ascending powers of \(x\) of the binomial expansion of
\[(2 - 3x)^5\]are given by
\[32 + px + qx^2 - 1080x^3\]where \(p\) and \(q\) are constants.
(a) Find the value of \(p\) and the value of \(q\) [2 marks]
(b) Hence find an approximation for \(1.94^5\)
Give your answer to five decimal places.
[2 marks]| Scheme | Marks | AO |
|---|---|---|
| Uses \({}^5C_1 \times 2^4 \times (-3x)\) or \(2^5 \times 5 \times \left(-\dfrac{3}{2}x\right)\) or \({}^5C_2 \times 2^3 \times (-3x)^2\) or \(2^5 \times 10 \times \left(-\dfrac{3}{2}x\right)^2\) with or without \(x\) PI by correct answer or \(-240x\) or \(720x^2\) | M1 | 1.1a |
| Obtains \(p = -240\) and \(q = 720\) Allow if seen in \(\ldots - 240x + 720x^2 \ldots\) Condone \(p = -240x\) and \(q = 720x^2\) | A1 | 1.1b |
| (2) |
Typical solution
\[p = {}^5C_1 \times 2^4 \times (-3)\]\[= -240\]\[q = {}^5C_2 \times 2^3 \times (-3)^2\]\[= 720\]| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(x = 0.02\) into \(32 + px + qx^2 - 1080x^3\) with their \(p\) and \(q\) PI by correct answer Ignore any additional terms | M1 | 2.2a |
| Obtains 27.47936 CAO ISW | A1 | 1.1b |
| (2) | ||
| (4 marks) |